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Differential Geometry of Singular Spaces and Reduction of Symmetry In this book, the author illustrates the power of the theory of subcartesian differential spaces for investigating spaces with singularities. Part I gives a detailed and comprehensive presentation of the theory of differential spaces, including integration of distributions on subcartesian spaces and the structure of stratified spaces. Part II presents an effective approach to the reduction of symmetries. Concrete applications covered in the text include the reduction of symmetries of Hamiltonian systems, non-holonomically constrained systems, Dirac structures and the commutation of quantization with reduction for a proper action of the symmetry group. With each application, the author provides an introduction to the field in which relevant problems occur. This book will appeal to researchers and graduate students in mathematics and engineering.

J. S´ NIATYCKI is a Professor in the Department of Mathematics and Statistics at the University of Calgary.

N E W M AT H E M AT I C A L M O N O G R A P H S Editorial Board Béla Bollobás, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry Simon, Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit www.cambridge.org/mathematics. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups J. B. Garnett and D. E. Marshall Harmonic Measure P. Cohn Free Ideal Rings and Localization in General Rings E. Bombieri and W. Gubler Heights in Diophantine Geometry Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. Wüstholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry

Differential Geometry of Singular Spaces and Reduction of Symmetry J . S´ N I AT Y C K I Department of Mathematics and Statistics University of Calgary Calgary, Alberta, Canada

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107022713 ´ c J. Sniatycki 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Sniatycki, Jedrzej. Differential geometry of singular spaces and reduction of symmetry / Jedrzej Sniatycki. pages cm. – (New mathematical monographs ; 23) ISBN 978-1-107-02271-3 (hardback) 1. Geometry, Differential. 2. Function spaces. 3. Symmetry (Mathematics) I. Title. QA641.S55 2013 516.3 6–dc23 2012047532 ISBN 978-1-107-02271-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface List of selected symbols 1 Introduction

page vii ix 1

PART I D I F F E R E N T I A L G E O M E T RY O F SI N G U L A R SPAC ES 2 Differential structures 2.1 Differential spaces 2.2 Partitions of unity

15 15 21

3 Derivations 3.1 Basic properties 3.2 Integration of derivations 3.3 The tangent bundle 3.4 Orbits of families of vector fields

25 25 31 37 44

4 Stratified spaces 4.1 Stratified subcartesian spaces 4.2 Action of a Lie group on a manifold 4.3 Orbit space 4.4 Action of a Lie group on a subcartesian space

52 52 56 67 81

5 Differential forms 5.1 Koszul forms 5.2 Zariski forms 5.3 Marshall forms

91 91 94 99

vi

Contents

PART II R E D U C T I O N O F S Y M M E T R I E S 6 Symplectic reduction 6.1 Symplectic manifolds with symmetry 6.1.1 Co-adjoint orbits 6.1.2 Symplectic manifolds 6.1.3 Poisson algebra 6.2 Poisson reduction 6.3 Level sets of the momentum map 6.4 Pre-images of co-adjoint orbits 6.5 Reduction by stages for proper actions 6.6 Shifting 6.7 When the action is free 6.8 When the action is improper 6.9 Algebraic reduction

105 105 105 108 110 111 114 125 126 129 134 135 136

7 Commutation of quantization and reduction 7.1 Review of geometric quantization 7.1.1 Prequantization 7.1.2 Polarization 7.1.3 Examples of unitarization 7.2 Commutation of quantization and singular reduction at J = 0 7.3 Special cases 7.3.1 The results of Guillemin and Sternberg 7.3.2 Kähler polarization without compactness assumptions 7.3.3 Real polarization 7.4 Non-zero co-adjoint orbits 7.5 Commutation of quantization and algebraic reduction 7.5.1 Quantization of algebraic reduction 7.5.2 Kähler polarization 7.5.3 Real polarization 7.5.4 Improper action

150 151 152 155 156 161 184 184 185 187 190 203 203 206 207 208

8 Further examples of reduction 8.1 Non-holonomic reduction 8.2 Dirac structures 8.2.1 Symmetries of the Pontryagin bundle 8.2.2 Free and proper action 8.2.3 Proper non-free action

211 211 214 215 217 223

References Index

228 233

Preface

My first encounter with differential spaces was in the mid 1980s. At a conference in Toru´n, I presented the notion of algebraic reduction of symmetries of a Hamiltonian system. After the lecture, Constantin Piron asked me if my reduced spaces were the differential spaces of Sikorski. I had to admit that I did not know what Sikorski’s differential spaces were. To this Piron replied something like ‘You should be ashamed of yourself! You are a Pole and you do not know what are differential spaces of Sikorski!’ During the lunch break I went to the library to consult Sikorski’s work. In the afternoon session, I told Piron that the spaces we were dealing with were not the differential spaces of Sikorski. At that time I did not realize that they were differential schemes. Around the same time, Richard Cushman was working out his examples of singular reduction. I was fascinated by his pictures of reduced spaces with singularities. However, I had not the faintest idea what he was really doing. Since Richard was spending a lot of time in Calgary working on his book with Larry Bates, I had a chance to ask him to explain singular reduction to me. It took me a long time to realize that he was talking the language of differential spaces without being aware of it. From conversations with Richard, it became clear that differential spaces provided a convenient language for the description of the reduction of symmetries for proper actions of symmetry groups. The next push in the direction of serious investigations of differential spaces came from Ryer Sjamaar and Eugene Lerman. In their Annals of Mathematics paper on reduction of symmetries of Hamiltonian systems, they proved a theorem using techniques that are natural to the theory of differential spaces. Studying their proof, I realized that it was very simple and that I could not think of an equally simple proof that would not utilize their techniques. It convinced me that the language of differential spaces facilitated obtaining new results, and I decided to investigate if reduction of symmetries could be completely formulated and analysed within the category of differential spaces.

viii

Preface

The theory of differential spaces is essentially differential geometry not restricted to smooth manifolds. Roman Sikorski, who is considered the father of the theory, called his book (in Polish) Wst˛ep do Geometrii Ró˙zniczkowej. This translates as ‘Introduction to Differential Geometry’. Originally, differential geometry meant the description, in terms of differentiable functions, of curves and surfaces in Rn . Singularities of curves or surfaces under consideration could also be described in terms of smooth functions. Differential geometry evolved in two different directions: the theory of manifolds and singularity theory. Manifolds are smooth spaces not presented as subsets of Rn . Singularity theory is the study of the failure of the manifold structure. Differential geometry in the sense of Sikorski is a reunification of the two theories. It contains the theory of manifolds and also allows the investigation of singularities. It is the investigation of geometry in terms of differentiable functions. Differential geometry, understood in this way, is analogous to algebraic geometry, which is the investigation of geometry in terms of polynomials. The difference between the two theories is in the choice of the space of functions. I am grateful to Constantin Piron for drawing my attention to Sikorski’s book. I greatly appreciate the support and encouragement of Hans Duistermaat. I would like to thank Larry Bates for his support and for bringing Richard Cushman to Calgary, and to thank Jordan Watts for his interest in my work. Above all, I want to thank Richard Cushman for his patience in explaining to me the foundations of his theory of singular reduction and his subsequent collaboration, encouragement and criticism. I also want to thank Cathy Beveridge and Leslie McNab for their help in editing the manuscript. Both Cathy and Leslie have worked hard to make sure that this book is written in proper English. However, I am sure that, in spite of their vigilance, I will have managed to slip in some phrases that go against the proper use of English. Last but not least, I want to thank my wife, Pamela Plummer, without whose support this book would not have been possible. Partial support from the National Science and Engineering Research Council of Canada is gratefully acknowledged.

List of selected symbols

Upper case Latin alphabet Ad ∗ B C C 0 (S) C ∞ (S) C ∞ (S)G D D D Der C ∞ (S) E Exp : T p P → P F F F FF FTCM G H H I J J0 K L L M

co-adjoint action open ball complex numbers continuous functions on S smooth functions on S G-invariant functions on S distribution; real part of polarization in Chapter 7 space of compactly supported sections dual of D space of derivations of C ∞ (S) family of Hamiltonian vector fields exponential map defined by connection function; polarization in Chapter 7 family of functions family of vector fields bundle of linear frames of F bundle of linear frames of T C M Lie group Lie group Hilbert space interval; inclusion map of co-adjoint orbit into g∗ in Chapter 7 momentum map ideal generated by components of J manifold manifold prequantization line bundle manifold; stratum

x M N NH N (S) N O O P P PH PH P(H ) P(R) Q Q R R R R(S) S Sp S ∞ (L) S F∞ (L) S ∞ (L)G T CM T S, T ∗ S Tϕ T p⊥ L U, V, W U X, Y, Z X( f ) X(S)

List of selected symbols

stratification manifold, stratum normalizer of H space of functions with vanishing restrictions to S stratification orbit partition by orbits manifold, differential space prequantization map set of points in P fixed by action of H set of points in P of symmetry type H set of points in P of orbit type H Poisson vector fields on R manifold quantization map manifold; differential space; orbit space real numbers linear representation space of restrictions to S of functions defined on a larger space manifold; differential space slice at p space of smooth sections of L space of smooth polarized sections of L space of G-invariant smooth sections of L complexified tangent bundle of M tangent and cotangent bundle spaces of S derived map of ϕ symplectic complement of T p L open subsets unitary representation global derivations; vector fields; sections of tangent bundle evaluation of X on f family of all vector fields on S

Lower case Latin alphabet a, b c c:I →S d

real numbers complex number curve in S differential

List of selected symbols

e exp : g → G exp(t X ) exp(t X )(x) f f −1 (I ) g g g∗ h h h∗ hor T P k m n p q s supp f t u, v ver T P w x, y z

group identity exponential map local one-parameter local group of diffeomorphisms defined by X point on maximal integral curve of X through x function inverse image of I under f element of group G Lie algebra of G dual of g function Planck’s constant divided by 2π Lie algebra of H dual of h horizontal distribution on P Riemannian metric Lie algebra Lie algebra of N point point parameter support of f parameter derivation at a point; vector vertical distribution on P derivation at a point; vector point point of L

Lower case Greek alphabet α β δi j ζ, η θ ϑ λ, μ, ν λ:L→ P ξ

1-form 1-form Kronecker δ elements of Lie algebra 1-form cotangent bundle projection elements of co-adjoint orbit complex line bundle projection element of Lie algebra

xi

xii π ρ ρ∗ ρ∗ σ σ∗ σ∗ τ τ∗ ϕ ϕ∗ ϕ∗ ω

List of selected symbols

projection map form; distributional symplectic form in Chapter 8 map pull-back by ρ push-forward by ρ section pull-back by σ push-forward by σ map; tangent bundle projection pull-back by τ map pull-back by ϕ push-forward by ϕ symplectic form

Upper case Greek alphabet kK (S) kM (S) kZ (S)

Lagrangian submanifold projection of G-invariant section restriction of section action action symplectic form of co-adjoint orbit space of Koszul k-forms on S space of Marshall k-forms on S space of Zariski k-forms on S

Non-alphabetic symbols ∇ · | ·

√ n |∧ F| [·, ·] {·, ·} |

(· | ·)

covariant derivative evaluation; sesquilinear form on a line bundle half-densities on F left interior product Lie bracket Poisson bracket restriction scalar product on a Hilbert space

1 Introduction

This book is written for researchers and graduate students in the field of geometric mechanics, especially the theory of systems with symmetries. A wider audience might include differential geometers, algebraic geometers and singularity theorists. The aim of the book is to show that differential geometry in the sense of Sikorski is a powerful tool for the study of the geometry of spaces with singularities. We show that this understanding of differential geometry gives a complete description of the stratification structure of the space of orbits of a proper action of a connected Lie group G on a manifold P. We also show that the same approach can handle intersection singularities; see Section 8.2. We assume here that the reader has a working knowledge of differential geometry and the topology of manifolds, and we use theorems in these fields freely without giving proofs or references. On the other hand, the material on differential spaces is developed from scratch. The results on differential spaces are proved in detail. This should make the book accessible to graduate students. The book is split into two parts. In Part I, we introduce the reader to the differential geometry of singular spaces and prove some results, which are used in Part II to investigate concrete systems. The technique of differential geometry presented here is fairly straightforward, and the reader might get a false impression that the scope of the theory does not differ much from that of the geometry of manifolds. However, the examples given in Part I will serve as warnings that such an impression is false. Part II is devoted to applications of the general theory. Each chapter in this part may be considered as an extensive example of the use of differential geometry to deal with singularities in concrete problems. Since these problems occur in various theories, each chapter begins with a section introducing elements of the underlying theory, in order to show the reader the relevance of the problem under consideration.

2

Introduction

The book contains no exercises, because the actual techniques involved are very simple. In addition to the standard techniques of the differential geometry of manifolds, we use techniques of algebraic geometry for rings of smooth functions. The fact that algebraically defined derivations of smooth functions admit integral curves is the main difference between differential and algebraic geometry. The technical details of the presentation are based on the TEX style file chosen for the preparation of this book. Displayed results are labelled by the number of the chapter, the number of the section in the chapter and the number of the result within the section. For example, ‘Lemma 2.1.3’ stands for Lemma 1.3 in Chapter 2; it can also be read as the third lemma in Section 2.1. Displayed equations are referenced by the number of the chapter and the number of the equation within the chapter. For example, ‘equation (3.21)’ stands for equation 21 in Chapter 3. This book is based on several years of research. Some of the results presented here were obtained by the author. Some other results have been taken directly from the work of other researchers. The remainder corresponds to an adaptation and reformulation of the work of other authors so that it fits into the theory presented here. In order to keep the flow of the presentation in the subsequent chapters free from obstructions, we give below a detailed description of the content of the book and the references to the literature. Part I is devoted to a comprehensive presentation of the current status of the differential geometry of singular spaces. A comprehensive bibliography of the literature on differential spaces during the period 1965–1992 was published in 1993 by Buchner, Heller, Multarzy´nski and Sasin (Buchner et al., 1993). According to these authors, the first paper on differential spaces was Sikorski (1967). In the same year, at a meeting of the American Mathematical Society, Aronszajn presented an extensive programme of differential-geometric study of subcartesian spaces in terms of singular charts. Aronszajn’s subcartesian spaces included arbitrary subspaces of Rn (see Aronszajn, 1967). In 1973, Walczak showed that subcartesian spaces are special cases of differential spaces (see Walczak, 1973). In Section 2.1, we describe the basic definitions and constructions of Sikorski’s theory following his book (see Sikorski, 1972). The fundamental notion of this theory is the differential structure C ∞ (S) of a space S, consisting of functions on S deemed to be smooth. The differential structure of a space carries all information about the geometry of the space. In particular, a map ϕ : S → T is smooth if it pulls back smooth functions to smooth functions. A diffeomorphism is an invertible smooth map with a smooth inverse. As in topology, subsets, products and quotients of differential spaces are differential

Introduction

3

spaces. However, the quotient differential space need not have the quotient topology. Proposition 2.1.11, which gives conditions for equivalence of the quotient differential-space topology and the quotient topology, is taken from the work of Pasternak-Winiarski (1984) . A differential space S is subcartesian if every point of S has a neighbourhood diffeomorphic to a subset of some Cartesian space Rn . The category of subcartesian differential spaces is the main object of our study. Manifolds are subcartesian spaces that are locally diffeomorphic to open subsets of Rn . If M is a manifold, the collection of all local diffeomorphisms to open subsets of Rn forms the maximal atlas on M. Differential geometry, understood as the study of the geometry of a space in terms of the ring of smooth functions on that space, naturally extends from manifolds to subcartesian spaces. We do not go beyond subcartesian spaces, because a differential space which is not subcartesian need not have a locally finite dimension. In Section 2.2, we show that subcartesian spaces admit partitions of unity. The importance of partitions of unity stems from the fact that they enable us to globalize collections of local data. The existence of partitions of unity on locally compact and paracompact differential spaces was first proved by Cegiełka (1974). Here, we follow the proof of Marshall (1975a). In Chapter 3, we discuss vector fields on subcartesian spaces. A vector field on a manifold M can be described either as a derivation of a ring C ∞ (M) of smooth functions on M or as a generator of a local one-parameter group of local diffeomorphisms of M. These two notions are equivalent if M is a manifold. However, they may be inequivalent on a subcartesian space S that is not a manifold. In Section 3.1, we study the basic properties of derivations of the differential structure C ∞ (S) of a subcartesian space S. We show that every derivation X of C ∞ (S) can be locally extended to a derivation of C ∞ (R n ). This result allows the study of ordinary differential equations on subcartesian spaces, which we discuss in Section 3.2. The existence and uniqueness theorem for integral ´ curves of derivations on a subcartesian space was first proved by Sniatycki (2003a). In Section 3.3, we discuss the tangent bundle space T S of S, defined as the space of derivations of C ∞ (S) at points of S. In the literature, T S is also called the tangent pseudobundle or the Zariski tangent bundle. Following Watts (2006), we define the regular component Sreg of S as the set of all points p of S at which dim T p S is locally constant, and prove that Sreg is open and dense in S and that the restriction T Sreg of T S to Sreg is locally spanned by global derivations; see Lusala et al. (2010). Example 3.3.12, taken from Epstein and

4

Introduction

´ Sniatycki (2006), shows that a differential space that is regular everywhere need not be a manifold. In Section 3.4, we study global derivations of S that generate local oneparameter groups of local diffeomorphisms. We call such global derivations vector fields. We show that the orbits of any family of vector fields on a subcartesian space S are smooth manifolds immersed in S. This result, first ´ proved by Sniatycki (2003b), is a generalization of some theorems of Sussmann (1973) and Stefan (1974). In particular, it implies that orbits of the family X(S) of all vector fields on S give a partition of S by smooth manifolds. Therefore, every subcartesian space S has a minimal partition by smooth manifolds. This result gives us an alternative interpretation of the strata of a minimal stratification of a subcartesian space, which we study in Chapter 4. In Chapter 4, we discuss stratified spaces, first investigated by Whitney (1955), who called them ‘manifold collections’. The term ‘stratification’ is due to Thom (1955–56). A stratified space is usually described as a topological space partitioned in a special way by smooth manifolds. Here, we restrict our considerations to stratified spaces that are also subcartesian differential spaces. In Section 4.1, we discuss stratified subcartesian spaces following the work ´ ´ of Sniatycki (2003b) and Lusala and Sniatycki (2011). A stratified space is, by definition, partitioned by smooth manifolds. The results of Chapter 3 show that a subcartesian space is also partitioned by smooth manifolds, which are orbits of the family of all vector fields. We show that if a stratified space S is subcartesian and the stratification of S is locally trivial, then the partition of S by orbits of the family of all vector fields is also a stratification of S. Moreover, this second stratification of S is coarser than the original stratification. If the original stratification is minimal, then it is the same as the stratification given by the orbits of the family of all vector fields. In other words, a minimal locally trivial stratification of a subcartesian space is completely determined by its differential structure. In Section 4.2, we describe the orbit type stratification M of a manifold P given by a proper action on P of a connected Lie group G. This stratification is not minimal, because the union of all the strata is the manifold P. The presentation adopted here borrows from the presentations of the same topic in the books by Cushman and Bates (1997), Duistermaat and Kolk (2000), and Pflaum (2001). Section 4.3 is devoted to a discussion of the structure of the orbit space R = P/G. We show that the projection to the orbit space R of the strata of M is a locally trivial and minimal stratification of R. This is called the orbit type stratification of the orbit space R. We also show that R is a subcartesian space.

Introduction

5

The material presented in Section 4.3 is based on the results of many authors. In particular, results of Bierstone (1975; 1980), Bochner’s Linearization Theorem (Duistermaat and Kolk, 2000), the Hilbert–Weyl Theorem (Weyl, 1946), Palais’s Slice Theorem (Palais, 1961), a theorem by Schwarz (1975) and the Tarski–Seidenberg Theorem (Abraham and Robbin, 1967). The form of pre´ sentation adopted here follows that of Cushman, Duistermaat and Sniatycki (2010). By combining the results of Sections 4.1 and 4.3, we conclude that the strata of the orbit type stratification of the orbit space R are orbits of the family of all vector fields on R. This result is the basis for the singular reduction of symmetries discussed in subsequent chapters. In Section 4.4, we study a proper action of a Lie group on a locally compact subcartesian space. Palais’s Slice Theorem applies to this case, and we prove that the space of orbits of the action is a locally compact differential space. We have no extension of Bochner’s Linearization Theorem to subcartesian spaces, and we can prove neither that the orbit space is subcartesian nor that it is stratified. Nevertheless, the result obtained here suffices to prove singular reduction by stages in Section 6.5. Chapter 5 is devoted to a discussion of differential forms on subcartesian spaces. We are led to three inequivalent notions of differential forms. Zariski differential forms on S are defined as alternating multilinear maps from spaces of pointwise derivations of C ∞ (S) to real numbers. Zariski differential forms can be pulled back by smooth maps. If S is not a manifold, then exterior differentials of Zariski differential forms are not defined. The second possibility is Koszul differential forms, defined as alternating multilinear maps from spaces of global derivations of C ∞ (S) to C ∞ (S). We can take exterior differentials of Koszul forms, but we cannot define their pull-backs by differential maps. The third possibility is Marshall forms, which agree with Zariski forms and Koszul forms on the regular component Sreg of S. Marshall forms allow pull-backs, as well as exterior differentials. The presentation adopted here follows a paper by Marshall (1975a), Watts’ theses (Watts, 2006; 2012) and his unpublished notes. In Part II, we apply the general theory introduced in Part I to the problem of reduction of the symmetries of various systems. In most cases, we make an assumption that the action of the symmetry group G on the phase space P of the system is proper. This assumption implies that the orbit space P/G is stratified, and the study of reduction involves an investigation of the interplay between the stratification structure of P/G and the geometric structure characterizing the system under consideration. There is no satisfactory theory of the structure of the space of orbits of an improper action of a Lie group on a manifold. However, if P is a symplectic

6

Introduction

manifold and the improper action of G on P is Hamiltonian, we can show that algebraic reduction, in terms of differential schemes, encodes a lot of information about the action of G on P. We also show that the information obtained by algebraic reduction may survive the process of quantization and may be decoded on the quantum level. The objective of symplectic reduction, discussed in Chapter 6, is to describe the structure of the space of orbits of a Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω). For a proper action, we know that the orbit space R = P/G is stratified, and we investigate the interaction between the stratification structure of R and the Poisson structure of R induced by the symplectic structure of P. We also discuss the case when the action of G on P fails to be proper. In Section 6.1, we give a brief review of Hamiltonian actions of a Lie group G on a symplectic manifold (P, ω), the properties of the momentum map J : P → g∗ , and the Poisson algebra structure of C ∞ (P) induced by the symplectic form ω on P. We begin with a discussion of the co-adjoint action of G on co-adjoint orbits in g∗ and describe the Kirillov–Kostant–Souriau symplectic form of a co-adjoint orbit (Kirillov, 1962; Kostant, 1966; Souriau, 1966). Moreover, we show that the momentum map for a co-adjoint orbit is the inclusion of the orbit in g∗ . This introductory material is included here in order to establish the notation and to introduce the problem to readers who might be unfamiliar with symplectic geometry. Symplectic reduction for a free and proper action was introduced by Meyer (1973) and Marsden and Weinstein (1974). It is known as regular reduction or Marsden–Weinstein reduction. The first study of the structure of the orbit space for a proper non-free Hamiltonian action of the symmetry group was the paper of Arms, Marsden and Moncrief (Arms et al., 1981), who showed that the zero level of the momentum map is stratified. The technique of singular reduction in terms of the Poisson algebra structure was initiated by Cushman (1983), and later formalized by Arms, Cushman and Gotay (Arms et al., 1991). The role of Sikorski’s theory of differential spaces in ´ singular reduction was first described by Cushman and Sniatycki (2001). Comprehensive presentations of singular reduction have been given in the books by Cushman and Bates (1997) and Ortega and Ratiu (2004). Our discussion of singular reduction is contained in Sections 6.2–6.6. Our presentation differs from the presentations in Cushman and Bates (1997) and Ortega and Ratiu (2004) because we have the general theory developed in Part I at our disposal. Nevertheless, it has many points in common with earlier approaches. In Section 6.2, we describe the structure of the orbit space R = P/G in terms of the structure of the ring C ∞ (R) of smooth functions on R. Using

Introduction

7

the results of Chapter 4, we describe strata of the orbit type stratifications of P/G as orbits of the family of all vector fields X(R) on R. For each stratum of R, the Poisson structure on C ∞ (R) induces the structure of a Poisson manifold. Since Poisson derivations of C ∞ (R) are vector fields on R, orbits of the family P(R) of all Poisson derivations of C ∞ (R) give foliations of strata of R by symplectic leaves. A proof that a Poisson manifold is singularly foliated by symplectic leaves was given in the book by Libermann and Marle (1987). In Section 6.3, we show that for each μ ∈ g∗ , the projection to R of the level set J −1 (μ) is a stratified space with symplectic strata, which are symplectomorphic to the corresponding symplectic leaves of strata of R. In Section 6.4, we obtain similar results for projections to R of J −1 (O), provided that the co-adjoint orbit O is locally closed.1 The main results obtained in Sections 6.3 and 6.4 are not new. However, the proofs of these results are new. In Section 6.5, we apply the results of Section 4.4 to the case when the symmetry group G of (P, ω) has a normal subgroup H . In this case, we can first reduce the action of H . The result is a stratified Poisson space P/H ´ with symmetry group G/H . Following Lusala and Sniatycki (to appear), we prove that the structure of the orbit space (P/H )/(G/H ) is isomorphic to that of P/G. This result is called ‘reduction by stages’ in the literature; see the book by Marsden, Misiołek, Ortega, Perlmutter and Ratiu (Marsden et al., 2007). In Section 6.6, we discuss the process of shifting, which gives rise to an equivalence between the reduction of J −1 (O) and the reduction at zero for a shifted momentum map on P × O, where O is a co-adjoint orbit. This is essential for the extension to non-zero co-adjoint orbits of the results on the commutation of quantization and reduction of J −1 (0) discussed in the next chapter. Shifting was introduced for a free and proper action by Guillemin and Sternberg (1984). For a proper non-free action, shifting was first proved by ´ Bates, Cushman, Hamilton and Sniatycki (Bates et al., 2009). In Section 6.7, we restrict singular reduction to the case when the action of G on P is free and proper. As a corollary, we obtain the results of the Marsden–Weinstein reduction (Marsden and Weinstein, 1974). In Section 6.8, we discuss the case when the action of G on P is not proper. In this case, the ring of G-invariant functions on P need not separate the orbits, and singular reduction is not applicable. At present, there is no satisfactory theory of the structure of the space of orbits of an improper 1 An example of a co-adjoint orbit which is not locally closed was first given by Pukanszky

(1971). Here, we do not study such co-adjoint orbits; however, they were discussed by Ortega and Ratiu (2004).

8

Introduction

action of a Lie group on a manifold. However, in our case, P is a symplectic manifold and the improper action of G on P is Hamiltonian, which allows algebraic reduction as discussed in Section 6.9. Algebraic reduction gives rise to a Poisson algebra defined in terms of differential schemes, which are differential-geometry analogues of schemes in algebraic geometry. The Poisson algebra of algebraic reduction encodes a lot of information about the action of G on P. The problem arises as to how to decode the information encoded in algebraic reduction and use it in applications. We return to this question in Chapter 7. Algebraic reduction of the zero level of the momentum map was introduced ´ by Sniatycki and Weinstein (1983). Algebraic reduction at non-zero co-adjoint orbits was introduced independently by Wilbour (1993) and Kimura (1993). Theorem 6.9.6 (the shifting theorem) was proved by Arms (1996). Example 6.9.4 was first investigated in the context of algebraic reduction by Arms, Gotay and Jennings (Arms et al., 1990). Example 6.9.7 was first outlined in ´ Sniatycki and Weinstein (1983); a full analysis of this example was given in ´Sniatycki (2005). Lemma 3.8.1 was proved by Bates (2007). Chapter 7 is devoted to the problem of commutation of geometric quantization and reduction. The term ‘geometric quantization’ is used in mechanics and in representation theory. In both cases, it describes essentially the same mathematical procedure, but its starting points and aims are different in the two cases. In representation theory, quantization is a technique for obtaining a unitary representation of a connected Lie group from its action on a symplectic manifold. In quantum mechanics, geometric quantization provides a geometric way to transition from the classical to the quantum description of a physical system. In physics, we often study a quantum subsystem of a classical system. This is usually done by starting with a classical description of the whole system and then imposing constraints to single out the subsystem, followed by reduction of spurious degrees of freedom and subsequent quantization. We expect that the physical results obtained will be the same as the results of a study of the subsystem in terms of quantization of the whole system. This expectation can be rephrased as the principle that quantization commutes with reduction. The importance of commutation of quantization and reduction was realized in the study of the quantization of gauge theories and general relativity. According to Noether’s Second Theorem (Noether, 1918), the presence of a gauge symmetry leads to a constraint in the theory, given by J = 0, where J is the momentum map for the gauge group action (Binz et al., 2006). In

Introduction

9

the studies by Bleuler (1950) and Gupta (1950) of the quantization of electrodynamics, these authors quantized the full space of the Cauchy data for the electromagnetic field and imposed an appropriate constraint on the space of quantum states. On the other hand, Dirac’s study of the quantization of gravity led to a distinction between first-class and second-class constraints (Dirac, 1964). First-class constraints could be imposed on the quantum level, whereas second-class constraints had to be imposed on the classical level. It is rather difficult to give a definite answer in the framework of quantum field theory to the question of whether quantization and reduction commute. Guillemin and Sternberg (1982) proved that geometric quantization commutes with reduction provided that some strong technical assumptions are satisfied. Their approach was formulated in the framework of the representation theory of Lie groups. Geometric quantization has its roots in the work of Kirillov (1962), Auslander and Kostant (1971), Kostant (1966; 1970) and Souriau (1966). A comprehensive bibliography was given in a book by Woodhouse (1992). We begin with a discussion of the significance of commutation of quantization and reduction in the framework of representation theory. In Section 7.1, ´ we give a review of geometric quantization following Sniatycki (1980). In Section 7.2, we discuss in general terms the problem of commutation of geometric quantization and singular reduction. This problem has been stud´ ied by Bates, Cushman, Hamilton and Sniatycki (Bates et al., 2009), using ´ an algebraic approach based on Sniatycki’s earlier results on commutation of ´ quantization and algebraic reduction (Sniatycki, 2012). The approach to the problem of commutation of geometric quantization and singular reduction, as well as many of the results presented in this section, is new. In Section 7.3, we discuss various special cases. We begin with the case of a Kähler quantization of a compact symplectic manifold (P, ω) with a Hamiltonian action of a compact connected Lie group G, investigated by Guillemin and Sternberg (1982) and by Sjamaar (1995). We discuss which of the results of Guillemin and Sternberg and of Sjamaar follow from our general approach, and which of these results are specific to the approach that they used. Our results also hold when the symplectic manifold P and the Lie group G are not compact, and agree with the results of Huebschmann (2006). Next, we discuss conditions for commutation of singular reduction and quantization with respect to a real polarization. For a free and proper action of G on P, these conditions ´ were first introduced by Sniatycki (1983), and subsequently studied by Duval, ´ Elhadad, Gotay, Sniatycki and Tuynman; see Duval et al. (1990; 1991) and the references therein.

10

Introduction

In Section 7.4, we discuss commutation of quantization and reduction at non-zero quantizable co-adjoint orbits using the shifting trick described in ´ Section 6.6. The approach adopted here follows Sniatycki (2012). In Section 7.5, we discuss the problem of commutation of geometric quantization and algebraic reduction. In fact, algebraic reduction was invented for this problem. In 1980, at a conference in Banff, Guillemin presented some unpublished results from his work with Sternberg. This lecture motivated the present author to investigate possible ways to generalize the results of Guillemin and Sternberg to singular momentum maps. In 1981, the author presented at a conference in Clausthal a paper discussing some examples in quantum mechanics which could be interpreted as quantum reduction of sin´ gular constraints (Sniatycki, 1983). Weinstein’s reaction to this lecture led to ´ a collaboration, which culminated in publication of a joint paper (Sniatycki and Weinstein, 1983). We discuss some special cases when the polarization is Kähler or real, and obtain results similar to the results for singular reduction. We conclude with some partial results on commutation of quantization and reduction for an improper action of the symmetry group. Chapter 8 contains two more examples of reduction of symmetry. In Section 8.1, we discuss reduction of symmetry for a proper action of the symmetry group G of a non-holonomically constrained Hamiltonian system. We begin with a description of the distributional Hamiltonian formulation of con´ strained dynamics, following Bates and Sniatycki (1993). Next, we reformulate the distributional Hamiltonian formulation in terms of the almost-Poisson formulation of van der Schaft and Maschke (1994). This encodes the distributional Hamiltonian structure of the theory in the structure of C ∞ (P). The space C ∞ (P)G of G-invariant functions is an almost-Poisson subalgebra of C ∞ (P). Since the differential structure C ∞ (P/G) of the orbit space P/G is isomorphic to C ∞ (P)G , it inherits an almost-Poisson algebra structure, which was first used to discuss reduction by Koon and Marsden (1998). The almost-Poisson bracket is a derivation and gives rise to a family P(P/G) of almost-Poisson vector fields on P/G. The orbits of this family are manifolds. Each orbit Q carries a generalized distribution D Q spanned by the restriction of P(P/G) to Q. Moreover, D Q carries a symplectic form Q defined by the almost-Poisson structure of C ∞ (Q). A comprehensive presentation of the current state of the geometry of non-holonomically constrained Hamiltonian systems can be found in a recent book by Cushman, Duistermaat ´ and Sniatycki (Cushman et al., 2010). In Section 8.2, we discuss reduction of symmetries for a proper action of the symmetry group G of a Dirac structure. A Dirac structure on a manifold P is a maximal isotropic subbundle D of the Pontryagin bundle P = T Q × Q T ∗ Q

Introduction

11

of a manifold Q. The notion of a Dirac structure was introduced by Courant and Weinstein (1988); see also Courant (1990) and Dorfman (1993). A proper action of a Lie group G on Q is a symmetry of D if the action of G lifted to P preserves D. Since the action of G on Q is proper, it follows that the action of G on P is proper. Moreover, the action of G on P preserves D and induces a proper action of G on D. Hence, the orbit space D/G is a stratified subcartesian space. The main problem of the reduction is to relate the stratification of P/G to stratifications of P/G and Q/G. We introduce a G-invariant Riemannian metric k on Q, and decompose T Q into its vertical component ver T Q, which is tangent to orbits of G, and its horizontal component hor T Q, which is k-orthogonal to ver T Q. We decompose T ∗ Q and P in a similar way. Even for a free and proper action of G, we may encounter intersection singularities because the intersection of D with T Q need not be clean. Therefore, we begin with a study of regular reduction for a free and proper action of G on Q. In this case Q is a left principal fibre bundle with structure group G. We show that the reduced Pontryagin bundle P/G is isomorphic to the direct sum of T (Q/G) ⊕ T ∗ (Q/G) and Q[g] ⊕ Q[g∗ ], where Q[g] and Q[g∗ ] are the adjoint and the co-adjoint bundle, respectively, of Q. The regular reduction gives rise to the quotient D/G in the form of the direct sum of T (Q/G) ⊕ T ∗ (Q/G) and Q[g] ⊕ Q[g∗ ], which can be interpreted as a generalized Dirac structure on Q/G. Next, we consider a proper non-free action of G on Q. For each stratum N of the orbit type stratification of Q, the quotient D/G defines a generalized Dirac structure on N . Moreover, D/G is uniquely determined by the collection of generalized Dirac structures on all strata of the orbit type stratification of Q. Reduction of the symmetries of Dirac structures was carried out for a free and proper Dirac action by Blankenstein and van der Schaft (2001) and Blankenstein (2000) in the context of generalized Poisson structures, and by Bursztyn, Cavalcanti and Gualtieri (Bursztyn et al., 2007) in the setting of Courant algebroids. Blankenstein and Ratiu (2004) treated a Dirac structure with symmetries as a generalized Poisson structure with a momentum map, and performed singular reduction at singular values of the momentum map. Jotz, ´ Ratiu and Sniatycki (Jotz et al., 2011) studied singular reduction completely within the Dirac category. In a recent paper, Jotz and Ratiu (2012) discussed the reduction of non-holonomic systems in terms of Dirac reduction. The authors of the references mentioned above were interested mainly in the horizontal reduced Dirac structure. The presence of a vertical Dirac structure was observed by Yoshimura and Marsden (2007) in an example in which the action of the symmetry group was free and proper and the horizontal

12

Introduction

reduced Dirac structure vanished identically. The presentation given here ´ follows that of Sniatycki (2011). The interdependence of the chapters is shown in the diagram below. Ch. 2

→

Ch. 3 ↓ Ch. 5

→

Ch. 4

→

Ch. 6 ↓ Ch. 8

→

Ch. 7

PART I Differential geometry of singular spaces

2 Differential structures

2.1 Differential spaces In this section, we describe the category of differential spaces, which includes finite-dimensional manifolds as a subcategory. Definition 2.1.1 A differential structure on a topological space S is a family C ∞ (S) of real-valued functions on S satisfying the following conditions: 1. The family { f −1 (I ) | f ∈ C ∞ (S) and I is an open interval in R} is a subbasis for the topology of S. 2. If f 1 , . . . , f n ∈ C ∞ (S) and F ∈ C ∞ (Rn ), then F( f 1 , . . . , f n ) ∈ C ∞ (S). 3. If f : S → R is a function such that, for every x ∈ S, there exist an open neighbourhood U of x and a function f x ∈ C ∞ (S) satisfying f x|U = f |U , then f ∈ C ∞ (S). Here, the subscript vertical bar | denotes a restriction. Functions f ∈ C ∞ (S) are called smooth functions on S. It follows from Condition 1 above that smooth functions on S are continuous. Condition 2 with F( f 1 , f 2 ) = a f 1 + b f 2 , where a, b ∈ R, implies that C ∞ (S) is a vector space. Similarly, taking F( f1 , f 2 ) = f 1 f 2 , we conclude that C ∞ (S) is closed under multiplication of functions. A topological space S endowed with a differential structure is called a differential space. In his original definition, Sikorski (1972) defined C ∞ (S) to be a family of functions satisfying Condition 2. Then, he used Condition 1 to define a topology on S. Finally, he imposed Condition 3 as a consistency condition. An example of a differential space is the Euclidean space Rn with the standard topology and the standard differential structure C ∞ (Rn ) as defined in

16

Differential structures

calculus. Another example is a smooth manifold M with the differential structure given by the ring C ∞ (M) of smooth functions on M. We also have some more exotic examples below. Example 2.1.2 S is an arbitrary set endowed with the trivial topology (the empty set and S are the only open sets), and its differential structure C ∞ (S) is the set of all constant functions on S. Example 2.1.3 S is an arbitrary set endowed with the discrete topology (every subset of S is open), and C ∞ (S) is the set of all functions on S. Let (R, C ∞ (R)) and (S, C ∞ (S)) be differential spaces. Definition 2.1.4 A map ϕ : R → S is smooth if ϕ ∗ f = f ◦ ϕ ∈ C ∞ (R) for every f ∈ C ∞ (S). A smooth map ϕ between differential spaces is a diffeomorphism if it is invertible and its inverse is smooth. Proposition 2.1.5 A smooth map between differential spaces is continuous. Proof Let ϕ : R → S be smooth, and let U be an open set in S. We need to show that ϕ −1 (U ) is open in R. Let x ∈ ϕ −1 (U ). By Condition 1, there exist functions f 1 , . . . , f n ∈ C ∞ (S) and open intervals I1 , . . . , In in R such that f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In ) is an open neighbourhood of ϕ(x) contained in U . Then x ∈ ϕ −1 ( f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In )) ⊆ ϕ −1 (U ). But ϕ −1 ( f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In )) = ( f 1 ◦ ϕ)−1 (I1 ) ∩ . . . ∩ ( f n ◦ ϕ)−1 (In ) = (ϕ ∗ f 1 )−1 (I1 ) ∩ . . . ∩ (ϕ ∗ f n )−1 (In ).

Since the functions ϕ ∗ f 1 , . . . , ϕ ∗ f n ∈ C ∞ (R), Condition 1 applied to C ∞ (R) implies that (ϕ ∗ f 1 )−1 (I1 ) ∩ . . . ∩ (ϕ ∗ f n )−1 (In ) is open in R. Thus, every point x ∈ ϕ −1 (U ) has an open neighbourhood contained in ϕ −1 (U ). Therefore ϕ −1 (U ) is open in R. Hence, ϕ : R → S is continuous. Corollary 2.1.6 A diffeomorphism of differential spaces is a homeomorphism of the underlying topological spaces. An alternative way of constructing a differential structure on a set S, also used by Sikorski (1972), goes as follows. Let F be a family of real-valued functions on S. Endow S with the topology generated by a subbasis { f −1 (I ) | f ∈ F and I is an open interval in R}.

(2.1)

2.1 Differential spaces

17

Define C ∞ (S) by the requirement that h ∈ C ∞ (S) if, for each x ∈ S, there exist an open subset U of S, functions f 1 , . . . , f n ∈ F, and F ∈ C ∞ (Rn ) such that h |U = F( f 1 , . . . , f n )|U .

(2.2)

Clearly, F ⊆ C ∞ (S). In Theorem 2.1.7 below, we show that C ∞ (S) defined here is a differential structure on S. We refer to it as the differential structure on S generated by F. Theorem 2.1.7 The family C ∞ (S) defined above is a differential structure on S. Proof Condition 1 of Definition 2.1.1 is satisfied by the choice of topology on S. To show that Condition 2 is satisfied, let h 1 , . . . , h n ∈ C ∞ (S) and F ∈ C ∞ (Rn ). By definition, for each x ∈ S, there exist an open neighbourhood U of x, functions Fi ∈ C ∞ (Rn i ) for i = 1, . . . , n, and functions fi ji ∈ F, where ji = 1, . . . , n i , such that h i|U = Fi ( f i1 , . . . , f ini )|U for every i = 1, . . . , n. Then, F(h 1 , . . . , h n )|U = F(h 1|U , . . . , h n|U ) = F(F1 ( f 11 , . . . , f 1n 1 )|U , . . . , Fn ( f n1 , . . . , f nn n )|U ) = F(F1 ( f 11 , . . . , f 1n 1 ), . . . , Fn ( f n1 , . . . , f nnn ))|U = F(F1 , . . . , Fn )(( f 11 , . . . , f 1n 1 ), . . . , ( fn1 , . . . , f nn n ))|U . Since F(F1 , . . . , Fn ) ∈ C ∞ (Rm ), where m = n 1 + . . . + n n and f i ji ∈ F, it follows that F(h 1 , . . . , h n ) ∈ C ∞ (S). Hence, Condition 2 is satisfied. To verify Condition 3, suppose that h : S → R is a function satisfying the assumption of Condition 3. In other words, for every x ∈ S, there exists an open neighbourhood U of x and h x ∈ C ∞ (S) such that h |U = h x|U . By the construction of C ∞ (S), there exist a neighbourhood Ux of x in S, functions f x1 , . . . , f xn ∈ F and a function Fx ∈ C ∞ (Rn ) such that h x|Ux = Fx ( f x1 , . . . , f xn )|Ux . Hence, h |U ∩Ux = h x|U ∩Ux = Fx ( f x1 , . . . , f xn )|U ∩Ux , which implies that h ∈ C ∞ (S). Hence, C ∞ (S) is a differential structure on S.

18

Differential structures

Let R be a differential space with a differential structure C ∞ (R), and let S be an arbitrary subset of R endowed with the subspace topology (open sets in S are of the form S ∩ U , where U is an open subset of R). Let R(S) = { f |S | f ∈ C ∞ (R)}.

(2.3)

In other words, R(S) is the space of restrictions to S of smooth functions on R. Proposition 2.1.8 The space R(S) of restrictions to S ⊆ R of smooth functions on R generates a differential structure C ∞ (S) on S such that the differential-space topology of S coincides with its subspace topology. In this differential structure, the inclusion map ι : S → R is smooth. Proof Theorem 2.1.7 ensures that C ∞ (S) is a differential structure on S. By assumption, the family { f −1 (I ) | f ∈ C ∞ (S) and I is an open interval in R} is a subbasis for the topology of R. Hence, { f −1 (I ) ∩ S | f ∈ C ∞ (R) and I is an open interval in R} = { f |S−1 (I ) | f ∈ C ∞ (R) and I is an open interval in R} = { f |S−1 (I ) | f |S ∈ R(S) and I is an open interval in R} is a subbasis for the subspace topology of S. Therefore, the differential-space topology of S coincides with its subspace topology. For each f ∈ C ∞ (R), the pull-back ι∗ f of f by the inclusion map is the restriction of f to S. Hence, ι∗ f ∈ R(S) ⊆ C ∞ (S) and the inclusion map ι : S → R is smooth. In the following, for every subset S of a differential space R, we use the differential structure C ∞ (S) on S described above. If we want to emphasize that S has the subspace topology and C ∞ (S) is generated by restrictions to S of functions in C ∞ (R), we say that S is a differential subspace of R. If S with the differential structure C ∞ (S) is a manifold, we say that S is a submanifold of the differential space R. We shall also encounter the situation in which S is a subset of a differential space R endowed with a topology T that is finer than the subspace topology. In this case, we consider the space C ∞ (S) of functions on S obtained from C ∞ (R) as follows. A function f ∈ C ∞ (S) if, for each set U ⊆ S that is open in the topology T , there exists a function h ∈ C ∞ (R) such that f |U = h |U . If C ∞ (S) satisfies Condition 1 of Definition 2.1.1, then we say that S is an immersed differential space. If, in addition, S with the differential structure C ∞ (S) is a manifold, then we say that S is an immersed manifold.

2.1 Differential spaces

19

differential structures C ∞ (S1 ) and S1 and π2 : S1 × S2 → S2 be the canonical projections on the first and the second factor, respectively. Consider the family π1∗ (C ∞ (S1 )) ∪ π2∗ (C ∞ (S2 )) of functions on S1 × S2 consisting of pull-backs to S1 × S2 of functions in C ∞ (S1 ) and functions in C ∞ (S2 ). Let S1 and S2 be differential spaces with C ∞ (S2 ), respectively. Let π1 : S1 × S2 →

Proposition 2.1.9 The family π1∗ (C ∞ (S1 )) ∪ π2∗ (C ∞ (S2 )) of functions on S1 × S2 generates a differential structure C ∞ (S1 × S2 ) on S1 × S2 such that the differential-space topology on S1 × S2 coincides with its product topology. In this differential structure, the projections π1 and π2 are smooth. Proof Theorem 2.1.7 ensures that C ∞ (S1 × S2 ) is a differential structure on S1 × S2 . By assumption, the families of sets { f i−1 (I ) | f i ∈ C ∞ (Si ) and I is an open interval in R}, where i = 1, 2, are subbases for the topologies of S1 and S2 , respectively. Hence, the family of sets { f 1−1 (I1 ) × S2 , S1 × f 2−1 (I2 )}, where f i ∈ C ∞ (Si ) and I1 and I2 are open intervals in R, is a subbasis for the product topology in S1 × S2 . On the other hand, f 1−1 (I1 ) × S2 = {(x1 , x2 ) ∈ S1 × S2 | f 1 (x1 ) ∈ I1 }

= {(x 1 , x2 ) ∈ S1 × S2 | π1∗ f 1 ((x1 , x2 )) ∈ I1 }.

Similarly, S1 × f 2−1 (I2 ) = {(x1 , x2 ) ∈ S1 × S2 | f 2 (x2 ) ∈ I2 }

= {(x 1 , x2 ) ∈ S1 × S2 | π2∗ f 2 ((x1 , x2 )) ∈ I2 }.

Therefore, the family of sets { f 1−1 (I1 )× S2 , S1 × f 2−1 (I2 )}, where fi ∈ C ∞ (Si ) and I1 and I2 are open intervals in R, is contained in the family { f −1 (I ) | f i ∈ C ∞ (S1 × S2 ) and I is an open interval in R}. Therefore, the differential-space topology on S1 ×S2 coincides with the product topology. By construction, for each i = 1, 2 and f i ∈ C ∞ (Si ), the pull-back πi∗ f i ∈ ∞ C (S1 × S2 ). This implies that the projections π1 and π2 are smooth. Now consider an equivalence relation ∼ on a differential space S with differential structure C ∞ (S). Let R = S/ ∼ be the set of equivalence classes of ∼, and let ρ : S → R be the map assigning to each x ∈ S its equivalence class ρ(s).

20

Differential structures

Theorem 2.1.10 The space of functions on R, given by C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (S)}, is a differential structure on R. In this differential structure, the projection map ρ : S → R is smooth. Proof The differential-space topology of R defined by C ∞ (R) has a subbasis consisting of sets f −1 (I ), where f ∈ C ∞ (R) and I is an open interval. Since ρ −1 ( f −1 (I )) = (ρ ∗ f )−1 (I ) and ρ ∗ f ∈ C ∞ (S), it follows that ρ −1 ( f −1 (I )) is open in S. Hence, f −1 (I ) is open in the quotient topology of S for every open interval I and each f ∈ C ∞ (R). Therefore, the quotient topology of S is finer than the topology defined by C ∞ (R). This implies that the projection map ρ : S → R is continuous. For f 1 , . . . , f n ∈ C ∞ (R) and F ∈ C ∞ (Rn ), ρ ∗ F( f 1 , . . . , f n ) = F(ρ ∗ f 1 , . . . , ρ ∗ f n ) ∈ C ∞ (S). This shows that C ∞ (S) satisfies Condition 2 of Definition 2.1.1. To verify Condition 3, suppose that we have a function f : R → R such that, for every y ∈ R, there exist a neighbourhood U y of y in R and a function f y ∈ C ∞ (R) satisfying f |U y = f y|U y . Hence, for every x ∈ S, there exist a neighbourhood ρ −1 (Uρ(x) ) in S and a function ρ ∗ f ρ(x) ∈ C ∞ (S) such that ρ ∗ f |ρ −1 (Uρ(x) ) = ρ ∗ f ρ(x)|ρ −1 (Uρ(x) ) . This implies that ρ ∗ f ∈ C ∞ (S). Hence, f ∈ C ∞ (R). Thus, C ∞ (R) is a differential structure on R. By definition, f ∈ C ∞ (R) implies that ρ ∗ f ∈ C ∞ (S). Hence, the projection map ρ : S → R is smooth. It should be emphasized that, in general, the quotient topology of R = S/ ∼ is finer than the differential-space topology defined by C ∞ (R). Waz˙ ewski (1934) gave an example of a free improper action of R1 on R2 such that the only invariant smooth functions on R2 are constant functions. Let R denote the space of orbits of this action, and let ρ : R2 → R denote the orbit map. In this case, the differential structure C ∞ (R) consists of constant functions and the differential-space topology of R is trivial. A condition for the differential-space topology to coincide with the quotient topology is given below. Proposition 2.1.11 We use the notation of Theorem 2.1.10. The topology of R induced by C ∞ (R) coincides with the quotient topology of R if, for each set U in R which is open in the quotient topology, and each y ∈ U , there exists a function f ∈ C ∞ (R) such that f (y) = 1 and f |R\U = 0, where R\U denotes the complement of U in R.

2.2 Partitions of unity

21

Proof Let U be a set in R which is open in the quotient topology of R. For each y ∈ U , there exists f ∈ C ∞ (R) such that f (y) = 1 and f |R\U = 0. Hence, y ∈ f −1 (0, 2) ⊆ U . Moreover, f ∈ C ∞ (R) implies that f −1 (0, 2) is open in the differential-space topology of R. Therefore, U is open in the differential-space topology of R. Thus, the differential-space topology of R is finer than its quotient topology. By Theorem 2.1.10, the quotient topology of R is finer than its differential-space topology. Hence, the quotient topology of R coincides with its differential-space topology. We can characterize smooth Hausdorff manifolds of dimension n as Hausdorff differential spaces S such that every point x ∈ S has a neighbourhood U diffeomorphic to an open subset V of Rn . Here, the differential structures on U and V are generated by restrictions of smooth functions of S and Rn , respectively. We can weaken this definition by not requiring that V is open in Rn , and allowing n to be an arbitrary non-negative integer. Definition 2.1.12 A differential space S is subcartesian if it is Hausdorff and every point x ∈ S has a neighbourhood U diffeomorphic to a subset V of Rn . It should be noted that V in Definition 2.1.12 may be an arbitrary subset of Rn , and n may depend on x ∈ S. As in the theory of manifolds, diffeomorphisms of open subsets of S onto subsets of Rn are called charts on S. The family of all charts is the complete atlas on S.1

2.2 Partitions of unity The aim of this section is to establish the existence of a partition of unity for locally compact, second countable Hausdorff differential spaces. Lemma 2.2.1 For every open subset U of a differential space S and every x ∈ U , there exists f ∈ C ∞ (S) satisfying f |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. Proof Let U be open in S, and let x ∈ U. It follows from Condition 1 of Definition 2.1.1 that there exist a map ϕ = ( f1 , . . . , f n ) : S → Rn , with f 1 , . . . , f n ∈ C ∞ (S), and an open set U˜ ⊆ Rn such that x ∈ ϕ −1 (U˜ ) ⊆ U. Since ϕ(x) ∈ U˜ ⊆ Rn , there exists F ∈ C ∞ (Rn ) such that F|V˜ = 1 for some neighbourhood V˜ of ϕ(x) in Rn contained in U˜ , and F ˜ = 0 for some open |W

1 The original definition of a subcartesian space, due to Aronszajn (1967), was formulated in

terms of charts.

22

Differential structures

set W˜ in Rn such that U˜ ∪ W˜ = Rn . Since ϕ is continuous, V = ϕ −1 (V˜ ) and W = ϕ −1 (W˜ ) are open in V . Moreover, ϕ −1 (U˜ ) ⊆ U and U˜ ∪ W˜ = Rn imply that U ∪ W = S. By Condition 2, f = F( f 1 , . . . , f n ) ∈ C ∞ (S). Furthermore, f |V = (F ◦ ϕ)|V = F|ϕ(V ) = F|V˜ = 1. Similarly, f |W = F|W˜ = 0, which completes the proof. It follows from Lemma 2.2.1 that, for a Hausdorff differential space S, functions in C ∞ (S) separate points. In other words, for every pair (x, y) of points in S, there exist disjoint open neighbourhoods V and W of x and y, respectively, and a function f ∈ C ∞ (S) such that f |V = 1 and f |W = 0. Lemma 2.2.2 Let S be locally compact, Hausdorff and second countable. Then every open cover {Uα } of S has a countable, locally finite refinement consisting of open sets with compact closures. Proof Since S is second countable, there exists a countable family {Un } of open sets in S, which forms a basis for the topology of S. Let {Un k } be a subcollection consisting of sets with compact closures. The assumption that S is Hausdorff and locally compact implies that {Un k } is a basis for the topology of S. This can be seen as follows. Since S is locally compact, given a point x ∈ S there exists a compact set C in S containing an open neighbourhood U of x. That is, x ∈ U ⊆ C. Since C is a compact subset of the Hausdorff space S, it is closed. Hence, U ⊆ C, which implies that U is compact. The assumption that {Un } is a basis implies that there exists n x such that x ∈ Un x ⊆ U . Hence, U n x is compact. Therefore, Un x ∈ {Un k }. We set V−1 = V0 = ∅ and take V1 = Un 1 . There exists a smallest integer k1 such that V 1 ⊆ Un 1 ∪. . .∪Un k1 . We now set V2 = V1 ∪ Un 2 ∪. . .∪Un k1 . Continuing in this way, we obtain a sequence of open sets V j = Un 1 ∪. . .∪Un k j for every j ∈ N, where n k j is the smallest integer such that V j−1 ⊆ V j−1 ∪ Un k j−1 +1 ∪. . .∪Un k j . For each j, the closure V j of V j is contained in V j+1 and ∪∞ j=1 V j = S. For each j ∈ N, the set V j \V j−1 is compact and is contained in the open set V j+1 \V j−2 . Let {Uα }α∈A be an arbitrary open cover of S. Hence, {Uα ∩ (V j+1 \V j−2 )}α∈A is an open cover of the compact set V j \V j−1 and it admits j j a finite subcover. We denote by {W1 , . . . , Wm j } the finite collection of sets in j

{Uα ∩ (V j+1 \V j−2 )}α∈A which cover V j \V j−1 . Each Wi is contained in a j Wi j Wi

j

compact set V j+1 . Hence, is compact. Moreover, for some α ∈ A, Wi = j Uα ∩(V j+1 \V j−2 ), so that ⊆ Uα . Moreover, Wi ∩ Wlk = ∅ if | j −k |> 4. mj j i Finally, ∪∞ j=1 ∪i=1 W j = S. Hence, the collection {Wi | i = 1, . . . , m j , j ∈ N}

2.2 Partitions of unity

23

is a countable, locally finite refinement of {Uα } and consists of open sets with compact closures. Definition 2.2.3 A countable partition of unity on a differential space S is a countable family of functions { f i } ⊆ C ∞ (S) such that: (a) The collection of their supports is locally finite. (b) fi (x) ≥ 0 for each i and each x ∈ S. ∞ (c) i=1 f i (x) = 1 for each x ∈ S. Let {Uα } be an open cover of S. A partition of unity { f i } is subordinate to {Uα } if, for each i, there exists α such that the support of fi is contained in Uα . Theorem 2.2.4 Let S be a differential space with differential structure C ∞ (S), and let {Uα } be an open cover of S. If S is Hausdorff, locally compact and second countable, then there exists a countable partition of unity { fi } ⊆ C ∞ (S), subordinate to {Uα } and such that the support of each fi is compact. Proof Let {Uα } be an open cover of a differential space S. Since S is Hausdorff, locally compact and second countable, there exists a family Vi , i = 0, 1, . . . , of open sets with compact closures in S such that V0 = ∅, V j ⊆ V j+1 and ∪∞ j=1 V j = S. For x ∈ S, let i x be the largest integer such that x ∈ S\V i x . Choose αx such that x ∈ Uαx . By Conditions 1 and 2 of Definition 2.1.1, there exist functions f 1 , . . . , f n ∈ C ∞ (S) such that the smooth map ϕx = ( f 1 , . . . , f n ) : S → Rn takes x to ϕx (x) = (0, . . . , 0) ∈ Rn and ϕx−1 ((−3, 3)n ) ⊆ Uαx ∩ (Vi x +2 \V i x ). Let F ∈ C ∞ (Rn ) be a non-negative function such that F|[−1,1]n = 1 and F|Rn \(−2,2)n = 0. Then f x = F( f 1 , . . . , f n ) is in C ∞ (S) and has compact support contained in ϕx−1 ((−3, 3)n ), and f x has value 1 in ϕx−1 ((−1, 1)n ). For each i ≥ 1, choose a finite set of points xi1 , . . . , xiki in S whose ((−1, 1)n ) cover V i \Vi−1 . The functions f xi j ∈ C ∞ (S) neighbourhoods ϕx−1 ij are non-negative and their supports form a locally finite family of sets in S. Hence, f =

ki ∞

f xi j

i=1 j=1

is a well-defined positive function on S and Condition 3 implies that f ∈ C ∞ (S). Each function h i j = f −1 f xi j has compact support, and the family {h i j } forms a partition of unity on S subordinate to the cover {Uα }. Condition 2 ensures that the functions h i j are in C ∞ (S).

24

Differential structures

Many problems that arise in differential geometry are easy to solve locally. Partitions of unity are used to construct global solutions from such local solutions. In particular, we have the following corollary to Theorem 2.2.4. Corollary 2.2.5 Let S be a Hausdorff, locally compact and second countable differential space with differential structure C ∞ (S), and let R be a closed subset of S. The differential structure C ∞ (R) induced by the inclusion map ι : R → S consists of restrictions to R of functions in C ∞ (S). In the following chapters we restrict our attention to Hausdorff, locally compact, second countable subcartesian spaces. Hence, we shall be able to rely on the existence of partitions of unity. Note that a subcartesian space is Hausdorff; see Definition 2.1.12. Moreover, Definition 2.1.12 and Condition 1 of Definition 2.1.1 imply that a subcartesian space is locally compact. Thus, we make the following assumption. Assumption 2.2.6 All subcartesian differential spaces considered here are second countable.

3 Derivations

In this chapter, for a differential space S, we study the properties of derivations of C ∞ (S). We show that if S is subcartesian, then derivations of C ∞ (S) admit maximal integral curves. We define vector fields on a subcartesian space to be derivations that generate local one-parameter groups of local diffeomorphisms of the space. We conclude the chapter with a proof showing that orbits of a family of vector fields on a subcartesian space are immersed manifolds.

3.1 Basic properties Definition 3.1.1 A derivation of C ∞ (S) is a linear map X : C ∞ (S) → C ∞ (S) : f → X ( f ) satisfying Leibniz’s rule X ( f1 f 2 ) = X ( f1 ) f2 + f 1 X ( f2 )

(3.1)

for every f 1 , f 2 ∈ C ∞ (S). We denote by Der C ∞ (S) the space of derivations of C ∞ (S). This has the structure of a Lie algebra, with the Lie bracket [X 1 , X 2 ] defined by [X 1 , X 2 ]( f ) = X 1 (X 2 ( f )) − X 2 (X 1 ( f )) for every X 1 , X 2 ∈ Der C ∞ (S) and f ∈ C ∞ (S). Moreover, Der C ∞ (S) is a module over the ring C ∞ (S), and [ f 1 X 1 , f 2 X 2 ] = f 1 f 2 [X 1 , X 2 ] + f 1 X 1 ( f 2 )X 2 − f 2 X 2 ( f 1 )X 1 for every X 1 , X 2 ∈ Der C ∞ (S) and f 1 , f 2 ∈ C ∞ (S).

26

Derivations

Definition 3.1.2 A derivation of C ∞ (S) at x ∈ S is a linear map v : C ∞ (S) → R such that v( f 1 f 2 ) = v( f 1 ) f 2 (x) + f 1 (x)v( f 2 )

(3.2)

for every f 1 , f2 ∈ C ∞ (S). We interpret derivations of C ∞ (S) at x ∈ S as tangent vectors to S at x. The set of all derivations of C ∞ (S) at x is denoted by Tx S and is called the tangent space to S at x.1 If X is a derivation of C ∞ (S), then for every x ∈ S we have a derivation X (x) of C ∞ (S) at x given by X (x) : C ∞ (S) → R : f → X (x) f = (X f )(x).

(3.3)

The derivation (3.3) is called the value of X at x. Clearly, the derivation X is uniquely determined by the collection {X (x) | x ∈ S} of its values at all points in S. In order to avoid confusion between derivations of C ∞ (S) and derivations of C ∞ (S) at a point in S, we shall often refer to the former as global derivations of C ∞ (S), or simply as global derivations if C ∞ (S) is understood. Lemma 3.1.3 If f ∈ C ∞ (S) is a constant function, then X ( f ) = 0 for all X ∈ Der C ∞ (S). Proof If f ∈ C ∞ (S) is identically zero, then f 2 = f = 0, and Leibniz’s rule implies that X ( f ) = X ( f 2 ) = 2 f X ( f ) = 0 for every X ∈ Der C ∞ (S). Similarly, if f is a non-zero constant function, that is, f (x) = c = 0 for all x ∈ S, then f 2 = c f , and the linearity of derivations implies that X ( f 2 ) = X (c f ) = cX ( f ). On the other hand, Leibniz’s rule implies that X ( f 2 ) = 2 f X ( f ) = 2cX ( f ). Hence, cX ( f ) = 2cX ( f ). Since c = 0, it follows that X ( f ) = 0. Lemma 3.1.4 If f ∈ C ∞ (S) vanishes identically in an open set U ⊆ S, then X ( f )|U = 0 for all X ∈ Der C ∞ (S). Proof If f ∈ C ∞ (S) vanishes identically in an open set U ⊆ S, then for each x ∈ U , by Lemma 2.2.1 there exist h ∈ C ∞ (S) satisfying h |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. This implies that h f = 0. Therefore, 0 = X (h f ) = h X ( f )+ f X (h) for every derivation X. Evaluating this identity at x, we obtain X ( f )(x) = 0 because f (x) = 0. Hence, X ( f )|U = 0. 1 The term ‘Zariski tangent space’ is also used.

3.1 Basic properties

27

Proposition 3.1.5 Let U be an open subset of a differential space S. A derivation X of C ∞ (S) defines a unique derivation X |U of C ∞ (U ), called the restriction of X to U , such that X |U ( f |U ) = (X ( f ))|U

(3.4)

for every f ∈ C ∞ (S). Conversely, if Y is a derivation of C ∞ (U ), then for each x ∈ U there exist an open neighbourhood V of x contained in U , and X ∈ Der C ∞ (S) such that Y|V = X |V . Proof Given a derivation X of C ∞ (S), we need to define its restriction X |U . The action of X |U on the restrictions to U of functions in C ∞ (S) is defined in equation (3.4). In general, given h ∈ C ∞ (U ), for every x ∈ U there exist f x ∈ C ∞ (S) and an open neighbourhood Vx of x in U such that h |Vx = f x|Vx . We define X |U (h) to be the function in C ∞ (S) such that (X |U (h))|V = (X ( f ))|V . First, we show that X |U (h) is well defined. Suppose that for each x ∈ U , there is another choice Vx and f x such that h |Vx = f x|V . Then, h |Vx ∩Vx = x f x|Vx ∩Vx = f x|Vx ∩V . Therefore, ( f x − f x )|Vx ∩Vx = 0 and Lemma 3.1.3 implies x that X ( f x − f x )|Vx ∩Vx = 0. Hence, X ( f x ) is equal to X ( f x ) on Vx ∩ Vx . Since {Vx ∩ Vx }x∈U covers U , it follows that X |U is well defined. Suppose now that Y is a derivation of C ∞ (U ). It follows from Lemma 2.2.1 that there exists f ∈ C ∞ (S) satisfying f |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. Then f |U Y is a derivation of C ∞ (U ), which vanishes on U \W . Hence, it extends to a smooth derivation X of C ∞ (S) such that X |V = Y|V . Let S and R be differential spaces with differential structures C ∞ (S) and respectively, and let ϕ be a diffeomorphism of S onto R. For each X ∈ Der C ∞ (S), the map

C ∞ (R),

ϕ∗ X : C ∞ (R) → C ∞ (R) : f → ϕ∗ X ( f ) = (ϕ −1 )∗ (X (ϕ ∗ f ))

(3.5)

is a derivation of C ∞ (R), called the push-forward of X by the diffeomorphism ϕ. Moreover, ϕ∗ : Der C ∞ (S) → Der C ∞ (R) : X → ϕ∗ X is a Lie algebra isomorphism. Suppose now that ϕ : S → R is a smooth map. Derivations X in Der C ∞ (S) and Y in Der C ∞ (R) are said to be ϕ-related if ϕ ∗ (Y ( f )) = X (ϕ ∗ f ) for every f ∈ C ∞ (R). In this case, we say that X pushes forward to a derivation Y on R and write Y = ϕ∗ X .

(3.6)

28

Derivations

Subcartesian spaces are differential spaces that are locally diffeomorphic to subsets of Rn . Therefore, local information about subcartesian spaces can be obtained by investigating differential structures of subsets of Rn . Taking Proposition 3.1.5 into account, we see that local information about derivations on subcartesian spaces can be completely determined by a study of derivations on subsets of Rn . Theorem 3.1.6 Let S be a differential subspace of Rn , and let X be a derivation of C ∞ (S). For each x ∈ S ⊆ Rn , there exist a neighbourhood U of x in Rn and a vector field Y on Rn such that X (F|S )|U ∩S = (Y (F))|U ∩S for every F ∈ C ∞ (Rn ). Proof Let u be a derivation of C ∞ (S) at x ∈ S ⊆ Rn . For each F ∈ C ∞ (Rn ), the restriction F|S of F to S is in C ∞ (S). It is easy to see that the map C ∞ (Rn ) → R : F → u(F|S ) is a derivation at x of C ∞ (Rn ). We denote the natural coordinate functions on Rn by x 1 , . . . , x n : Rn → R. Every derivation Y of C ∞ (Rn ) is of the form n

Fi

i=1

∂ , ∂xi

where F i = Y (x i ) for i = 1, . . . , n. Let X be a derivation of C ∞ (S), and let F ∈ C ∞ (Rn ). For each x ∈ S, the derivation X (x) of C ∞ (S) at x gives a derivation of C ∞ (Rn ) at x. Hence, X (F|S )(x) = X (x)(F|S ) =

n ∂F i (x)(X (x)(x|S )) ∂xi i=1

=

n i=1

∂F i (x)(X (x|S ))(x). ∂xi

This is valid for every x ∈ S. Hence, n ∂F i (X (x|S )). X (F|S ) = ∂xi i=1

i ) are in C ∞ (S). Since S is a differFor i = 1, . . . , n, the coefficients X (x|S n ential subspace of R , for each x ∈ S there exist a neighbourhood U of x in i ) i Rn and functions F 1 , . . . , F n ∈ C ∞ (Rn ) such that X (x |S |U ∩S = F|U ∩S for each i = 1, . . . , n. Hence,

3.1 Basic properties X (F|S )|U ∩S =

n i=1

∂F F ∂xi

29

i

. |U ∩S

Since the F 1 , . . . , F n are smooth functions on Rn , it follows that Y =

n

Fi

i=1

is a vector field on

∂ ∂xi

Rn .

We can rephrase Theorem 3.1.6 by saying that every derivation on a differential subspace S of Rn can be locally extended to a vector field on Rn . Suppose that S is closed. In this case, we can use a partition of unity on Rn to extend every derivation of C ∞ (S) to a global vector field on Rn . Conversely, suppose that Y =

n i=1

Fi

∂ ∂xi

Rn .

is a vector field on We want to know under what conditions Y restricts to a derivation of S. Since S is a differential subspace of Rn , its differential structure C ∞ (S) is generated by the ring R(S) = {F|S | F ∈ C ∞ (Rn )},

(3.7)

of smooth functions on Rn

in the sense of Propoconsisting of restrictions to S sition 3.1.8. Let N (S) denote the ideal of functions in C ∞ (Rn ) that vanish identically on S: N (S) = {F ∈ C ∞ (Rn ) | F|S = 0}.

(3.8)

We identify R(S) with the quotient C ∞ (Rn )/N (S). Proposition 3.1.7 Every derivation of R(S) at x extends to a unique derivation of C ∞ (S) at x. Proof Let w be a derivation of R(S) at x ∈ S. Consider f ∈ C ∞ (S). There exist an open neighbourhood U of x in Rn and a function Fx ∈ C ∞ (Rn ) such ( f ) = w(F x |S ). Let V be another open neighthat f |U ∩S = Fx |U ∩S . Set w bourhood of x in Rn , and let Hx ∈ C ∞ (Rn ) be a function such that f |V ∩S = Hx |V ∩S . Now U ∩ V ∩ S is an open subset of S, and Fx |U ∩V ∩S = Hx |U ∩V ∩S . Therefore (Fx − Hx )|U ∩V ∩S = 0, i.e. (Fx − Hx )|S ∈ R(S) ⊂ C ∞ (Rn ) vanishes identically on the open subset U ∩V ∩S of S. Hence, w(Fx |S −Hx |S ) = 0. This proves that the extension w is a well-defined derivation of C ∞ (S) that extends the derivation w of R(S) at x. Finally, it is clear that such an extension w of w is uniquely defined.

30

Derivations

Proposition 3.1.8 A derivation w of C ∞ (Rn ) at x ∈ S ⊆ Rn defines a derivation of C ∞ (S) at x if and only if w annihilates N (S); that is, w(F) = 0 for all F ∈ N (S). Proof It follows from Proposition 3.1.7 that derivations at x of C ∞ (S) are determined by derivations of R(S) at x. But R(S) = C ∞ (Rn )/N (S). Hence, a derivation w of C ∞ (Rn ) at x passes to the quotient if and only if w(F) = 0 for all F ∈ N (S). Corollary 3.1.9 A smooth vector field Y on Rn restricts to a derivation of C ∞ (S) if Y preserves the null ideal N (S) of S. In other words, Y (F) ∈ N (S) for every F ∈ N (S). Proof By Proposition 3.1.8, if Y restricts to a derivation of C ∞ (S), then for every x ∈ S and F ∈ N (S), we have Y (F)(x) = 0. Hence, Y (F)|S = 0, which implies that Y (F) ∈ N (S). Conversely, suppose that Y preserves the null ideal of N (S) of S. If F, H ∈ C ∞ (Rn ) are such that F − H ∈ N (S), then Y (F) − Y (H ) ∈ N (S) and Y (F)|S = Y (H )|S . Thus, the map F|S → Y (F)|S is a well-defined derivation of R(S). By Proposition 3.1.7, it extends to a derivation of C ∞ (S). Example 3.1.10 Let S = {(x, y) ∈ R2 | x y = 0}. In other words, S is the union of the x-axis and the y-axis in R2 . Since S is a closed subset of R2 , smooth functions on S are restrictions to S of smooth functions on R2 . In other words, C ∞ (S) = R(S). Corollary 3.1.9 implies that a vector field Y = Yx

∂ ∂ + Yy ∂x ∂y

on R2 restricts to a derivation of C ∞ (S) if and only if Y preserves the null ideal N (S) of S. Hence, Y (x y) = Yx (x, y)y + Y y (x, y)x = 0 if x y = 0, which implies Yx (0, y) = 0 and Y y (x, 0) = 0 for all x, y ∈ R.

(3.9)

Conversely, if F(x, y) ∈ N (S), then F(x, 0) = F(0, y) = 0 for all x, y ∈ R. Hence, Y (F)(x, y) = Yx (x, y)

∂F ∂F (x, y) + Y y (x, y) (x, y). ∂x ∂y

3.2 Integration of derivations

31

In particular, using equation (3.9), we obtain Y (F)(0, y) = Yx (0, y)

∂F ∂F (0, y) + Y y (0, y) (0, y) = 0, ∂x ∂y

(3.10)

Y (F)(x, 0) = Yx (x, 0)

∂F ∂F (x, 0) + Y y (x, 0) (x, 0) = 0 ∂x ∂y

(3.11)

for every x, y ∈ R. Equations (3.10) and (3.11) imply that Y (F) ∈ N (S). Thus, equation (3.9) ensures that Y restricts to a derivation of C ∞ (S). By Theorem 3.1.6, every derivation X of C ∞ (S) extends locally to a derivation of C ∞ (R2 ). Hence, X (0, 0) = 0 for every derivation X of C ∞ (S). On the other hand, if ∂ ∂ + vy v = vx ∂x ∂y is a non-zero derivation of C ∞ (R2 ) at the origin (0, 0), then for every F ∈ N (S), v(F) = vx

∂F ∂F (0, 0) + v y (0, 0) = 0, ∂x ∂y

because F(x, 0) = F(0, y) = 0 implies ∂∂ Fx (0, 0) = ∂∂Fy (0, 0) = 0. Therefore, v defines a derivation of C ∞ (S) at (0, 0) ∈ S, which is not a value of a global derivation X .

3.2 Integration of derivations We show here that derivations on a subcartesian space admit unique maximal integral curves. Let c : I → S be a smooth map of an interval I in R to a differential space S. If I has a non-empty interior, we say that c is an integral curve of a derivation X of C ∞ (R) if d f (c(t)) = X ( f )(c(t)) (3.12) dt for every f ∈ C ∞ (S) and every t ∈ I . We extend the notion of an integral curve to the case when the interior of I is empty by saying that if I is a single point in R, then c : I → S is an integral curve of every derivation of C ∞ (S). Integral curves of a given derivation X of C ∞ (S) can be ordered by inclusion of their domains. In other words, if c1 : I1 → S and c2 : I2 → S are two integral curves of X and I1 ⊆ I2 , then c1 c2 . An integral curve c1 : I → S of X is maximal if c1 c2 implies that c1 = c2 .

32

Derivations

Theorem 3.2.1 Let S be a subcartesian space, and let X be a derivation of C ∞ (S). For every x ∈ S, there exists a unique maximal integral curve c of X such that c(0) = x. Proof (i) Local existence. For x ∈ S, let ϕ be a diffeomorphism of a neighbourhood V of x in S onto a differential subspace R of Rn . Let Z = ϕ∗ X |V be a derivation of C ∞ (R) obtained by pushing forward the restriction of X to V by ϕ. In other words, Z ( f ) ◦ ϕ = X |V ( f ◦ ϕ) for all f ∈ C ∞ (R). Without loss of generality, we may assume that there is an extension of Z to a vector field Y on Rn . Let z = ϕ(x), and let c0 be an integral curve in Rn of the vector field Y such that c0 (0) = z. Let I x be the connected component of c0−1 (R) containing 0, and let c : I x → R be the curve in R obtained by the restriction of c0 to I x . Clearly, c(0) = z. For each t0 ∈ Ix and each f ∈ C ∞ (R), there exist a neighbourhood U of c(t0 ) in R and a function F ∈ C ∞ (Rn ) such that f |U = F|U . Therefore, d d f (c(t))|t=t0 = F(c(t))t=t0 = (Y (F))(c(t0 )) dt dt = (Y (F))|U (c(t0 )) = (Z ( f ))(c(t0 )), which implies that c : I x → R is an integral curve of Z through z. Since Ix is a connected subset of R containing 0, it is an interval (possibly the single point {0}). Then cx = ϕ −1 ◦ c : Ix → V ⊆ S satisfies cx (0) = ϕ −1 (c(0)) = ϕ −1 (z) = x. Moreover, for every t ∈ I x and h ∈ C ∞ (S), f = h ◦ ϕ −1 ∈ C ∞ (R) and d d d h(cx (t)) = h(ϕ −1 (c(t))) = (h ◦ ϕ −1 )(c(t)) dt dt dt d = ( f (c(t)) = Z ( f )(c(t)) dt = Z (h ◦ ϕ −1 )(ϕ ◦ cx (t)) = X (h)(cx (t)). Thus, cx : I x → S is an integral curve of X through x. (ii) Smoothness. It follows from the theory of differential equations that the integral curve c0 in Rn of a smooth vector field Y is smooth. Hence, c = c0|Ix is smooth. Since ϕ is a diffeomorphism of a neighbourhood of x in S to R, its inverse ϕ −1 is smooth, and the composition cx = ϕ −1 ◦ c is smooth. (iii) Local uniqueness. This follows from the local uniqueness of the solutions of first-order differential equations in Rn . (iv) Maximality. Suppose that p ≤ 0 ≤ q are the end points of the domain I of the integral curve c of X through x obtained in section (i) of the proof.

3.2 Integration of derivations

33

If q ∈ I , q = ∞ or limt→q − c(t) does not exist, then the curve c does not extend beyond q. If x 1 = limt→q − c(t) exists, then it is unique because S is Hausdorff and we can repeat the construction of section (i) beginning from the point x1 . In this way, we obtain an integral curve c1 : I1 → S of X with the initial condition c1 (0) = x 1 . Let I˜1 = I ∪ {t = q + s | s ∈ I1 ∩ [0, ∞)}, and let c˜1 : I˜1 → S be given by c˜1 (t) = c(t) if t ∈ I and c˜1 (t) = c1 (t − q) if t ∈ {q + s | s ∈ I1 ∩ [0, ∞)}. Clearly, c˜1 is continuous. Moreover, since x1 = limt→q − c(t), it follows that the lower end point p1 of I1 is strictly less than zero. Hence, the restriction of c to (max( p, p1 ) + q, q) differs from the restriction of c1 to (max( p, p1 ), 0) by the reparametrization t → t − q. Since c and c1 are smooth, it follows that c˜1 is smooth. Let q1 be the upper limit of I1 . If q1 ∈ I1 , q1 = ∞ or limt→q − c1 (t) does not exist, then the curve c1 does 1 not extend beyond q1 . Otherwise, we can extend c˜1 by an integral curve c2 of X through x2 = limt→q − c1 (t). Continuing the process, we obtain a maximal 1 extension for t ≥ 0. We can construct a maximal extension for t ≤ 0 in a similar way. (v) Global uniqueness. Let c : I → S and c : I → S be two maximal integral curves of X through x, and let T + = {t ∈ I ∩ I | t > 0 and c(t) = c (t)}. Suppose that T + = ∅. Since T + is bounded from below by 0, there exists a greatest lower bound l of T + . This implies that c(t) = c (t) for 0 ≤ t ≤ l and, for every ε > 0, there exists tε ∈ T + such that l < tε < l+ε and c(tε ) = c (tε ). Let xl = c(l) = c (l), and let cl : Il → S be an integral curve of X through xl constructed as in section (i). We denote by ql the upper end point of the interval Il . If ql > 0, the local uniqueness implies that c(t) = c (t) = cl (t − l) for all l ≤ t ≤ l + ql . Hence, we obtain a contradiction with the assumption that l is the greatest lower bound of T + . If ql = 0, then there is no extension of cl to t > 0. Let q and q be the upper end points of I and I , respectively. Since c and c are maximal integral curves of X , it follows that q = q = l. Hence, the set T + is empty. A similar argument shows that T − = {t ∈ I ∩ I | t < 0 and c(t) = c (t)} = ∅. Therefore, c(t) = c (t) for all t ∈ I ∩ I . If I = I , then we obtain a contradiction with the assumption that c and c are maximal. Hence, I = I and c = c . Let X be a derivation of C ∞ (S). We denote by (exp t X )(x) the point on the maximal integral curve of X through x corresponding to the value t of the parameter. Given x ∈ S, (exp t X )(x) is defined for t in an interval I x

34

Derivations

containing zero, and (exp 0X )(x) = x. If t, s and t + s are in Ix , if s ∈ I(exp t X )(x) , and if t ∈ I(exp s X )(x) , then (exp(t + s)X )(x) = (exp s X )((exp t X )(x)) = (exp t X )((exp s X )(x)). In the case when S is a manifold, the map exp t X is a local one-parameter group of local diffeomorphisms of S. For a subcartesian space S, the map exp t X might fail to be a local diffeomorphism. Definition 3.2.2 A vector field on a subcartesian space S is a derivation X of C ∞ (S) such that for every x ∈ S, there exist an open neighbourhood U of x in S and ε > 0 such that for every t ∈ (−ε, ε), the map exp t X is defined on U , and its restriction to U is a diffeomorphism from U onto an open subset of S. In other words, X is a vector field on S if exp t X is a local one-parameter group of local diffeomorphisms of S. Example 3.2.3 Consider S = [0, ∞) ⊆ R with the structure of a differential subspace of R. Let (X f ) = f (x) for every f ∈ C ∞ ([0, ∞)) and x ∈ [0, ∞). Note that the derivative at x = 0 is the right derivative; it is uniquely defined by f (x) for x ≥ 0. For this X , the map exp t X is given by (exp t X )(x) = x + t whenever x and x + t are in [0, ∞). In particular, for every neighbourhood U of 0 in [0, ∞), there exists δ > 0 such that [0, δ) ⊆ U . Moreover, exp t X maps [0, δ) onto [t, δ + t), which is not an open neighbourhood of t = (exp t X )(0) in [0, ∞). Hence, the derivation X is not a vector field on [0, ∞). On the other hand, for every f ∈ C ∞ [0, ∞) such that f (0) = 0, the derivation f X is a vector field, because 0 is a fixed point of f X . A subset A of a topological space T is locally closed if every point x ∈ A has a neighbourhood Ux in T such that A ∩ Ux is closed in Ux . The closure of A ∩ U x in Ux is the intersection A ∩ Ux , where A is the closure of A in T . Therefore, A ∩ Ux = A ∩ U x for every x ∈ A. Taking the union over all x ∈ A, we obtain A= A ∩ Ux = A ∩ Ux = A ∩ Ux . x∈A

x∈A

x∈A

Hence, A is an intersection of an open and a closed set or, equivalently, A is open in A. We extend the notion of local closedness to subcartesian spaces as follows. Definition 3.2.4 A subcartesian space S is locally closed if every point of S has a neighbourhood diffeomorphic to a locally closed subset of Rn . Next, we give a simple characterization of vector fields on locally closed subcartesian spaces.

3.2 Integration of derivations

35

Lemma 3.2.5 Let S be a differential subspace of Rn . If U and V are open subsets of Rn and ϕ : U → V is a diffeomorphism such that ϕ(U ∩ S) = V ∩ S, then the restriction of ϕ to U ∩ S is a diffeomorphism ψ of U ∩ S onto V ∩ S. Proof By assumption, S is a topological subspace of Rn , the mapping ϕ : U → V is a homeomorphism and ϕ(U ∩ S) = V ∩ S. Hence, for every open subset W of Rn , ϕ −1 (W ∩ (V ∩ S)) is open in U ∩ S and ϕ(W ∩ (U ∩ S)) is open in V ∩ S. Thus, ϕ induces a homeomorphism ψ : U ∩ S → V ∩ S. Moreover, ϕ induces a diffeomorphism of the open differential subspaces U and V of Rn . We want to show that f ∈ C ∞ (V ∩ S) implies that ψ ∗ f ∈ C ∞ (U ∩ S). Given x ∈ U ∩ S, let y = ψ(x) ∈ V ∩ S. Since S is a differential subspace of Rn and f ∈ C ∞ (V ∩ S), there exist a neighbourhood W of y in V and a function f W ∈ C ∞ (V ) such that f |W ∩S = f W |W ∩S . Moreover, ϕ −1 (W ) is a neighbourhood of x in U , ϕ ∗ f W is in C ∞ (U ), and (ψ ∗ f )|ϕ −1 (W )∩S = ( f ◦ ψ)|ϕ −1 (W )∩S = f ◦ ϕ|ϕ −1 (W )∩S = f |W ∩S = f W |W ∩S = f ◦ ϕ|ϕ −1 (W )∩S = f W ◦ ϕ|ϕ −1 (W )∩S = (ϕ ∗ f W )|ϕ −1 (W )∩S . Thus, for every x ∈ U ∩ S, there exist a neighbourhood ϕ −1 (W ) of x in U and a function ϕ ∗ f W in C ∞ (U ) such that (ψ ∗ f )|ϕ −1 (W )∩S = (ϕ ∗ f W )|ϕ −1 (W )∩S . This implies that ψ ∗ f ∈ C ∞ (U ∩ S). It follows that ψ is smooth. We can prove in a similar manner that ψ −1 is smooth. Hence, ψ is a diffeomorphism. Proposition 3.2.6 Let S be a locally closed subcartesian space. A derivation X of C ∞ (S) is a vector field on S if the domain of every maximal integral curve of X is open in R. Proof Consider first the case when S is a locally closed differential subspace of Rn . That is, S = O ∩ C, where O is open and C is closed in Rn . Let X be a derivation on S such that domains of all its integral curves are open in R. In other words, for each x ∈ S, the domain I x of the map t → (exp t X )(x) is an open interval in R. We need to show that the map x → (exp t X )(x) is a local diffeomorphism of S. Given x0 ∈ S, there exists an open neighbourhood W of x0 such that the restriction of X to W extends to a vector field Y on Rn . We show first that the restriction of X to W generates a local one-parameter group of local diffeomorphisms of W . Since open sets in S are intersections of open sets in Rn with S, we can write without loss of generality W = U ∩ C, where U is an open set in Rn contained in O. Let exp tY denote the local one-parameter group of local diffeomorphisms of U generated by Y . For each x ∈ W , we denote by Jx

36

Derivations

the maximum interval in R such that (exp t X )(x) = exp(tY )(x) for all t ∈ Jx . Note that Jx is the intersection of I x and K x = {t ∈ R | (exp tY )(x) ∈ U }. Since exp tY is a local one-parameter group of local diffeomorphisms of an open subset U in Rn , it follows that K x is open in R. Hence, the assumption that I x is an open interval implies that Jx is also an open interval. Given x ∈ W = U ∩ C, there exist ε > 0 and a neighbourhood U of x in U such that, for every t ∈ (−ε, ε), the map exp tY is defined on U , and its restriction to U is a diffeomorphism from U onto an open subset of U. In view of Lemma 3.2.5, it suffices to show that there exist δ ∈ (0, ε] and a neighbourhood U of x in U such that exp tY maps U ∩ C to ((exp tY )(U )) ∩ C for all t ∈ (−δ, δ). Suppose that there are no U and δ satisfying this condition. This means that for every neighbourhood U of x in U and every δ ∈ (0, ε], there exist a point y ∈ U ∩ C and s ∈ (−δ, δ) such that (exp sY )(y) ∈ / ((exp tY )(U )) ∩ C. Since (exp sY )(y) ∈ (exp tY )(U ) for every t ∈ (−ε, ε), it follows that (exp sY )(y) ∈ / C. Hence, s is not in the domain I y of the maximal integral curve of X through y. If s > 0, let u be the infimum of the set {t ∈ [0, s] | (exp tY )(y) ∈ / C}. Then, (exp t X )(y) ∈ C for all t ∈ [0, u). Since t → (exp t X )(y) is continuous and C is closed, it follows that (exp u X )(y) ∈ C. Moreover, for every v > u, there exists t ∈ (u, v) such that (exp t X )(y) ∈ / C. This implies that [0, ∞) ∩ Jy = [0, u]. Since Jy is open if the domain I y of the maximal integral curve of X through y is open, it follows that I y is not open in R, contrary to the assumption of the theorem. Hence, the case s > 0 is excluded. Similarly, we can show that the case s < 0 is inconsistent with the assumption that the domains of all maximal integral curves of X are open. We have shown that there exist δ ∈ (0, ε] and a neighbourhood U of x in U such that exp tY maps U ∩ C to ((exp tY )(U )) ∩ C for all t ∈ (−δ, δ). This implies that exp t X (z) = exp tY (z) is defined for every t ∈ (−δ, δ) and each z ∈ U . By Lemma 3.2.5, it follows that exp t X restricted to U ∩ W is a diffeomorphism onto (exp tY )(U ) ∩ W . Since this holds for every x ∈ W , we conclude that exp t X is a local one-parameter group of local diffeomorphisms of W . Hence, X |W is a vector field on W . Since for every x0 ∈ S there is an open neighbourhood W of x0 in S such that X |W is a vector field on W , it follows that X is a vector field on S. Now consider the case of a general locally closed subcartesian space S. Let X be a derivation on S such that the domains of all maximal integral curves of X are open. For each x ∈ S, the function I x → S : t → (exp t X )(x), where I x is an open interval in R, is continuous.

3.3 The tangent bundle

37

For every x ∈ S, there exist a neighbourhood W of x in S and a diffeomorphism χ of W onto a locally closed subspace U ∩ C of Rn . By the first part of the proof, the push-forward of X by the diffeomorphism χ is a vector field on U ∩C. Since χ is a diffeomorphism, it follows that there exist a neighbourhood W of x in W ⊆ S and ε > 0 such that, for every t ∈ (−ε, ε), the map exp t X is defined on W , and its restriction to W is a diffeomorphism from W onto an open subset of W ⊆ S. Hence, X is a vector field on S. The following example shows that the assumption that S is locally closed is essential in Proposition 3.2.6. Example 3.2.7 The set S = {(x1 , x2 ) ∈ R2 | x12 + (x2 − 1)2 < 1 or x2 = 0} is not locally closed at (0, 0). The vector field Y =

∂ ∂ x1

on R2 restricts to a derivation X of C ∞ (S). For every (x1 , x2 ) ∈ R2 , (exp tY )(x1 , x 2 ) = (x1 +t, x2 ) for all t ∈ R. All integral curves of X have open domains. Nevertheless, exp t X fails to be a local one-parameter local group of diffeomorphisms of S.

3.3 The tangent bundle Recall that a derivation of C ∞ (S) at x ∈ S is a linear map v : C ∞ (S) → R such that v( f 1 f 2 ) = v( f 1 ) f 2 (x) + f 1 (x)v( f 2 ) for every f 1 , f 2 ∈ C ∞ (S); see Definition 3.1.2. The existence of integral curves of global derivations established in the preceding section justifies the interpretation of derivations of C ∞ (S) at x ∈ S as tangent vectors to S at x. We denote by Tx S the set of all derivations of C ∞ (S) at x. This is a vector space, referred to as the tangent space to S at x. Definition 3.3.1 The tangent bundle of a differential space S is the union T S of all tangent spaces to S at all points in S. In other words, Tx S. TS = x∈S

The tangent bundle projection is the map τ : T S → S that assigns to each v ∈ T S the point x ∈ S such that v ∈ Tx S.

38

Derivations

We want to describe the differential structure of T S. First, consider the case when S is a differential subspace of Rn . We denote by q1 , . . . , qn the restrictions to S of the canonical coordinate functions (x 1 , . . . , xn ) on Rn . For every function f ∈ C ∞ (S) and x ∈ S, there exist a neighbourhood U of x in Rn and F ∈ C ∞ (Rn ) such that f |U ∩S = F(q1 , . . . , qn )|U ∩S .

(3.13)

Consider v ∈ Tx S, and let yi = v(qi ) for i = 1, . . . , n. Equation (3.13) yields v( f ) = (∂1 F)v(q1 ) + . . . + (∂n F)v(qn ) = y1 ∂1 F + . . . + yn ∂n F,

(3.14)

where ∂i f =

∂F . ∂qi

Equation (3.14) shows that v ∈ Tx S can be identified with a vector (y1 , . . . , yn ) ∈ Rn . Since Tx S has the structure of a vector space, the set Vx = {(v(q1 ), . . . , v(qn )) ∈ Rn | v ∈ Tx S}

(3.15)

is a vector subspace of Rn . The tangent bundle T S is the subset of R2n given by T S = {(x, y) = (x 1 , . . . , xn , y1 , . . . , yn ) ∈ R2n | x ∈ S and y ∈ Vx }. (3.16) We now consider the case of a general differential space S. For every f ∈ C ∞ (S), the differential of f is a function d f : T S → R given by d f (v) = v( f )

(3.17)

for every v ∈ T S. In the following discussion, we use the notation d f | v = d f (v)

(3.18)

to simplify complicated formulae. Definition 3.3.2 The differential structure C ∞ (T S) of T S is generated by the family of functions { f ◦ τ, d f | f ∈ C ∞ (S)}. Proposition 3.3.3 The tangent bundle projection τ : T S → S is smooth. Proof For every f ∈ C ∞ (S), the pull-back τ ∗ f = f ◦ τ ∈ C ∞ (T S), by the definition of C ∞ (T S). Hence, τ is smooth. Definition 3.3.4 A smooth section of τ : T S → S is a smooth map σ : S → T S such that τ ◦ σ = identity.

3.3 The tangent bundle

39

Proposition 3.3.5 If X is a derivation of C ∞ (S), then the map S → T S : x → X (x) is a smooth section of τ . Conversely, if σ : S → T S is a smooth section of the tangent bundle projection τ , then for each f ∈ C ∞ (S), the function X ( f ) such that X ( f )(x) = σ (x)( f ) for every x ∈ S is smooth, and the map X : C ∞ (S) → C ∞ (S) : f → X ( f ) is a derivation of C ∞ (S). Proof A section σ : S → T S of τ is smooth if, for every smooth function h ∈ C ∞ (T S), its pull-back σ ∗ h by σ is in C ∞ (S). It suffices to check this condition on the set {τ ∗ f, d f | f ∈ C ∞ (S)} of functions generating the differential structure of T S. For every f ∈ C ∞ (S), σ ∗ (τ ∗ f ) = σ ∗ ( f ◦ τ ) = f ◦ (τ ◦ σ ) = f ∈ C ∞ (S). Also, for every x ∈ S, σ ∗ (d f )(x) = σ (x)( f ) = X ( f )(x), which implies that σ ∗ (d f ) = X ( f ). Hence, σ ∗ (d f ) is smooth if and only if X ( f ) is smooth. Moreover, for every x ∈ S, X ( f 1 f 2 )(x) = σ (x)( f 1 f 2 ) = (σ (x)( f 1 )) f 2 (x) + f 1 (x)(σ (x)( f 2 (x)) = (X ( f 1 )(x)) f 2 (x) + f 1 (x)(X ( f 2 )(x)). Hence, X ( f 1 f 2 ) = X ( f 1 ) f 2 + f 1 X ( f 2 ). For every x ∈ S, σ (x) maps C ∞ (S) linearly to R. Hence, the map X : C ∞ (S) → C ∞ (S) is linear. Therefore, X is a derivation of C ∞ (S), which completes the proof. Definition 3.3.6 The derived map of a smooth map ϕ : S → R is the map T ϕ : T S → T R such that, for each x ∈ S and u ∈ Tx S, we have T ϕ(u) ∈ Tϕ(x) R and T ϕ(u)( f ) = u( f ◦ ϕ) for every f ∈ C ∞ (R). Proposition 3.3.7 If ϕ : S → R is smooth, then T ϕ : T S → T R is smooth. Proof Let τ S : T S → S, and let τ R : T R → R denote the tangent bundle projections. It follows from Definition 3.3.2 that it suffices to show that if f ∈ C ∞ (R), then (T ϕ)∗ (d f ) and (T ϕ)∗ (τ R∗ f ) are in C ∞ (T S). Since T ϕ maps Tx S to Tϕ(x) R, it follows that τ R ◦ T ϕ = ϕ ◦ τS .

40

Derivations

Hence, (T ϕ)∗ (τ R∗ f ) = (T ϕ)∗ ( f ◦ τ R ) = f ◦ τ R ◦ T ϕ = f ◦ (ϕ ◦ τ S ) = τ S∗ (ϕ ∗ f ) ∈ C ∞ (T S) because ϕ ∗ f ∈ C ∞ (S). Using equation (3.18), for every f ∈ C ∞ (R) and u ∈ T S, ((T ϕ)∗ (d f ))(u) = d f | T ϕ(u) = T ϕ(u)( f ) = u( f ◦ ϕ) = u(ϕ ∗ f ) = dϕ ∗ ( f ) | u . Hence, (T ϕ)∗ (d f ) = dϕ ∗ ( f ) ∈ C ∞ (S). It follows from Definition 3.3.6 that the following diagram commutes: TS

Tϕ

/ TR

τS

τR

S

ϕ

/ R.

Moreover, if ϕ is a diffeomorphism of S onto R, then T ϕ is a diffeomorphism of T S onto T R. In this case, if we identify derivations X and ϕ∗ X of C ∞ (S) and C ∞ (R), respectively, with the corresponding sections of τ S and τ R , we obtain ϕ∗ X = T ϕ ◦ X ◦ ϕ −1 . The derived map T ϕ is also referred to as the tangent map of ϕ. Proposition 3.3.8 For every derivation X of the differential structure C ∞ (S) of a subcartesian space and a diffeomorphism ϕ : S → R, exp(tϕ∗ X ) = ϕ ◦ exp(t X ) ◦ ϕ −1 . Proof

For each f ∈ C ∞ (R) and y = ϕ(x) ∈ R,

d d f ((ϕ ◦ exp(t X ) ◦ ϕ −1 )(y)) = f (ϕ ◦ exp(t X ))(x) dt dt d d = Tϕ exp(t X )(x) (f) = exp(t X )(x) (ϕ ∗ f ) dt dt ∗ by equation (3.12) = X (ϕ f )(exp(t X )(x)) = ϕ ∗ (ϕ∗ X ( f ))(exp(t X )(x))

by equation (3.5)

= (ϕ∗ X ( f ))(ϕ(exp(t X )(x))) = (ϕ∗ X ( f ))(ϕ(exp(t X )(ϕ −1 (y))) = (ϕ∗ X ( f ))(ϕ ◦ (exp(t X )) ◦ ϕ −1 )(y). Hence, t → (ϕ◦(exp(t X ))◦ϕ −1 )(y) is an integral curve of ϕ∗ X through y.

3.3 The tangent bundle

41

Definition 3.3.9 Let S be a subcartesian space. The structural dimension of S at a point x ∈ S is the smallest integer n such that for some open neighbourhood U ⊆ S of x, there is a diffeomorphism of U onto a subset V ⊆ Rn . Theorem 3.3.10 For a subcartesian space S, the structural dimension at x is equal to dim Tx S. Proof Let n be the structural dimension of S at x. There is a neighbourhood U of x in S and a diffeomorphism ϕ : U → V , where V is a differential subspace V of Rn . By Theorem 3.1.6, every derivation of C ∞ (S) can be extended locally to a derivation of C ∞ (Rn ). Hence, dim Tx S ≤ dim Rn = n. Now assume that dim Tx S < n. Then there exists a derivation u ∈ Tx Rn that is not the value of a local extension of a derivation of C ∞ (V ). This implies, by Corollary 3.1.9, that there is a function f ∈ N (V ) such that u( f ) = 0. In this case, if x 1 , . . . , x n are the canonical coordinate functions on Rn , then ∂f (ϕ(x)) = 0 ∂x j for some j ∈ {1, . . . , n}. Hence, there is a neighbourhood W ⊆ f −1 (0) of ϕ(x) that is a submanifold of Rn of dimension (n − 1). Therefore, there is a neighbourhood W˜ of ϕ(x) in W diffeomorphic to an open set in Rn−1 . Since f ∈ N (V ), it follows that V ⊆ f −1 (0). Therefore, there is a neighbourhood of ϕ(x) in V diffeomorphic to a subset of Rn−1 , which contradicts the assumption that n is the structural dimension of S at x. This completes the proof that dim Tx S is equal to the structural dimension of S at x. A point x ∈ S is a manifold point of S if there is an open neighbourhood U of x in S that is a manifold. In this case, dim Ty S = dim Tx S for all y ∈ U . It is convenient to weaken this condition as follows. Definition 3.3.11 A point x ∈ S is regular if there is a neighbourhood U of x in S such that dim Ty S = dim Tx S for all y ∈ U . The set of all regular points of S is called the regular component of S and is denoted by Sreg . The complement of Sreg in S is called the singular set of S. It is denoted Ssing , and its elements are called singular points of S. Example 3.3.12 The Koch curve is a subset K of R2 defined as follows. The set K 0 = {(0, 0), (1, 0)} consists of the end points of the line segment C0 = [0, 1] × {0} ∈ R2 . Construct a set C1 by removing the middle third from the segment C0 and replacing it with two equal segments that would form an equilateral triangle (pointing, say, upwards) with the removed piece. The resulting √ four-sided zigzag has vertices K 1 = {(0, 0), (0, 13 ), ( 12 , 63 ), ( 23 , 0), (1, 0)}.

42

Derivations

Next, construct a set C2 by applying the same construction to each line segment of the set C1 . We denote the set of vertices of C2 by K 2 . Continuing in this way, we obtain a sequence of piecewise linear sets Cn and the sets K n of their vertices. Let K ∞ be the union of all sets K n , i.e. K ∞ = ∪∞ n=0 K n . The Koch curve K is the topological closure of K ∞ . Since K is a closed subset of R2 , its differential structure C ∞ (K ) consists of the restrictions to K of smooth functions on R2 . We can show that dim Tx K = 2 for each x ∈ K . Hence, every point of K is regular in the sense of Definition 3.3.11. Lemma 3.3.13 Let n be the maximum of the structural dimensions of S at the points of an open subset V ⊂ S. If every open subset contained in V has a point at which the structural dimension is n, then V consists of regular points. Proof By Theorem 3.3.10, the structural dimension of S at x is equal to dim Tx S. Hence, the assumption implies that the subset W = {x ∈ V : dim Tx S = n} is dense in V . For each x ∈ V , let O x be an open neighbourhood of x in V diffeomorphic to a subset of Rn . Take y ∈ V \ W . Then dim Ty S < n (by Theorem 3.3.10 and the definition of the structural dimension). Let O y be an open neighbourhood of y in V diffeomorphic to a subset of Rn y . Since W is dense in V , there exists x ∈ W ∩ O y . So, Ox ∩ O y is diffeomorphic to a subset of Rn y . But n is the minimum of all m such that a neighbourhood of x is diffeomorphic to a subset of Rm . Since O x ∩O y is a neighbourhood of x diffeomorphic to a subset of Rn y , we have n ≤ n y . But n y < n, by assumption. Therefore, V \ W is empty; that is, the dimension of S at a point of the open subset V is n. This implies that every point in V is regular. Theorem 3.3.14 The regular component Sreg of a subcartesian space S is open and dense in S. Proof Let x ∈ Sreg . There exists an open neighbourhood U ⊆ S of x such that for every y ∈ U , dim Ty S = dim Tx S. Hence, U ⊆ Sreg . Therefore, Sreg is an open subset of S. Now, suppose that Sreg is not dense in S. In this case, there exists a nonempty open subset U ⊆ S such that U contains no regular points. Without loss of generality, we assume that U is locally diffeomorphic to a differential subspace of Rn for some n > 0. In other words, we assume that for x ∈ U , there is a neighbourhood V1 of x in U such that V1 is diffeomorphic to a subset of Rn . Suppose first that n = 0. This means that V1 is a non-empty set of isolated points. Since V1 is an open subset of U , and U is open in S, it follows that the points of V1 are isolated in S. Hence, dim Ty S = 0 for each y ∈ V1 . Therefore V1 ⊆ Sreg , which is a contradiction with the supposition that U has no regular points.

3.3 The tangent bundle

43

Define S i = {y ∈ S : dim Ty S ≤ i}. By construction, V1 ⊂ S n . Since V1 ⊆ U is open, it follows that V1 contains infinitely many points at which the structural dimensions are at least two different integers from the set {0, 1, . . . , n}. Let n 1 be the maximum of these structural dimensions at the points in V1 . By Lemma 3.3.13, there exists an open subset V2 ⊂ V1 such that the maximum of the structural dimensions of S at the points in V2 is n 2 < n 1 . Similarly, there exists an open subset V3 ⊂ V2 with a maximum of the structural dimensions at its points n 3 < n 2 . Thus, by continuing this process, we obtain the following decreasing sequence: n1 > n2 > n3 > · · · > ni , which stops at some n i ≥ 0. We reach some open subset Vi ⊂ U such that the structural dimension at all points of Vi is n i ≥ 0. Hence, all points of Vi are regular points. This contradicts the assumption that U contains no regular points. Therefore, the set Ssing of singular points of S has an empty interior, which means that Sreg is dense in S. Proposition 3.3.15 Let S be a subcartesian space. Then the restriction of the tangent bundle projection τ : T S −→ S to T Sreg is a locally trivial fibration over Sreg . For each x ∈ Sreg with structural dimension n, there are a neighbourhood W of x in S and a family X 1 , . . . , X n of global derivations of C ∞ (S) such that TW S = τ −1 (W ) is spanned by the restrictions X 1 , . . . , X n to V . Proof Let x ∈ Sreg , with dim Tx S = n. Since Sreg is open, there exists a neighbourhood U ⊂ Sreg of x such that dim Ty S = n for all y ∈ U . Since S is a subcartesian space and n is the structural dimension of S at x, we may assume without loss of generality that there is a diffeomorphism ϕ : U → V , where V is a subset of Rn . We first prove that T V , the set of all pointwise derivations of C ∞ (V ), is a trivial bundle. Let R(V ) consist of the restrictions to V of all smooth functions on Rn , and let N (V ) be the space of functions on Rn that vanish on V . We identify R(V ) with C ∞ (Rn ) modulo N (V ). Since n = dimx S, it follows that there are n linearly independent derivations of C ∞ (V ) at ϕ(x). Therefore, by Proposition 3.1.8, ∂i f |V = 0 for every i = 1, . . . , n and each f ∈ N (V ). This implies that the ∂1 , . . . , ∂n are derivations of C ∞ (V ). Hence, there are n sections X 1 , . . . , X n of the tangent bundle projection τV : T V −→ V such that X i (h mod N (V ))(y) = (∂i h)(y) for every i = 1, . . . , n, h ∈ R(V ) and y ∈ V . Now we need to prove that the sections X 1 , . . . , X n are smooth. Let q1 , . . . , qn be the restrictions to V of the

44

Derivations

coordinate functions on Rn . For i = 1, . . . , n, we denote by dqi the function on T V such that dqi (w) = w(qi ) for every w ∈ T V . The differential structure of T V is generated by the functions (τV∗ q1 , . . . , τV∗ qn , dq1 , . . . , dqn ) in the sense that every function f ∈ C ∞ (T V ) is of the form f = F(τV∗ q1 , . . . , τV∗ qn , dq1 , . . . , dqn ) for some F ∈ C ∞ (R2n ). In order to show that X i : V → T V is smooth, it suffices to show that for every f ∈ C ∞ (T V ), the pull-back X i∗ f is in C ∞ (V ). Since

1 if i = j dqi ◦ X j = δi j = 0 if i = j, it follows that X i∗ f = f ◦ X i = F(τV∗ q1 , . . . , τ V∗ qn , dq1 , . . . , dqn ) ◦ X i = F(τV∗ q1 ◦ X i , . . . , τV∗ qn ◦ X i , dq1 ◦ X i , . . . , dqn ◦ X i ) = F(q1 ◦ τV ◦ X i , . . . , qn ◦ τV ◦ X i , δ1i , . . . , δni ) = F(q1 , . . . , qn , δ1i , . . . , δni ). Hence, X i∗ f is in C ∞ (V ). This implies that the tangent bundle T V is globally spanned by n linearly independent smooth sections X 1 , . . . , X n . Thus, T V is a trivial bundle. We can choose an open neighbourhood W of y contained in V such that its closure W is also in V . Using a partition of unity, we can construct derivations of C ∞ (S) that extend restrictions of X 1 , . . . , X n to W . Hence, T W is spanned by the restrictions to W of global derivations of C ∞ (S). This completes the proof.

3.4 Orbits of families of vector fields In this section, we prove that orbits of families of vector fields are immersed manifolds. Let F be a family of vector fields on a subcartesian space S, and let x0 be a point in S. Let X 1 , . . . , X n be vector fields in F. Consider a piecewise smooth curve given by first following the integral curve of X 1 through x0 for a time τ1 , next following the integral curve of X 2 through x 1 = (exp τ1 X 1 )(x0 ) for a time τ2 , after that following the integral curve of X 3 through x2 = (exp τ2 X 2 )(x1 )

3.4 Orbits of families of vector fields

45

for some τ3 , and so on. For each i = 1, . . . , n, let Ji be the closed interval in R with end points 0 and τi . In other words, Ji = [0, τi ] if τi > 0 and Ji = [τi , 0] if τi < 0. Note that τi < 0 means that the integral curve of X i is followed in the negative time direction. Clearly, for every i, Ji is contained in the domain I xi−1 of the maximal integral curve of X i originating at xi−1 . The range of the curve is n {(exp ti X i )(xi−1 ) ∈ S | t ∈ Ji }. i=1

Definition 3.4.1 The orbit through x 0 of the family F is the union O x0 of the ranges of all the curves described above. In other words, Ox0 =

∞

n

{(exp ti X i )(xi−1 ) ∈ S | ti ∈ Ji },

(3.19)

n=1 X 1 ,...,X n J1 ,...,Jn i=1

where the vector fields X 1 , . . . , X n are in F and, for each i = 1, . . . , n, the interval Ji ⊂ I xi−1 is either [0, τi ] or [τi , 0], with xi = (exp τi X i )(xi−1 ). Proposition 3.4.2 Let Ox0 be the orbit through x0 of a family F of vector fields on a subcartesian space S. For each X ∈ F and f ∈ C ∞ (S), the integral curve of f X through x0 is contained in Ox0 . Similarly, if X, Y are in F, then the integral curve of (exp X )∗ Y through x0 is contained in O x0 . Proof For f ∈ C ∞ (S) and X ∈ F, the integral curves of X and f X differ by parametrization, provided f = 0. The integral curves of f X originating at the points for which f = 0 are constant. Hence, the integral curves of f X originating at x0 are contained in the orbit Ox0 of F. By Proposition 3.3.8, we have the equality exp(t (exp X )∗ Y )(x0 ) = exp(X ) ◦ exp(tY ) ◦ exp(−X )(x0 ) whenever both sides are defined. Let I0 ⊆ R1 be the domain of the maximal integral curve of (exp X )∗ Y through x 0 . Since exp(−X )(x 0 ) is defined, it follows that −1 is in the domain I1 of the maximal integral curve of X through x 0 . Let I2 be the domain of the maximal integral curve of Y through exp(−X )(x 0 ). There exists s ∈ I0 ∩ I2 such that 1 is contained in the domain of the maximal integral curve of X through exp(sY )[exp(−X )(x 0 )]. Hence, the curves c1 : [0, 1] → S : t → exp(−t X )(x0 ), c2 : [0, s] → S : t → exp(tY )[exp(X )(x0 )], c3 : [0, 1] → S : t → exp(t X )[exp(sY )[exp(−X )(x0 )]]

46

Derivations

are well defined and contained in Ox0 . Moreover, the point exp(s(exp X )∗ Y ))(x0 ) = {exp(X ) ◦ exp(sY ) ◦ exp(−X )}(x 0 ) can be obtained by first following c1 from x 0 to exp(−X )(x 0 ), next following c2 from exp(−X )(x 0 ) to [exp(sY )[exp(−X )(x0 )]], and finally following c3 from the point [exp(sY )[exp(−X )(x 0 )]] to the point exp((exp s X )∗ Y ))(x 0 ). Therefore, exp((exp s X )∗ Y ))(x 0 ) is contained in Ox0 . Definition 3.4.3 A family F of vector fields on S is locally complete if, for every X, Y ∈ F and x ∈ S, there exist an open neighbourhood U of x and Z ∈ F such that ((exp X )∗ Y )|U = Z |U . For example, a family consisting of a single vector field X is locally complete because (exp t X )∗ X (x) = X (x) at all points x ∈ S for which (exp t X )(x) is defined. Proposition 3.4.4 Every family F of vector fields on a subcartesian space S can be extended to a locally complete family F with the same orbits. Proof If F is not locally complete, we can find vector fields X and Y in F and an x0 ∈ S such that there do not exist a neighbourhood U of x0 and a Z ∈ F satisfying ((exp X )∗ Y )|U = Z |U . Since X is a vector field on S, there is a neighbourhood V of x0 in S such that exp X restricts to a diffeomorphism of V onto its image. Hence, (exp X )∗ Y is well defined on V . There exist f ∈ C ∞ (S) and open neighbourhoods U1 and U2 of x0 in S such that U 1 ⊂ U2 ⊂ V , f |U1 = 1 and f |S\U2 = 0. Define a vector field Z by Z |V = f (exp X )∗ Y and Z |S\U2 = 0. By Proposition 3.3.2, the integral curves of Z through x ∈ S are contained in the orbit O x of F. Hence, orbits of the family F1 = F ∪ {Z } are the same as orbits of F. Continuing this process, we obtain a locally complete family F of vector fields on S such that orbits of F coincide with orbits of F. Theorem 3.4.5 Each orbit O of a family F of vector fields on a subcartesian space S is a manifold. Moreover, in the manifold topology of O, the differential structure on O induced by its inclusion in S coincides with its manifold differential structure. Proof By Proposition 3.3.4, there exists a locally complete family of vector fields on S with the same set of orbits as F. Hence, without loss of generality, we may assume that the family F is locally complete. (i) Notation. In order to simplify the presentation, we introduce the following notation. For k > 0, let X = (X 1 , . . . , X k ) ∈ Fk , t = (t1 , . . . , tk ) and

3.4 Orbits of families of vector fields

47

exp(t X)(x) = exp(tk X k )◦ . . . ◦ exp(t1 X 1 ) (x). Given X, the expression for exp(t X)(x) is defined for all (t, x) in an open subset (X) of Rk × S. Let t (X) denote the set of all x ∈ S such that (t, x) ∈ (X). In other words, t (X) is the set of all x for which exp(t X)(x) is defined. In addition, we denote by xX ⊆ Rk the set of t ∈ Rk such that exp(t X)(x) is defined, and set expx X : xX → S : t → exp(t X)(x).

(3.20)

By construction, if x ∈ O, then expx X is smooth and its range is contained in O. (ii) Rank of a locally complete family of vector fields. For each x ∈ O, the rank of F at x, denoted by rank Fx , is the number of vector fields X 1 , . . . , X m in F such that X 1 (x), . . . , X m (x) form a basis of the subspace of Tx S spanned by the values at x of vector fields in F. Since linear independence is an open property, it follows that if X 1 , . . . , X m are linearly independent at x, then they are linearly independent in a neighbourhood of x. The assumption that the family F is locally complete ensures that the rank of F is constant on O. This can be seen as follows. Suppose that rank Fx0 = m. This implies that there is a basis X = (X 1 , . . . , X m ) of the linear span of F at x0 . For each i = 1, . . . , m, T exp(t X)(X i (x0 )) = T exp(tm X m )◦ . . . ◦ exp(t1 X 1 ) (X i (x0 )) = T (exp(tm X m )(. . . (T exp(t1 X 1 )(X i (x0 ))))) = exp(tm X m 1 )∗ (. . . (exp(t1 X 1 )∗ X i ))(exp(t X)(x0 )) = (exp(t X)∗ X i )((exp(t X)(x0 )). Since F is locally complete, for each i = 1, . . . , m there is a vector field Z i in F which is equal to exp(t X)∗ X i in a neighbourhood of exp(t X)(x 0 ). Each map exp(t X ) : x → exp(t X )(x) is a local diffeomorphism. Hence, the composition exp(t X 1 ) = exp(tm 1 X 1m 1 )◦ . . . ◦ exp(t1 X 11 ) is also a local diffeomorphism. This implies that T exp(t X) : Tx0 S → Texp(t X)(x0 ) S is a vector space isomorphism. Therefore, the linear independence of the vector fields X 11 , . . . , X 1m 1 at x 0 implies that the vector fields Z 1 , . . . , Z m 1 are linearly independent at exp(t X)(x0 ). Since m = rank Fx0 is the maximum number of vector fields X 01 , . . . , X 0m in F which are linearly independent at x0 , it follows that rank Fx0 ≤ rank Fexp(t X)(x0 ) . On the other hand, x0 = exp(−t X)(exp(t X)(x 0 )). Repeating the same argument as above, we obtain the result that rank Fexp(t X)(x0 ) ≤ rank Fx0 . Therefore,

48

Derivations

rank Fexp(t X)(x0 ) = rank Fx0 . Since the orbit O is connected, it follows that rank F is constant on O. (iii) Covering of the orbit by manifolds. Given x ∈ O, there exist X = (X 1 , . . . , X m ) ∈ Fm and a neighbourhood V of x in S such that {X 1 , . . . , X m } is a maximal linearly independent subset of F|V . For each i = 1, . . . , m, and u ∈ R, d u exp(t X i )(x) = u X i (exp(t X i )(x)). dt Hence, for each u = (u 1 , . . . , u m ) ∈ Rm , T expx X(u) = u 1 X 1 (x) + . . . + u m X m (x). The vectors X 1 (x), . . . , X m (x) are linearly independent, which implies that the derived map T expx X : Rm → Tx S is one-to-one. Since S is subcartesian, we may assume without loss of generality that there exists a smooth map ϕ : V → Rn that induces a diffeomorphism of V onto its image ϕ(V ) ⊆ Rn . By Theorem 3.1.6, for every i = 1, . . . , m, the vector field ϕ∗ X i on ϕ(V ) extends locally to a vector field Y i on Rn . Shrinking V if necessary, we may assume that all vector fields ϕ∗ X i are restrictions to ϕ(V ) of vector fields Y i on Rn . Let y = ϕ(x) and Y = (Y1 , . . . , Ym ). As before, we denote by exp y Y the map from the neighbourhood of 0 ∈ Rm to Rn given by exp y (Y )(t) = exp(tm Y m )◦ . . . ◦ exp(t1 Y 1 ) (y). The linear independence at x of the vector fields X 1 , . . . , X m implies that the vector fields Y1 , . . . , Yn are linearly independent at y. Hence, there exists a neighbourhood W of 0 in Rm such that exp y Y (W ) is a submanifold of Rn and that exp y Y , restricted to W , gives a diffeomorphism exp y Y |W : W → exp y Y (W ). Since y = ϕ(x) ∈ ϕ(V ) and, for i = 1, . . . , m, the restriction to ϕ(V ) of Y i gives the vector field ϕ∗ X i on ϕ(V ), the set exp y Y (W ) is contained in ϕ(V ), and it is the image of W ⊆ Rm under the map expϕ(x) (ϕ∗ X) : W → ϕ(V ) : t → exp(tm ϕ∗m X 1 )◦ . . . ◦ exp(t1 ϕ∗ X 1 ) (ϕ(x)). In other words, exp y Y (W ) = expϕ(x) (ϕ∗ X)(W ) ⊆ ϕ(V ).

(3.21)

The differential structure of exp y Y (W ) is generated by restrictions to exp y Y (W ) of smooth functions in C ∞ (Rm ). The differential structure of ϕ(V ) is also generated by restrictions to ϕ(V ) of smooth functions in C ∞ (Rm ). Hence, equation (3.21) implies that expϕ(x) (ϕ∗ X)(W ) is a manifold in the differential structure generated by restrictions of smooth functions on

3.4 Orbits of families of vector fields

49

ϕ(V ). We say that expϕ(x) (ϕ∗ X)(W ) is a submanifold of ϕ(V ). Moreover, expϕ(x) (ϕ∗ X)|W : W → expϕ(x) (ϕ∗ X)(W ) is a diffeomorphism. Since ϕ is a diffeomorphism of V on its image ϕ(V ) and V is open in S, it follows that expx (X)(W ) is a submanifold of S and expx (X)|W is a diffeomorphism of W onto expx (X)(W ). The construction above can be repeated for each point x in the orbit O, a finite collection X of vector fields in F and a neighbourhood W of 0 ∈ Rm , where m is the number of vector fields in X that satisfy the assumptions made above. In this way, we obtain a family of sets expx X(W ) in O covering O. In other words, O= expx X(W ). (3.22) x∈O

X

W

Each expx X(W ) is a submanifold of S diffeomorphic to W . (iv) Topology of the orbit. We have shown that the orbit O is covered by a family {expx X(W )} of subsets of O, where x ∈ O, X = (X 1 , . . . , X m ) ∈ Fm is a frame field for F in a neighbourhood of x, and W is a neighbourhood of 0 ∈ Rm such that expx X is a diffeomorphism of W on its image. We want to take this family of subsets of O to be a basis for the topology of O. For this definition to make sense, we must verify that if x0 ∈ expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ), then there exist a frame field X 0 for F in a neighbourhood of x 0 and an open neighbourhood W0 of 0 in Rm such that expx0 X 0 (W0 ) ⊆ expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ).

(3.23)

First, we show that for every point x0 ∈ expx1 X 1 (W1 ) and every X ∈ F, there is an open neighbourhood I of 0 ∈ R such that the integral curve of X through x 0 with domain I is contained in expx1 X 1 (W1 ). Let c : s → x(s) be an integral curve of X such that x(0) = x0 . It suffices to show that there is a curve s → t(s) = (t1 (s), . . . , tm (s)) in W1 such that x(s) = exp x X(t(s)) = exp(tm (s)X m )◦ . . . ◦ exp(t1 (s)X 1 )(x1 ) for all s in a neighbourhood I of 0 ∈ R. Differentiating this equation with respect to s at x(s), we obtain X (x(s)) = (T expx X 1 )(t(s))

d (t(s)) ds

dtm dtm−1 + exp(tm (s)X 1m )∗ X 1m−1 (x(s)) + ... ds ds dt1 . (3.24) + exp(tm (s)X 1m ) ◦ . . . ◦ exp(t2 (s)X 12 )∗ X 11 (x(s)) ds

= X 1m (x(s))

50

Derivations

Since expx X is a diffeomorphism of W onto its image expx X(W ), it follows that T (expx X)(t(s)) is a vector space isomorphism of Tt Rm onto Texpx X(t(s)) expx X(W ). In particular, the vectors X m (x(s)), . . . , (exp(tm (s)X m ) ◦ . . . ◦ exp(t2 (s)X 2 ))∗ X 1 (x(s)) are linearly independent. Hence, equation (3.24) is a system of linear differential equations of first order for t(s), and it has a unique smooth solution for s in a neighbourhood I of 0 ∈ R. Repeating this argument for X 11 , . . . , X 1m , we obtain the result that there is a neighbourhood W0 of 0 ∈ Rm such that expx0 X 1 is a diffeomorphism of W0 onto expx0 X 1 (W0 ) ⊂ expx1 X 1 (W1 ). By shrinking W0 if necessary, we can obtain expx0 X 1 (W0 ) ⊂ expx2 X 2 (W2 ), which proves the inclusion (3.23). Therefore, we can take the family {expx X(W )} of subsets of the orbit O, where x ∈ O, X = (X 1 , . . . , X m ) ∈ Fm is a frame field for F in a neighbourhood of x, and W is a neighbourhood of 0 ∈ Rm such that expx X is a diffeomorphism of W on its image, as a basis of a topology T on O. In this topology, O is a connected topological space locally homeomorphic to Rm . Note that the topology T of O may be finer than its subspace topology. (v) Differential structure of the orbit. The orbit O is covered by open sets {expx X(W )}, each of which is diffeomorphic to an open neighbourhood of 0 ∈ Rm . Moreover, if the intersection U12 = expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ) is not empty, then it is an open subset of O, and expx1 X 1 ◦ (expx2 X 2 )−1 is a diffeomorphism of expx2 X 2 (U12 ) onto expx1 X 1 (U12 ). Hence, O is a smooth manifold diffeomorphic to Rm . For each function f ∈ C ∞ (O) and each x ∈ O, the restriction of f to expx X(W ) is smooth. But expx X(W ) is a submanifold of S. This means that if f ∈ C ∞ (O), then for each x ∈ O there exist a neighbourhood U = expx X(W ) of x in O and a function h ∈ C ∞ (S) such that f |U = h |U . Conversely, let f : O → R be such that for each x ∈ O, there exist an open neighbourhood U of x in O and h ∈ C ∞ (S) such that f |U = h |U . Consider expx0 X 0 (W0 ) ⊆ O. By hypothesis, for each x ∈ expx0 X 0 (W0 ), there exist an open neighbourhood U of x in O and h ∈ C ∞ (S) such that f |U = h |U . In particular, the restrictions of f and h to the open neighbourhood U ∩ expx0 X 0 (W0 ) of x in expx0 X 0 (W0 ) coincide. Hence, the restriction of f to expx0 X 0 (W0 ) is smooth. This holds for every open set expx0 X 0 (W0 ) of our covering of O by manifolds. Hence, f is smooth in the manifold differential structure C ∞ (O) of the orbit. We have shown that the manifold differential structure of the orbit O coincides with the differential structure of O induced by its inclusion into S; see Proposition 2.1.8. This completes the proof.

3.4 Orbits of families of vector fields

51

We can restate the results obtained above in terms of distributions. Definition 3.4.6 (i) A distribution on a subcartesian space S is a subset D ⊆ T S such that for each x ∈ S, the intersection Dx = D ∩ Tx S is a vector subspace of T S. (ii) A distribution D on S is smooth if there is a family F of vector fields on S that spans D. In other words, for every x ∈ S, Dx = span {X (x) ∈ Tx S | X ∈ F}. We denote by X(S) the family of all vector fields on a subcartesian space S. Given a smooth distribution D on S, the family F D , defined by F D = {X ∈ X(S) | X (x) ∈ Dx for each x ∈ S}, is the maximal family of vector fields on S which spans D. An integral manifold of a distribution D is a connected manifold M in S such that Tx M = Dx for every x ∈ M. We say that M is a maximal integral manifold of D through x ∈ S if M is an integral manifold of D and contains x, and every other integral manifold of D containing x is an open submanifold of M. Definition 3.4.7 A distribution D on a subcartesian space S is integrable if, for every x ∈ S, there exists a maximal integral manifold of D containing x. Theorem 3.4.5 ensures that the orbits of every family F of vector fields on S are manifolds. However, the orbit of F passing through x ∈ S need not be an integral manifold through x of the distribution D spanned by F. According to Proposition 3.4.4, orbits of F are the same as orbits of its locally complete extension F. Therefore, in the proof of Theorem 3.4.5, we could assume that our family F of vector fields was locally complete. We are led to the following corollary. Corollary 3.4.8 If F is a locally complete family of vector fields on a subcartesian space S, then the orbits of F are integral manifolds of the distribution D on S spanned by F. Clearly, the family X(S) of all vector fields on a subcartesian space S is locally complete. Therefore, the distribution on S spanned by X(S) is integrable. It is the maximal integrable distribution on S. Its maximal integral manifolds, that is, orbits of X(S), give a partition O of S by smooth manifolds. This partition will be discussed further in Chapter 4.

4 Stratified spaces

Stratified spaces are examples of singular spaces and can be analysed in the framework of differential geometry. In this chapter, we describe stratified subcartesian spaces and develop a differential-geometric approach to their study. We shall use this approach in subsequent sections to describe the singular reduction of symmetries.

4.1 Stratified subcartesian spaces A stratification of a subcartesian space S is a partition of S by a locally finite family M of locally closed connected submanifolds M, called strata of M, which satisfy the following condition.1 Frontier Condition 4.1.1 For M, M ∈ M, if M ∩ M = ∅, then either M = M or M ⊂ M\M. Here, M denotes the topological closure of M in S and M\M the complement of M in M. The pair (S, M) is called a stratified space. Local finiteness of M means that, for each point x ∈ S, there exists a neighbourhood U of x in S that intersects only a finite number of manifolds M ∈ M. A subset M of S is locally closed if, for each x ∈ M, there exists a neighbourhood U of x in S such that M ∩ U is closed in U. If S is a manifold, an injectively immersed submanifold M of S is embedded if and only if M is locally closed in S. 1 There are a variety of definitions of the notion of ‘stratification’ in the literature. For example,

Mather (1973) defined a stratification of a topological space S as a map from S to a sheaf of germs of manifolds satisfying certain conditions. Our definition is equivalent to Mather’s in the case when S is a differential space. It is more convenient because it does not require the introduction of sheaves.

4.1 Stratified subcartesian spaces

53

In Chapter 3, we showed that every subcartesian space S admits a partition O by orbits of the family X(S) of all vector fields on S. It is of interest to see under what conditions this partition of S is a stratification. Proposition 4.1.2 The partition O of a subcartesian space S by orbits of the family X(S) of all vector fields on S satisfies Frontier Condition 4.1.1. Proof Let O and O be orbits of X(S). Suppose that x ∈ O ∩ O, with O = O. We first show that O ⊂ O. Note that the orbit O is invariant under the family of one-parameter local groups of local diffeomorphisms of S generated by vector fields. Since x ∈ O, it follows that, for every vector field X on S, exp(t X )(x) is in O if it is defined. But O is the orbit of X(S) through x. Hence, O ⊂ O. Corollary 4.1.3 The partition O of S by orbits of X(S) is a stratification of S if and only if it is locally finite and the orbits of X(S) are locally closed. Stratified subcartesian spaces form a category with morphisms ϕ : (S1 , M1 ) → (S2 , M2 ) given by smooth maps ϕ : S1 → S2 such that, for each M1 ∈ M1 , the range ϕ(M1 ) is contained in M2 ∈ M2 . If ϕ is a diffeomorphism of S1 onto S2 and M2 = {ϕ(M) | M ∈ M1 }, then ϕ is an isomorphism of (S1 , M1 ) onto (S2 , M2 ). Stratifications of S can be partially ordered by inclusion. If M1 and M2 are two stratifications of S, we say that M1 is a refinement of M2 and write M1 ≥ M2 if, for every M1 ∈ M1 , there exists M2 ∈ M2 such that M1 ⊆ M2 . We say that M is a minimal (coarsest) stratification of S if it is not a refinement of a different stratification of S. If S is a manifold, then the minimal stratification of S consists of a single manifold M = S. If (S, M) is a stratified subcartesian space and N is a manifold, the product S × N is stratified by the family M S×N = {M × N | M ∈ M}. If U is an open subset of a stratified space (S, M), we can consider a family MU = {M ∩ U | U ∈ M}. In general, MU need not be a stratification of U . Definition 4.1.4 A stratification M of a subcartesian space S is locally trivial if, for every M ∈ M and each x ∈ M: (i) there exists an open neighbourhood U of x in S such that MU is a stratification of U ; (ii) there exists a subcartesian stratified space (S , M ) with a distinguished point y ∈ S such that the singleton {y} ∈ M ; and (iii) there is an isomorphism ϕ : (U, MU ) → ((M ∩ U ) × S , M(M∩U )×S ) such that ϕ(x) = (x, y).

54

Stratified spaces

Let M be a stratification of a subcartesian space S. We say that M admits local extensions of vector fields if, for each M ∈ M, each vector field X M on M and each point x ∈ M, there exist a neighbourhood V of x in M and a vector field X on S such that X |V = X M|V . In other words, the vector field X is an extension to S of the restriction of X M to V . Proposition 4.1.5 Every locally trivial stratification of a subcartesian space S admits local extensions of vector fields. Proof Let X M be a vector field on M ∈ M. Given x 0 ∈ M, by Definition 4.1.4, there exist a neighbourhood U of x0 in M, a stratified differential space (S , M ) with a distinguished point y ∈ S such that the singleton {y0 } ∈ M , and an isomorphism ϕ : U → (M ∩ U ) × S of stratified subcartesian spaces such that ϕ(x0 ) = (x0 , y0 ). Let exp(t X M ) be the local one-parameter group of local diffeomorphisms of M generated by X M , and let X (M∩U )×S be a derivation of C ∞ ((M ∩ U ) × S ) defined by (X (M∩U )×S h)(x, y) =

d h(exp(t X M )(x), y)|t=0 , dt

for every h ∈ C ∞ ((M ∩ U ) × S ) and each (x, y) ∈ (M ∩ U ) × S . Since X (M∩U )×S is defined in terms of a local one-parameter group (x, y) → (exp(t X M )(x), y) of diffeomorphisms, it is a vector field on (M ∩ U ) × S . We can use the inverse of the diffeomorphism ϕ : U → (M ∩U )×S to push forward X (M∩U )×S to a vector field X U = (ϕ −1 )∗ X (M∩U )×S on U . Choose a function f 0 ∈ C ∞ (S) with support in U and such that f (x) = 1 for x in some neighbourhood U0 of x0 contained in U . Let X be a derivation of C ∞ (S) extending f 0 X U by zero outside U . In other words, for every f ∈ C ∞ (S), if / U0 , then (X f )(x) = 0. x ∈ U , then (X f )(x) = f 0 (x)(X U f )(x), and if x ∈ Clearly, X is a vector field on S extending the restriction of X M to M ∩U0 . Theorem 4.1.6 Let M be a stratification of a subcartesian space S admitting local extensions of vector fields. The partition O of S by orbits of the family X(S) of all vector fields is a stratification of S, and M is a refinement of O. Moreover, if M is minimal, then M = O. Proof Let M be a stratification of S admitting local extensions of vector fields. Since every vector field X M on a manifold M ∈ M extends locally to a vector field on S and M is connected, it follows that M is contained in an orbit O ∈ O. Every orbit O ∈ O is a union of strata of M. Since M is locally finite, for each x ∈ O there exists a neighbourhood V of x in S which intersects only a finite number of strata M1 , . . . , Mk of M. Hence, V intersects only a finite

4.1 Stratified subcartesian spaces

55

number of orbits in O. Moreover, since strata of M form a partition of S, it follows that V =

k

Mi ∩ V.

(4.1)

i=1

Consider x ∈ M1 . Since M1 is locally closed, there exists a neighbourhood U of x contained in V , and such that M1 ∩ U is closed in U . We can relabel the manifolds M1 , . . . , Mk so that O ∩U =

l

Mi ∩ U

i=1

for some l ≤ k. Without loss of generality, we may assume that x ∈ M i for each i = 2, . . . , l. We want to find out whether O ∩ U is closed in U . Suppose we have a sequence (yk ) in O ∩U convergent to y ∈ U . Since O ∩U is a finite union of disjoint manifolds, there must be a subsequence of (yk ) contained in one of them. Without loss of generality, we may assume that each yk ∈ Mi for some i = 1, . . . , l. We want to show that the limit y = limk→∞ yk ∈ O ∩ U . If y ∈ Mi , then y ∈ Mi ∩ U ⊆ O ∩ U . If y ∈ M i \Mi , then y ∈ M j for some j = 1, . . . , k. By assumption, y ∈ U , and U intersects only the strata that have / O ∩ U implies x in their closure. If M j ⊆ O, then y ∈ O ∩ U . Therefore, y ∈ that M j is not contained in O. We have shown that O ∩ U is not closed in U only if there exist strata Mi and M j such that Mi ⊆ O, M j O, M j ⊆ M i and x ∈ M j ∩ M i . To prove that this is inconsistent with our assumptions, consider a neighbourhood of x in O of the form expx X(W ), where W is an open neighbourhood of 0 ∈ Rm , X = (X 1 , . . . , X m ) ∈ X(S)m is an m-tuple of vector fields on S, and m = dim O. By the construction in the proof of Theorem 3.4.5, expx X(W ) is an m-dimensional locally closed submanifold of S. Let U0 be an open neighbourhood of x in U such that U0 ∩ expx X(W ) is closed in U0 . As before, we consider a sequence (yk ) in Mi ∩ U0 ∩ expx X(W ) ⊆ O ∩ U0 which converges / U0 ∩expx X(W ) ⊆ U0 ∩ O. to y ∈ M j ∩U0 . Since M j O, it follows that y ∈ This contradicts the fact that U0 ∩expx X(W ) is closed in U0 . Therefore, O ∩U is closed in U . Since x is an arbitrary point of the orbit O, it follows that O is locally closed. We have shown that the partition O of S by orbits of the family X (S) of all vector fields on S is locally finite and that each orbit in O is locally closed. Corollary 4.1.3 asserts that O is a stratification of S. By construction, every stratum of the original stratification M is contained in a stratum of O. This implies that M ≥ O. If M is minimal, then M = O.

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4.2 Action of a Lie group on a manifold We are now going to apply the results of the preceding section to describe the structure of the space of orbits of a proper action of a connected Lie group G on a manifold P. We begin with a review of proper actions. An action : G × P → P : (g, p) → (g, p) ≡ g ( p) ≡ gp of a locally compact connected Lie group on a manifold P is proper if, for every convergent sequence ( pn ) in P and a sequence (gn ) in G such that the sequence (gn pn ) is convergent, the sequence (gn ) has a convergent subsequence (gn k ) and lim pn k . lim (gn k pn k ) = lim gn k k→∞

k→∞

k→∞

The isotropy group G p of a point p ∈ G is G p = {g ∈ G | gp = p}. Proposition 4.2.1 Isotropy groups of a proper action are compact. Proof Let (gn ) be a sequence in the isotropy group G p of p ∈ P. Then gn p = p for all n, and the sequence gn p converges to p. By the definition of a proper action, there exists a subsequence (gn k ) in G converging to g such that gp = limk→∞ (gn k p) = limk→∞ p = p. Hence, g ∈ G p , which implies that G p is compact. The orbit of G through p ∈ P is the set Gp = {gp | g ∈ G}. It is a manifold diffeomorphic to the quotient G/G p . Let H be a compact subgroup of G. We denote by P H the set of points in P fixed by the action of H . In other words, P H = { p ∈ P | gp = p ∀ g ∈ H } = { p ∈ P | H ⊆ G p }. We denote the set of points p ∈ P with isotropy group H by PH , and the set of points with an isotropy group conjugate to H by P(H ) . The set PH = { p ∈ P | G p = H } is usually called the subset of P of isotropy type H , and P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G} is called the subset of P of orbit type H .

4.2 Action of a Lie group on a manifold

57

Definition 4.2.2 A slice through p ∈ P for an action of G on P is a submanifold S p of P containing p such that: 1. S p is transverse and complementary to the orbit Gp of G through p. In other words, T p P = T p S p ⊕ T p (Gp). 2. For every p ∈ S p , the manifold S p is transverse to the orbit Gp ; that is, T p P = T p S p + T p (Gp ). 3. S p is G p -invariant. 4. Let p ∈ S p . If gp ∈ S p , then g ∈ G p . Given a G-invariant Riemannian metric k on P, we can construct slices for the action of G on P as follows. We denote by ver T P the generalized distribution on P consisting of vectors tangent to G-orbits in P, and by hor T P the k-orthogonal complement of ver T P. The tangent bundle space T P can be expressed as the product of ver T P and hor T P : T P = ver T P × hor T P,

(4.2)

where the product is taken in the category of differential spaces. The Ginvariance of k implies that this product structure of T P is invariant under the derived action of G on T P. Let p ∈ P and H = G p . We denote by H : H × P → P : (g, p) → H g ( p) = gp the restriction of to H . Since H leaves p fixed, for each g ∈ H , the derived map T gH : T P → T P preserves T p P, and it defines a linear action p : H × T p P → T p P : (g, v) → T gH (v) of H on T p P. Let Exp p be the exponential map from a neighbourhood of 0 in T p P to P corresponding to the metric k. In other words, Exp p (v) is the value at 1 of the geodesics of k originating from p in the direction v. It intertwines the linear action p of H on T p P and the action H of H on P. For each v in the domain of Exp p ∈ T p P and g ∈ H , g Exp p (v) = Exp p (T gH (v)).

(4.3)

Since T p P is a vector space, for each u ∈ T p P we have the natural identification of Tu (T p P) with T p P. With this identification, T0 Exp p : T p P → T p P : v → v. In other words, T0 Exp p is the identity on T p P.

(4.4)

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Proposition 4.2.3 2 (i) There is an open ball B in hor T p P centred at 0 such that S p = Exp p (B) is a slice through p for the action of G on P. (ii) The restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action p of H on hor T p S p and the action of H on S p . For each g ∈ H and q ∈ S p , β( p (g, v)) = g (β(v)) = g(β(v)). Proof The exponential map Exp p corresponding to the metric k is a diffeomorphism of a neighbourhood V of 0 in T p P onto a neighbourhood of p in P. Therefore, Exp p maps submanifolds of T p P contained in V onto submanifolds of P. In particular, if B is a ball in hor T p P centred at 0 and contained in V , then S p = Exp p (B) is a submanifold of P containing p. Let β : B → S p : u → β(u) = Exp p (u) be the diffeomorphism obtained by restricting Exp p to B. Equation (4.4) implies that T p S p = Tβ(T0 B) = Tβ(hor T p P) = hor T p P. Taking Equation (4.2) into account, we obtain T p P = ver T p P × hor T p P = T p (Gp) × T p S p = T p (Gp) ⊕ T p S p . Hence, Condition 1 of Definition 4.2.2 is satisfied. Moreover, this shows that the restriction |G×S p of : G × P → P to G × S p is a submersion at (e, p), where e is the identity in G. Hence, it is a submersion at (e, p ) for all p in a neighbourhood of p in S p . By shrinking B if necessary, we can redefine S p so that |G×S p is a submersion. This implies Condition 2 of Definition 4.2.2. Since k is G-invariant and p ∈ PH , it follows that Exp p intertwines the linear action p of H on T p P and the action of H on P. Moreover, the open ball B in hor T p P defined in terms of the metric k is invariant under the action of H in T p P. Therefore, S p is H -invariant and, for each g ∈ H and v ∈ B, β( p (g, v)) = g (β(v)). This proves the second part of Proposition 4.2.3 and shows that Condition 3 of Definition 4.2.2 is satisfied. It remains to show that we can choose the radius r of B so that Condition 4 of Definition 4.2.2 is met. We prove this by contradiction. Suppose that there is no r > 0 such that p ∈ S p and gp ∈ S p ; then g ∈ H . Taking r = n1 , we obtain a sequence ( pn ) of points in S p converging to p and a sequence (gn ) in G\H such that the sequence (gn pn ) in S p converges to p. Since the action of 2 This proposition is known as Bochner’s Linearization Lemma (see Duistermaat and Kolk

(2000)).

4.2 Action of a Lie group on a manifold

59

G on P is proper, there exists a subsequence of (gn m ) converging to g ∈ G. Without loss of generality, we may assume that gn → g as n → ∞. Since lim pn = gp, p = lim gn pn = lim gn n→∞

n→∞

n→∞

it follows that g ∈ H . Moreover, g −1 gn → e as n → ∞. Let g = h ⊕ m be a decomposition of the Lie algebra g of G into the direct sum of the Lie algebra h of H and a subspace m of g complementary to h. Consider the map : h ⊕ m → G : (η, ξ ) → (exp η)(exp ξ ),

(4.5)

where exp denotes the exponential map from the Lie algebra g to G. The derived map T(0,0) of at (0, 0) ∈ h ⊕ m is the inclusion of h ⊕ m into g. Hence, there is a neighbourhood V × W of (0, 0) in h ⊕ m such that defines a diffeomorphism of V × W onto (V × W ) ⊆ G. Since g −1 gn → e as n → ∞, it follows that g −1 gn ∈ (V × W ) for sufficiently large n. Hence, there exist unique ηn ∈ V ⊆ h and ξn ∈ W ⊆ m such that g −1 gn = (exp ηn )(exp ξn ). Since g −1 gn → e, it follows that ξn → 0. Since Gp = G/G p = G/H , it follows that exp : g → G induces a diffeomorphism of a neighbourhood of 0 in m onto a neighbourhood of p in Gp. Consider a map m × S p → P : (ξ, q) → (exp ξ, q). Condition 1 of Definition 4.2.2 implies that it gives a diffeomorphism of a neighbourhood of (0, p) in m × S p onto its image. By shrinking W ⊆ m if necessary, we can find a neighbourhood U of p in S p such that ϕ : W × U → ϕ(W × U ) ⊆ G : ϕ(ξ, q) → (exp ξ, q)

(4.6)

is a diffeomorphism. For each n, qn = gn pn = g(exp ηn )(exp ξn ) pn ∈ S p . Since g(exp ηn ) ∈ H and S p is H -invariant, it follows that qn = (exp ξn ) pn = (exp(−ηn ))g −1 qn ∈ S p . Moreover, pn → p and ξn → 0 imply that, for sufficiently large n, we have pn ∈ U , ξn ∈ W and qn = (exp ξn ) pn ∈ ϕ(U × W ) ∩ U . Since pn , qn ∈ U and ξn ∈ W , we can write ϕ(ξn , pn ) = (exp ξn ) pn = qn = (exp 0)qn = ϕ(0, qn ). But ϕ is a diffeomorphism of the product W × U onto its image. Therefore, ϕ(ξn , pn ) = ϕ(0, qn ) implies that ξn = 0 and pn = qn . Hence, for sufficiently large n, gn = g(exp ηn )(exp ξn ) = g(exp ηn ) ∈ H , / H . This completes the proof. which contradicts the assumption that gn ∈

60

Stratified spaces

Remark 4.2.4 (i) Let g = h ⊕ m be a decomposition of the Lie algebra g of G into the direct sum of the Lie algebra h of H and a subspace m of g complementary to h. For p ∈ PH , there exists a slice S p through p, a neighbourhood W of 0 in m and a neighbourhood U of p in P such that ϕ : W × S p → ϕ : U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. (ii) The set G S p = {gq | g ∈ G and q ∈ S p } is a G-invariant open neighbourhood of p in P. Proof (i) Since Gp = G/G p = G/H , it follows that the exponential map exp : g → G induces a diffeomorphism of a neighbourhood of 0 in m onto a neighbourhood of p in Gp. Let S p = Exp p (B p ) be a slice through p for an action of G on P. Condition 1 of Definition 4.2.2 implies that there are neighbourhoods of U and V of p in P and S p , respectively, such that ϕ : W × V → U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. If V = S p , we can find a ball B centred at 0 ∈ T p P, contained in B and such that S p = Exp p (B) ⊆ V . There exist W ⊆ W and U ⊆ U such that ϕ : W × S p → U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. (ii) Clearly, G S p is G-invariant. Note that exp W is a subset of G diffeomorphic to a neighbourhood of p of the orbit Gp. It follows from part (i) that U = (exp W )S p = {gq | g ∈ exp W and q ∈ S p } ⊆ G S p is open in P. For each g ∈ G, the diffeomorphism g : P → P maps S p to a slice Sgp through gp. In other words, gS p = Sgp . Hence, gU = g(exp W )S p = g(exp W )g −1 Sgp ⊆ G S p is an open subset of P. Therefore, G Sp =

gU

g∈G

is open in P. We are now in a position to discuss the manifold properties of the subsets P H , PH and P(H ) of P. We say that a subset S of P is a local submanifold of P if each connected component of S is a submanifold. To verify that S is a

4.2 Action of a Lie group on a manifold

61

local submanifold of P, it suffices to show that, for each p ∈ S, there exists a neighbourhood U of p in P such that S ∩ U is a submanifold of P. Proposition 4.2.5 The set P H of fixed points of the action of a compact Lie group H on P is a local submanifold of P. Moreover, for each p ∈ P H , the tangent bundle space of P H at p consists of H -invariant vectors in T p P: T p P H = (T p P) H = {v ∈ T p P | p (g, v) = v ∀ g ∈ H }. Proof For p ∈ P H , let Exp p be the exponential map from a neighbourhood of 0 in T p P to P corresponding to the metric k. This defines a diffeomorphism of a neighbourhood V of 0 ∈ T p P onto a neighbourhood U of p in P. The vectors in T p P which are fixed by the action p form a subspace (T p P) H of T p P. Therefore, V ∩(T p P) H is a submanifold of T p P, and Exp p (V ∩(T p P) H ) is a submanifold of U . Since Exp p intertwines the linear action of H on V and the action of H on U , it follows that Exp p (V ∩ (T p P) H ) = U ∩ P H . Thus, every p ∈ P H has a neighbourhood U ∩ P H in P H which is a submanifold of P. Hence, P H is a local submanifold of P. Moreover, T Exp p restricted to T0 V = T0 (T p P) = T p P is the identity transformation. Hence, T p P H = (T p P) H . The normalizer N H of H in G is a closed subgroup of G given by N H = {g ∈ G | g H g −1 = H }. For each p ∈ P and g ∈ G, G gp = gG p g −1 . Hence, the action of g ∈ G on P preserves PH if and only if g H g −1 = H . In other words, g preserves PH if and only if g ∈ N H . Let L be a connected component of PH . The group N L = {g ∈ N H | gp ∈ L ∀ p ∈ L} is a closed subgroup of N H containing H as a normal subgroup. Hence, G L = N L /H

(4.7)

is a Lie group. For each g ∈ N L , we denote the class of g in G L by [g]. Since H acts trivially on L, the action of N L on L induces an action L of G L on L, given by L : G L × L → L : ([g], p) −→ [g] p = gp. Proposition 4.2.6 (i) The set PH = { p ∈ P | G p = H }

62

Stratified spaces

of points with isotropy group H is a local submanifold of P. Moreover, connected components of PH are open in P H . (ii) For each p ∈ PH , the tangent space T p PH consists of H -invariant vectors in T p P. (iii) For every vector field X on P, let T g ◦ X ◦ g −1 dμ(g) XH = H

be the H -average of X . For any H -invariant Riemannian metric k on P, the restriction of X H to PH is k-orthogonal to the restriction of X ⊥ H = X − X H to PH . Thus, the restriction of X to PH has a unique decomposition into its component tangent to PH and its component normal to PH , which is independent of the choice of an H -invariant Riemannian metric on P. (iv) For each connected component L of PH , the action of G L = N L /H on L is free and proper. (v) For each p ∈ L, Gp ∩ L = G L p, where Gp is the orbit through p of the action of G on P, and G L p is the orbit through p of the action of G L on L. Proof (i) It follows from the definition that PH ⊆ P H . Let p ∈ PH , so that G p = H , and let S p be the slice through p for the action of G on P that satisfies the conditions of Proposition 4.2.3. Hence, there exist neighbourhoods W of 0 in m and U of p in P such that ϕ : W × S p → U : (ξ, q) → (exp ξ, q) is a diffeomorphism. Let q ∈ U ∩ P H . Then there exists (ξ, q ) ∈ m × S p such that q = (exp ξ, q ) = (exp ξ )q . Hence, G q = (exp ξ )G s (exp(−ξ )). However, q ∈ P H implies that G q ⊇ H . On the other hand, q ∈ S p implies that G q ⊆ H . Therefore, H ⊆ G q = (exp ξ )G q (exp(−ξ )) ⊆ (exp ξ )H (exp(−ξ )), so that (exp(−ξ ))H (exp ξ ) ⊆ H.

(4.8)

Hence, the Lie algebra of (exp(−ξ ))H (exp ξ ) is contained in h. Since the conjugation H → (exp(−ξ ))H (exp ξ ) is an isomorphism, it follows that the dimension of the Lie algebra of (exp(−ξ ))H (exp ξ ) is the same as the dimension of h. Therefore, the Lie algebra of (exp(−ξ ))H (exp ξ ) is equal to h. This

4.2 Action of a Lie group on a manifold

63

implies that the connected component of the identity in (exp(−ξ ))H (exp ξ ) coincides with the connected component He of the identity in H . Thus, the connected component of g in (exp(−ξ ))H (exp ξ ) is g He , which is the same as the connected component of g in H . Hence, (exp(−ξ ))H (exp ξ ) = H . We have shown that for each q ∈ U ∩ P H , the isotropy group G q is equal to H , so that q ∈ PH . Therefore, U ∩ P H ⊆ U ∩ PH . But PH ⊆ P H . Hence, U ∩ P H = U ∩ PH . Therefore, PH is open in P H . Since P H is a local submanifold of P, it follows that PH is a local submanifold of P. (ii) If u ∈ T p PH ⊆ T p P, let t → c(t) be a curve in PH such that u = c(0). ˙ For every f ∈ C ∞ (P) and each g ∈ H , we have T g (u)( f ) = d f | T g (u) = d f ◦ T g | u = d( f ◦ g ) | u

d d = u(( f ◦ g )) = ( f (gc(t)))|t=0 = ( f (c(t)))|t=0 = u( f ). dt dt Hence, T g (u) = u for every g ∈ H . Conversely, suppose that u ∈ T p P is H -invariant. By Proposition 4.2.5, u ∈ T p P H . Since connected components of PH are open in P H , it follows that T p P H = T p PH . Therefore, every H -invariant vector in T p P is contained in T p PH . (iii) Let X H = H T g ◦ X ◦ g −1 dμ(g) be the H -average of a vector field X on P. Since X H is H -invariant, it follows from part (ii) that X H is tangent to PH . For p ∈ PH and u ∈ T p P, the average u H = H T g (u) dμ(g) of u over H is in T p PH . Suppose that k is an H -invariant Riemannian metric on P. For every u, v ∈ T p P, the H -invariance of k implies that k(T g (u), T g (v) = k(u, v) for all g ∈ H . Hence, if v ∈ T p PH , then v is H -invariant, and T g (u) dμ(g), v = k(T g (u), v) dμ(g) k(u H , v) = k H H k T g (u), T g (v) dμ(g) = k(u, v) = k(u, v). = H

H

If u is k-orthogonal to T p PH , then k(u H , v) = 0 for all v ∈ T p PH . Since u H is in T p PH , it follows that u H = 0. Conversely, suppose that u H = 0. Then, k(u, v) = 0 for all v ∈ T p PH , which implies that u is k-orthogonal to T p PH . If X is a vector field on P, then the H -average of X ⊥ H = X − X H van( p) is k-orthogonal to T p PH . Thus the ishes. Therefore, for every p ∈ PH , X ⊥ H

64

Stratified spaces

decomposition X |PH = X H |PH + X ⊥ H |PH is k-orthogonal for every H -invariant Riemannian metric k on P. (iv) Let [g1 ] and [g2 ] be elements of G L such that [g1 ] p = [g2 ] p for some p ∈ L. This means that g1 and g2 are in N L and g1 p = g2 p. Hence, h = g1−1 g2 ∈ H , and g2 = g1 h. Thus, [g2 ] = [g1 ]. Therefore, the action of G L on L is free. We are now going to show that L is proper. Consider a convergent sequence ( pn ) in L with limit p ∈ L. Let ([g]n ) be a sequence in G L such that [g]n pn converges to p ∈ L. For each n ∈ N, let gn ∈ N L be such that [g]n = [gn ]. Then [g]n pn = gn pn converges to p ∈ L. Since the action of G on P is proper, there exists a subsequence (gn k ) of (gn ) converging to g ∈ G such that p = lim (gn k pn k ) = g lim pn k = gp. k→∞

k→∞

By assumption, p and p are in L, which implies that g ∈ N L . Since the quotient map N L → G L = N L /H is continuous, the sequence ([gn k ]) converges to [g]. Thus the action of G L on L is proper. (v) For p ∈ L, consider a point gp ∈ L ⊆ PH , and g H g −1 = H , which / NL . implies that g ∈ N H . We want to show that g ∈ N L . Suppose that g ∈ Then there exists p ∈ L such that gp ∈ / L. Let U = { p ∈ L | gp ∈ L} and V = { p ∈ L | gp ∈ L}. The map g|L : L → PH , obtained as a restriction of g to the domain L, with codomain PH , is continuous. Since L is a connected component of PH , −1 it follows that U = −1 g|L (L) and V = g|L (PH \L) are both open in L. By construction, U ∩ V = ∅ and U ∪ V = L. Hence, if U and V are not empty, then the sets U and V form a disconnection of L. But L is connected, and we have assumed that p ∈ U . Thus, V = ∅, which implies that g ∈ N L , so that [g] ∈ G L and gp = [g] p. Therefore, Gp ∩ L ⊆ G L p. Conversely, G L p = {[g] p | [g] ∈ G L } = {gp | g ∈ N L ⊆ G} ⊆ Gp, which completes the proof. Proposition 4.2.7 The set P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G} of points of orbit type H is a local submanifold of P.

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65

Proof Let p ∈ PH , and let S p be a slice at p for the action of G on P. As in Remark 4.2.4, let m be a subspace of g complementary to h, and let ϕ : W × S p → U : (ξ, q) → (exp ξ, q) be a diffeomorphism of an open neighbourhood W × S p of (0, p) in m × S p onto an open neighbourhood U of p in P. We can take U = Exp p (V ) for an open neighbourhood of 0 in T p P. We have shown in the proof of Proposition 4.2.6 that PH ∩ U = P H ∩ U . Observe that P(H ) = G PH . Hence, U ∩ P(H ) = {(exp ξ, q) | ξ ∈ W, q ∈ S p ∩ PH }. But S p ⊆ U , and so S p ∩ PH = S p ∩ P H = S pH is the set of H -invariant points in S p . By Proposition 4.2.5, S pH is a local submanifold of S p . Hence, W × S pH is a local submanifold of W × S p and U ∩ P(H ) = ϕ −1 (W × S pH ) is a local submanifold of U . Therefore, P(H ) is a local submanifold of P. In the remainder of this section, we are going to show that the family M of connected components of P(H ) as H varies over compact subgroups of G is a stratification of P. In order to prove this claim, we have to show that this family is locally finite and that it satisfies Frontier Condition 4.1.1. Connected components of P(H ) are locally closed, since they are submanifolds of P. Proposition 4.2.8 The family M of connected components of P(H ) as H varies over compact subgroups of G is locally finite. Proof We prove this result by induction on the dimension of P. If dim P = 0, then P is discrete. The assumption that the action of G on P is continuous and G is connected implies that every point of P is a fixed point of G. Hence, orbits of G in P are singletons. This implies that we have only one orbit type P(G) = PG = P. The assumption that the action is proper implies that G is compact. Since P has the discrete topology, for every p ∈ P the singleton { p} is open and intersects only one connected component of P(G) = P. Therefore, M is locally finite for dim P = 0. Suppose now that dim P = m and that we have proved local finiteness of M for all proper actions G × P → P such that dim P < m. Consider a point p ∈ P and a slice S p through p as in Proposition 4.2.3. That is, S p = Exp p (B), where Exp p is a G p -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a G p -invariant open ball in hor T p P centred at the origin. For sufficiently small r > 0, the sphere Sr = {v ∈ hor T p P | k(v, v) = r 2 }

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is contained in B. Since k is G p -invariant, it follows that the action p on T p P preserves Sr and induces a proper action of G p on Sr . By construction, dim Sr = dim(hor T p P) − 1 < m = dim P. Therefore, by the induction hypothesis, the family of connected components of (Sr )(H ) as H varies over compact subgroups of G p is locally finite. Here we have denoted by (Sr )(H ) the subset of Sr of orbit type (H ). Since the action p is linear, for every t ∈ R and g ∈ G p , we have (tv) = t(v), and the family of connected components of subsets (T p S p )(H ) of T p S p as H varies over compact subgroups of G p is locally finite. Therefore, the family of connected components of subsets (S p )(H ) of S p is locally finite. Since the group G is connected and G S p is a G-invariant neighbourhood of p in P, it follows that, for each compact subgroup H of G, the intersection with S p of every connected component of P(H ) is a connected component of (S p )(H ) . Hence, the family M of connected components of P(H ) as H varies over compact subgroups of G is locally finite. Proposition 4.2.9 The family M of connected components of P(H ) as H varies over compact subgroups of G satisfies Frontier Condition 4.1.1. Proof Let H and K be compact subgroups of G such that K ⊆ H . Suppose that P(H ) ∩ P (K ) = ∅, and let p ∈ PH ∩ P (K ) . Consider a slice S p at p for the action of G on P constructed as in Proposition 4.2.3. That is, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in hor T p P to a neighbourhood of p in P, and B is an H -invariant open ball in hor T p P centred at the origin. Since p ∈ PH , there is an action of H on S p . We have a linear action of H on hor T p P. For v ∈ hor T p P, we denote by Hv the isotropy group of v. Clearly, (hor T p P) H = {v ∈ hor T p P | Hv = H } = {v ∈ hor T p P | gv = v for all g ∈ H } = (hor T p P) H . Let (hor T p P) K = {v ∈ hor T p P | Hv = K }. Since the action of H on hor T p P is linear, for every g ∈ K ⊆ H , v ∈ (hor T p P) K and w ∈ (hor T p ) H , we have g(v + w) = gv + gw = v + w. Hence, K ⊆ Hv+w . On the other hand, if g ∈ Hv+w , then we also have g(v + w) = gv + gw = v + w, so that gv = v because w ∈ (T p S p ) H implies that gw = w. Therefore Hv+w = K , and (hor T p P) K = (hor T p P) K + (hor T p P) H . This implies that (hor T p H ) H = (hor T p P) H ⊆ (hor T p P) K .

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Now, B is an H -invariant open ball in hor T p P centred at 0, B H = B ∩ (hor T p H ) H and B K = B ∩ (hor T p P) K . Therefore, B ∩ B K = B ∩ (hor T p P) K ⊇ B ∩ (hor T p P) H = B H . Since Exp p : B → S p is a diffeomorphism intertwining the linear action of H on T p S p and the action of H on S p , it follows that (S p ) K = Exp p (B K ), (S p ) H = Exp p (B H ) and (S p ) H ⊆ S p ∩ (S p ) K . Furthermore, G S p is a G-invariant neighbourhood of p in P, P(H ) = G PH and P(K ) = G PK . Hence, (G S p ) ∩ P(H ) = (G S p ) ∩ (G PH ) = G(S p ∩ PH ) = G(S p ) H is a G-invariant neighbourhood of p in P(H ) . Similarly, (G S p ) ∩ P(K ) = (G S p ) ∩ (G PK ) = G(S p ∩ PK ) = G(S p ) K is a G-invariant open subset of P(K ) such that (G S p ) ∩ P(K ) = G(G S p ∩ P(K ) ) = G(G S p ∩ (G S p ∩ PK )) = G(G S p ∩ (G S p ) K ) ⊇ G(G S p ) H = G(G S p ∩ PH ) = G S p ∩ P(H ) . Hence, the component M of P(H ) containing p is contained in the closure N of the component N of P(K ) such that p ∈ N . We have shown that the family M of connected components of P(H ) as H varies over compact subgroups of G gives a stratification of P. This stratification is called the orbit type stratification of P.3

4.3 Orbit space In this section, we are going to show that the projection to the orbit space R = P/G of the orbit type stratification M of P is a minimal stratification N of R. Hence, strata of N coincide with orbits of the family of all vector fields on R. 3 It should be noted that the orbit type stratification M of P is not minimal, because P is a

smooth manifold and the union of all strata of M.

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We endow the orbit space R with the quotient topology. In other words, a subset V of R is open if U = ρ −1 (V ) is open in P, where ρ : P → R is the canonical projection (the orbit map). Let C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}.

(4.9)

By Theorem 2.1.10, C ∞ (R) is a differential structure on R. We denote by C ∞ (P)G the ring of G-invariant smooth functions on P. For each f ∈ C ∞ (R), its pull-back ρ ∗ f is G-invariant. Conversely, if f˜ is a G-invariant smooth function on P, then it pushes forward to a function f = ρ∗ f˜ in C ∞ (R). Since ρ ∗ ◦ ρ∗ = identity, it follows that f˜ = ρ ∗ f . This means that the pull-back ρ ∗ : C ∞ (R) → C ∞ (P) induces a ring isomorphism between C ∞ (R) and the ring C ∞ (P)G of G-invariant functions in C ∞ (P). Proposition 4.3.1 The topology of R induced by C ∞ (R) coincides with the quotient topology. Proof In view of Proposition 2.1.11, it suffices to show that, for each set V in R which is open in the quotient topology, and each y ∈ R, there exists f ∈ C ∞ (R) such that f (y) = 1 and f |R\V = 0. Let y ∈ V ⊆ R, where V is open in the quotient topology. Consider a slice S p through a point p ∈ ρ −1 (V ) ⊆ P such that ρ( p) = y. The intersection W = ρ −1 (V ) ∩ S p is a neighbourhood of p in S p . Moreover, it is G p -invariant because ρ −1 (V ) and S p are invariant. There exists a compactly supported nonnegative function h ∈ C ∞ (S p ) such that h( p) = 1 and the support of h is contained in W . Since G p is compact, we may average h over G p , obtaining a G p -invariant function ˜h = ∗g h dμ(g), Gp

where dμ(g) is the Haar measure on G p normalized so that vol G p = 1. Since G p and the support of h are compact, W is G p -invariant, and the support of ˜ denoted by supp h, ˜ is compact and h ⊆ W , it follows that the support of h, contained in W . The set G S p is a G-invariant open neighbourhood of p in P. We can define a G-invariant function f˜1 on G S p as follows. For each p ∈ G S p , there exists g ∈ G such that p = gp for p ∈ S p , and we set ˜ p ). f˜1 ( p ) = h( Since h˜ is G p -invariant, the function f˜1 is well defined on U and is Ginvariant. Moreover, the support of f˜1 is contained in U ∩ ρ −1 (V ). The sets ˜ form a G-invariant open cover of P. We G S p ∩ ρ −1 (V ) and P\G(supp h)

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can extend f˜1 to a smooth G-invariant function f˜ on P, which vanishes on ˜ Since f˜ is G-invariant, there exists a function f ∈ C ∞ (R) P\ρ −1 (ρ(supp h)). such that f˜ = ρ ∗ f . We have ˜ p) = h( p) = 1. f (y) = f (ρ( p)) = f˜( p) = f˜1 ( p) = h( ˜ ⊆ G S p ∩ ρ −1 (V ) ⊆ ρ −1 (V ), it follows that Moreover, since G(supp h) −1 ˜ and f˜|P\ρ −1 (V ) = 0. Hence, f |R\V = 0, which P\ρ (V ) ⊆ P\G(supp h) completes the proof. A left principal fibre bundle with structure group G is a manifold P with an action of G on P such that the orbit space R = P/G is a manifold and the orbit map ρ : P → R is a locally trivial fibration. Moreover, for each x ∈ R, there exist an open neighbourhood V of x in R and a diffeomorphism ψV : π −1 (V ) → G × V , which intertwines the action of G on π −1 (V ) and the action of G on G ×V given by multiplication on the left. In this case, the action of G on P is free; that is, gp = p implies that g is the identity element of G. Theorem 4.3.2 If the action of G on P is free and proper, then P is a left principal fibre bundle with structure group G. Proof We consider the orbit space R = P/G with the differential structure given by equation (4.9). By Proposition 4.3.1, R is a differential space with the quotient topology. Since the action is also free, for each p ∈ P, the orbit Gp through p is diffeomorphic to G. Given p ∈ P, there exists a submanifold S of P containing p and satisfying the condition Ts P = Ts (Gs) ⊕ Ts S

(4.10)

for every s ∈ S ⊂ P. Consider a map of : G × S → P obtained by the restriction of to G × S. For each (g, s) ∈ G × S, we have (g, s) = (g, s) = s (g) = gs. We show first that T(e,s) , where e is the identity in G, is bijective. For each (ξ, v) ∈ Te G × Ts S, we have T(e,s) (ξ, v) = T s (ξ ) + v. If T(e,s) (ξ, v) = 0 for some (ξ, v) ∈ Te G × Ts S, then T s (ξ ) + v = 0, which implies that v = −T s (ξ ) ∈ Ts (Gs). By assumption, v ∈ Ts S, so that v ∈ Ts (Gs) ∩ Ts S. But, by equation (4.10), Ts (Gs) ∩ Ts S = 0, which implies that v = 0 and T s (ξ ) = 0. Hence, T s (tξ ) = 0 for all t ∈ R. Therefore, s is a fixed point of the action restricted to the one-parameter subgroup exp tξ

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of G. The assumption that the action is free implies that ξ = 0, so that T(e,s) . Equation (4.10) ensures that rank T(e,s) = dim G + dim S = dim P. Hence, T(e,s) is surjective. Thus, T(e,s) is bijective. Since (g, s) = g ◦ (e, s), it follows that T(g,s) = Ts g ◦ T(e,s) . Since T(e,s) is bijective and g is a diffeomorphism, we conclude that T(g,s) is bijective for each g ∈ G and s ∈ S. Therefore, : G × S → P is a local diffeomorphism. The next step in the proof is to show that, shrinking S if necessary, we can obtain a diffeomorphism of G × S onto its image in P. We prove this by contradiction. Suppose that there is no neighbourhood S0 of p in S such that | G × S0 is one-to-one. Then, there exists a nested neighbourhood base {Sk } of p in S with the following property. For each k, there are a point sk ∈ Sk and an element gk ∈ G such that gk sk ∈ Uk but gk is bounded away from e. Therefore, gk sk → p as k → ∞. By the properness of , there is a subsequence gkl → g such that gm = m. This contradicts the assumption that the action is free. We have shown that, for each p ∈ P, there exists a submanifold S of P through p satisfying equation (4.10) and the condition that : G × S → P is a diffeomorphism onto its image. It follows from the construction that intertwines the action of G on G × S, given by multiplication on the left, and the action of G on P. In order to show that R is a manifold, observe that each point x ∈ R has a neighbourhood V = ρ(S), where S is a manifold through p ∈ ρ −1 (x) satisfying the conditions in the preceding paragraph. Since the restriction of the orbit map ρ to S is one-to-one, the map ρ|S : S → V is a diffeomorphism of S onto V . Thus, each point x of R has a neighbourhood V diffeomorphic to an open subset of Rdim S . Hence, R is a manifold. Since ρ(G × S) = ρ(S) = V , it follows that maps G × S onto ρ −1 (V ). Hence, ψV : ρ −1 (V ) → G × V = G × ρ(S) : p → (id, ρ) ◦ −1 ( p) is a diffeomorphism intertwining the action of G on ρ −1 (V ) and the action of G on G × V . We now consider a more general case in which the action of G on P need not be free. Let H be a compact subgroup of G, and let p ∈ PH . Consider

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a slice S p = Exp p (B), where Exp p is an H -equivariant diffeomorphism of a neighbourhood of 0 in T p P onto a neighbourhood of p in P, and B is an open ball in hor T p P centred at the origin. Since H is compact, the action of H on S p is proper. Hence, the orbit space S p /H endowed with the differential structure C ∞ (S p /H ) = { f ∈ C 0 (S p /H ) | π ∗ f ∈ C ∞ (S p )}, where π : S p → S p /H is the orbit map, is a differential space with the quotient topology. Lemma 4.3.3 G S p /G is diffeomorphic to S p /H . Proof For every orbit of G in G S p , its intersection with S p is an orbit of the action of H on S p . This gives a natural bijection δ : G S p /G → S p /H. We need to verify that δ is a diffeomorphism of a differential subspace G S p /G of R = P/G onto the quotient of S p by H , where S p is a differential subspace P. Let h be a function in C ∞ (S p /H ). Then, π ∗ h is an H -invariant function in C ∞ (S p ). For q ∈ S p , let W and W be H -invariant neighbourhoods of q in S p such that W ⊆ W ⊆ W ⊆ S p . Let k ∈ C ∞ (S p ) be a function such that k|W = 1 and k|S p \W = 0. The function ∗g (kπ ∗ h) dμ(g) f¯S p = H

on S p , where dμ(g) is the Haar measure on H normalized so that vol H = 1, is smooth and H -invariant. Moreover, the function f¯S p coincides with π ∗ h on W and vanishes on the complement of W in S p . We can extend f¯S p to a G-invariant function f˜ on G S p . Let f be a function on P such that f |G S p = f˜ and f |P\GW = 0. Since f¯S p vanishes on the complement S p \W of W in S p , it follows that f˜ vanishes on G(S p \W ) = G S p \GW . Hence, f is well defined, smooth and G-invariant. Therefore, f pushes forward to a function ρ∗ f in C ∞ (R) such that ρ ∗ (ρ∗ f ) = f. Since W is an H -invariant open neighbourhood of q ∈ S p , W/H is a neighbourhood of π(q) in S p /H and GW/G is a neighbourhood of ρ(q) in R. For each p ∈ W , we have ρ( p ) ∈ GW/G ⊆ R, π( p ) ∈ W/H ⊆ S p /H and δ(ρ( p )) = π( p ). Moreover, ρ∗ f (ρ( p )) = f ( p ) = f˜( p ) = f¯S p ( p ) = k( p ) f S p ( p ) = π ∗ h( p ) = h(π( p )),

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which implies that (δ ∗ h)|GW/G = (ρ∗ f )|GW/G . Since p ∈ S p is arbitrary, Condition 3 of Definition 2.1.1 implies that δ ∗ h ∈ C ∞ (G S p /G). To show that δ −1 is smooth, consider a function h ∈ C ∞ (R). The restriction h |G S p /G of h to G S p /G is in C ∞ (G S p /G). The pull-back ρ ∗ h is a smooth G-invariant function on P. Its restriction to G S p is a smooth H invariant function (ρ ∗ h)|G S p on G S p . Hence, (ρ ∗ h)|G S p pushes forward to a smooth function π∗ ((ρ ∗ h)|G S p /H ) on G S p /H . Moreover, for each p ∈ G S p , β(ρ( p )) = π( p ) implies ρ( p ) = δ −1 (π( p )) and h |G S p /G (δ −1 (π( p ))) = h |G S p /G (ρ( p )) = ρ ∗ h( p ) = (ρ ∗ h)|G S p ( p ) = π∗ ((ρ ∗ h)|G S p )(π( p )). Hence, (δ −1 )∗ h |G S p /G = π∗ ((ρ ∗ h)|G S p ) ∈ C ∞ (S p /H ). Therefore, δ is a diffeomorphism. The next step is to prove that the orbit space R = P/G of a proper action of G on P is subcartesian. The proof uses several classical results, references to which are given in Chapter 1. Theorem 4.3.4 The orbit space R = P/G of a proper action of G on P with the differential structure C ∞ (R) given by Equation (4.9) is subcartesian. Proof (i) Hausdorff property. If R were not Hausdorff, there would be two distinct points x and x in R which cannot be separated by open sets. We choose points p and p in ρ −1 (x) and ρ −1 (x ), respectively. Let (Wk ) be a nested neighbourhood basis of p, and let (Wk ) be a nested neighbourhood basis of p . The assumption that x and x cannot be separated by open sets implies that, for each k ∈ N, there exists pk ∈ GWk ∩ GWk . Hence, there exist gk and gk in G such that gk pk ∈ Wk and gk pk ∈ Wk . By construction, gk pk converges to p and gk pk converges to p . Therefore, (gk gk−1 )(gk pk ) = gk pk converges to p . Since the action of G on P is proper, there exists a subsequence gk n gk−1 in G n that is convergent in G, and −1 )(g p )} = lim (g g ) lim (g p ) p = lim {(gk n gk−1 k k k k n n n n k n kn n n→∞ n→∞ n→∞ −1 = lim (gkn gkn ) p. n→∞

This implies that p and p are in the same G-orbit, which contradicts the assumption that x = x . (ii) Choice of neighbourhood. Next, we need to show that, for each x ∈ R, there exists a neighbourhood of x in R that is diffeomorphic to a subset of Rn , for some n ∈ N. Choose a point p ∈ ρ −1 (x), and let S p be a slice through p constructed as in Proposition 4.2.3. Let H be the isotropy group of p. We have

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shown in Lemma 4.3.3 that G S p /G is diffeomorphic to S p /H . We are going to show that S p /H is diffeomorphic to a subset of Rn . (iii) Differential structure of S p /H . By construction, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a ball in hor T p P centred at the origin. The action of H on T p P is linear, and it leaves hor T p P invariant. Hence, it gives rise to a linear action of H on hor T p P. Moreover, the restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action of H on hor T p P and the action of H on S p . Therefore, S p /H is diffeomorphic to B/H . Since B is an H -invariant open subset of hor T p P and the action of H on hor T p S p is linear, by a theorem of G.W. Schwarz, smooth H -invariant functions on S p are smooth functions of algebraic invariants of the action of H on hor T p P. Let R[hor T p P] H denote the algebra of H -invariant polynomials on hor T p P. Hilbert’s Theorem ensures that R[hor T p P] H is finitely generated. Let σ1 , . . . , σn be a Hilbert basis for R[hor T p P] H consisting of homogeneous polynomials. The corresponding Hilbert map σ : hor T p P → Rn : v → σ (v) = (σ1 (v), . . . , σn (v)) induces a monomorphism σ˜ : (hor T p P)/H → Rn : H v → σ (v), where H v is the orbit of H through v ∈ hor T p P treated as a point in (hor T p P)/H . Let Q be the range of σ . By the Tarski–Seidenberg Theorem, Q is a semi-algebraic set in Rn . Let ϕ : (hor T p P)/H → Q ⊆ Rn be the bijection induced by σ˜ . We want to show that ϕ is a diffeomorphism. Smoothness of σ implies that ϕ is smooth. To show that ϕ −1 : Q → (hor T p P)/H is smooth, consider a function f ∈ C ∞ ((hor T p P)/H ). Let π : hor T p P → (hor T p P)/H denote the orbit map. Then, π ∗ f ∈ C ∞ (hor T p P) H . By the theorem of Schwarz quoted above, there exists F ∈ C ∞ (Rn ) such that, for all v ∈ T p S p , f (π(v)) = π ∗ f (v) = F(σ (v)) = F|Q (σ (v)) = F|Q (ϕ(π(v))). Hence, f = ϕ ∗ F|Q , which implies that F|Q = (ϕ −1 )∗ f . Thus, for every f ∈ C ∞ ((hor T p P)/H ), (ϕ −1 )∗ f ∈ C ∞ (Q). Therefore ϕ : (hor T p P)/H → Q is a diffeomorphism. Since B is an H -invariant open neighbourhood of 0 in hor T p S p , it follows that B/H is open in (hor T p P)/H . Hence, B/H is in the domain of the diffeomorphism ϕ : (hor T p P)/H → Q, which induces a diffeomorphism of B/H onto ϕ(B/H ) ⊆ Q ⊆ Rn . Thus, B/H is diffeomorphic to a subset of Rn . But B/H is diffeomorphic to S p /H , and S p /H is diffeomorphic to G S p /G. Therefore, G S p /G is diffeomorphic to a subset of Rn .

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The arguments given above hold for each point p ∈ P. Therefore, the orbit space R = P/G is subcartesian. Now that we know that the orbit space is subcartesian, we want to show that it is stratified. In the preceding section, we described the orbit type stratification M of P given by connected components of P(H ) = { p ∈ P | ∃g ∈ G such that G p = g H g −1 } as H varies over compact subgroups of G. We are going to show that the orbit map ρ : P → R maps M to a stratification N of R. Theorem 4.3.5 The family N = {ρ(M) ⊆ R | M ∈ M}, where M is the orbit type stratification of P and ρ : P → R is the orbit map, is a stratification of the orbit space R. Proof (i) Strata. Let H be a compact subgroup of G. Consider a point p0 ∈ PH ⊆ P(H ) , and let M be the connected component of P(H ) that contains p0 . We begin with a claim that L = M ∩ PH is the connected component containing p0 , and that M = G L. Clearly, L is an open subset of PH , and p0 ∈ L. Moreover, we can observe that P(H ) = G PH and that the connectedness of G implies that G M = M. Suppose now that L is disconnected. Let (U1 , U2 ) be a disconnection of L. In other words, U1 , U2 are open sets in L such that U1 ∩ U2 = ∅ and U1 ∪ U2 = L. Then, (GU1 , GU2 ) is a disconnection of M, which contradicts the assumption that M is connected. On the other hand, if the connected component L 0 of PH that contains p0 contains L as a proper subset, then G L 0 is a connected open subset of P(H ) , and G L 0 contains M as a proper subset, which contradicts the assumption that M is a connected component of P(H ) . It follows from Proposition 4.2.6 that L is a submanifold of P. Moreover, M = G L is a connected component of P(H ) , and it is a stratum of the orbit type stratification M of P. We want to show that ρ(M) is a submanifold of R. We showed in Proposition 4.2.6 that the action of G L on L is free and proper. It follows that the quotient L/G L is a manifold and that the orbit map π : L → L/G L is a (left) principal fibre bundle projection. Moreover, for every p ∈ L, we have Gp = G L p. Hence, there is a map γ : L/G L → ρ(M), given as follows. Each point in L/G L is of the form π( p), where p ∈ L. Since L ⊆ M ⊆ P, the projection ρ( p) to R is contained in ρ(M). We set γ (π( p)) = ρ( p) ∈ ρ(M). Conversely, consider ρ( p ) ∈ ρ(M) for some p ∈ M. Since M = G L , there exists p ∈ L such that ρ( p) = ρ( p ). Then, γ (π( p)) = ρ( p ). Hence, γ : L/G L → ρ(M) is a bijection. An argument analogous to the proof of Lemma 4.3.3 ensures that γ is a diffeomorphism of the quotient

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differential space L/G L , where L is a differential subspace of M = G L ⊆ P, and the differential subspace ρ(M) of the quotient R = P/G. Since ρ(M) is a differential subspace of R diffeomorphic to the manifold L/G L , it follows that ρ(M) is a submanifold of R. (ii) Frontier condition. Suppose M and M are strata of M such that ρ(M )∩ ρ(M) = ∅. Since M is G-invariant, it follows that ρ(M) = ρ(M). Therefore, M ∩ M = ∅, which implies that either M = M or M ⊂ M\M. Hence, either ρ(M ) = ρ(M) or ρ(M ) ⊂ ρ(M\M) = ρ(M)\ρ(M). Thus, the family N satisfies Frontier Condition 4.1.1. (iii) Local finiteness. For each p ∈ P, there is a neighbourhood V of p in P intersecting a finite number of manifolds M ∈ M. Hence, GV is a Ginvariant neighbourhood of p which intersects a finite number of manifolds M ∈ M. Therefore, ρ(GV ) is a neighbourhood of ρ( p) which intersects a finite number of manifolds ρ(M) ∈ N. This implies that N is locally finite. We have shown that the family N = {N = ρ(M) ⊆ R | M ∈ M} is a locally finite partition of R by submanifolds and satisfies the frontier condition. Hence, N is a stratification of R. The stratification N is called the orbit type stratification of R. Our ultimate goal in this section is to prove that the stratification N of R coincides with the partition O of R by orbits of the family X(R) of all vector fields on R. By Proposition 4.1.5 and Theorem 4.1.6, it suffices to show that N is locally trivial and minimal. First, we show that N of R is locally trivial. According to Definition 4.1.4, N is locally trivial if, for every N ∈ N and each x ∈ N , (i) there exists a neighbourhood U of x in R such that NU is a stratification of U ; (ii) there exists a subcartesian stratified space (S , M ) with a distinguished point y ∈ S such that the singleton {y} ∈ M ; and (iii) there is an isomorphism ϕ : (U, NU ) → ((N ∩ U ) × S , M(N ∩U )×S ) such that ϕ(x) = (x, y). For x ∈ R, there exists p ∈ P such that x = ρ( p). Let H be the isotropy group of p. Consider a slice S p through p constructed as in Proposition 4.2.3. The set U = ρ(G S p ) = G S p /G is an open neighbourhood of x in R. Lemma 4.3.2 ensures that U = G S p /G is diffeomorphic to S p /H . Hence, the restriction to U of the orbit type stratification N of R is a stratification NU isomorphic to the orbit type stratification of S p /H . By construction, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a ball in

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hor T p P centred at the origin. The action of H on T p P is linear, and it leaves hor T p P invariant. Hence, it gives rise to a linear action of H on hor T p P. Moreover, the restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action of H on hor T p P and the action of H on S p . Therefore, S p /H is diffeomorphic to B/H , and the orbit type stratification of S p /H is isomorphic to the orbit type stratification of B/H . Thus, the stratification NU is isomorphic to the orbit type stratification of B/H , where B is an open ball invariant under a linear action of H in a vector space hor T p P. We begin with a discussion of the orbit type stratification of the space of orbits of a linear action of a compact group on a vector space. Lemma 4.3.6 Consider a linear action of a compact Lie group H on a vector space E endowed with an H -invariant metric k. Let E H be the space of H invariant vectors in E, and let F be the k-orthogonal complement of E H in E. The linear action of H on E induces a linear action of H on F. Let N E/H and N F/H be the orbit type stratifications of E/H and F/H , respectively. Then there is a stratified space isomorphism ψ : (E/H, N E/H ) → (E H × F, M E H ×F/H ) such that ϕ(0) = {0×0}, where M E H ×F/H is the product of the single-stratum stratification {E H } of E H and the orbit type stratification N F/H of F/H . Proof Since F is the k-orthogonal complement of E H in E, we obtain an H -equivariant product structure E = E H × F. For each compact subgroup K of G, E K = E H × FK and E (K ) = H E K = E H × H FK = E H × F(K ) . The product structure E = E H × F induces an isomorphism of the quotient spaces ψ : E/H → (E H /H ) × (F/H ). Since 0 ∈ E is H -invariant, it follows that the quotient map E → E/H maps 0 ∈ E to 0 ∈ E H . Moreover, 0 ∈ F is the only H -invariant vector in F, and the quotient F/H is a cone with vertex 0. Therefore, ψ(0) = (0, 0). For each proper compact subgroup K of H , we have ψ(E (K ) /H ) = E H × (F(K ) )/H. Since E H is connected, it follows that each stratum of N E/H of the orbit type stratification of E/H is mapped by ϕ to the product of E H and a stratum of the orbit type stratification N F/H of F/H . Therefore, ψ is an

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isomorphism of the stratified space (E/H, N E/H ) onto the stratified space (E H × F/H, M E H ×F/H ), where M E H ×F/H is the product of the singlestratum stratification {E H } of E H and the orbit type stratification N F/H of F/H . Next, we can use the local results obtained in Lemma 4.3.6 to show that the stratification N of R is locally trivial. Proposition 4.3.7 The orbit type stratification N of the space R of orbits of a proper action of a connected Lie group G on a manifold P is locally trivial. Proof Let p ∈ PH . Consider a slice S p through p for the action of G on P constructed as in Proposition 4.2.3. The set U = ρ(G S p ) = G S p /G is an open neighbourhood of x in R, and U = G S p /G is diffeomorphic to S p /H . Hence, the restriction to U of the orbit type stratification N of R is a stratification NU isomorphic to the orbit type stratification of S p /H . Moreover, NU is isomorphic to the orbit type stratification of B/H , where B is an open ball invariant under a linear action of H in a vector space hor T p P. In order to use the results of Lemma 4.3.6, we set hor T p P = E so that B is an H -invariant open ball in E. Let B and B be open balls in E H and F, respectively, both centred at the origin and such that B × B ⊆ B. Moreover, we assume that B is invariant under the action of H in F. Note that the H -invariance of B is self-evident because B consists of fixed points. The product B × B is an H -invariant neighbourhood of 0 ∈ E. Since multiplication of vectors in E commutes with the action of H , it follows that (B × B )(K ) = B × (B ∩ F(K ) ) = B × B(K )

for each compact proper subgroup K of H . The isomorphism ψ : E/H → (E H /H ) × (F/H ) restricted to (B × B )/ H gives ψ((B × B )/H ) = (B /H ) × (B /H ) = B × (B /H ). Therefore, for each compact proper subgroup K of H ,

ψ((B × B )(K ) /H ) = B × (B(K ) /H ). Let χ = ψ|B ×B be the restriction of ψ to B × B . This is an isomorphism of the stratified space (B × B , N(B ×B )/H ), where N(B ×B )/H is the orbit type stratification, and the stratified space (B × (B /H ), M B ×(B /H ) ), where M B ×(B /H ) is the product of the single-stratum stratification {B } of B and the orbit type stratification N B /H of B /H .

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Recall that S p = Exp p (B), and U = ρ(S p ) is a neighbourhood of x = ρ( p) ∈ R such that the orbit type stratification N of R restricts to a stratification NU of U , which is isomorphic to the orbit type stratification of B/H . Restricting the neighbourhood of x to U = ρ(B × B ), we obtain an isomorphism ϕ of the stratification NU of U that is induced by the stratification NU of U to the stratification (B × (B /H ), M B ×(B /H ) ) of B × B , which satisfies the conditions of Definition 4.1.4. Since this construction can be performed for every x ∈ R, it follows that the orbit type stratification N of R is locally trivial. Next, we show that the orbit type stratification N of R = P/G is minimal. As before, we begin with the case of a linear action of a compact Lie group. Consider a linear action of a compact Lie group H on a vector space E endowed with an H -invariant metric k. Let E H be the space of H -invariant vectors in E, and let F be the k-orthogonal complement of E H in E. The linear action of H on E induces a linear action of H on F. Moreover, E = E H × F is an H -equivariant product. As in the proof of Theorem 4.3.4, let R[E] H = R[E H × F] H denote the finitely generated algebra of H -invariant polynomials on E. Since E = E H × F, polynomials on E are sums of products of polynomials on E H and on F. We consider a Hilbert basis (σ1 , . . . , σn ) for R[E H × F] H consisting of homogeneous polynomials. For each i = 1, . . . , n, we denote by di the degree of σi . We choose σ1 , . . . , σl , where l = dim E H , to be degree-1 polynomials on E H . These polynomials are H -invariant, since each point of E H is fixed by the action of H on E. Moreover, H -invariant polynomials on E H of degree ≥ 2 are polynomials in σ1 , . . . , σl . Hence, the remaining elements of the basis can be chosen to be polynomials on F. Since F does not contain H -invariant non-zero vectors, there are no H -invariant polynomials on F of degree 1. Therefore, σl+1 , . . . , σn are homogeneous polynomials on F of degree ≥ 2. Moreover, for every v ∈ F, we can set σl+1 (v) = k(v, v), where k is the H -invariant metric on E. Let S = {v ∈ F | k(v, v) = 1} be the unit sphere in F. For each i ≥ l + 2, −1 let Ci be the maximum of |σi (v)| for v ∈ S. For v = 0, |v| v ∈ S, where √ |v| = [k(v, v)] = σl+1 (v) is the norm of v. Moreover, σi (v) = σi (|v| (|v|−1 v)) = |v|di σi (|v|−1 v) ≤ Ci |v|di implies the inequality |σi (v)| ≤ Ci [σl+1 (v)]di /2 , which is also valid for v = 0. Therefore, the image of the Hilbert map σ : E = E H × F → Rn : (u, v) → σ (u, v) = (σ1 (u), . . . , σl (u), σl+1 (v), . . . , σn (v))

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is contained in the set = {(σ1 , . . . , σn ) ∈ Rn | σl+1 ≥ 0 and |σi | ≤ Ci [σl+1 ]di /2 ∀ l + 2 ≤ i ≤ n}. In the description of given here, we have treated σ1 , . . . , σn as coordinates in Rn . Lemma 4.3.8 Let γ : I → σ (E) ⊆ Rn : t → σ (t), where I is an open interval in R containing 0, be a curve in σ (E), differentiable at t = 0 and such that σl+1 (v(0)) = 0. Then σi (0) = 0 for each i = l + 1, . . . , n. Proof The inclusion γ (I ) ⊆ σ (E) and the inequality |σi | ≤ Ci [σl+1 ]di /2 , for all l + 2 ≤ i ≤ n, imply that σl+1 (t) ≥ 0 for each t ∈ I . The assumption that σl+1 (0) = 0 implies that 0 is a local minimum of t → σl+1 (t). Hence, (0) = 0 and σl+1 σl+1 (t) →0 t as t → 0. Moreover, for each i = l + 2, . . . , n, and t ∈ I \{0}, |σi (t)| [σl+1 (t)]d/2 σl+1 (t) . ≤ Ci = Ci [σl+1 (t)](di /2−1) |t| |t| t For i = l +2, . . . , n, the degree di ≥ 2 and the map t → σi (t) is differentiable and continuous at t = 0. This implies that [σl+1 (t)](di /2−1) is bounded as t → 0. Since σl+1 (t) →0 t as t → 0, it follows that |σi (t)| →0 |t| as t → 0. Hence, σi (0) = 0 for i = l + 2, . . . , n. We showed earlier that (0) = 0. Therefore, σ (0) = 0 for i = l + 1, . . . , n. σl+1 i It follows from Lemma 4.3.8 that, if σ (E) is a submanifold of Rn , then the dimension of T0 σ (E) is l, which is equal to the dimension of the space E H of fixed points of H . This implies that E = E H and that the orbit type stratification of E/H has only one stratum. In other words, if the orbit type stratification of E/H has more than one stratum, it cannot be a refinement of a coarser stratification. Thus, the orbit type stratification of E/H is minimal. Theorem 4.3.9 Let R = P/G be the space of G-orbits of the proper action of a connected Lie group G on a manifold P. Then the orbit type stratification of R is minimal.

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Proof Let B be an H -invariant open ball in E centred at the origin. The orbit type stratification N E/H of E/H restricts to the orbit type stratification N B of B. Since B contains the origin in E, the origin in Rn is contained in the range σ (B) of the Hilbert map σ : E → Rn . Hence, the argument above applies to σ (B) and ensures that the orbit type stratification of B/H is minimal. In Proposition 4.2.3, we constructed a slice S p through p for the action of G on P as the image of an open ball B in a vector space E = hor T p P under a local H -equivariant diffeomorphism Exp p of a neighbourhood of 0 in E onto a neighbourhood of p ∈ P, where H = G p . Therefore, the orbit type stratification of S p /H is isomorphic to the orbit type stratification of B/H . Hence, the orbit type stratification of S p /H is minimal. In Lemma 4.3.3, we showed that the orbit type stratification of G S p /G is isomorphic to the orbit type stratification of S p /H . Therefore, the orbit type stratification of S p /H is minimal. However, G S p /G is an open subset of R = P/G and the orbit type stratification of G S p /G is given by the restriction to G S p /P of the orbit type stratification of R. Therefore, the orbit type stratification of R is minimal. Combining the results of this section with Theorem 4.1.6, we obtain the desired result. Theorem 4.3.10 The space R = P/G of G-orbits of the proper action of a connected Lie group G on a manifold P endowed with the differential structure C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}, where ρ : P → R is the orbit map, is a subcartesian space with the quotient space topology. The orbit type stratification of P projects to an orbit type stratification of R. Strata of the orbit type stratification of R are orbits of the family of all vector fields on R. Proof According to Proposition 4.3.1, the differential-space topology of R coincides with the quotient topology. Theorem 4.3.4 ensures that R is subcartesian. In Theorem 4.3.4 we showed that the family N = {ρ(M) ⊆ R | M ∈ M}, where M is the orbit type stratification of P, is a stratification of the orbit space R, called the orbit type stratification of R. According to Proposition 4.3.7, N is locally trivial. Proposition 4.1.5 states that a locally trivial stratification of a subcartesian space R admits local extensions of vector fields. According to Theorem 4.1.6, if N is a stratification of a subcartesian space R which admits local extensions of vector fields, then the partition O of R by orbits of the family X(R) of all vector fields is a stratification of R, and N is a refinement of O. Moreover, if N is minimal, then N = O. Minimality of N has been proven in

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Theorem 4.3.9. Hence, strata of the orbit type stratification of R are orbits of the family of all vector fields on R. Theorem 4.3.10 provides a justification for the technique of singular reduction of symmetries in geometric mechanics. We shall discuss singular reduction of symmetries of symplectic manifolds in Chapter 6.

4.4 Action of a Lie group on a subcartesian space In this section, we show that the space of orbits of a proper action of a Lie group on a locally compact subcartesian space is a Hausdorff, locally compact differential space with the quotient topology. We begin with a review of the original Palais formulation for Hausdorff, locally compact topological spaces. Next, we specialize it to locally compact subcartesian spaces. We consider a continuous action : G × P → P : (g, p) → g ( p) = gp of a Lie group G on a Hausdorff, locally compact topological space P. We assume that P is a proper G-space. This means that each point p ∈ P has a neighbourhood U such that for every q ∈ P, there exists a neighbourhood V of q for which the closure of the set {g ∈ G | gU ∩ V = ∅} is compact. Let H be a closed subgroup of G. A subset S of P is an H -kernel if there exists an equivariant map f : G S → G/H such that f −1 (H ) = S. The following two theorems are quoted, without proof, from Palais (1961). Theorem 4.4.1 Let H be a closed subgroup of G. If S is an H -kernel in P, then: 1. S is closed in G S. 2. S is invariant under H . 3. gS ∩ S = ∅ implies that g ∈ H . If H is compact, then in addition: 4. S has a neighbourhood U in P such that the set {g ∈ G | gU ∩ U = ∅} has compact closure. Conversely, if the above conditions hold, then H is compact and S is an H kernel in P. Proof

See Theorem 2.14 in Palais (1961).

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Let a subset S of P be an H -kernel. If G S is open in P, the set S is called an H -slice in P. If in addition G S = P, then S is a global H -slice in P. Since P is a proper G-space, the isotropy group G p = {g ∈ G | gp = p} of p is compact for each p ∈ P. We say that a subset S of P is a slice at p if S is a G p -slice containing p. In the following, we shall denote a slice at p ∈ P by S p . Theorem 4.4.2 If P is a proper G-space, then for every point p ∈ P there exists a slice at p. Proof

See Proposition 2.3.1 in Palais (1961).

Next, we show that the notions of a ‘proper action’ and a ‘proper G-space’ are equivalent. Proposition 4.4.3 A Hausdorff, locally compact topological space P is a proper G-space if and only if the action of G on P is proper. Proof Given p0 ∈ P, let U be a neighbourhood of p0 in P with compact closure. Take any q0 ∈ P, and let V be a neighbourhood of q0 with compact closure. We want to show that the set W = {g ∈ G | gU ∩ V = ∅} has compact closure. In other words, if gn is a sequence of points in W , then there exists a convergent subsequence. Each gn ∈ W is the limit point of a sequence gn,m ∈ W . That is, for each n, m, there exists pn,m ∈ U such that gn,m pn,m ∈ V . Since V has compact closure, there exists a subsequence gn,m j pn,m j convergent to some qn ∈ V . Similarly, since U has compact closure, there exists a subsequence of pn,m k convergent to pn ∈ U in the limit as k → ∞. Without loss of generality, we may assume that gn,m pn,m → qn and pn,m → pn as m → ∞. By construction, gn,m → gn as m → ∞. The assumption that the action is proper implies that qn = gn pn for every n ∈ N. Let us now consider sequences gn ∈ W , pn ∈ U and qn ∈ V such that qn = gn pn for all n ∈ N. Since U and V are compact, we may assume without loss of generality that the sequences pn and qn are convergent to p ∈ U and q ∈ V , respectively. The properness of the action of G on P implies that there is a subsequence of gn convergent to g ∈ G such that q = gp. However, W is closed, which implies that g ∈ W . Hence, W is compact. Conversely, suppose that P is a proper G-space. Let pn be a sequence of points in P convergent to p, and let gn be a sequence in G such that the sequence gn pn converges to q ∈ P. Let U and V be neighbourhoods of p and q, respectively, such that U , V and W are compact, where W = {g ∈ G | gU ∩ V = ∅}. Since pn → p and gn pn → q, there exists N > 0 such that gn ∈ W ⊆ W for all n > N . The compactness of W ensures that the sequence

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gn has a convergent subsequence gn m with limit g ∈ W . Since the action of G on P is continuous, it follows that lim pn m = gp. q = lim gn pn = lim gn m pn m = lim gn m n→∞

m→∞

m→∞

m→∞

This implies that the action of G on P is proper. From now on, we assume that P is a locally compact subcartesian space and that the action of G on P is smooth. Let H be a compact subgroup of G. We begin with a lemma, which will be needed in what follows. Lemma 4.4.4 Consider an action : H × P → P : (g, p) → g ( p) = gp of a compact Lie group H on a subcartesian differential space P. Let dμ be a Haar measure on H , normalized so that the total volume of H is 1. For each f ∈ C ∞ (P), the H -average ∗g f dμ(g) f˜ = H

is a smooth function on P. Proof The pull-back ∗ f of f ∈ C ∞ (P) by the action is a smooth function on H × P such that ∗ f (g, p) = f (gp) = g f ( p). For each p ∈ P, the function g → g f ( p) on H is smooth. Hence, the integral ∗g f ( p) dμ(g) = ∗ f (g, p) dμ(g) f˜( p) = H

H

exists and f˜ is a function on P. We need to show that f˜ is smooth. Since P is subcartesian, for each p ∈ P there exist a neighbourhood V p of p and a diffeomorphism ϕ p of V p onto a subset of Rn p . Hence, id × ϕ p : H × V p → H × ϕ p (V p ) : (g, q) → (g, ϕ(q)) ∈ H × Rn p is a diffeomorphism. This implies that there exists a function F p ∈ C ∞ (H × Rn p ) such that ((id × ϕ p )−1 )∗ (∗ f |H ×V p ) = F p|H ×ϕ p (V p ) . Therefore, for every (g, q) ∈ H × V p , ∗ f (g, q) = F p (g, ϕ p (q)). Integrating this equation over H , we obtain the following for each q ∈ V p : ∗ f (g, q) dμ(g) = F p (g, ϕ p (q)) dμ(g). f˜|V p (q) = H

H

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Since F p ∈ C ∞ (H × Rn p ) and H is compact, it follows that f˜|V p is smooth. Therefore, there exists a function h p ∈ C ∞ (P) such that f˜|V p = h p|V p . This holds for every p ∈ P, which ensures that f˜ ∈ C ∞ (P). Let be an H -slice at p for the action of G on P. By the definition of a slice, is invariant under the action of H on P. Hence, we have an action of H on H × → : (g, s) → gs = g s.

(4.11)

The differential structure C ∞ () of is generated by restrictions to of smooth functions on P. We denote the space of H -orbits in by /H and the orbit map by ρ : → /H . The orbit space /H is a differential space with a differential structure ∗ C ∞ (/H ) = { f : → R | ρ f ∈ C ∞ ()}.

By the definition of a slice, the space G = {gs ∈ P | g ∈ G and s ∈ } is an open G-invariant neighbourhood of p ∈ P. Its differential structure is generated by the restrictions to G of smooth functions on P. We denote the space of G-orbits in G by G/G and the orbit map by ρG : G → G/G. The differential structure C ∞ (G/G) of G/G consists of functions ∗ f ∈ C ∞ (G). f : G/G → G such that ρG Let ι : → G be the inclusion map. For each s ∈ , the H -orbit H s extends to the unique G-orbits through s. Thus, we have a one-to-one map β : /H → G/G : H s → Gs. Moreover, every G-orbit in G intersects along a unique H -orbit, which implies that β is invertible. We have the following commutative diagram:

ι

ρ

/H

/ G ρG

β

/ G/G.

Theorem 4.4.5 The bijection β : /H → G/G : H s → Gs is a diffeomorphism. Proof For every G-invariant function f ∈ C ∞ (G), the restriction of f to is H -invariant. This implies that β : /H → G/G is smooth.

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In order to demonstrate that β −1 : G/G → /H is smooth, we have to show that every H -invariant function h on extends to a G-invariant function on G. Since each point q ∈ G can be presented as q = gs for some g ∈ G and s ∈ , we can define a function f on G by f (gs) = h(s).

(4.12) g2 s2 , then s2 = g2−1 g1 s1 , of h implies that h(s2 ) =

If (g1 , s1 ) and (g2 , s2 ) ∈ are such that g1 s1 = which implies that g2−1 g1 ∈ H . The H -invariance h(g2−1 g1 s1 ) = h(s1 ). Hence, f is well defined by equation (4.12). Next, we need to show that f is smooth. For each ξ in the Lie algebra g of G, let X ξ be the vector field on P corresponding to the action of exp tξ ξ on P. Since G is G-invariant, the restriction X |G of X ξ to G is a vector field on G. By assumption, P is subcartesian, which implies that G is subcartesian. Hence, for each q ∈ G, there exists an open neighbourhood Uq of q in G and a diffeomorphism ϕq of Uq onto a subset of Rn q . For each ξ ξ ∈ g, ϕq∗ (X |Uq ) is a vector field on ϕ(Uq ). Consider the following system of differential equations on ϕ(Uq ) for functions Fq ∈ C ∞ (Rn q ): ξ

ϕq∗ (X |Uq )(Fq ) = 0 ∀ ξ ∈ g, Fq|ϕq (Uq ∩) = (ϕq−1 )∗ ( f˜|Uq ∩ ). Since every G-orbit in G intersects , there exists a unique solution of this system of equations, and this solution satisfies the condition ϕq∗ (Fq )|Uq ∩Uq = ϕq∗ (Fq )|Uq ∩Uq for every q, q ∈ G. Hence, there exists a unique smooth function on G which coincides with ϕq∗ (Fq )|Uq for every q ∈ G. It is easy to see that this function is the function f defined above. Next, we show that the quotient and differential-space topologies of our orbit spaces coincide. We begin with the action of an isotropy group H of p ∈ P on the slice at p. Proposition 4.4.6 The differential-space topology of C ∞ (/H ) coincides with the quotient topology. Proof Taking Proposition 2.1.11 into account, in order to prove that the topology of the orbit space /H induced by C ∞ (/H ) coincides with the quotient topology, it suffices to show that for each set V in /H which is open in the quotient topology, and each y ∈ V , there exists h ∈ C ∞ (/H ) such that h(y) = 0 and h |(/H )\V = 0. For y ∈ /H , choose q ∈ /H such that ρ (q) = y. Since G is open in P, and P is locally compact and Hausdorff, it follows that there exists an open neighbourhood W of q in G with its closure W contained in ρ −1 (V ),

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where ρ : P → P/G is the orbit map. Moreover, there exists a non-negative function f ∈ C ∞ (G) such that f (q) > 0 and f |P\W = 0, where G\W denotes the complement of W in G. By Lemma 4.4.4, the H -average g f dμ(g) f˜ = H

is in C ∞ (G). The assumption that

f is non-negative and f (q) > of f over H 0 implies that f˜(q) > 0. Since f |G\W = 0, it follows that f˜|G\H W = 0. The compactness of W and H implies that the union H W of all H -orbits through W is compact, and that H W = H W , where H W is the union of all H -orbits through W . Moreover, the assumption that W ⊆ ρ −1 (V ) and the H -invariance of ρ −1 (V ) ensure that H W = H W ⊆ ρ −1 (V ). Thus, f˜ is an H -invariant smooth function on G such that f˜( p) > 0 and f˜ vanishes on G\ρ −1 (V ). Let f˜| be the restriction of f˜ to . Since the differential structure C ∞ () is induced by the restrictions to of smooth functions on P, it follows that f˜| is smooth. Moreover, f˜| (q) = f˜(q) > 0, because q ∈ . On the other hand, f˜ vanishes on G\ρ −1 (V ). Hence, f˜| vanishes on −1 (G\ρ −1 (V )) ∩ = \(ρ −1 (V ) ∩ ) = \ρ (V ).

Furthermore, f˜| is H -invariant because f˜ and are H -invariant. By the definition of the differential structure C ∞ (P/H ) of the orbit space, there exists a function h ∈ C ∞ (P/H ) such that f˜| = ρ ∗ h. Clearly,

h(y) = h(ρ(q)) = ρ h(q) = f˜(q) > 0 ∗

and ∗ 0 = ( f˜| )|\ρ −1 (V ) = (ρ h)|\ρ −1 (V ) = h |ρ ()\V = h |(/H )/V ,

which ensures that the quotient topology and the differential-space topology of /H coincide. Using Theorem 4.4.5, we can extend the result above to the space of orbits of a proper action of G on P. Theorem 4.4.7 For a proper action : G × P → P of a Lie group G on a locally compact, subcartesian differential space P, the differential-space topology of C ∞ (P/G) coincides with the quotient topology. Proof Let V be a neighbourhood of y ∈ P/G that is open in the quotient topology. Choose p ∈ P such that ρ( p) = y. The set ρ −1 (V ) is an open G-invariant neighbourhood of p in P.

4.4 Action of a Lie group on a subcartesian space

87

Let be a slice through p for the action of G on P. Then G is an open G-invariant neighbourhood of p in P. We denote the isotropy group of p by H , and the orbit map by ρ : → /H . Since P is locally compact and Hausdorff, there exists an open neighbourhood W of p in P with a compact closure W contained in ρ −1 (V ) ∩ G. Without loss of generality, we may assume that W is H -invariant; see the proof of Proposition 4.4.6. Then, the set ρ −1 (ρ(W )) ∩ ρ −1 (V ) ∩ G is an open G-invariant neighbourhood of p in G. Hence, ρ −1 (ρ(W ) ∩ V ) ∩ = ρ −1 (ρ(W )) ∩ (ρ −1 (V ) ∩ G) ∩ is an H -invariant open neighbourhood of p in . Thus, ρ (ρ(W ) ∩ ρ −1 (V ) ∩ ) is an open neighbourhood of ρ ( p) in the quotient topology of /H . By Proposition 4.4.6, the differential-space topology of C ∞ (/H ) coincides with the quotient topology. Therefore, there exists a smooth function h ∈ C ∞ (/H ) that vanishes on the complement of ρ (ρ −1 (ρ(W ) ∩ V ) ∩ ) in /H and is such that h(ρ ( p)) = 1. Since G-orbits in G intersect along orbits of H in , and W is H -invariant, it follows that ρ −1 (ρ(W )) ∩ = W ∩ . Therefore, our function h vanishes on the complement of ρ (ρ −1 (V )∩W ∩). By Theorem 4.4.5, the map β : /H → G/G : H s → Gs is a diffeomorphism. Therefore, (β −1 )∗ h ∈ C ∞ (G/G), and ρ ∗ (β −1 )∗ h is a G-invariant smooth function on G. By construction, ρ ∗ (β −1 )∗ h( p) = (β −1 )∗ h(ρ( p)) = h(β −1 (ρ( p))) = h(ρ ( p)) = 1, because ρ|G ◦ ι = β ◦ ρ , where ι : → P is the inclusion map. On the other hand, suppose that q ∈ G is in the complement of ρ −1 (ρ(W ) ∩ V ) ∩ G. Hence, ρ(q) is in the complement of ρ(W ) ∩ V ∩ ρ(G) = ρ(W ) ∩ V ∩ (G/G)ρ(W ∩ ρ −1 (V ) ∩ ) in G/G. Since β −1 : G/G → /H is a diffeomorphism, β −1 (ρ(q)) is in the complement of β −1 ◦ ρ(W ∩ ρ −1 (V ) ∩ ) = ρ (W ∩ ρ −1 (V ) ∩ ) in /H . But h vanishes on the complement of ρ (ρ −1 (V ) ∩ W ∩ ) in /H . Therefore, ρ ∗ (β −1 )∗ h = h ◦ β −1 ◦ ρ vanishes on the complement of

88

Stratified spaces

ρ −1 (ρ(W ) ∩ V ) ∩ G in G. Hence, the support of ρ ∗ (β −1 )∗ h is a closed set contained in ρ −1 (ρ(W ) ∩ V ) ∩ G, which is open in P. Now consider a point q in the boundary ∂(G) of G in P. Since G is open in P, it follows that ∂(G) = G\G. We want to show that there exists an open neighbourhood U of q in P such that U ∩ G is contained in the complement of ρ −1 (ρ(W ) ∩ V ) ∩ G. Suppose that there is a sequence Un of neighbourhoods of q such that ∩∞ n=1 Un = {q} and Un ∩ G ∩ ρ −1 (ρ(W ) ∩ V ) = ∅. Then, for each n there is a point qn ∈ Un ∩ G ∩ ρ −1 (ρ(W ) ∩ V ), and the sequence qn convarges to q. Moreover, there exists gn ∈ G such that gn qn ∈ W . Since W is compact, there exists a subsequence gn k qn k convergent to q¯ = limk→∞ gn k qn k ∈ W . By the properness of the action, we may assume without ¯ loss of generality that the sequence gn k is convergent to g¯ ∈ G and that q¯ = gq. This implies that W ∩ ∂(G) = ∅. But, by assumption, W ⊆ G, and G is open in P, so that G ∩ ∂(G) = ∅. Hence, we obtain a contradiction. This implies that there exists a function f ∈ C ∞ (P) such that f |G = ρ ∗ (β −1 )∗ h and f |P\G = 0. Clearly, f is G-invariant, and it pushes forward to a function ρ∗ f ∈ C ∞ (P/G). By construction, ρ∗ f (y) = 1 and (ρ∗ f )|(P/G)\V = 0. The argument above is valid for each point y ∈ P/G and each neighbourhood V of y in P/G that is open in the quotient topology. Hence, the differential-space topology of P/G coincides with its quotient topology. We now show that the orbit space P/G is Hausdorff. First, we observe that the orbit map ρ : P → P/G is open. This can be seen as follows. Let U be an open subset of P. For each g ∈ G, gU = {gp ∈ P | p ∈ P} is open and GU = Ug∈G gU is open. Hence, ρ(U ) = ρ(GU ) is open in P/G. Next, consider the relation ϒ = {( p, q) ∈ P × P | q = gp for some g ∈ G} defined by the partition of P by orbits of G. A convergent sequence of points in ϒ can be written as ( pn , qn ) = ( pn , gn pn ), where the sequences ( pn ) and (gn pn ) converge in P. Since the action of G on P is proper, there exists a convergent subsequence (gn k ) in G, and limn→∞ (gn pn ) = (limk→∞ gn k )(limn→∞ pn ). Therefore, limn→∞ ( pn , gn pn ) ∈ R, which implies that ϒ is closed in P × P. This ensures that P/G is Hausdorff.4 In Theorem 4.4.5, we showed that the bijection β : /H → G/G : H s → Gs is a diffeomorphism. If W is an H -invariant open subset of , then ρ (W ) is an open subset of /H that consists of H -orbits contained in W . 4 See Theorem 11 in Chapter 3 of Kelley (1955).

4.4 Action of a Lie group on a subcartesian space

89

On the other hand, since G is an open subset of P, the set GW , consisting of G-orbits intersecting W, is a G-invariant open subset of G, and ρ(GW ) is an open subset of P/G. Moreover, ρ(W ) = ρ(GW ) = β(ρ (W )). We shall use the above equalities in the arguments below. Proposition 4.4.8 Let G × P → P be a proper action of a Lie group G on a locally compact subcartesian differential space P. The space P/G of G-orbits in P is locally compact. Proof Proposition 4.4.6 ensures that P/G is a differential space with the quotient topology. For p ∈ P, let H be the isotropy group of p and let V be an open neighbourhood of ρ( p) ∈ P/G. We want to show that there exists an open neighbourhood of ρ( p) in P/G that has a compact closure contained in V . Let be the slice at p for the action of G on P. By definition, G is an open G-invariant neighbourhood of p in P. Without loss of generality, we may assume that ρ −1 (V ) ⊆ G. Hence, we may consider V as an open subset of G/G. By Theorem 4.4.5, β : /H → G/G : H s → Gs is a diffeomorphism. Hence, β −1 (V ) is open in /H . Since P is Hausdorff and locally compact, there exists a neighbourhood U of p with a compact closure U contained in ρ −1 (V ). Let g(U ∩ ) = H (U ∩ ). W = {gs ∈ | g ∈ H and s ∈ U ∩ } = g∈H

Since U ∩ is open in and the action of H on is continuous, it follows that g(U ∩ ) is open in for each g ∈ H . Hence, W is an open H -invariant neighbourhood of p in . Therefore, ρ (W ) is an open neighbourhood of ρ ( p) contained in β −1 (V ). This implies that ρ(W ) = β(ρ (W )) is an open neighbourhood of ρ( p) in G/G contained in V . The closure W of W is the set of limit points of sequences in W . Suppose a sequence (gn sn ) in W converges to q ∈ W . Since the sequence (sn ) is contained in U ∩ ⊆ U ∩ and U is compact, there exists a subsequence of sn convergent to q¯ in U ∩ . Compactness of H implies that there is a subsequence of (gn ) convergent to g¯ ∈ H , and q = g¯ q¯ ∈ H (U ∩ ). Conversely, every point of H (U ∩ ) can be presented as a limit of a sequence gn sn for gn ∈ H and sn ∈ U ∩ . Hence, W = H (U ∩ ).

90

Stratified spaces

Since U ⊆ ρ −1 (V ) ⊆ G, it follows that W = H (U ∩ ) ⊆ ρ −1 (V ) ∩ . The action : G × P → P is continuous. Its restriction H : H × P → P to an action of H on P is also continuous. Moreover, the set H (U ∩ ) = H (H × (U ∩ )). Furthermore, U ∩ is compact as a closed subset of a compact set U . Since a product of compact sets is compact and the image of a compact set under a continuous map is compact, it follows that W is compact. Thus, W is an H -invariant neighbourhood of p in such that its closure W is compact and contained in ρ −1 (V ) ∩ . We have shown above that orbit maps of a proper action are open. Hence, ρ (W ) is an open neighbourhood of ρ ( p) in /H . Moreover, ρ (W ) is a compact subset of /H contained in β −1 (V ). Hence, ρ (W ) is closed in /H and contains the closure of ρ (W ). Since every point s ∈ W is the limit of a convergent sequence sn in W , it follows that ρ (s) = limn→∞ ρ (sn ) ∈ ρ (W ). Therefore, ρ (W ) = ρ (W ), which implies that the closure of ρ (W ) is compact. Since β : /H → G/G : H s → Gs is a diffeomorphism, β(ρ (W )) is an open neighbourhood of ρG ( p) in G/G with a compact closure contained in V ⊆ G/G. But G is open in P, so that G/G = ρ(G) ⊆ P/G. Thus, β(ρ (W )) is an open neighbourhood of ρ( p) with a compact closure contained in V ⊆ P. This implies that P/G is locally compact. We have shown that the space of orbits of a proper action of a Lie group G on a locally compact subcartesian space P is a Hausdorff, locally compact differential space P/G with the quotient topology. This result is somewhat disappointing if we compare it with the wealth of information we have about spaces of orbits of proper actions of Lie groups on manifolds. In both cases (manifolds and differential spaces), the starting point is an application of the Slice Theorem, which we have discussed here. In the case of smooth manifolds, the next step is Bochner’s Linearization Lemma. It would be of interest to find a class of subcartesian spaces for which there is an analogue of Bochner’s Linearization Lemma.

5 Differential forms

As in the case of differential forms on a manifold, we can define differential forms on a differential space S either as alternating multilinear maps from the space of derivations of C ∞ (S), called Koszul forms, or pointwise, as maps associating to each point of x ∈ S an alternating multilinear form on Tx S, which we call Zariski forms. These definitions are inequivalent on the singular part of S. Moreover, Koszul forms admit an exterior differential but not a pullback, whereas Zariski forms admit a pull-back but not an exterior differential. There is a third definition, given by Marshall, which leads to forms that admit both pull-backs and exterior differentials. All three definitions agree on the level of 1-forms. It is of interest to see how these forms appear in applications.

5.1 Koszul forms Recall that, for a differential space S, the space of all derivations Der C ∞ (S) is a module over C ∞ (S). Definition 5.1.1 For each k ∈ N, a Koszul k-form is an alternating map ω from (Der C ∞ (S))k to C ∞ (S) that is k-linear over C ∞ (S). We denote the space of Koszul k-forms on S by kK (S). For ω ∈ kK (S), the value of ω on X 1 , X 2 , . . . , X k ∈ Der C ∞ (S) is denoted by ω(X 1 , . . . , X k ). By definition, for every permutation σ of (1, . . . , k), ω(X σ (1) , . . . , X σ (k) ) = (sign σ ) ω(X 1 , . . . , X k ), where sign σ denotes the signature of σ , which is 1 if σ is even and (−1) if σ is odd. Moreover, for every f ∈ C ∞ (S), ω( f X 1 , . . . , X k ) = f ω(X 1 , . . . , X k ).

92

Differential forms

In the remainder of this section, we shall refer to Koszul forms simply as forms. For k = 0, we identify 0-forms with smooth functions; that is, 0K (S) = C ∞ (S). For each f ∈ C ∞ (S), we denote by d f a 1-form given by d f (X ) = X ( f ), for every X ∈ Der C ∞ (S). We shall also use the notation d f (X ) = d f | X , which is convenient if d f or X is given by a lengthy expression. We shall prove in the next section that, for every x in the regular part Sreg of S, there exists an open neighbourhood U of x in Sreg such that for every 1-form θ , there exist functions f 1 , . . . , f n , h 1 , . . . , h n ∈ C ∞ (S), where n is the structural dimension of S at x, such that θ|U =

n

(h i d f i )|U ;

(5.1)

i=1

see the discussion following Proposition 5.2.2. More precisely, for every X ∈ Der C ∞ (S), n h i d fi | X . θ (X )|U = |U

i=1

Since θ (X ) and h i d f i | X are smooth functions on S and the regular component Sreg is dense in S, one can use continuity arguments to find values of θ (X ) at points of the singular component Ssing . For θ1 , . . . , θk ∈ 1K (S), we denote by θ1 ∧ . . . ∧ θk the k-form given by1 (θ1 ∧ . . . ∧ θk )(X 1 , . . . , X k ) = det( θi | X j ),

(5.2)

for every X 1 , . . . , X k ∈ Der C ∞ (S). If f 1 , . . . , f n are local coordinates on U ⊆ Sreg , then the restriction of a k-form ω to U can be given locally by (h i1 ...i k d f i1 ∧ . . . ∧ d fik )|U . (5.3) ω|U = i 1 ,...,i k

We introduce the notation K (S) =

∞

nK (S).

n=0

The sum above is finite if the structural dimension of S is bounded. We have several natural operations on K (S). For each n ≥ 0, n-forms can be added and multiplied by smooth functions. Hence, nK (S) is a module over C ∞ (S). 1 This definition follows the convention adopted by Warner (1971).

5.1 Koszul forms

93

n For ω ∈ m K (S) and ∈ K (S), the wedge product ω ∧ is an (m+n)form in such that for each (X 1 , . . . , X n+m ) ∈ Der C ∞ (S),

(ω ∧ )(X 1 , . . . , X n+m ) (sgn π )ω(X π(1) , . . . , X π(m) ) (X π(m+1) , . . . , X π(m+n) ), =

(5.4)

π

where the summation is taken over permutations π of {1, . . . , m + n} such that π(1) < . . . < π(m) and π(m + 1) < . . . < π(m + n). The next operation on forms is the exterior differential d : K (S) → K (S), which is defined as follows. For X 0 , . . . , X m ∈ Der C ∞ (S), dω(X 0 , . . . , X m ) =

m

(−1)k X k (ω(X 0 , . . . , X k , . . . , X m ))

(5.5)

k=0

+

(−1)k+l ω([X k , X l ], X 0 , . . . , Xk, . . . , X l , . . . , X m ), k

where the terms marked , although written down here, must be omitted. n Proposition 5.1.2 For ω1 ∈ m K (S) and ω2 ∈ K (S),

d(ω1 ∧ ω2 ) = (dω1 ) ∧ ω2 + (−1)m ω1 ∧ dω2 , d 2 ω = ddω = 0. Proof

We leave the proof as an exercise for the reader.

Since d 2 = 0, we can introduce the cohomology for a subcartesian space, which we refer to as the Koszul cohomology of the differential space. However, we shall not pursue this topic here. For a discussion of the Koszul cohomology in the context of algebraic geometry, the reader may consult Aprodu and Nagel (2010) and the references therein. There are two more standard operations on forms that involve derivations of C ∞ (S). If ω is a k-form on S and X is a derivation of C ∞ (S), then the left interior product of X and ω is a (k −1)-form X ω, which is defined as follows. For each x ∈ S and X 1 , . . . , X k−1 ∈ Der C ∞ (S), (X ω)(X 1 , . . . , X k−1 ) = ω(X, X 1 , . . . , X k−1 ). The Lie derivative of a differential form with respect to a derivation X is the anticommutator of the left interior product and the exterior differential £ X ω = X dω + d(X ω). Proposition 5.1.3 For every X ∈ C ∞ (S) and ω1 , ω2 ∈ , £ X (ω1 ∧ ω2 ) = £ X ω1 ∧ ω2 + ω1 ∧ £ X ω2 .

94

Proof

Differential forms

We leave the proof as an exercise for the reader.

Recall that the regular component Sreg of S is open and dense in S and that T Sreg is globally spanned by global derivations; see Theorem 3.3.14 and Proposition 3.3.15. Consider x ∈ Sreg . Given a k-form ω, for every X 1 , . . . , X k ∈ Der C ∞ (S) and each f ∈ C ∞ (S), ω( f X 1 , . . . , X k )(x) = f (x)ω(X 1 , . . . , X k )(x). This implies that there is an alternating k-linear function ω(x) on Tx Sreg such that ω(X 1 , . . . , X k )(x) = ωx (X 1 (x), . . . , X k (x)). In other words, the value of ω(X 1 , . . . , X k ) at a regular point x of Sreg can be expressed as the evaluation of a k-linear function ωx on the derivations X 1 (x), . . . , X k (x) ∈ Tx Sreg . We refer to ωx as the value of ω at x ∈ S. Example 5.1.4 Let S = {(x, y) ∈ R2 | x y = 0}. The singular part of S is the origin (0, 0). We showed in Example 3.1.10 that every derivation X of C ∞ (S) vanishes at the origin (0, 0) ∈ S. Therefore, for any Koszul 1-form ω, the value ω(x,y) is defined for every (x, y) = (0, 0), and ω(0,0) is not defined. Let ϕ : R → S : x → ϕ(x) = (x, 0) be the inclusion map of the x-axis, denoted by R, into S. Clearly, R is a manifold and ϕ is smooth. The derived map T ϕ : T R → T S is an inclusion. If ω is a Koszul 1-form on S, then for each x = 0, we can pull back ω(x,0) to a linear function ω(x,0) ◦ T ϕ on Tx R. In this way, we obtain a 1-form on the complement of 0 in R. Since ω(0,0) is not defined, we do not obtain a 1-form on R. Example 5.1.4 shows that Koszul forms need not have values at singular points. In the next section, we discuss a different type of differential forms, called Zariski forms, which have well-defined values at singular points. However, as we shall see, Zariski forms do not allow exterior differentials.

5.2 Zariski forms Definition 5.2.1 A Zariski k-form on a differential space S is an alternating k-linear smooth map ω from T S to R. We can reformulate this definition as follows. The differential structure of T S, given in Definition 3.3.2, induces a differential structure on the product (T S)k ; see Proposition 2.1.9. The fibre product T k S = {(u 1 , . . . , u k ) ∈ (T S)k | τ (u 1 ) = . . . = τ (u k )}

5.2 Zariski forms

95

is a subset of the differential space (T S)k . Hence, T k S is a differential space with a differential structure generated by restrictions to T k S of smooth functions on (T S)k ; see Proposition 2.1.8. A Zariski k-form on S is a map ω : T k S → R such that ω(u 1 , . . . , u i , . . . , u j , . . . , u k ) = −ω(u 1 , . . . , u j , . . . , u i , . . . , u k ) for every i, j = 1, . . . , k, and ω(u 1 , . . . , au i + bwi , . . . , u k ) = aω(u 1 , . . . , u i , . . . , u k ) + bω(u 1 , . . . , wi , . . . , u k ) for every i = 1, . . . , k, and all a, b ∈ R. Moreover, ω is smooth in the differential structure of T k S described here. We denote by kZ (S) the space of Zariski k-forms on S. As in the case of Koszul forms, the space Z (S) =

∞

kZ (S)

k=0

is a graded module over C ∞ (S), and it admits the following well-defined operations: (i) the wedge product (ω1 , ω2 ) → ω1 ∧ ω2 , such that (ω1 ∧ ω2 )(u 1 , . . . , u n+m ) (sgn π )ω1 (u π(1) , . . . , u π(m) )ω2 (u π(m+1) , . . . , u π(m+n) ); = π

(ii) the left interior product of X ∈ Der C ∞ (S) and ω ∈ kZ (S), i.e. X ω ∈ k−1 Z (S), defined by (X ω)(u 1 , . . . , u k−1 ) = ω(X (x), u 1 , . . . , u k−1 ) for every x ∈ S and u 1 , . . . , u k−1 ∈ Tx S. Let χ : Q → S be a smooth map from a differential space Q to a differential space S, and let ω be a Zariski k-form on S. The pull-back of ω by χ is the Zariski k-form χ ∗ ω on Q such that, for each q ∈ Q and every v1 , . . . , vk ∈ Tq Q, χ ∗ ω(v1 , . . . , vk ) = ω(T χ (v1 ), . . . , T χ (vk )).

(5.6)

It can easily be verified that χ ∗ (ω1 ∧ ω2 ) = χ ∗ ω1 ∧ χ ∗ ω2 for every ω1 , ω2 ∈ kZ (S).

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Differential forms

A subcartesian space is locally diffeomorphic to an open subset of Rn for some positive integer n that depends on x. We can use this fact to describe Zariski forms on S in terms of differential forms on Rn . Proposition 5.2.2 For a subcartesian space S, an alternating k-linear map ω from T S to R is a Zariski differential form on S if and only if, for each x ∈ S, there exists a smooth map ϕ : U → Rn , where U is an open neighbourhood of x in S, that restricts to a diffeomorphism of U onto ϕ(U ) ⊆ Rn , and a differential form on Rn , such that ω(u 1 , . . . , u k ) = (T ϕ(u 1 ), . . . , T ϕ(u k ))

(5.7)

for every (u 1 , . . . , u k ) ∈ T k U . Proof Since S is subcartesian, for each x ∈ S, there exists a smooth map ϕ : W → Rn , where W is an open neighbourhood x in S, into Rn that restricts to a diffeomorphism of W onto ϕ(W ) ⊆ Rn . The derived map T ϕ : T W → T Rn induces a smooth map T k ϕ : T k W → T k Rn = Rkn × Rn : (u 1 , . . . , u k ) → (T ϕ(u 1 ), . . . , T ϕ(u k )). Since ϕ is a diffeomorphism of W on ϕ(W ) ⊆ Rn , and T ϕ is linear on the fibres of the tangent bundle projection, it follows that the range of T k ϕ is Rnk ×ϕ(W ). Moreover, the restriction of T k ϕ to T k ϕ(T k W ) = Rnk ×ϕ(W ) is invertible. Abusing the notation, we denote by T k ϕ −1 : T k ϕ(T k W ) → T k W the map such that T k ϕ −1 ◦ T k ϕ = identity. Let ω be an alternating k-linear map from T S to R. Equation (5.7) implies that ω is smooth. Hence, ω is a Zariski form on S. Conversely, let ω be a Zariski form on S. We denote by ω|W the pull-back of ω by the inclusion map W → S. We may treat ω|W as a smooth function on T k W . By composing ω|W with T k ϕ −1 , we obtain a smooth function (ω|W ) ◦ T k ϕ −1 on T k ϕ(T k W ) = Rnk ×ϕ(W ). The differential structure of Rnk ×ϕ(W ) is induced by the inclusion map Rnk × ϕ(W ) → Rkn × Rn . Therefore, there exists a neighbourhood V of (0, ϕ(x)) ∈ Rnk × ϕ(W ) and a function F ∈ C ∞ (T k Rn ) = C ∞ (Rkn ×Rn ) such that ((ω|W )◦T k ϕ −1 )|V = F|V . We can pull back this equality to a neighbourhood V˜ = (T k ϕ −1 )(V ) of (0, x) in T k (W ), obtaining (ω|W )|V˜ = (F ◦ T ϕ k )|V˜ = ((T ϕ k )∗ F)|V˜ ,

(5.8)

where ω|W is treated as a smooth function on T k W . Let o : W → T k W be the zero section of T k W , and let U be a neighbourhood of x in W such that o(U ) ⊆ V˜ . Then T k U is the restriction of T k W to

5.2 Zariski forms

97

base points in U . For each z ∈ U , we denote by ωz the restriction of ω|W to the fibre Tzk U . In other words, for each (u 1 , . . . , u k ) ∈ Tzk U , ωz (u 1 , . . . , u k ) = ω|W (u 1 , . . . , u k ). This notation emphasizes that z ∈ U is fixed. For each i = 1, . . . , k, we let πi : T k U → T U : (u 1 , . . . , u i , . . . , u k ) → u i be the projection on the ith factor. The fibres of the tangent bundle projection map τ : T U → U are linear. Hence, for a fixed z ∈ U , ωz (u 1 , . . . , u i , . . . , u k ) is a homogeneous linear function of u i . For every vi ∈ Tz U , we denote by vi ∂iz the derivative of ωz (u 1 , . . . , u i , . . . , u k ) with respect to u i in the direction vi . In other words, d ωz (u 1 , . . . , u i + tvi , . . . , u k )|t=0 dt = ωz (u 1 , . . . , vi , . . . , u k ).

vi ∂iz (ωz (u 1 , . . . , u i , . . . , u k )) =

Therefore, for v1 , . . . , vk ∈ Ty U , vk ∂kz (. . . (v2 ∂2z (v1 ∂1z (ωz (u 1 , . . . , u k )))) . . .) = ωz (v1 , . . . , vk ).

(5.9)

Thus, ωz (v1 , . . . , vk ) is determined by the derivatives of ωz (u 1 , . . . , u i , . . . , u k ) along the image o(U ) of the zero section o of T k U . In the following, we denote variables in T k Rn by boldface symbols. In particular, z = ϕ(z), and ui = Tz ϕ(u i ) for i = 1, . . . , k. Moreover, v i ∂ i = Tz ϕ(vi ∂iz ) denotes the partial derivative given by (v i ∂ i )(F)(u1 , . . . , ui , . . . , uk ) =

d F(u1 , . . . , ui + tv i , . . . , uk )|t=0 . dt

Equations (5.8) and (5.9) give ωz (v1 , . . . , vk ) = vk ∂1z (. . . (v1 ∂1z (ωz (u 1 , . . . , u k ))) . . .)|u i =0 = vk ∂kz (. . . (v1 ∂1z (F ◦ (Tz ϕ(u 1 ), . . . , Tz ϕ(u k )))) . . .)|u i =0 = Tz ϕ(vk ∂kz )(. . . (Tz ϕ(v1 ∂1z )F(Tz ϕ(u 1 ), . . . , Tz ϕ(u k ))) . . .)|Tz ϕ(u i )=0 = (v k ∂ k )(. . . ((v 1 ∂ 1 )F(u1 , . . . , uk )) . . .)|ui =0 .

98

Differential forms

Since ωz is skew-symmetric in the arguments v1 , . . . , vk , the equation will still hold if we antisymmetrize the right-hand side. Therefore, ωz (v1 , . . . , vk ) sgn(π )(v π(k) ∂ π(k) )(. . . (v π(1) ∂ π(1) )F(u1 , . . . , uk ) . . .)|u1 =···=uk =0 , = π

(5.10) where the summation is taken over all permutations π of the ordered set (1, . . . , k). The right-hand side of this equation is an alternating k-linear function of the variables v1 , . . . , v k ∈ Tz Rn that depends smoothly on z ∈ R. In other words, the right-hand side of equation (5.10) is the value on (v 1 , . . . , v k ) of a differential k-form on R, which we denote by . Therefore, equation (5.10) implies that ω|U = ϕ ∗ |U , which is equivalent to equation (5.7). By Proposition 3.3.15, the tangent bundle T Sreg of the regular component of a subcartesian space S is locally spanned by global derivations. Hence, the restriction of a Zariski form on S to Sreg defines a Koszul form on Sreg . Therefore, Proposition 5.2.2 provides a justification for equation (5.1) in the preceding section. The exterior differential of a Zariski form is not defined. For a subcartesian space, we might want to define d : kZ → k+1 Z (S) using Proposition 5.2.2 as follows. If the restriction of ω to Ux is given by equation (5.7), then for y ∈ Ux and u 0 , u 1 , . . . , u k ∈ Ty U x , dω(u 0 , u 1 , . . . , u k ) = d(T ϕx (u 0 ), T ϕx (u 1 ), . . . , T ϕx (u k )),

(5.11)

where d is the exterior differential of the form on Rn . However, this condition does not determine dω uniquely, as can be seen in the following example. Example 5.2.3 Let S = {(x, y) ∈ R2 | x y = 0}, and let ι : S → R2 be the inclusion map. Take 1 = 0 on R2 . Then ω1 = ι∗ = 0 and dω1 = 0. On the other hand, let 2 = y d x on R2 . The pull-back ω2 = ι∗ 2 is identically zero on T S, because y vanishes on the x-axis and d x vanishes on the y-axis. However, the pull-back of d2 = dy ∧ d x by the inclusion map ι : S → R2 does not vanish at the origin (0, 0), because T(0,0) S " R2 . Thus, dω given by equation (5.11) depends on the choice of .

5.3 Marshall forms

99

5.3 Marshall forms We continue with the assumption that S is a subcartesian space. Definition 5.3.1 For each m, we denote by Mm (S) the subset of mZ (S) consisting of Zariski m-forms ω such that, for each x ∈ S, there exists a differential (m − 1)-form " x on Rn , an open neighbourhood Ux of x in S and a smooth map ϕx : U → Rn with the property that ϕx∗ d"x = ω|Ux , ϕx∗ "x

= 0.

(5.12) (5.13)

The conditions (5.12) and (5.13) can be rewritten explicitly in the form ω(u 1 , . . . , u m ) = d"x (T ϕx (u 1 ), . . . , T ϕx (u m )), 0 = "x (T ϕx (u 1 ), . . . , T ϕx (u m−1 )),

(5.14) (5.15)

for every y ∈ U and u 1 , . . . , u m ∈ Ty S. It follows from Example 5.1.4 that Mk (S) need not vanish. Let Mm (S). M(S) = m

Proposition 5.3.2 M(S) is closed under the operations of addition, multiplication by functions in C ∞ (S), the wedge product and the differentiation d given by equation (5.11). Proof Since equations (5.14) and (5.15) are linear homogeneous equations in (ω, "x ), it follows that M is closed under addition and multiplication by numbers. For each ω ∈ Mk (S) and f ∈ C ∞ (S), equation (5.14) yields f (y)ω(u 1 , . . . , u k ) = f (y) d"x (T ϕx (u 1 ), . . . , T ϕx (u k )) = ((ϕx∗ f ) d"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = (d((ϕx∗ f )"x ) − d(ϕx∗ f ) ∧ "x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = d((ϕx∗ f )"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) because equation (5.15) implies that (d(ϕx∗ f )∧"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = 0. Moreover, equation (5.15) also implies that ((ϕx∗ f )"x )(T ϕx (v1 ), . . . , T ϕx (vk−1 )) = 0 for every v1 , . . . , vk−1 ∈ Ty S. Hence, f ω ∈ Mk (S).

100

Differential forms

Suppose now that ω1 ∈ Mm (S) and ω2 ∈ nZ (S). Equation (5.14) yields (ω1 ∧ ω2 )(u 1 , . . . , u n+m ) = (d"x1 ∧ "x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )) = d("x1 ∧ "x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )) − ("x1 ∧ d"x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )). On the other hand, ("x1 ∧ d"x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m−1 )) = 0 because "x1 (T ϕx (v1 ), . . . , T ϕx (vm−1 )) = 0 for every v1 , . . . , vm−1 ∈ Ty S. This implies that ω1 ∧ ω2 ∈ Mm+n (S). Suppose that ω ∈ Mm (S), which implies equations (5.14) and (5.15). Assume also that dω satisfies equation (5.11). Taking equations (5.7) and (5.11) into account, we obtain dω(u 0 , u 1 , . . . , u k ) = d(d"x )(T ϕx (u 0 ), T ϕx (u 1 ), . . . , T ϕx (u k )) = 0. Hence, dω = 0 ∈ M. Definition 5.3.3 Marshall k-forms on S are elements of the quotient kM (S) = kZ (S)/Mk (S). We denote by μ the projection map from the space Z of Zariski forms to the space M (S) = kM (S) k

of Marshall forms. In other words, the Marshall form corresponding to a Zariski form ω is denoted by μ(ω). Below, we show that all of the usual operations on differential forms on manifolds extend to Marshall forms on subcartesian spaces. Proposition 5.3.4 M (S) is a graded module over C ∞ (S). The operation of the wedge product ∧ on Z (S) induces a wedge product and an exterior differential on mM (S) such that μ(ω1 ) ∧ μ(ω2 ) = μ(ω1 ∧ ω2 ). Proof This proposition is a direct consequence of Proposition 5.3.2 and standard properties of differential forms on Rn .

5.3 Marshall forms

101

Proposition 5.3.5 There is an exterior derivation operator d : M (S) → M (S) such that for every Zariski form ω, dμ(ω) = μ(dω),

(5.16)

where dω satisfies equation (5.11). In particular, d2 = 0

(5.17)

and, if ω1 ∈ kZ and ω2 ∈ mZ , then d(μ(ω1 ) ∧ μ(ω2 )) = dμ(ω1 ) ∧ μ(ω2 ) + (−1)k μ(ω1 ) ∧ dμ(ω2 ).

(5.18)

Proof Recall that if ω is a Zariski form, then equation (5.11) defines dω up to an element of M(S). By Proposition 5.3.2, d maps M(S) to zero. Hence, μ(dω) is uniquely determined by μ(ω) for every Zariski form ω. Equations (5.17) and (5.18) are direct consequences of Proposition 5.3.2 and standard properties of differential forms on Rn . Let χ : Q → S be a smooth map from a subcartesian space Q to a subcartesian space S. Lemma 5.3.6 The pull-back χ ∗ : Z (S) → Z (Q) of Zariski forms, defined by equation (5.6), maps M(S) to M(Q). Proof Let ω ∈ Mk (S). For each q ∈ Q and every v1 , . . . , vk ∈ Tq Q, equations (5.6), (5.14) and (5.15) give χ ∗ ω(v1 , . . . , vk ) = ω(T χ (v1 ), . . . , T χ (vk )) = d"x (T ϕx (T χ (v1 )), . . . , T ϕx (T χ (vk ))) = d"x (T (ϕx ◦ χ )(v1 ), . . . , T (ϕx ◦ χ )(vk )) and "x (T ϕx (T χ (v1 )), . . . , T ϕx (T χ (vk−1 ))) = "x (T (ϕx ◦ χ )(v1 ), . . . , T (ϕx ◦ χ )(vk−1 )) = 0, where x = χ (z). Hence, ϕx ◦ χ : χ −1 (Ux ) → Rn is a map of a neighbourhood χ −1 (Ux ) of q in Q that satisfies equations (5.14) and (5.15). Therefore, χ ∗ ω ∈ M(Q). Corollary 5.3.7 The pull-back χ ∗ : Z (S) → Z (Q) of Zariski forms induces a pull-back χ ∗ : M (S) → M (Q) of Marshall forms such that μ(χ ∗ ω) = χ ∗ (μ(ω)) for every ω ∈ Z (S).

102

Differential forms

The notion of Marshall forms has been extended to general differential spaces by Sasin (1986). Some examples of the de Rham cohomology, defined in terms of Marshall forms, were studied by Marshall (1975b). Koszul forms, Zariski forms, Marshall forms and sections of appropriate bundles were compared by Watts (2006). In particular, Marshall and Koszul forms coincide on the set Sreg of regular points of S and are smooth sections of the corresponding bundle. This suggests that, in some sense, Marshall forms are extensions of Koszul forms to the singular part Ssing of S. The nature of this extension requires further investigation.

PART II Reduction of symmetries

6 Symplectic reduction

In Chapter 4, we studied the orbit type stratification N of the orbit space R = P/G of a proper action of a Lie group G on a manifold P. Here, we consider a special case in which P is a symplectic manifold and the action of G on P is Hamiltonian. We show that the symplectic structure of P gives rise to a Poisson structure on the orbit space. Moreover, each stratum of N is a Poisson manifold singularly foliated by symplectic manifolds. We apply our results to Hamiltonian systems with symmetry and show that symplectic reduction leads to a reduced Hamiltonian system in each symplectic leaf of every stratum of N. Since the reduced Hamiltonian systems have a smaller number of degrees of freedom than the original Hamiltonian system, the process of reduction helps us to analyse the equations of motion of the original system. The approach to symplectic reduction of a proper action of the symmetry group presented here is called singular reduction. It was initiated in Cushman (1983). If the action of the symmetry group is free and proper, singular reduction leads to the regular reduction that was introduced by Meyer (1973) and by Marsden and Weinstein (1974). We also discuss algebraic reduction, which is applicable even for an improper action of the symmetry group.

6.1 Symplectic manifolds with symmetry 6.1.1 Co-adjoint orbits The orbits of the co-adjoint action of a connected Lie group are fundamental examples of symplectic manifolds with symmetry. The adjoint action of a Lie group G on its Lie algebra g is Ad : G × g → g : (g, ξ ) → Adg ξ =

d g(exp tξ )g −1 |t=0 , dt

106

Symplectic reduction

where exp tξ is the one-parameter subgroup of G generated by ξ ∈ g. Let g∗ be the dual of g. The co-adjoint action of G on g∗ is Ad ∗ : G × g∗ → g∗ : (g, μ) → Adg∗ μ, where Adg∗ μ | ξ = μ | Adg −1 ξ

for every ξ ∈ g. For each ξ, ζ ∈ g and μ ∈ g∗ , d d ∗ Adexp μ | Adexp(−tζ ) ξ |t=0 tζ μ | ξ |t=0 = dt dt = μ | −ζ ξ + ξ ζ

= μ | [ξ, ζ ] .

(6.1)

The isotropy group of μ ∈ g∗ is G μ = {g ∈ G | Adg∗ μ = μ}. The co-adjoint orbit through μ, given by Oμ = { Adg∗ μ ∈ g∗ | g ∈ G}, is a manifold diffeomorphic to G/G μ , and the inclusion map Oμ → g∗ is an immersion. Hence, Oμ is an immersed submanifold of g∗ . However, Oμ need not be an embedded submanifold; see Pukanszky (1971). We denote by X ξ the vector field on Oμ induced by the co-adjoint action of exp tξ on g∗ . In other words, d ∗ (6.2) f (Adexp tξ λ)|t=0 dt for every f ∈ C ∞ (Oμ ) and λ ∈ Oμ . Since G acts transitively on Oμ , for every vector u ∈ Tλ Oμ there exists ξ ∈ g such that X ξ (λ) = u. Let μ be a 2-form on Oμ such that, for each λ ∈ Oμ and ξ, ζ ∈ g, X ξ ( f )(λ) =

μ (X ξ (λ), X ζ (λ)) = − λ | [ξ, ζ ] .

(6.3)

The form μ is known as the Kirillov–Kostant–Souriau form of the co-adjoint orbit Oμ . Proposition 6.1.1 Equation (6.3) gives a well-defined 2-form μ , which is non-degenerate and closed. Proof Suppose that, for ξ ∈ g and λ ∈ Oμ , X ξ (λ) = 0. That is, d ∗ dt Adexp tξ λ|t=0 = 0. Hence, for each ζ ∈ g, d d ∗ Adexp(−tξ ) ζ Adexp = λ | ζ = 0. λ | [ξ, ζ ] = − λ | tξ dt dt |t=0 |t=0

6.1 Symplectic manifolds with symmetry

107

This ensures that μ is well defined. Conversely, if X ξ (λ) μ = 0, then for each ζ ∈ g, d ∗ λ | ζ = λ | [ξ, ζ ] = −(X ξ (λ), X ζ (λ)) = 0, Adexp tξ dt |t=0 which implies that X ξ (λ) = 0. Therefore, μ is non-degenerate. Next, we show that μ is closed. For each ξ0 , ξ1 , ξ2 ∈ g, dμ (X ξ0 , X ξ1 , X ξ2 ) = X ξ0 (μ (X ξ1 , X ξ2 )) − X ξ1 (μ (X ξ0 , X ξ2 )) + X ξ2 (μ (X ξ0 , X ξ1 )) − μ ([X ξ0 , X ξ1 ], X ξ2 ) + μ ([X ξ0 , X ξ2 ], X ξ1 ) − μ ([X ξ1 , X ξ2 ], X ξ0 ). For each λ ∈ Oμ , − μ ([X ξ0 , X ξ1 ], X ξ2 )(λ) + μ ([X ξ0 , X ξ2 ], X ξ1 )(λ) − μ ([X ξ1 , X ξ2 ], X ξ0 )(λ) = λ | [[ξ0 , ξ1 ], ξ2 ] − λ | [[ξ0 , ξ2 ], ξ1 ] + λ | [[ξ1 , ξ2 ], ξ0 ]

= λ | [[ξ0 , ξ1 ], ξ2 ] + [[ξ1 , ξ2 ], ξ0 ] + [[ξ2 , ξ0 ], ξ1 ] = 0. On the other hand, d ∗ (μ (X ξ j , X ξk ))(Adexp tξi λ)|t=0 dt d ∗ = − Adexp tξi λ | [ξ j , ξk ] |t=0 dt d = − λ | Adexp −tξi ([ξ j , ξk ]) |t=0 dt = λ | [ξi [ξ j , ξk ]] .

X ξi (μ (X ξ j , X ξk ))(λ) =

Hence, dμ (X ξ0 , X ξ1 , X ξ2 )(λ) = λ | [ξ0 , [ξ1 , ξ2 ]]−[ξ1 , [ξ0 , ξ2 ]]+[ξ2 , [ξ0 , ξ1 ]] = 0. Therefore, dμ = 0. Let I : Oμ → g∗ be the inclusion map. Recall that I is an immersion but need not be an embedding. For each ξ ∈ g, the evaluation Iξ = I | ξ

is a smooth function on Oμ . Proposition 6.1.2 For each ξ ∈ g, X ξ μ = −d Iξ .

108

Proof

Symplectic reduction For each ξ ∈ g and λ ∈ Oμ , Iξ (λ) = λ | ξ . Hence, for every ζ ∈ g,

d Iξ | X ζ (λ) = X ζ (Iξ )(λ) =

d d ∗ ∗ I (Adexp Adexp tζ λ) | ξ |t=0 = tζ λ | ξ |t=0 dt dt

d λ | Adexp(−tζ ) ξ |t=0 = − λ | [ζ, ξ ] = μ (X ζ , X ζ )(λ) dt = −μ (X ξ , X ζ )(λ) = − (X ξ μ ) | X ζ (λ). =

Since ζ ∈ g and λ ∈ Oμ are arbitrary, this implies that X ξ μ = −d Iξ . Corollary 6.1.3 If G is connected, then the form μ on Oμ is invariant under the co-adjoint action of G on Oμ . Proof Since G is connected, it suffices to show that £ X ξ μ = 0 for each ξ ∈ g. But £ X ξ μ = X ξ dμ + d(X ξ μ ) = 0, because μ is closed and (X ξ μ ) = −d Iξ is exact.

6.1.2 Symplectic manifolds Symplectic manifolds are generalizations of co-adjoint orbits. Let P be a manifold. A symplectic form on P is a closed, non-degenerate 2-form on P. Non-degeneracy of ω implies that for every f ∈ C ∞ (P), there exists a unique vector field X f such that X f ω = −d f.

(6.4)

The vector field X f is called the Hamiltonian vector field of f .1 For each f ∈ C ∞ (P), the one-parameter local group exp t X f of local diffeomorphisms of P preserves the symplectic form ω, because £ X f ω = X f dω + d(X f ω) = 0. Diffeomorphisms of P that preserve the symplectic form ω are called symplectomorphisms of (P, ω). Proposition 6.1.4 For every f1 and f 2 in C ∞ (P), [X f1 , X f2 ] ω = −d X f1 ( f 2 ) = −dω(X f1 , X f2 ). Proof [X f1 , X f2 ] ω = (£ X f1 X f2 ) ω = £ X f1 (X f2 ω) − X f2 £ X f1 ω = £ X f1 (−d f 2 ) = −d(X f1 ( f 2 )). 1 We follow the notation and sign convention of Sniatycki ´ (1980) here.

6.1 Symplectic manifolds with symmetry

109

Let G be a connected Lie group, and let : G × P → P : (g, p) → g ( p) = gp be a Hamiltonian action of G on P. This means that the action is symplectic. That is, ∗g ω = ω for every g ∈ G. Moreover, there exists an Ad ∗ -equivariant momentum map J : P → g∗ with the following property: for each ξ ∈ g, the action on P of the one-parameter subgroup exp tξ of G is given by translations along the integral curves of X Jξ , where Jξ = J | ξ . The function Jξ is called the momentum corresponding to ξ . For every f ∈ C ∞ (P), d ∗ f |t=0 = X Jξ ( f ). (6.5) dt exp tξ Proposition 6.1.5 The map g → X(P) : ξ → X Jξ is an antihomomorphism of Lie algebras. In other words, [X Jξ , X Jζ ] = −X J[ξ,ζ ] for every ξ, ζ ∈ g. Proof By assumption, the momentum map J : P → g∗ is Ad ∗ -equivariant; that is, ∗ ∗exp tξ J = Adexp tξ ◦ J.

Evaluating this equation on ζ ∈ g, we obtain ∗ ∗exp tξ Jζ = Adexp tξ ◦ J | ζ = J | Adexp −tξ ζ .

Taking equations (6.1) and (6.5) into account, we obtain X Jξ (Jζ ) = J | [ζ, ξ ] = J[ζ,ξ ] .

(6.6)

By Proposition 6.1.4, [X Jξ , X Jζ ] = X X Jξ (Jζ ) = X J[ζ,ξ ] = −X J[ξ,ζ ] , which completes the proof. Comparing the definitions presented above with Section 6.1.1, we see that the Kirillov–Kostant–Souriau form μ on a co-adjoint orbit Oμ is symplectic. Moreover, for each ξ ∈ g, the vector field X ξ given by equation (6.2) is the Hamiltonian vector field of Iξ , so that the inclusion I : O → g∗ is the momentum map of the co-adjoint action of G on O.

110

Symplectic reduction

6.1.3 Poisson algebra The assignment f → X f gives a linear map of the space C ∞ (P) of smooth functions on P into the space X(P) of smooth vector fields on P. If P is connected, the kernel of this map consists of constant functions on P. The symplectic form ω on P induces a bracket on C ∞ (P), called the Poisson bracket, such that for each f 1 , f 2 ∈ C ∞ (P), { f 1 , f2 } = −X f1 f 2 = X f2 f 1 = −ω(X f1 , X f2 ).

(6.7)

The Poisson bracket (6.7) is bilinear and antisymmetric, acts as a derivation { f 1 , f 2 f 3 } = f 2 { f 1 , f 3 } + f 3 { f 1 , f 2 },

(6.8)

and satisfies the Jacobi identity {{ f 1 , f 2 }, f 3 } + {{ f 2 , f 3 }, f 1 } + {{ f 3 , f 1 }, f 2 } = 0.

(6.9)

The associative algebra C ∞ (P) endowed with the Poisson bracket (6.7) is called the Poisson algebra of (P, ω). The map C ∞ (P) → X(P) : f → X f is an antihomomorphism of the Lie algebra structure of C ∞ (P) to the Lie algebra of vector fields on P. In other words, X { f 1 , f2 } = −[X f1 , X f2 ]

(6.10)

for all f 1 , f 2 ∈ C ∞ (P). Proposition 6.1.6 The map g → C ∞ (P) : ξ → Jξ is a homomorphism of Lie algebras. Proof For each ξ, ζ ∈ g, equations (6.6) and (6.7) give J[ζ,ξ ] = X Jξ (Jζ ) = {Jζ , Jξ }. The action of G on P gives rise to the action G × C ∞ (P) → C ∞ (P) : (g, f ) → ∗g f . For each ξ ∈ g, the infinitesimal action of exp tξ on C ∞ (P) is given by d f ◦ exp tξ |t=0 = X Jξ ( f ) = { f, Jξ }. f → dt Since the action of G on P is symplectic, it follows that its action on C ∞ (P) is Poisson. That is, it preserves the Poisson bracket. For each g ∈ G and f 1 , f 2 ∈ C ∞ (P), ∗g { f 1 , f 2 } = {∗g f 1 , ∗g f 2 }.

(6.11)

Therefore, the space C ∞ (P)G of G-invariant smooth functions on P is a Poisson subalgebra of P.

6.2 Poisson reduction

111

6.2 Poisson reduction We assume here that the action of G on P is proper. As before, we denote the space of G-orbits on P by R = P/G and the orbit map by ρ : P → R. The differential structure of R is C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}. In Section 4.4, we showed that the projection of the orbit type stratification M of P to the orbit space is a stratification N of R, and that the strata of N coincide with orbits of the family X(R) of all vector fields on R. Here, we begin a discussion of the additional structure of R induced by the symplectic structure on P and the existence of the Ad G∗ -equivariant momentum map J : P → g∗ . The pull-back map ρ ∗ : C ∞ (R) → C ∞ (P)G is an isomorphism of associative algebras. Since C ∞ (P)G is a Poisson algebra, we can use ρ ∗ to pull back the Poisson bracket from C ∞ (P)G to C ∞ (R). For each f 1 , f 2 ∈ C ∞ (R), we define { f 1 , f 2 } ∈ C ∞ (R) by ρ ∗ { f 1 , f 2 } = {ρ ∗ f 1 , ρ ∗ f 2 }.

(6.12)

With this definition, C ∞ (R) is a Poisson algebra isomorphic to C ∞ (P)G . Given f ∈ C ∞ (R), let X f ∈ Der C ∞ (R) be defined by X f (h) = {h, f }

(6.13)

for each h ∈ C ∞ (R). We refer to X f as the Poisson derivation or Poisson vector field of f . In Proposition 6.2.2 below, we show that this terminology is consistent with the definition of vector fields on subcartesian spaces in Definition 3.2.2. We denote the family of all Poisson vector fields on R by P(R) = {X f | f ∈ C ∞ (R)}.

(6.14)

Since P(R) ⊆ X(R), it follows that for each stratum N ⊆ R and every x ∈ N , the value at x of the Poisson bracket { f 1 , f 2 } of functions in C ∞ (R) depends only on the restrictions f 1|N and f 2|N to N of f 1 and f 2 , respectively. Hence, the space R(N ) = { f |N | f ∈ C ∞ (R)} of the restrictions to N of smooth functions on R inherits the structure of a Poisson algebra from C ∞ (R). The Poisson bracket on R(N ) is given by { f 1|N , f2|N } = { f 1 , f2 }|N for every f 1 , f 2 ∈ C ∞ (R). By the definition of a stratification, the strata N ∈ N are locally closed connected submanifolds N of R. Proposition 2.1.8 ensures

112

Symplectic reduction

that every f ∈ C ∞ (N ) coincides locally with a function in R(N ). Hence, the Poisson algebra structure of R(N ) extends to C ∞ (N ). Thus, we have proved the following result. Proposition 6.2.1 Each stratum N of the orbit type stratification N of R is a Poisson manifold. In the next proposition, we prove that the Poisson vector fields on R defined by equation (6.13) are vector fields in the sense of Definition 3.2.2, and discuss their relation to Hamiltonian vector fields of G-invariant functions on P. Proposition 6.2.2 For each f ∈ C ∞ (R), the Poisson derivation X f is the push-forward of the Hamiltonian vector field X ρ ∗ f on P by the orbit map ρ : P → R. Moreover, X f is a vector field on R. That is, exp t X f is a local one-parameter local group of diffeomorphisms of R. Proof For f ∈ C ∞ (R), its pull-back ρ ∗ f by ρ is in C ∞ (P)G , and X ρ ∗ f is a G-invariant Hamiltonian vector field on P. Moreover, for each h ∈ C ∞ (R), equation (6.7) implies that X ρ ∗ f ρ ∗ h = {ρ ∗ h, ρ ∗ f } = ρ ∗ {h, f } = ρ ∗ (X f (h)). Thus, X f is the push-forward of X ρ ∗ f by the orbit map ρ. In other words, Tρ ◦ X ρ ∗ f = X f ◦ ρ.

(6.15)

Since X ρ ∗ f generates a local one-parameter local group exp t X ρ ∗ f of diffeomorphisms of P, it follows that translations along integral curves of X f give rise to a local one-parameter local group exp t X f of diffeomorphisms of R such that ρ ◦ (exp t X ρ ∗ f ) = (exp t X f ) ◦ ρ.

(6.16)

Hence, X f is a vector field on R. By Theorem 3.4.5, the orbits of P(R) are smooth manifolds immersed in R. Let Q be the orbit of P(R) through x ∈ R. For each f ∈ C ∞ (R), the restriction X f |Q of the Poisson vector field of f to Q is a vector field on Q, and T Q = {X f (x) | x ∈ Q, f ∈ C ∞ (R)}. This implies that C ∞ (Q) inherits from C ∞ (R) a Poisson algebra structure such that { f 1|Q , f2|Q } = { f 1 , f 2 }|Q

(6.17)

for each f 1 , f 2 ∈ C ∞ (R). Proposition 6.2.3 Each orbit Q of the family P(R) of Poisson vector fields on R is a symplectic manifold with a unique symplectic form ω Q on Q such that ω Q (X f1 |Q , X f2 |Q ) = −{ f 1 , f 2 }|Q

(6.18)

6.2 Poisson reduction

113

for every f 1 , f 2 ∈ C ∞ (R). Moreover, for each p ∈ ρ −1 (Q), ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p).

(6.19)

Proof Let N denote the stratum of R containing Q. In order to see that ω Q is well defined by equation (6.18), take x ∈ Q and consider f 2 , f 3 ∈ C ∞ (R) such that X f 2 (x) = X f3 (x). For each f ∈ C ∞ (R), equation (6.13) gives { f 2 , f }(x) = X f2 ( f )(x) = (X f2 (x))( f ) = (X f3 (x))( f ) = X f3 ( f )(x) = { f 3 , f }. Hence, ω Q is well defined. Clearly, ω Q is bilinear and smooth. Hence, it is a 2-form on Q. Suppose that X f0 (x0 ) ω Q = 0 for f 0 ∈ C ∞ (R) and x0 ∈ Q. Then, X f0 (x0 )( f ) = 0 for each f ∈ C ∞ (R). Since Q is immersed in N and N is a submanifold of R, for each f Q ∈ C ∞ (Q) there exists a neighbourhood U of x in Q, open in the topology of Q, and a function f ∈ C ∞ (R) such that f |U = f Q|U . Therefore, X f0 (x0 )( f Q ) = 0 for every f Q ∈ C ∞ (Q). Therefore, X f0 (x0 ) = 0 ∈ Der C ∞ (Q), and this implies that ω Q is non-degenerate. We have dω Q (X f0 , X f1 , X f2 ) = X f0 (ω Q (X f1 , X f2 )) − X f1 (ω Q (X f0 , X f2 )) + X f2 (ω Q (X f0 , X f1 )) − ω Q ([X f0 , X f1 ], X f2 ) + ω Q ([X f0 , X f2 ], X f1 ) − ω Q ([X f1 , X f2 ], X f0 ). Taking equations (6.7) and (6.10) into account, we obtain dω Q (X f0 , X f1 , X f2 ) = X f0 { f 2 , f 1 } − X f1 { f 2 , f 0 } + X f2 { f 1 , f 0 } + ω Q (X { f0 , f 1 } , X f2 ) − ω Q (X { f0 , f 2 } , X f1 ) + ω Q (X { f 1 , f 2 } , X f0 ) = −{ f 0 , { f 2 , f 1 }} + { f 1 , { f 2 , f 0 }} − { f2 , { f 1 , f 0 }} − {{ f 0 , f 1 }, f 2 } + {{ f 0 , f 2 }, f 1 } − {{ f 1 , f 2 }, f 0 } = 0. Hence, dω Q = 0. Therefore, ω Q is symplectic. Moreover, the definition of ω Q and equations (6.12) and (6.7) imply that for each p ∈ ρ −1 (Q), ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = −{ f 1 , f 2 }|Q (ρ( p)) = −{ f 1 , f2 }(ρ( p)) = −ρ ∗ { f 1 , f 2 }( p) = −{ρ ∗ f 1 , ρ ∗ f 2 }( p) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p), which completes the proof.

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In Proposition 6.2.3, we showed that Q is a Poisson manifold contained in R. Moreover, the Poisson structure on Q is equivalent to the Poisson structure of C ∞ (Q) defined by the symplectic form ω Q . We shall refer to these facts by saying that Q is a symplectic submanifold of R.

6.3 Level sets of the momentum map We continue with the assumption that the action of G on P is proper. In Hamiltonian mechanics, we often perform the reduction procedure by investigating the structure of the quotient J −1 (μ)/G μ , where G μ = {g ∈ G | Adg∗ μ = μ} is the isotropy group of μ. Since J : P → g∗ is continuous, it follows that J −1 (μ) is a closed subset of P. The local compactness of P implies that J −1 (μ) is locally compact. Moreover, the action G μ on J −1 (μ) is proper because the action of G on P is proper. By the results of Section 4.4, J −1 (μ)/G μ is a locally compact differential space with the quotient space topology and the differential structure C ∞ (J −1 (μ)/G μ ) = { f ∈ C 0 (J −1 (μ)/G μ ) | ρμ∗ f ∈ C ∞ (J −1 (μ))}, where ρμ : J −1 (μ) → J −1 (μ)/G μ is the orbit map. Our aim in this section is to describe the geometric structure of J −1 (μ), the orbit space J −1 (μ)/G μ and the projection ρ(J −1 (μ)) to the orbit space R = P/G. An important tool in this task will be the family E(P) = {X f | f ∈ C ∞ (P)G } of Hamiltonian vector fields of G-invariant smooth functions on P. For each p ∈ P, we denote by E(P) p = span {X f ( p) ∈ T p P | f ∈ C ∞ (P)G }

(6.20)

the subspace of T p P defined by the generalized distribution span E(P) on P spanned by E(P). Proposition 6.3.1 For each p ∈ P, the orbit through p of the family E(P) of Hamiltonian vector fields of G-invariant functions on P is contained in the set PH = {x ∈ P | G x = H }, where H = G p is the isotropy group of p.

6.3 Level sets of the momentum map

115

Proof For f ∈ C ∞ (P)G , let exp t X f denote the local one-parameter group of local diffeomorphisms generated by the Hamiltonian vector field X f of f . The G-invariance of X implies that for each g ∈ G, g ◦ exp t X f = (exp t X f ) ◦ g . Let x = (exp t X f )( p), and let g ∈ G x . Then x = g x implies (exp t X f )( p) = (g ◦ (exp t X f ))( p). Hence, p = ((exp t X f )−1 ◦ g ◦ (exp t X f ))( p) = ((exp t X f )−1 ◦ (exp t X f ) ◦ g )( p) = g p, and g ∈ G p . Thus, G x ⊆ G p = H . In a similar way, we can show that H = G p ⊆ G x . Hence, G x = H , which ensures that the orbit of X f through p is contained in PH . But this is valid for every f ∈ C ∞ (P)G . Therefore, the orbit through p of the family {X f | f ∈ C ∞ (P)G } of all Hamiltonian vector fields of G-invariant functions on P is contained in PH . It is worth noticing that Proposition 6.3.1 does not require properness of the action of G on P. We denote by ker d J the generalized distribution on P given by ker d J = {u ∈ T P | u(J ) = 0}. Hence, for each p ∈ P, ker p d J = {u ∈ T p P | u(J ) = 0}. Theorem 6.3.2 (i) Assume that the action of G on P is proper. Then, for each p ∈ P, E(P) p = ker p d J ∩ T p PG p ,

(6.21)

and the orbit of E(P) through p is the connected component of J −1 (J ( p)) ∩ PG p that contains p.2 (ii) For each compact subgroup H of G, the connected components of PH are symplectic manifolds. (iii) In particular, if p ∈ PH , μ = J ( p) and L is the connected component of PH that contains p, then the connected component of J −1 (μ) ∩ L that contains p is a manifold and its tangent bundle is spanned by Hamiltonian vector fields of G-invariant functions. Proof

(i) The map ω# : T P → T ∗ P : u → ω# (u) = u ω

2 This theorem, due to Ortega and Ratiu, is the foundation of their theory of optimal reduction

(Ortega and Ratiu, 2002).

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is a vector bundle isomorphism that intertwines the projection maps of τ : T P → P and ϑ : T ∗ P → P. We denote the inverse of ω# by ω$ : T ∗ P → T P. With this notation, we can write, for every f ∈ C ∞ (P), X f = −ω$ ◦ d f. Hamiltonian vector fields X f preserve the symplectic form ω. Therefore, they also preserve the tensor fields ω# and ω$ . Given p ∈ P, for every u ∈ E(P) p there exists h ∈ C ∞ (P)G such that u = X h ( p). Hence, E(P) p = span {X f ( p) ∈ T p P | f ∈ C ∞ (P)G } = {X f ( p) ∈ T p P | f ∈ C ∞ (P)G } = {ω$ ◦ d f ( p) | f ∈ C ∞ (P)G } = ω$ ({d f ( p) | f ∈ C ∞ (P)G }).

(6.22)

Let Gp denote the orbit of G through p. We have T p (Gp) = {X Jξ ( p) | ξ ∈ g}. We denote by T p (Gp)0 the annihilator of T p (Gp) in T p∗ (G P); that is, T p (Gp)0 = {q ∈ T p∗ (G P) | q | X Jξ ( p) = 0 for all ξ ∈ g}. If f ∈ C ∞ (P)G , then d f ( p) | X Jξ ( p) = X Jξ ( f )( p) = 0 for all ξ ∈ g. Therefore, d f ( p) ∈ T p (Gp)0 whenever f ∈ C ∞ (P)G . Let H = G p be the isotropy group of p, and let h be the Lie algebra of H . Since f ∈ C ∞ (P)G and H ⊆ G, for each g ∈ H , we have d f = d∗g f . For every u ∈ T p P, we have T g u ∈ T p P and d f ( p) | u = (d∗g f )( p) | u = u(∗g f ) = T g (u)( f ) = d f ( p) | T g (u) = T Tg (d f ( p)) | u . Hence, d f ( p) is invariant under the linear action of H on T p∗ P induced by the action of H on P. Therefore, d f ( p) ∈ (T p (Gp)0 ) H ∀ f ∈ C ∞ (P)G .

(6.23)

The next step is to show that if q ∈ (T p (Gp)0 ) H , it follows that there exists f ∈ C ∞ (P)G such that d f ( p) = q. Let S p be a slice through p for the action of G on P. There exists a compactly supported function f S on S such that

6.3 Level sets of the momentum map

117

d f S ( p) = q |T p S p . As in the proof of Proposition 4.3.1, we can average f S over the action of H on S p , obtaining an H -invariant function ∗g f S dμ(g). f˜S = H

Since f S has compact support and H is compact, it follows that f˜S has compact support. For each g ∈ H and u ∈ T p S p , we have (d∗g f S )( p) | u = u(∗g f S ) = T g (u)( f S ) = d f S ( p) | T g (u)

= T Tg (d f S ( p)) | u = T Tg (q) | u = q | u

because q is assumed to be H -invariant. This implies that d f˜S ( p)|T p S p = q |T p S p . Following the proof of Proposition 4.3.1, we can extend f˜S to a G-invariant function f on P with support contained in ρ −1 (ρ(S)). By construction, d f ( p)|T p S p = d f˜S ( p)|T p S p = q |T p S p . Since f is G-invariant, it follows that d f ( p)|T p (Gp) = 0. On the other hand, q ∈ T p (Gp)0 is equivalent to q||T p (Gp) = 0. But T p P = T p S p + T p Gp, which implies that d f ( p) = q. Therefore, {d f ( p) | f ∈ C ∞ (P)G } = (T p (Gp)0 ) H .

(6.24)

Combining equations (6.22) and (6.24), we obtain E(P) p = ω$ ({d f ( p) | f ∈ C ∞ (P)G }) = ω$ ((T p (Gp)0 ) H ) = (ω$ (T p (Gp)0 ) H ) = {u ∈ T p P | ω(u, v) = 0 ∀ v ∈ T p (Gp)} H = {u ∈ T p P | ω(u, X Jξ ( p)) = 0 ∀ ξ ∈ g} H = {u ∈ T p P | u(Jξ )( p) = 0 ∀ ξ ∈ g} H = (ker p d J ) H = (ker p d J ) ∩ T p PH , as required. Since this result holds for every p ∈ P, we have proved the first statement of the theorem. (ii) In Proposition 4.2.6, we showed that each connected component of PH is a submanifold of P. It remains to show that, for each p ∈ PH , the form ω p on T p PH is non-degenerate. Note that T p PH coincides with the space (T p P) H of H -invariant vectors in T p P. Since the action of G on P is proper, there exists a G-invariant Riemannian metric k on P. Using k, we can define a G-invariant map j : T P → T P such that, for each p ∈ P and u, v ∈ T p P, k(u, v) = ω(u, j (v)).

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For a point p ∈ PH , the restriction of k to T p PH is a positive definite bilinear form on T p PH . Moreover, the group H acts on T p P, and the restriction j p to T p P is H -invariant. Hence, j p maps T p PH = (T p P) H to itself. Hence, for every non-zero u ∈ T p PH , we have j p (u) ∈ T p PH and ω(u, j p (u)) = k(u, u) > 0, which implies that the restriction of ω to T p PH is non-degenerate. Hence, the symplectic form ω on P pulls back to a symplectic form on each connected component of PH . (iii) For every f ∈ C ∞ (P)G , equation (6.21) implies that the orbit of E through p is contained in the connected components of the level set J −1 (μ), where μ = J ( p), and of PH , where H = G p . Let L be the connected component of PH that contains p. By Proposition 4.2.6, L is a submanifold of P. For each f ∈ C ∞ (P)G , the Hamiltonian vector field X f is tangent to L. Hence, it defines a vector X f |L on L, which we call the restriction of X f to L. Consider the family E(P)|L = {X f |L | f ∈ C ∞ (P)G }. The orbits of E(P)|L are the orbits of E(P) that are contained in L. On the other hand, (ker d J ) ∩ T L = ker d J|L , where J|L is the restriction of J : P → g∗ to L . Equation (6.21) gives span E(P)|L = ker d J|L , where span E(P)|L is the distribution spanned by E(P)|L . Hence, ker d J|L is a smooth distribution, and integral manifolds of d J|L are connected components of level sets of J|L . If μ = J ( p), then (J|L )−1 (μ) = J −1 (μ) ∩ L. Hence, the connected component of (J|L )−1 (μ) that contains p is the same as the connected component of J −1 (μ) ∩ L that contains p. Since L is the connected component of PH that contains p, it follows that the connected component of J −1 (μ) ∩ L that contains p is the same as the connected component of J −1 (μ) ∩ PH that contains p. This completes the proof. In Proposition 6.2.3, we showed that the orbits of the family P(R) of Poisson vector fields on R are symplectic manifolds. In the proposition below, we show that they are projections to R of intersections of connected components of level sets of J with submanifolds of P with a fixed isotropy group.

6.3 Level sets of the momentum map

119

Proposition 6.3.3 Assume that the action of a connected Lie group G on a symplectic manifold (P, ω) is Hamiltonian and proper. Given p0 ∈ P, let μ = J ( p0 ) and let H = G p0 be the isotropy group of p0 . The connected component K of J −1 (μ) ∩ PH is a submanifold of P, and the projection Q = ρ(K ) is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K , where ρ K : K → Q is the restriction of the orbit map ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Proof Let L be the connected component of PH that contains p0 . According to Theorem 6.3.2, the connected component K of J −1 (μ) ∩ L that contains p0 is the orbit through p0 of the family E(P) of Hamiltonian vector fields of G-invariant functions on P. Hence, the space G C ∞ (P)|K = {h |K | h ∈ C ∞ (P)G }

of restrictions to K of G-invariant functions in C ∞ (P) is a Poisson algebra with a Poisson bracket {h 1|K , h 2|K } = (X h 2 (h 1 ))|K = {h 1 , h 2 }|K . In Proposition 6.2.2, we established that the Poisson vector field X f of f ∈ C ∞ (R) is the push-forward of the Hamiltonian vector field X ρ ∗ f of the pullback ρ ∗ f of f by the orbit map ρ : P → R, and that ρ ◦ (exp t X ρ ∗ f ) = (exp t X f ) ◦ ρ for every f ∈ C ∞ (R); see equation (6.16). Every G-invariant function h ∈ C ∞ (P) pushes forward to a function f = ρ∗ h ∈ C ∞ (R) such that ρ ∗ f = h. Therefore, the projection to R of the orbit K of the family E(P) of Hamiltonian vector fields of G-invariant functions on P is an orbit Q = ρ(K ) of the family P(R) of Poisson vector fields on R. Since Q is an orbit of P(R), it follows that Q is a Poisson manifold with a Poisson bracket such that { f1|Q , f 2|Q } = { f 1 , f 2 }|Q for every f 1 , f 2 ∈ C ∞ (R). Thus, the restriction f → f |Q is a Poisson algebra homomorphism from C ∞ (R) to C ∞ (Q). In Proposition 6.2.3, we established that Q is a symplectic manifold with a symplectic ω Q such that ω Q (X f1 |Q , X f2 |Q ) = −{ f 1 , f 2 }|Q

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Symplectic reduction

for every f 1 , f 2 ∈ C ∞ (R). Let ρ K : K → Q be the restriction of ρ : P → R to the domain K and codomain Q, and let ω K = ρ K∗ ω Q be the pull-back of ω Q by ρ K : K → Q. For each p ∈ K , we have T p (K ) = {X ρ ∗ f ( p) | f ∈ C ∞ (R)}, and ω K is uniquely determined by ω K (X ρ ∗ f1 , X ρ ∗ f2 )( p) for all f 1 , f 2 ∈ f ∈ C ∞ (R) and p ∈ K . But ω K (X ρ ∗ f1 , X ρ ∗ f2 )( p) = ω K (X ρ ∗ f1 ( p), X ρ ∗ f2 ( p)) = ω Q (Tρ K (X ρ ∗ f1 ( p)), Tρ K (X ρ ∗ f2 ( p))) = ω Q (X f1 (ρ( p)), X f2 (ρ( p))) by equation (6.15) = ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p) by equation (6.19) = ω|K (X ρ ∗ f1 , X ρ ∗ f2 )( p). Since the chain of equalities above is valid for every p ∈ K and all f 1 , f 2 ∈ C ∞ (R), it follows that ρ K∗ ω Q = ω K , which completes the proof. In Chapter 4, we described the orbit type stratification M of P given by a proper action of a connected Lie group G on P, and the corresponding orbit type stratification N of the orbit space R = P/G. Our next task is to describe the intersections of the level sets of J with the strata of M and their projections to R. Recall that the strata of M are connected components of P(H ) = { p ∈ P | G p is conjugate to H }. We are now in a position to give a description of the structure of the orbit space induced by the momentum map. For each μ ∈ g∗ , the inverse image J −1 (μ) is a differential subspace of P with a differential structure generated by inclusions into J −1 (μ) of smooth functions on P. Similarly, ρ(J −1 (μ)) is a differential subspace of R with a differential structure generated by the ring R(ρ(J −1 (μ))) = {h |ρ(J −1 (μ)) | h ∈ C ∞ (R)}. Theorem 6.3.4 We assume that the action of G on P is proper, and denote orbit type stratifications of P and R = P/G by M and N, respectively. (i) For each μ ∈ g∗ , the family of sets Mμ = {connected components of J −1 (μ) ∩ M | M ∈ M}

(6.25)

is a stratification of the level set J −1 (μ). The inclusion map J −1 (μ) → P is a morphism of stratified spaces.

6.3 Level sets of the momentum map

121

(ii) The connected components of the sets ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ)) ∩ N , where N = ρ(M), are symplectic orbits of the family P(R) of Poisson vector fields on R. (iii) The family of sets Nμ = {connected components of ρ(J −1 (μ)) ∩ N | N ∈ N} is a stratification of ρ(J −1 (μ)) with symplectic strata. The restriction ρ|J −1 (μ) of ρ to J −1 (μ) is a morphism of stratified spaces. Proof

(i) For each p ∈ P, rank d J p = dim T p P − dim T p Gp = dim P − dim G/G p = dim P − dim G + dim G p .

Let M be a stratum in M. If p1 , p2 ∈ M, then G p1 and G p2 are conjugate. Hence, dim G p is constant on M. Therefore, the restriction J|M : M → g∗ of the momentum map J to M has constant rank. By the Rank Theorem, −1 (μ) are submanifolds for every μ ∈ g∗ , the connected components of J|M

−1 (μ) = J −1 (μ) ∩ M. Therefore, the connected compoof M. Moreover, J|M nents of J −1 (μ) ∩ M are submanifolds of M. Moreover, different connected components of J −1 (μ) ∩ M have empty intersections, and the collection of all connected components is a covering of J −1 (μ) ∩ M. Hence, the family Mμ gives a partition of J −1 (μ) by smooth manifolds. The family Mμ is locally finite, because M is locally finite. Moreover, for each M ∈ M, the connected components of J −1 (μ) ∩ M are locally closed, because J −1 (μ) ∩ M is closed in M and M is locally closed. Finally, the frontier condition for M implies the frontier condition for Mμ . Hence, Mμ is a stratification of J −1 (μ). Moreover, each stratum of Mμ is a connected component of J −1 (μ) ∩ M for some M ∈ M, and J −1 (μ) ∩ M ⊆ M. Therefore, the inclusion map J −1 (μ) → P is a morphism of stratified spaces. (ii) Given p ∈ J −1 (μ) ∩ PH , let L be the connected component of PH that contains p and let M be the connected component of P(H ) that contains p0 . Both M and L are submanifolds of P. Moreover, we have shown in the proof of Theorem 4.3.5 that G L = M, so that ρ(M) = ρ(L). For each g ∈ G, gL is the connected component of Pg H g−1 that contains gp. Then ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ) ∩ G L) = ρ(J −1 (μ) ∩ gL). g∈G

By construction, p ∈ J −1 (μ) ∩ PH . Since J : P → g ∗ is AdG∗ -equivariant, it follows that J (gp) = Adg∗ (J ( p)) = Adg∗ μ. Hence, J −1 (μ) ∩ gL = ∅ only for g ∈ G μ = {g ∈ G | Adg∗ μ = μ}. If g ∈ G μ , then the action g : P → P

122

Symplectic reduction

restricts to a diffeomorphism of J −1 (μ) ∩ L onto J −1 (μ) ∩ gL. Moreover, ρ(J −1 (μ) ∩ L) = ρ(J −1 (μ) ∩ gL) for every g ∈ G μ . Therefore, ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ) ∩ gL) = ρ(J −1 (μ) ∩ L). g∈G μ

In Proposition 6.3.3, we established that connected components of ρ(J −1 (μ)∩ L) are symplectic orbits of the family P(R) of Poisson vector fields on R. (iii) We have Nμ = {connected components of ρ(J −1 (μ)) ∩ N ) | N ∈ N} = {connected components of ρ(J −1 (μ) ∩ M) | M ∈ M} = {connected components of ρ(J −1 (μ) ∩ L) | L = PH M, M ∈ M} = {orbits of P(R) that are contained in ρ(J −1 (μ)) ∩ N ) | N ∈ N}. Since the family Mμ is a stratification of J −1 (μ), it follows that the family Nμ is a partition of ρ(J −1 (μ)). The proof that this partition is a stratification of ρ(J −1 (μ)) is analogous to the proof that Mμ is a stratification of J −1 (μ). We have shown in part (ii) that the strata of Nμ are symplectic manifolds. Since ρ|J −1 (μ) maps connected components of J −1 (μ)∩ M to connected components of ρ(J −1 (μ)) ∩ ρ(M), and N = ρ(M) ∈ N, it follows that ρ|J −1 (μ) : J −1 (μ) → ρ(J −1 (μ)) is a morphism of stratified spaces. We now proceed to investigate the structure of the orbit space J −1 (μ)/G μ with the quotient differential structure C ∞ (J −1 (μ)/G μ ) = { f ∈ C 0 (J −1 (μ)/G μ ) | ρμ∗ f ∈ C ∞ (J −1 (μ))}, where ρμ : J −1 (μ) → J −1 (μ)/G μ is the orbit map. Theorem 6.3.5 There exists a diffeomorphism l : J −1 (μ)/G μ → ρ(J −1 (μ)) such that the following diagram commutes: J −1 (μ) ρu

i

/P ρ

J −1 (μ)/G μ ? R ?? ?? ?? ?? l ??? j ?? −1 ρ(J (μ)),

(6.26)

6.3 Level sets of the momentum map

123

where i : J −1 (μ) −→ P and j : ρ(J −1 (μ)) −→ R denote the inclusion maps. The stratification of ρ(J −1 (μ)) gives rise to a stratification of J −1 (μ)/G μ such that the orbit map ρμ: : J −1 (μ) → J −1 (μ)/G μ is a morphism of stratified spaces. Proof Since ρ(J −1 (μ)) is a subset of R, for each p ∈ J −1 (μ), the orbit G μ p ⊆ J −1 (μ) through p is mapped by the inclusion map i : J −1 (μ) → P into the orbit Gp in P. Hence, ρ(G μ p) in R depends only on ρμ ( p) ∈ J −1 (μ)/G μ . Moreover, ρ(G μ p) is in ρ(J −1 (μ)). Hence, there exists a welldefined map l : J −1 (μ)/G μ → ρ(J −1 (μ)) such that the diagram commutes. We first show that l is invertible. Clearly, l is one-to-one. To show that l maps J −1 (μ)/G μ onto ρ(J −1 (μ)), consider x ∈ ρ(J −1 (μ)). There exists p ∈ J −1 (μ) such that x = ρ( p). Its projection ρμ ( p) is in J −1 (μ)/G μ , and l(ρμ ( p)) = x. Hence, l is a bijection. We want to show that l is a diffeomorphism. Consider first the restriction h |ρ(J −1 (μ)) of h ∈ C ∞ (R) to ρ(J −1 (μ)). For p ∈ J −1 (μ) and x = ρ( p), l ∗ h |ρ(J −1 (μ)) (ρμ ( p)) = h |ρ(J −1 (μ)) (l(ρμ ( p))) = h(x) = h(ρ( p)) = (ρ ∗ h)( p) = (ρ ∗ h)|J −1 (μ) ( p). Since ρ ∗ h ∈ C ∞ (P) is G-invariant, it follows that (ρ ∗ f h)|J −1 (μ) is G μ invariant and that it pushes forward to a smooth function on J −1 (μ)/G μ . Hence, l ∗ h |ρ(J −1 (μ)) ∈ C ∞ (J −1 (μ)/G μ ). This holds for the restriction to ρ(J −1 (μ)) of every function h ∈ C ∞ (R). Proposition 2.1.8 implies that l ∗ maps C ∞ (ρ(J −1 (μ))) to C ∞ (J −1 (μ)/G μ ). Hence, l : J −1 (μ)/G μ → ρ(J −1 (μ)) is smooth. In order to show that l −1 is smooth, it suffices to show that l ∗ maps C ∞ (ρ(J −1 (μ))) onto C ∞ (J −1 (μ)/G μ ). Consider f ∈ C ∞ (J −1 (μ)/G μ ). It follows that ρμ∗ f is a G μ -invariant function in J −1 (μ). The differential structure of J −1 (μ) is generated by its inclusion into P. Hence, for each p ∈ J −1 (μ), there exist a neighbourhood U of p in P and k ∈ C ∞ (P) such that k|U ∩J −1 (μ) = (ρμ∗ f )|U ∩J −1 (μ) . Let S p be a slice for the action of G on P contained in U . The slice S p is invariant under the action on P of the isotropy group H of p. Hence, H ∩G μ acts on S p ∩ J −1 (μ), and the restriction (ρμ∗ f )|S p ∩J −1 (μ) is invariant under this action. Let W1 and W2 be H -invariant neighbourhoods of p in S p such that W 1 ⊆ W2 . There exists a non-negative function h ∈ C ∞ (S p ) such that h |W 1 = 1 and the support of h is contained in W2 . Since H is compact, we may average hk|S p over H , obtaining an H -invariant function

124

Symplectic reduction

k˜ S p =

H

∗g hk|S p dμ(g),

where dμ(g) is the Haar measure on H normalized so that vol H = 1. Since H is compact, W2 is H -invariant and the support of h is contained in W2 , it follows that the support of k˜ is compact and contained in W2 . Now, G S p is a G-invariant neighbourhood of p in P. Since k˜ S p is H -invariant, it extends to a unique smooth G-invariant function kG S p on G S p . Let k P be a function on P such that

k G S p ( p ) if p ∈ G S p . kP(p ) = 0 if p ∈ / GW2 Since k G S p ( p ) = 0 for all p ∈ G S p \GW2 , it follows that k P is well defined. Also, k P is smooth and G-invariant. Moreover, the restriction of k P to J −1 (μ) ∩ W1 coincides with the restriction of ρμ∗ f to W1 ∩ J −1 (μ). Since k P is in C ∞ (P)G , it pushes forward to ρ∗ k P ∈ R. For each p ∈ −1 J (μ) ∩ W1 , we have ρ( p ) ∈ ρ(J −1 (μ)) and f (ρμ ( p )) = ρμ∗ f ( p ) = k P ( p ) = (ρ∗ k P )(ρ( p )) = ρ∗ k P (l(ρμ ( p ))). Hence, f |ρ(J −1 (μ))∩ρ(GW1 ) = l ∗ (ρ∗ k P )|ρ(J −1 (μ))∩ρ(GW1 ) , where (ρ∗ k P )|ρ(J −1 (μ)) is in C ∞ (ρ(J −1 (μ))). A similar result holds for a neighbourhood of every p ∈ J −1 (μ). Using a partition-of-unity argument, we conclude that f is in the range of l ∗ . This implies that l ∗ maps C ∞ (ρ(J −1 (μ))) onto C ∞ (J −1 (μ)/G μ ). Hence, l −1 is smooth. Thus, l is a diffeomorphism. By Theorem 6.3.4, the orbit type stratification N of R induces a stratification {ρ(J −1 (μ)) ∩ N | N ∈ N} of ρ(J −1 (μ)). The diffeomorphism l : J −1 (μ)/G μ → ρ(J −1 (μ)) defines a family {l −1 (ρ(J −1 (μ)) ∩ N ) | N ∈ N} of manifolds contained in J −1 (μ)/G μ , which stratifies J −1 (μ)/G μ . For each M ∈ M, it follows from the commutativity of the diagram (6.26) that ρμ (J −1 (μ)∩ M) = ρ(J −1 (μ)∩ M) = ρ(J −1 (μ))∩ρ(M) = ρ(J −1 (μ))∩ N , where N = ρ(M) ∈ N. Hence, ρμ (J −1 (μ) ∩ M) = l −1 (ρ(J −1 (μ)) ∩ N ), which implies that ρμ is a morphism of stratified spaces. We know that the orbits of the family P(R) of Poisson vector fields on R are symplectic manifolds and that they are strata of the stratification of ρ(J −1 (μ)). Since l : J −1 (μ)/G μ → ρ(J −1 (μ)) is an isomorphism of stratified spaces, for each stratum Q = ρ(J −1 (μ)) ∩ N of ρ(J −1 (μ)), the pull-back l ∗ ω Q of

6.4 Pre-images of co-adjoint orbits

125

the symplectic form ω Q on Q, defined by equation (6.18), is a symplectic form on the stratum ρμ (J −1 (μ) ∩ M) = l −1 (ρ(J −1 (μ)) ∩ N ) of J −1 (μ)/G μ .

6.4 Pre-images of co-adjoint orbits We continue with the assumption that the action of G on P is proper. Let Oμ = { Adg∗ μ | g ∈ G} be the co-adjoint orbit through μ. Since ρ −1 (ρ(J (μ)) is the G-orbit of J −1 (μ), it follows that

ρ −1 (ρ(J −1 (μ))) = G J −1 (μ) = g∈G g J −1 (μ) = g∈G J −1 (Adg∗ μ) = J −1 (Oμ ). Hence, ρ(J −1 (μ)) = J −1 (Oμ )/G. In the following discussion, we identify these two spaces. An analysis of the structure of J −1 (Oμ ) requires caution because, in general, a co-adjoint orbit need not be locally closed in g∗ ; see ?. For example, if the action of G on P is free and proper, but Oμ is not locally closed in g∗ , then J −1 (Oμ ) is only immersed in P. Thus, in discussing the differential structure of J −1 (Oμ ) induced by its inclusion in P, we have to allow for the possibility that the topology of J −1 (Oμ ) may be finer than the subspace topology. Consider the diagram J −1 (Oμ )

i

ρ

ρ Oμ

J −1 (Oμ /G)

/P

j

(6.27)

/ R,

where i : J −1 (Oμ ) → P and j : J −1 (Oμ )/G → R are inclusion maps and ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is the orbit map. The differential structure of J −1 (Oμ ) is defined by its inclusion in P. That is, h : J −1 (Oμ ) → R is in C ∞ (J −1 (Oμ )) if, for each p ∈ J −1 (Oμ ), there exist a neighbourhood V of p in J −1 (Oμ ) and a function h˜ ∈ C ∞ (P) such that h˜ |V = h |V . If the topology of J −1 (Oμ ) is finer than the subspace topology, the neighbourhood

126

Symplectic reduction

V of p in J −1 (Oμ ) need not be equal to the intersection of J −1 (Oμ ) with a neighbourhood of p in P. Nevertheless, the inclusion map i : J −1 (Oμ ) → P is smooth. The orbit space J −1 (Oμ )/G has two differential structures because it is the quotient space of J −1 (Oμ ) and a subset of R. In general, these two differential structures need not coincide. Here, we consider J −1 (Oμ )/G with the differential structure induced by the inclusion map j : J −1 (Oμ )/G = ρ(J −1 (μ)/G μ ) → R. Proposition 6.4.1 The orbit map ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is smooth in the differential structure C ∞ (J −1 (Oμ )/G) generated by the ring R(J −1 (Oμ )/G) = { f |J −1 (Oμ )/G | f ∈ C ∞ (R)}. Proof As before, it suffices to show that for each f ∈ C ∞ (R), the pull∗ f −1 (O )/G is in back ρ O −1 (O )/G of the restriction f |J −1 (O )/G of f to J μ μ μ μ |J C ∞ (J −1 (Oμ )). But

∗ ρO f −1 (Oμ )/G = (ρ ∗ f )|J −1 (Oμ ) ∈ C ∞ (J −1 (Oμ )). μ |J

Hence, ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is smooth. The analogue of Theorem 6.3.5 is trivial, because μ ∈ Oμ implies that J −1 (Oμ )/G = ρ(J −1 (Oμ )) = ρ(J −1 (μ)). For the same reason, the analogues of part (ii) of Theorem 6.3.4 and of the first statement of part (iii) of Theorem 6.3.2 are true. At present, we have no proof that J −1 (Oμ ) is a stratified space, but we have no counterexample either. Hence, we cannot make a statement analogous to part (i) of Theorem 6.3.2.

6.5 Reduction by stages for proper actions In many applications, the symmetry group G of (P, ω) has a normal subgroup H , and it is convenient to perform reduction by stages. First, one passes to the space of H -orbits in P, obtaining a stratified Poisson differential space P/H . The quotient group G/H acts on P/H by isomorphisms of its structure. In reducing the symmetries of P/H , we can no longer use the results obtained for manifolds, because P/H may have more than one stratum. Since P/G = (P/H )/(G/H ), we may expect reduction by stages to give results equivalent to the results of direct reduction when we pass from P to the stratified Poisson differential space P/G.

6.5 Reduction by stages for proper actions

127

We denote the space of G-orbits in P by R = P/G and the space of H orbits in Q by Q = P/H ; these have orbit maps ρ : P → R and η : P → Q, respectively. The action of the quotient group K = G/H on Q is K : K × Q → Q : ([g], η( p)) → [g]η( p) = η(gp),

(6.28)

where [g] = H g is the coset of g in K = G/H . If g = hg is another representative of [g], then η(g p) = η(hgp) = η(gp) because the orbit map η : P → Q = P/H is constant on H -orbits in P. Similarly, if η( p ) = η( p), then p = hp for some h ∈ H . Since H is a normal subgroup of G, for each g ∈ G and h ∈ H there exists h ∈ H such that gh = h g. Hence, Therefore,

K

η(gp ) = η(ghp) = η(h gp) = η(gp). is well defined.

Proposition 6.5.1 If the action of G on P is proper, it follows that the action K of K = G/H on P/H is proper. Proof Let (η( pn )) be a sequence of points in Q convergent to η( p), and let ([gn ]) be a sequence of elements of K such that the sequence ([gn ]η( pn )) = η(gn pn ) is convergent. Since P is a manifold, there exists a diffeomorphism of a neighbourhood U of p that maps p to the origin in Rd and takes U to an open ball Br of radius r > 1, centred at the origin in Rd . Using the Euclidean metric of Rd , we can construct a nested family Vm of neighbourhoods of p in P diffeomorphic to open balls in Rd of radius 1/m centred at the origin. Since η( pn ) → η( p) as n → ∞, for each m ∈ N there exists n m ∈ N such that the orbit H pn m intersects Um . Therefore, there exists h n m ∈ H such that h n m pn m ∈ Vm . This implies that the sequence h n m pn m converges to p. Hence, η(h n m pn m ) = η( pn m ) and h n m pn m → p as m → ∞. Similarly, the assumption that η(gn pn ) is convergent implies that there exists a sequence h n k such that h n k gn k pn k is convergent. Without loss of generality, we may assume that pn → p and gn pn → p as n → ∞. The properness of the action of G on P implies that there exists a convergent subsequence gnl such that lim (gnl pnl ) = lim gnl lim pnl = lim gnl p. l→∞

l→∞

l→∞

l→∞

Since the projection G → G/H is continuous, it follows that lim [gnl ] = lim gnl , l→∞

l→∞

128

Symplectic reduction

so that the sequence [gnl ] is convergent. The continuity of the orbit map η : P → P/H implies that lim η(gnl pnl ) = η lim (gnl pnl ) = η lim gnl p = lim gnl η( p).

l→∞

l→∞

l→∞

l→∞

Hence, the action of K = G/H on Q = P/H is proper. The assumption that the action of G on (P, ω) is Hamiltonian and proper implies that the orbit spaces R = P/G and Q = P/H are stratified Poisson subcartesian spaces. Let S = Q/K be the space of K -orbits in Q, and let σ : Q → S be the corresponding orbit map. Since the action of K on Q is proper and Q is a locally compact subcartesian space, it follows from the discussion in Section 4.4 that S is a Hausdorff, locally compact differential space with the quotient topology and the differential structure C ∞ (S) = { f S : C 0 (S) | σ ∗ f S ∈ C ∞ (Q)}. Proposition 6.5.2 The orbit space S is a Poisson differential space, and there is a unique diffeomorphism ϕ : R → S such that the diagram P

ρ

/R

η

Q

ϕ

σ

/S

commutes and ϕ ∗ : C ∞ (S) → C ∞ (R) is a Poisson algebra isomorphism. Proof The orbit map ρ : P → R is an epimorphism. Given r ∈ R, choose p ∈ ρ −1 (r ) ⊆ P, and set ϕ(r ) = σ (η( p)). If a point p ∈ P is such that ρ( p ) = ρ( p), then there exists g ∈ G satisfying p = gp. Equation (6.28) gives η( p ) = η(gp) = [g]η( p). The fibre of σ : Q → S over η( p) is the K -orbit K η( p) through η( p). Hence, σ (η( p )) = σ ([g]η( p)) = σ (η( p)), which implies that ϕ is well defined. Since the orbit maps η and σ are epimorphisms, it follows that ϕ maps R onto S. Moreover, if ϕ(r ) = ϕ(r ), then there exists p ∈ ρ −1 (r ) such that σ (η( p)) = σ (η( p )). This means that η( p) and η( p ) are in the same fibre of σ . In other words, there exists g ∈ G such that η( p ) = [g]η( p) = η(gp). This implies that there exists h ∈ H such that p = hgp. Therefore, p and p are in the same G-orbit, so that ρ( p ) = ρ( p). This proves that ϕ is one-to-one. To prove that ϕ is smooth, observe that if f S ∈ C ∞ (S), then σ ∗ f S ∈ ∞ C (Q) K ⊆ C ∞ (Q), which implies that η∗ (σ ∗ f S ) ∈ C ∞ (Q) H . Since σ ∗ f S

6.6 Shifting

129

is constant on orbits of K = G/H in Q = P/H , and η∗ (σ ∗ f S ) is constant on orbits on H in P, it follows that η∗ (σ ∗ f S ) is constant on G-orbits in P. Hence, η∗ (σ ∗ f S ) = ρ ∗ f R for a function f R ∈ C ∞ (R). But σ ◦ η = ϕ ◦ ρ implies that ρ ∗ f R = η∗ (σ ∗ f S ) = (σ ◦ η)∗ f S = (ϕ ◦ ρ)∗ f S = ρ ∗ (ϕ ∗ f S ). Since ρ : P → R is an epimorphism, it follows that ρ ∗ : C ∞ (R) → C ∞ (P) is a monomorphism. Therefore, ϕ ∗ f S = f R ∈ C ∞ (R). Hence, ϕ : R → S is smooth. In order to prove that ϕ −1 is smooth, it suffices to show that ϕ ∗ is an epimorphism. Consider f R ∈ C ∞ (R). Then ρ ∗ f R is a G-invariant function on P. Hence, ρ ∗ f R is H -invariant, which implies that ρ ∗ f R = η∗ f Q for f Q ∈ C ∞ (Q). Moreover, the G-invariance of η∗ f Q implies that f Q is K -invariant. Therefore, f Q = σ ∗ f S for a function f S ∈ C ∞ (S). As before, ρ ∗ f R = η∗ (σ ∗ f S ) = ρ ∗ (ϕ ∗ f S ), which implies that f R = ϕ ∗ f S . This proves that ϕ −1 is smooth. Hence, ϕ is a diffeomorphism. It remains to show that ϕ ∗ is a morphism of the Poisson algebra structures of ∞ C (S) and C ∞ (R). For f 1R and f 2R in C ∞ (R), their Poisson bracket satisfies the identity ρ ∗ { f 1R , f 2R } = {ρ ∗ f 1R , ρ ∗ f 2R }. Similarly, the Poisson bracket of f 1S and f 2S in C ∞ (S) satisfies the identities η∗ {σ ∗ f 1S , σ ∗ f 2S } = {η∗ (σ ∗ f 1S ), η∗ (σ ∗ ( f 2S ))} = {ρ ∗ (ϕ ∗ f 1S ), ρ ∗ (ϕ ∗ f 2S )} = ρ ∗ {ϕ ∗ f 1S , ϕ ∗ f 2S }. Hence, if f i R = ϕ ∗ f i S for i = 1, 2, then {ϕ ∗ f 1S , ϕ ∗ f 2S } = { f 1R , f 2R }, which implies that ϕ ∗ is a morphism of Poisson algebras. Corollary 6.5.3 For every p ∈ P, the diffeomorphism ϕ : R → S induces a symplectomorphism of the symplectic leaf of R that contains ρ( p) onto the symplectic leaf of S through σ (η( p)). This result can be paraphrased by saying that for a proper action of a Lie group on a symplectic manifold, the symplectic reduction of the symmetries by stages is equivalent to the symplectic reduction of all symmetries in one stage.

6.6 Shifting The case when μ = 0 is particularly simple because G μ = G and Oμ = {0}, so that the spaces J −1 (μ)/G μ and J −1 (Oμ )/G are identical. Therefore, if μ = 0, it is useful to be able to modify the system so that J −1 (μ) becomes the

130

Symplectic reduction

zero level of the momentum map of the modified system. For a free and proper action, the modification described here is usually called the ‘shifting trick’ of Guillemin and Sternberg. Consider the product P˜ = P × Oμ , with projections π1 : P˜ → P and π2 : P˜ → Oμ and a symplectic form ω˜ = π1∗ ω ⊕ (−π2∗ μ ). The action of G on P˜ is given by ˜ : G × P˜ → P˜ : (g, ( p, λ)) → ˜ g ( p, λ) = (g ( p), Adg∗ λ). This action has an Ad ∗ -equivariant momentum map J˜ = π1∗ J − π2∗ I . ˜ We denote the space of G-orbits in P˜ by R˜ = P/G and the corresponding ˜ ˜ orbit map by ρ˜ : P → R. The restriction of the domain of π1 : P˜ → P : ( p, λ) → p to J˜−1 (0) gives a map from J˜−1 (0) to P with range J −1 (Oμ ), because J˜( p, λ) = 0 implies J ( p) = λ ∈ Oμ . Thus, we obtain a map : J˜−1 (0) → J −1 (Oμ ) : ( p, J ( p)) → p.

(6.29)

Now consider a map P → P × g∗ : p → ( p, J ( p)). Restricting the domain to J −1 (Oμ ), we obtain a map from J −1 (Oμ ) to P × g∗ with range J˜−1 (0). Hence, we have a map % : J −1 (Oμ ) → J˜−1 (0) : p → ( p, J ( p)).

(6.30)

For each p ∈ J −1 (Oμ ), (%( p)) = ( p, J ( p)) = p. Similarly, for each ( p, J ( p)) ∈ J˜−1 (0), %(( p, J ( p)) = %( p) = ( p, J ( p)). Therefore, % = −1 . For each g ∈ G, (gp, Adg∗ J ( p)) = (gp, J (gp)) = gp and %(gp) = ˜ g %( p). Thus, and % are G-equivariant, (gp, J (gp)) = (gp, Adg∗ J ( p)) = and they induce maps of the corresponding orbit spaces

and

γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) : ρ( ˜ p, J ( p)) → ρ( p)

(6.31)

˜ J˜−1 (0)) : ρ( p) → ρ( ˜ p, J ( p)), δ : ρ(J −1 (Oμ )) → ρ(

(6.32)

which are called shifting maps. Proposition 6.6.1 If Oμ is an embedded submanifold of g∗ , then the shifting ˜ J˜−1 (0)) are maps γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) and δ : ρ(J −1 (Oμ )) → ρ( diffeomorphisms.

6.6 Shifting

131

Proof The map : J˜−1 (0) → J −1 (Oμ ) is obtained from π1 : P˜ = P × Oμ → P by restricting its domain to J˜−1 (0) and its codomain to J −1 (Oμ ). Since Oμ is a submanifold of g∗ , we take the subspace topology on J −1 (Oμ ). By Proposition 2.1.8, in order to prove the smoothness of , it suffices to show that for each f ∈ C ∞ (P), the map ∗ f |J −1 (Oμ ) = f |J −1 (Oμ ) ◦ : J˜−1 (0) → R is smooth. Since ( p, λ) ∈ J˜−1 (0) implies that J ( p) = λ ∈ Oμ , we have f |J −1 (Oμ ) ◦( p, λ) = f |J −1 (Oμ ) ◦π1 ( p, λ) = π1∗ f ( p, λ) = (π1∗ f )| J˜−1 (0) ( p, λ). Therefore, ∗ f |J −1 (Oμ ) = (π1∗ f )| J˜−1 (0) ∈ C ∞ ( J˜−1 (0)), which implies the smoothness of . We show smoothness of % : J −1 (Oμ ) → J˜−1 (0) in a similar manner. ˜ We need to show that %∗ f ˜−1 is smooth. For each Consider f˜ ∈ C ∞ ( P). | J (0) p ∈ J −1 (Oμ ), %∗ f˜| J˜−1 (0) ( p) = f˜| J˜−1 (0) ◦ %( p) = f˜| J˜−1 (0) ( p, J ( p)) = f˜( p, J ( p)) = f˜ ◦ (id P × J )( p) = (id P × J )∗ f˜( p) = ((id P × J )∗ f˜)|J −1 (O ) ( p). μ

Hence, %∗ f˜| J˜−1 (0) = ((id P × J )∗ f˜)|J −1 (Oμ ) ∈ C ∞ (J −1 (Oμ )). Since Oμ is an embedded submanifold of g∗ , it is locally compact. Hence, −1 J˜ (0) and J −1 (Oμ ) are locally compact differential spaces. Moreover, the actions of G on J˜−1 (0) and J −1 (Oμ ) are proper. By Proposition 4.4.6 and Theorem 4.4.7, the orbit spaces ρ( ˜ J˜−1 (0)) and ρ(J −1 (Oμ )) are locally compact differential spaces endowed with the quotient topologies. This implies that the shifting maps γ and δ are smooth. Since they are inverses of each other, it follows that they are diffeomorphisms. It follows from Theorem 6.3.4 that for each stratum N of the orbit type stratification of R, the intersection N ∩ ρ(J −1 (Oμ )) coincides with a symplectic leaf Q of N . Hence, for each x ∈ ρ(J −1 (Oμ )) ⊆ R and every pair f 1 , f 2 of functions in C ∞ (R), the Poisson bracket { f 1 , f 2 }(x) depends only on the restrictions of f 1 and f 2 to Q = N ∩ ρ(J −1 (Oμ )), where N is the stratum of R through x. Hence, { f 1 , f 2 }(x) depends on the restrictions of f 1 and f 2 to ρ(J −1 (Oμ )). This implies that ρ(J −1 (Oμ )) is a Poisson differential space; that is, C ∞ (ρ(J −1 (Oμ ))) has the structure of a Poisson algebra. Similarly, ρ( ˜ J˜−1 (0)) is a Poisson differential space. ˜ J˜−1 (0))) and δ ∗ : We want to show that γ ∗ : C ∞ (ρ(J −1 (Oμ ))) → C ∞ (ρ( ∞ −1 ∞ −1 ˜ C (ρ( ˜ J (0))) → C (ρ(J (Oμ ))) are isomorphisms of Poisson algebras. Since the symplectic form ω˜ on P˜ = P×Oμ is given by ω˜ = π1∗ ω⊕(−π2∗ μ ), → P : ( p, λ) → p is Poisson. Moreover, π1 is it follows that π1 : P

132

Symplectic reduction

G-equivariant. Hence, it induces a Poisson map π1 : R˜ → R such that the following diagram commutes: P

π1

/P ρ

ρ˜

˜ R

π1

/ R.

˜ is a Poisson algebra homomorphism. The The map π1∗ : C ∞ (R) → C ∞ ( R) −1 ˜ is also a Poisson map. Therefore, the cominclusion map ι : ρ( ˜ J (0)) → R −1 ˜ J˜ (0)) → R is a Poisson map. Note that π1 ◦ ι has values position π1 ◦ ι : ρ( in ρ(J −1 (Oμ )), and γ is obtained by the restriction of the codomain of π1 ◦ ι −1 to ρ(J (Oμ )). The differential structure of ρ(J −1 (Oμ )) is generated by the ring R(ρ(J −1 (Oμ ))) = { f |ρ(J −1 (Oμ )) | f ∈ C ∞ (R)}, which we identify with C ∞ (R)/N (ρ(J −1 (Oμ ))), where N (ρ(J −1 (Oμ ))) = { f ∈ C ∞ (R) | f |ρ(J −1 (Oμ )) = 0} is the null ideal of ρ(J −1 (Oμ )) in C ∞ (R). Lemma 6.6.2 The associative ideal N (ρ(J −1 (Oμ ))) in C ∞ (R) is a Poisson ideal. In other words, if f ∈ N (ρ(J −1 (Oμ ))) and h ∈ C ∞ (R), then { f, h} ∈ N (ρ(J −1 (Oμ ))). Proof

If h ∈ C ∞ (R), then ρ ∗ h ∈ C ∞ (P)G and, for every ξ ∈ g, X ρ ∗ h Jξ = −X Jξ ρ ∗ h = 0.

Hence, X ρ ∗ h preserves the level sets of the momentum map J. In particular, X ρ ∗ h preserves J −1 (Oμ ). If f ∈ N (ρ(J −1 (Oμ ))), then ρ ∗ f vanishes on J −1 (Oμ ) and X ρ ∗ h (ρ ∗ f ) also vanishes on J −1 (Oμ ). Since ρ{ f, h} = {ρ ∗ f, ρ ∗ h} = X ρ ∗ h (ρ ∗ f ), it follows that π ∗ { f, h} vanishes on J −1 (Oμ ). Thus, the bracket { f, h} vanishes on ρ(J −1 (Oμ )). This implies that N (ρ(J −1 (Oμ ))) is a Poisson ideal in C ∞ (R). Since N (ρ(J −1 (Oμ ))) is a Poisson ideal in C ∞ (R), it follows that the quotient R(ρ(J −1 (Oμ ))) = C ∞ (R)/N (ρ(J −1 (Oμ )))

6.6 Shifting

133

inherits the structure of a Poisson algebra. For every f1 , f 2 ∈ C ∞ (R), the Poisson bracket of the restrictions f 1|ρ(J −1 (Oμ )) and f 2|ρ(J −1 (Oμ )) of f 1 and f 2 , respectively, to ρ(J −1 (Oμ )), evaluated at x ∈ ρ(J −1 (Oμ )) ⊆ R, is given by { f 1|ρ(J −1 (Oμ )) , f2|ρ(J −1 (Oμ )) }(x) = { f 1 , f 2 }(x).

(6.33)

Every function in C ∞ (ρ(J −1 (Oμ ))) coincides locally with a function in R(ρ(J −1 (Oμ ))). Hence, it follows that equation (6.33) specifies uniquely the Poisson bracket on C ∞ (ρ(J −1 (Oμ ))). ˜ J˜−1 (0))) vanishes on Next, we show that ( π1 ◦ ι)∗ : C ∞ (R) → C ∞ (ρ( the null ideal N (ρ(J −1 (Oμ ))). Let f ∈ C ∞ (R) be a function vanishing on ρ(J −1 (Oμ )). For each ( p, J ( p)) ∈ J˜−1 (0), ( π1 ◦ ι)∗ f (ρ( ˜ p, J ( p))) = f ( π1 (ι(ρ( ˜ p, J ( p))))) = f ( π1 (ρ( ˜ p, J ( p)))) = f (ρ(π1 ( p, J ( p)))) = f (ρ( p)) = 0, because ( p, J ( p)) ∈ J˜−1 (0) implies that p ∈ J −1 (Oμ ) and f vanishes on ρ(J −1 (Oμ )). The vanishing of ( π1 ◦ ι)∗ : C ∞ (R) → C ∞ (ρ( ˜ J˜−1 (0))) on the null −1 ∗ π1 ◦ ι) induces a Poisson algebra homoideal N (ρ(J (Oμ ))) implies that ( morphism C ∞ (R)/N (ρ(J −1 (Oμ ))) → C ∞ (ρ( ˜ J˜−1 (0))). Recall that the ∞ −1 quotient C (R)/N (ρ(J (Oμ ))) is the subspace C ∞ (ρ(J −1 (O))) consisting of smooth functions on ρ(J −1 (O)), which extend to smooth functions on R. Hence, the Poisson homomorphism induced by ( π1 ◦ ι)∗ is the restriction of ∗ ∞ −1 ∞ −1 ∞ ˜ γ : C (ρ(J (Oμ ))) → C (ρ( ˜ J (0))) to C (R)/N (ρ(J −1 (Oμ ))). ∗ In order to show that γ is a Poisson algebra homomorphism, note that for every f 1 , f 2 ∈ C ∞ (ρ(J −1 (Oμ ))) and x ∈ ρ( ˜ J˜−1 (0)), there exist a neighbourhood U of γ (x) in R and h 1 , h 2 ∈ C ∞ (R) such that for i = 1, 2, h i|U ∩ρ(J −1 (Oμ )) = f i|U ∩ρ(J −1 (Oμ )) and {γ ∗ h 1|ρ(J −1 (Oμ )) , γ ∗ h 2|ρ(J −1 (Oμ )) }(x) = γ ∗ {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(x), because h i|ρ(J −1 (O)) ∈ C ∞ (R)/N (ρ(J −1 (Oμ ))). Therefore, {ρ ∗ f 1 , ρ ∗ f 2 }(x) = {ρ ∗ h 1|ρ(J −1 (Oμ )) , ρ ∗ h 2|ρ(J −1 (Oμ )) }(x) = γ ∗ {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(x)

= {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(γ (x)) = { f 1 , f2 }(γ (x)) = γ ∗ { f 1 , f 2 }(x).

Hence, γ ∗ is a Poisson algebra homomorphism. Since γ is a diffeomorphism and δ = γ −1 , it follows that γ ∗ and δ ∗ are Poisson algebra isomorphisms. We have proved the following result.

134

Symplectic reduction

Theorem 6.6.3 If Oμ is an embedded submanifold of g∗ , then the shifting ˜ J˜−1 (0)) are maps γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) and δ : ρ(J −1 (Oμ )) → ρ( diffeomorphisms of Poisson differential spaces.

6.7 When the action is free In the case of a free and proper action of G on P, the reduction described here is called regular reduction or Marsden–Weinstein reduction (Marsden and Weinstein, 1974). In this case, the stratification N of R consists of a single stratum P, the stratification M of P also consists of a single stratum R and we can rephrase the results of the preceding sections as follows. Corollary 6.7.1 Let J : P → g∗ be an equivariant momentum map for a free and proper action of a connected Lie group G on a symplectic manifold (P, ω). (i) The orbit space R = P/G is a Poisson manifold, and the orbit map ρ : P → R is a submersion. The pull-back map ρ ∗ : C ∞ (R) → C ∞ (P) gives rise to a Poisson algebra isomorphism from C ∞ (R) to C ∞ (P)G . For each f ∈ C ∞ (R), the Poisson vector field X f on R is ρ-related to the Hamiltonian vector field X ρ ∗ f on P; that is, ρ∗ X ρ ∗ f = X f . (ii) The kernel of d J : T P → g∗ is spanned by Hamiltonian vector fields of G-invariant functions. The connected components of level sets of J are orbits of the family {X h | h ∈ C ∞ (P)G } of Hamiltonian vector fields of G-invariant functions on P. (iii) Each orbit Q of the family {X f | f ∈ C ∞ (R)} of Poisson vector fields on R is endowed with a symplectic form ω Q on Q induced by the Poisson structure of C ∞ (Q). The pre-image ρ −1 (Q) is a connected component of J −1 (μ), where μ is the constant value of J on ρ −1 (Q). In other words, Q is a connected component of ρ(J −1 (μ)). The pull-back of ω Q to ρ −1 (Q) coincides with the restriction of ω to ρ −1 (Q). (iv) For each μ ∈ g∗ , each connected component of the orbit space J −1 (μ)/G μ , where G μ = {g ∈ G | Adg∗ μ = μ}, is a symplectic manifold symplectomorphic to the corresponding connected component of ρ(J −1 (μ)). (v) If the co-adjoint orbit Oμ through μ is locally closed,3 consider the ˜ ω), symplectic manifold ( P, ˜ where P˜ = P × Oμ

and

ω˜ = π1∗ ω ⊕ (−π2∗ μ ),

3 Regular reduction of the pre-images of co-adjoint orbits that are not locally closed was

discussed by Ortega and Ratiu (2004).

6.8 When the action is improper

135

and μ is the Kirillov–Kostant–Souriau symplectic form on Oμ . The ˜ given by action of G on P, ˜ : G × P˜ → P˜ : (g, ( p, ν)) → ˜ g ( p, ν) = (g ( p), Adg∗ ν), is Hamiltonian with the momentum map J˜ : (P × Oμ ) → g∗ : ( p, ν) → (J ( p), −ν). The manifolds J˜−1 (0), J −1 (Oμ )/G, ρ(J −1 (Oμ )), ρ(J −1 (μ)) and J −1 (μ)/G μ are symplectic manifolds, and they are symplectomorphic to each other.

6.8 When the action is improper If the action of G on P is not proper, then invariant functions need not separate G-orbits. This implies that the differential-space topology of the orbit space R = P/G may not be Hausdorff. Also, the differential-space topology may differ from the quotient topology. We illustrate this fact with a simple example. Example 6.8.1 We consider here an example of an action of G = R on P = R2 # (x, y), given by : R × P → P : (t, (x, y)) → (x, y + xt). This action is Hamiltonian with respect to the canonical symplectic form ω = d x ∧ dy on R2 , and the corresponding momentum map is J (x, y) = 12 x 2 . The action describes a free motion of a particle with one degree of freedom and unit mass, but this fact is not relevant here. Each point of the y-axis is fixed by the action of R. Hence, the fixed-point set is PR = {(0, y) | y ∈ R} = J −1 (0). Since R is not compact, it follows that the action is not proper. For μ = 0, the level set is given by J −1 (μ) = {(x, y) | x 2 = 2μ, y ∈ R}. Thus, J −1 (μ) consists of two lines in R2 parallel to the y-axis. These lines tend to the y-axis as μ → 0. It is easy to see that the action is free and proper on the complement of the fixed-point set. Hence, the orbit space is {(a, y) | y ∈ R} {(0, y)} . R= a=0 y

136

Symplectic reduction

The orbit map ρ : R2 → R is given by

{(x, y) | y ∈ R} ρ(x, y) = {(0, y)}

if x = 0 if x = 0.

Let f ∈ C ∞ (P) be an R-invariant function. The restriction of f to R) depends only on x. Hence, by continuity, the value of f is constant on the fixed-point set J −1 (0). Thus, the differential-space topology of R is not Hausdorff. In fact, the differential-space topology does not even satisfy the separation axiom T1 , because every open set containing a fixed point contains all fixed points. The quotient topology of R is also not Hausdorff. However, the quotient topology is T1 because, for every pair of distinct points (x 1 , y1 ) and (x2 , y2 ), there exists an open set V ∈ R containing ρ(x 1 , y1 ) but not ρ(x 2 , y2 ). This is obvious if x 1 = 0 and x2 = 0. If x 1 = x2 = 0, the complement U of the point (0, y2 ) in R2 is open and R-invariant, and contains the point (0, y1 ). Hence, V = ρ(U ) is open in R, contains ρ(0, y1 ) and does not contain ρ(0, y2 ). This shows that the quotient topology is different from the differential-space topology. R2 \({0} ×

In most applications to dynamics, the action of the symmetry group is proper except for some exceptional values of the momentum map. Hence, in the case of an improper action, it suffices to define the reduction for fixed values of the momentum map. Thus, we have arrived at the following problem. Let G be a connected Lie group with a Hamiltonian action on a symplectic manifold (P, ω) corresponding to an Ad ∗ -equivariant momentum map J : P → g∗ . Suppose that for μ ∈ g∗ , the action of the isotropy group G μ of μ on the μ-level J −1 (μ) is improper. In this case, we can use the algebraic reduction described below.

6.9 Algebraic reduction For μ ∈ g∗ , let Jμ be the ideal in C ∞ (P) generated by the components of J − μ : P → g∗ . Thus,

k ∞ Jμ = J − μ | ξi fi | ξ1 , . . . , ξk ∈ g and f 1 , . . . , f k ∈ C (P) , i=1

where (ξ1 , . . . , ξk ) is a basis in g. If our functions were polynomials, we would refer to the μ-level of J as the variety generated by the ideal Jμ . We also know that some problems, especially problems related to multiplicities of roots, have

6.9 Algebraic reduction

137

no solutions in terms of varieties, but can be solved in terms of schemes. Extrapolating this insight from algebraic geometry to differential geometry, we proceed as follows. We denote by C ∞ (P)/Jμ the quotient of C ∞ (P) by the ideal Jμ . For each f ∈ C ∞ (P), we denote by [ f ]μ the class of f in C ∞ (P)/Jμ . Since the momentum map J is Ad ∗ -equivariant, it follows that for every g ∈ G μ , k k ∗g J − μ | ξi fi = ∗g ( J | ξi − μ | ξi ) ∗g f i i=1

i=1

=

k

( ∗g J | ξi ∗g f i − μ | ξi )∗g f i

i=1

=

k

( J | Adg ξi ∗g f i − μ | Adg ξi )∗g f i

i=1

=

k

J − μ | Adg ξi ∗g f i .

i=1

Hence, Jμ is G μ -invariant. This implies that the action of G on P induces an action ∗ μ : G μ ×(C ∞ (P)/Jμ ) → C ∞ (P)/Jμ : (g, [ f ]μ ) → μ g [ f ]μ = [g −1 f ]μ

of G μ on C ∞ (P)/Jμ . We denote by (C ∞ (P)/Jμ )G μ the set of G μ -invariant elements of C ∞ (P)/Jμ ; that is, (C ∞ (P)/Jμ )G μ = {[ f ]μ ∈ C ∞ (P)/Jμ ) | [∗g−1 f ]μ = [ f ]μ ∀ g ∈ G μ }. It follows from the definition that [ f ]μ ∈ (C ∞ (P)/Jμ )G μ ⇐⇒ ∗g−1 f − f ∈ Jμ ∀ g ∈ G μ .

(6.34)

In particular, [ f ]μ ∈ (C ∞ (P)/Jμ )G μ =⇒ X Jξ ( f ) ∈ Jμ ∀ ξ ∈ gμ , where gμ is the Lie algebra of G μ . If G μ is connected, then the reverse implication holds. Proposition 6.9.1 The Poisson algebra structure in C ∞ (P) induces a Poisson algebra structure in (C ∞ (P)/Jμ )G μ , with a Poisson bracket {[ f 1 ]μ , [ f 2 ]μ } such that {[ f 1 ]μ , [ f 2 ]μ } = [{ f 1 , f 2 }]μ .

(6.35)

138

Proof

Symplectic reduction For f ∈ C ∞ (P) and h =

{ f, h} =

k

=

i=1

i=1 J

{ f, J − μ | ξi h i } =

i=1 k

k

− μ | ξi h i ∈ Jμ ,

k k { f, Jξi }h i + J − μ | ξi { f, h i } i=1

X Jξi ( f )h i +

k

i=1

J − μ | ξi { f, h i }.

i=1

Hence, { f, h} ∈ Jμ , provided [ f ]μ ∈ (C ∞ (P)/Jμ )G μ . In particular, if f ∈ Jμ , then [ f ]μ = 0 ∈ (C ∞ (P)/Jμ )G μ , so that { f, h} ∈ Jμ for every h ∈ Jμ . Let f 1 , f 2 be functions in C ∞ (P) such that [ f 1 ]μ , [ f 2 ]μ ∈ (C ∞ (P)/Jμ )G μ . Equation (6.34) implies that for each g ∈ G μ , there exist h 1 , h 2 ∈ Jμ such that ∗g { f 1 , f 2 } = {∗g f 1 , ∗g f 2 } = { f 1 + h 1 , f 2 + h 2 } = { f 1 , f 2 } + { f 1 , h 2 } − { f 2 , h 1 } + {h 1 , h 2 }. Therefore, ∗g { f 1 , f 2 } − { f 1 , f2 } ∈ Jμ , which implies that { f 1 , f 2 } is in (C ∞ (P)/Jμ )G μ . The same calculation shows that for every h 1 , h 2 ∈ Jμ , { f 1 + h 1 , f 2 + h 2 } − { f 1 , f 2 } ∈ Jμ . Hence, the class [{ f 1 , f 2 }]μ depends only on [ f 1 ]μ and [ f 2 ]μ , which implies that the bracket {[ f 1 ]μ , [ f 2 ]μ } on (C ∞ (P)/Jμ )G μ is well defined by equation (6.35). Since the Poisson bracket on C ∞ (P) is bilinear and antisymmetric, acts as a derivation, and satisfies the Jacobi identity, it follows that the bracket {[ f 1 ]μ , [ f 2 ]μ } has the same properties. Therefore, {[ f 1 ]μ , [ f 2 ]μ } is a Poisson bracket on (C ∞ (P)/Jμ )G μ . Definition 6.9.2 Let G × P → P be a Hamiltonian action of a connected Lie group on a symplectic manifold corresponding to an Ad ∗ -equivariant momentum map J : P → g∗ . Algebraic reduction assigns the Poisson algebra (C ∞ (P)/Jμ )G μ to each level J −1 (μ) of J . For μ ∈ g∗ , we refer to (C ∞ (P)/Jμ )G μ as the Poisson algebra of algebraic reduction at μ. It should be noted that algebraic reduction at μ encodes not only information about the level set J −1 (μ) but also some information about its inclusion in (P, ω). It is of interest to compare algebraic reduction with other reduction techniques. Proposition 6.9.3 Let G × P → P be a free and proper Hamiltonian action of a connected Lie group on a symplectic manifold corresponding to an Ad ∗ equivariant momentum map J : P → g∗ . For each μ ∈ g∗ , the Poisson algebra (C ∞ (P)/Jμ )G μ is naturally isomorphic to the Poisson algebra of the symplectic manifold (Q, ω Q ), where Q = ρ(J −1 (μ)) and ρ ∗ ω Q is the restriction of ω to J −1 (μ); see Proposition 6.3.3.

6.9 Algebraic reduction

139

Proof The assumption that the action of G on P is free and proper implies that for each p ∈ P, the differential d J ( p) maps T p P onto g∗ . This can be seen as follows. If d J ( p) : T p P → g∗ is not a submersion, there exists ξ ∈ g such that d J (v) | ξ = 0 for all v ∈ T p P. This is equivalent to X Jξ ( p) ω = 0. Since ω is non-degenerate, this implies that X Jξ ( p) = 0. Hence, p is a fixed point of the action of the one-parameter subgroup exp tξ of G. This contradicts the assumption that the action of G on p is free. For each μ in the range of the momentum map J : P → g∗ , the level set J −1 (μ) is a submanifold of P. Moreover, every function that vanishes on J −1 (μ) is in the ideal Jμ . That is, N (J −1 (μ)) = Jμ and R(J −1 (μ)) = C ∞ (P)/N (J −1 (μ)) = C ∞ (P)/Jμ . Furthermore, J −1 (μ) is a closed subset of P, which implies that C ∞ (J −1 (μ)) = R(J −1 (μ)) = C ∞ (P)/Jμ . Since Q is a closed subset of R, each function f ∈ C ∞ (Q) extends to a function h ∈ C ∞ (R); that is, f = h |Q . The pull-back ρ ∗ h is in C ∞ (P)G , which implies that its class [ρ ∗ h]μ in C ∞ (P)/Jμ is G μ -invariant. Thus, [ρ ∗ h]μ ∈ (C ∞ (P)/Jμ )G μ . On the other hand, (ρ|J −1 (μ) )∗ h |Q = (ρ ∗ h)|J −1 (μ) , where ρ|J −1 (μ) : J −1 (μ) → Q is the restriction of the orbit map ρ : P → R to J −1 (μ). Therefore, [ρ ∗ h]μ = (ρ|J −1 (μ) )∗ (ρ ∗ h)|J −1 (μ) = h |Q = f . For h 1 , h 2 ∈ C ∞ (R), f 1 = h 1|Q and f 2 = h 2|Q are in C ∞ (Q). The Poisson bracket in (C ∞ (P)/Jμ )G μ is given by {[ρ ∗ h 1 ]μ , [ρ ∗ h 2 ]μ } = [{ρ ∗ h 1 , ρ ∗ h 2 }]μ = (ρ|J −1 (μ) )∗ {ρ ∗ h 1 , ρ ∗ h 2 }|J −1 (μ) = (ρ|J −1 (μ) )∗ (ρ ∗ {h 1 , h 2 })|J −1 (μ) = {h 1 , h 2 }|Q = {h 1|Q , h 2|Q } = { f 1 , f 2 }. It follows from equation (6.18) that the Poisson bracket in C ∞ (Q) is given by the symplectic form ω Q on Q. It follows from Proposition 6.9.3 that for a free and proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω), algebraic reduction is equivalent to regular reduction. However, when the action of G on (P, ω) is proper but not free, then algebraic reduction and singular reduction need not be equivalent. This can be seen in the following example. Example 6.9.4 Let P be the space of 2 × 2 complex matrices z = (z i j ) with the symplectic form ω = i(d z¯ 11 ∧ dz 11 + d z¯ 21 ∧ dz 21 + d z¯ 12 ∧ dz 12 + d z¯ 22 ∧ dz 22 ).

140

Symplectic reduction

The group G = SU (2) of unitary matrices g = (gi j ) with determinant 1 acts on P by multiplication on the left. Consider a basis i 1 0 0 0 2i 2 2 , , (ξ1 , ξ2 , ξ3 ) = i 0 − 2i 0 − 12 0 2 of the Lie algebra su(2) of SU (2). The vector fields X ξi corresponding to the action on P of the one-parameter subgroup exp tξi of SU (2) are X ξ1 =

1 (z 21 ∂11 + z¯ 21 ∂¯11 − z 11 ∂21 − z¯ 11 ∂¯21 ) 2 1 + (z 22 ∂12 + z¯ 22 ∂¯12 − z¯ 12 ∂¯22 − z 12 ∂22 ), 2

X ξ2 =

i (z 21 ∂11 + z 22 ∂12 + z 11 ∂21 + z 12 ∂22 ) 2 i ¯ z¯ 21 ∂11 + z¯ 22 ∂¯12 + z¯ 11 ∂¯21 + z¯ 12 ∂¯22 − 2

and X ξ3 =

i (z 11 ∂11 + z 12 ∂12 − z 21 ∂21 − z 22 ∂22 ) 2 i ¯ z¯ 11 ∂11 + z¯ 12 ∂¯12 − z¯ 21 ∂¯21 − z¯ 22 ∂¯22 . − 2

Each X ξi is the Hamiltonian vector field of a function Ji , where i J1 = − [¯z 11 z 21 − z¯ 21 z 11 + z¯ 12 z 22 − z¯ 22 z 12 ], 2 1 J2 = − [z 21 z¯ 11 + z 22 z¯ 12 + z 11 z¯ 21 + z 12 z¯ 22 ] 2 and 1 [−¯z 11 z 11 − z¯ 12 z 12 + z¯ 21 z 21 + z¯ 22 z 22 ]. 2 The functions J1 , J2 , J3 are components of a momentum map J : P → su(2)∗ with respect to the basis in su(2)∗ dual to (ξ1 , ξ2, ξ3 ). The ideal J0 is generated by J1 , J2 , and J3 . Consider the following functions: J3 =

i K 1 = − [¯z 11 z 12 − z¯ 12 z 11 + z¯ 21 z 22 − z¯ 22 z 21 ], 2 1 K 2 = − [z 12 z¯ 11 + z¯ 12 z 11 + z 22 z¯ 21 + z¯ 22 z 21 ] 2 and K3 =

1 [−z 11 z¯ 11 + z¯ 12 z 12 − z 21 z¯ 21 + z¯ 22 z 22 ]. 2

6.9 Algebraic reduction

141

It is easy to check that these functions are SU (2)-invariant and that they satisfy the identity J12 + J22 + J32 = K 12 + K 22 + K 32 . Hence, they vanish on the zero level set of J . However, none of the functions K 1 , K 2 and K 3 is contained in the ideal J0 in C ∞ (P) generated by J1 , J2 , J3 . Hence, the Poisson algebra (C ∞ (P)/J0 )G of algebraic reduction at 0 ∈ su(2)∗ contains non-zero elements [K 1 ]0 , [K 2 ]0 and [K 3 ]0 , which correspond to functions that vanish identically on J −1 (0). The fact that in the case of a proper non-free action the algebraic reduction need not be equivalent to the singular reduction is somewhat disturbing. We shall see in the following chapter, using examples, that quantization of the two reductions leads to equivalent results. We need further examination of this phenomenon. We now return to the case in which the action of the group G on P is not proper. In this case, since singular reduction does not apply, we have no alternative to algebraic reduction at present. We give a simple example of algebraic reduction for an improper action; this example is a continuation of Example 6.8.1. Example 6.9.5 Consider the symplectic manifold with symmetry discussed in Example 6.8.1. We describe here the algebraic reduction of the action of G = R on P = R2 at 0 ∈ g∗ = R. The ideal J0 is given by J0 = {x 2 f (x, y) | f ∈ C ∞ (R2 )}. Hence, C ∞ (P)/J0 can be parametrized as C ∞ (P)/J0 = {h(y) + xk(y)0 | h, k ∈ C ∞ (R)}, where the functions on the right-hand side have been chosen as representatives of their J0 -equivalence classes. The action of R on C ∞ (P)/J0 associates to t ∈ R and h(y) + xk(y) ∈ C ∞ (P) the class [h(y − xt) + xk(y − xt)]0 = [h(y) − xth (y) + xk(y)]0 in C ∞ (P)/J0 . Hence, [h(y) + xk(y)]0 is R-invariant if h (y) = 0. This means that (C ∞ (P)/J0 )R = {c + xk(y) | c ∈ R and k ∈ C ∞ (R)}. Recall that the zero level of the momentum map is the fixed-point set J −1 (0) = {(0, y) | y ∈ R}.

142

Symplectic reduction

Given y1 = y2 ∈ R, there exists a function k ∈ C ∞ (R) separating y1 from y2 . Hence, the elements of (C ∞ (P)/J0 )R separate points in J −1 (0). For each f 1 , f 2 ∈ C ∞ (R2 ), we have { f1 , f 2 } = −X f1 f 2 =

∂ f1 ∂ f2 ∂ f1 ∂ f2 − . ∂y ∂x ∂x ∂y

In particular, {c1 + xk1 (y), c2 + xk2 (y)} = xk1 (y)k2 (y) − k1 (y)xk2 (y). Hence, the Poisson bracket in (C ∞ (P)/J0 )R is {c1 + xk1 (y), c2 + xk2 (y)} = x(k1 (y)k2 (y) − k1 (y)k2 (y)). Thus, algebraic reduction at 0 gives rise to a non-trivial Poisson algebra.

As in the case of singular reduction, we have a shifting theorem for algebraic reduction, proved by Arms (1996), which we state without proof. Theorem 6.9.6 If G and G μ are connected and Oμ is an embedded submanifold of g∗ , then the Poisson algebra (C ∞ (P)/Jμ )G μ is isomorphic to the G ˜ Poisson algebra (C ∞ ( P)/J 0) . Example 6.9.7 Let P = R4 # (x, y), where x = (x1 , x 2 ) and y = (y1 , y2 ) are in R2 , and let ω = d x1 ∧ dy1 + d x 2 ∧ dy2 . The corresponding Poisson bracket is 2 ∂ f1 ∂ f2 ∂ f1 ∂ f2 { f1 , f2} = (6.36) − ∂ yi ∂ xi ∂ xi ∂ yi i=1

for all f 1 , f 2 ∈ C ∞ (R4 ). We consider an action of R on P given by : R × P → P : (t, (x, y)) → t (x, y) = (x, y + t x). This action is Hamiltonian, with the momentum map given by the kinetic energy J (x, y) = 12 x 2 = 12 ((x1 )2 + (x 2 )2 ). As in the case of one degree of freedom, the action is improper. The zero level of the momentum map P0 ≡ J −1 (0) = {(0, 0, y1 , y2 ) | (y1 , y2 ) ∈ R2 } ∼ = R2 consists of fixed points of the action. Its complement P\P0 is G-invariant, and the induced action of R on P\P0 is free and proper. Hence, every R-invariant function f ∈ C ∞ (P) restricted to P\P0 can be expressed as a smooth function of the algebraic invariants x1 , x2 and j = x 1 y2 − x 2 y1 .

6.9 Algebraic reduction

143

In the remainder of this section, we split the discussion of Example 6.9.7 into a sequence of lemmas, propositions etc., which are numbered from 6.9.7/1 to 6.9.7/7. Lemma 6.9.7/1 For each f ∈ C ∞ (P)R , − j x2 j x1 , f (x1 , x2 , y1 , y2 ) = f x1 , x2 , (x1 )2 + (x 2 )2 (x1 )2 + (x2 )2 whenever J = 12 (x1 )2 + (x 2 )2 = 0.

(6.37)

Proof For (x1 , x2 ) = (0, 0), the action is free and proper. Moreover, f is constant along the line x1 y2 − x2 y1 = j. The line x 1 y2 − x2 y1 = j has a unique point − j x2 j x1 , (y1 , y2 ) = (x1 )2 + (x2 )2 (x1 )2 + (x 2 )2 that is closest to the origin. Substituting this expression for (y1 , y2 ) into f (x 1 , x2 , y1 , y2 ), we obtain equation (6.37). Proposition 6.9.7/2 For each f ∈ C ∞ (P)R , the restriction of f to P0 is a constant function. Proof If f ∈ C ∞ (P)R , then f ((x1 , x2 , y1 +t x 1 , y2 +t x2 ) = f (x 1 , x2 , y1 , y2 ) for all t ∈ R. Differentiating with respect to t and setting x 2 = 0, we obtain x1

∂f (x1 , 0, y1 , y2 ) = 0 ∂ y1

for all x1 = 0 and (y1 , y2 ) ∈ R2 . By continuity, ∂f (0, 0, y1 , y2 ) = 0 ∂ y1 for all (y1 , y2 ) ∈ R2 . Similarly, ∂f (0, 0, y1 , y2 ) = 0 ∂ y2 for all (y1 , y2 ) ∈ R2 . Hence, the restriction of f to P0 is constant. It follows from Proposition 6.9.7/2 that the R-invariant functions on P do not separate points in P0 . Hence, singular reduction does not apply. The ideal J0 is given by 1 2 2 ˜ ∞ ˜ J0 = (x 1 ) + (x2 ) f (x, y) | f ∈ C (P) . 2

144

Symplectic reduction

In order to describe the equivalence class [ f ]0 ∈ C ∞ (P)/J0 of a function f ∈ C ∞ (P), we introduce the variables z = x1 + i x2 and z¯ = x1 − i x 2 , and the corresponding derivations ∂ ∂ 1 ∂ −i = ∂z 2 ∂ x1 ∂ x2

and

∂ 1 = ∂ z¯ 2

∂ ∂ +i ∂ x1 ∂ x2

.

We have z z¯ = (x 1 )2 + (x2 )2

and

z

∂ ∂ ∂ ∂ + x2 . + z¯ = x1 ∂z ∂ z¯ ∂ x1 ∂ x2

Therefore, the Taylor series for f with respect to x = (x1 , x 2 ) can be written as follows: ∞ 1 d n f (t x, y) |t=0 n! dt n n=0 ∞ 1 ∂ ∂ n = + x2 f (x, y) |x=0 x1 n! ∂ x1 ∂ x2 n=0 ∞ 1 ∂ n ∂ = f (x, y) |x=0 + z¯ z n! ∂z ∂ z¯ n=0 ∞ n 1 n! ∂ n−k ∂ k z¯ = f (x, y) |x=0 z n! k!(n − k)! ∂z ∂ z¯

f (x, y) = f (t x, y) |t=1 ∼

n=0

k=0

∞ n z k z¯ n−k = f k(n−k) (y), k!(n − k)!

(6.38)

n=0 k=0

where f mn (y) =

∂m ∂n f (x, y) |x=0 . ∂z m ∂ z¯ n

(6.39)

Proposition 6.9.7/3 A function f ∈ C ∞ (R4 ) is divisible by z z¯ = (x1 )2 + (x2 )2 if and only if the coefficients f n0 (y) and f 0n (y) vanish for every n = 0, 1, 2, . . . , and y = (y1 , y2 ) ∈ R2 . Proof If f ∈ C ∞ (R4 ) is divisible by (x 1 )2 + (x2 )2 , then f = ((x1 )2 + (x2 )2 )h for some h ∈ C ∞ (R4 ), and the Taylor series for f is the product of (x 1 )2 + (x2 )2 = z z¯ and the Taylor series for h. Hence, f n0 (y) and f 0n (y) vanish identically. Conversely, suppose that f n0 (y) and f 0n (y) vanish for every n = 0, 1, 2, . . . , and that y = (y1 , y2 ) ∈ R2 . Hence, the Taylor series for f is

6.9 Algebraic reduction

145

divisible by z z¯ = (x 1 )2 + (x2 )2 . By the Borel Extension Lemma, there exists a function h ∈ C ∞ (R4 ) with its Taylor series in x given by ∞ n−1 k−1 n−1−k z z¯ f k,n−k (y). k!(n − k)! n=1 k=1

Hence, f and ((x1 )2 + (x2 )2 )h have the same Taylor series, which implies that f − ((x1 )2 + (x2 )2 )h vanishes at x = 0 together with all partial derivatives with respect to x1 and x2 . This implies that f − ((x1 )2 + (x 2 )2 )h is divisible by (x1 )2 + (x2 )2 . Hence, f is divisible by (x 1 )2 + (x2 )2 , as required. It follows from Proposition 6.9.7/3 that for every f ∈ C ∞ (R4 ), the representative of f in C ∞ (P)/J0 is the formal power series [ f ]0 = f 0,0 (y) +

∞ 1 f n,0 (y)z n + f 0,n (y)¯z n , n!

(6.40)

n=1

where the coefficients fmn are given by equation (6.39). Since f is real-valued, it follows that f 00 (y) = f (0, y) is real and f n0 (y) = f 0n (y). By the Borel Extension Lemma, for every formal power series a(y) +

∞

(a¯ n (y)z n + an (y)¯z n ),

(6.41)

n=1

where a(y) and an (y) are smooth functions of y, and a0 (y) is real, there exists a function f ∈ C ∞ (R)4 such that a = f 00 and a¯ n = f n0 /n!. The behaviour of Taylor series under addition and multiplication of functions implies that the map

f → [ f ]0 = f 00 (y) +

∞ 1 f n0 (y)z n + f 0n (y)¯z n n! n=1

is a homomorphism of associative algebras. Proposition 6.9.7/4 [ f ]0 ∈ (C ∞ (P)/J0 )R if and only if ∂ ∂ z + z¯ f (x, y) ∂w ∂ w¯ is in J0 , where ω = y1 + i y2

and

ω¯ = y1 − i y2 .

146

Symplectic reduction

Proof Since R is connected, it follows that the class [ f ]0 is G-invariant if there exists a function λ such that X J f = λJ . But, ∂f ∂f d (x, y) + x2 (x, y) f (x, y + t x)|t=0 = x1 dt ∂ y1 ∂ y2 i 1 ∂f ∂f (x, y) − (z − z¯ ) (x, y) = (z + z¯ ) 2 ∂ y1 2 ∂y 2 ∂ ∂ ∂ ∂ z z¯ −i +i = f (x, y) + f (x, y) 2 ∂ y1 ∂ y2 2 ∂ y1 ∂ y2 ∂ ∂ + z¯ f (x, y). = z ∂w ∂ w¯

X J f (x, y) =

Hence, [ f ]0 ∈ (C ∞ (P)/J0 )R if and only if ∂ ∂ z + z¯ f (x, y) ∈ J0 . ∂w ∂ w¯ Proposition 6.9.7/5 An element [ f ]0 of C ∞ (P)/J0 is R-invariant if and only if the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 . Proof

For f ∈ C ∞ (P), let

∂ ∂ h(x, y) = z + z¯ ∂w ∂ w¯ Equation (6.39) implies that

∂ ∂ + z¯ h 00 (y) = h(0, y) = z ∂w ∂ w¯ and ∂n ∂n h 0n (y) = n h(x, y) |x=0 = n ∂ z¯ ∂ z¯ =

f (x, y). f (x, y)|x=0 = 0,

∂ ∂ + z¯ z ∂w ∂ w¯

f (x, y) |x=0

∂ ∂ n−1 ∂ f 0,n−1 (y) f (x, y)|x=0 = n−1 ∂ w¯ ∂ z¯ ∂ w¯

for n ≥ 1. By Proposition 6.9.7/3, h ∈ J0 if and only if the coefficients h n0 (y) and h 0n (y) vanish identically in y. If [ f ]0 is R-invariant, then h ∈ J0 by Proposition 6.9.7/4, and h 0n (y) =

∂ f 0,n−1 (y) = 0 ∂ w¯

for all y ∈ R2 and n = 1, 2, . . . . Hence, all the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 .

6.9 Algebraic reduction

147

Suppose that the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 . Then h 0n (y) = 0 for n = 0, 1, . . . and all y. Since h is a real-valued function, it follows from equation (6.39) that h n0 (y) = h 0n (y) = 0. Hence, h ∈ J0 , which implies that f mod J0 is G-invariant. Corollary 6.9.7/6 (C ∞ (P)/J0 )R consists of formal power series a+

∞

(a¯ n (w)z n + an (w)¯z n ),

n=1

where a ∈ R and, for every n ∈ N , an (w) is an entire analytic function of w. Next we want to describe the Poisson bracket in (C ∞ (P)/J0 )G . A straightforward but tedious computation from the definition yields 2 ∂ f ∂h ∂ f ∂h { f, h} ≡ (6.42) − ∂ xi ∂ yi ∂ yi ∂ xi i=1 ∂ f ∂h ∂ f ∂h ∂ f ∂h ∂ f ∂h + − − . =2 ∂z ∂ w¯ ∂ z¯ ∂w ∂ w¯ ∂z ∂w ∂ z¯ A calculation using equation (6.39) gives m m! ∂ ∂ { f, h}m,0 (y) = 2 h m−k,0 + f k1 h m−k,0 f k+1,0 k!(m − k)! ∂ w¯ ∂w k=0 m m! ∂ ∂ −2 f m−k,0 + h k1 f m−k,0 . h k+1,0 k!(m − k)! ∂ w¯ ∂w k=0

(6.43) Similarly, { f, h}0,m (y) = 2

m

m! k!(m − k)!

k=0 m

−2

k=0

f 0,k+1

∂ ∂ h 0,m−k + f 1,k h 0,m−k ∂w ∂ w¯

m! ∂ ∂ f 0,m−k + h 1,k f 0,m−k . h 0,k+1 k!(m − k)! ∂w ∂ w¯ (6.44)

Therefore, [{ f, h}]0 = { f, h}0,0 +

∞ 1 { f, h}n,0 z n + { f, h}0,n z¯ n , n! n=1

where { f, h}0,0 = { f, h}(0, y) and the coefficients of z n and z¯ n are given by equations (6.43) and (6.44), respectively.

148

Symplectic reduction

For G-invariant [ f ]0 and [h]0 , Proposition 6.9.7/5 implies ∂ fl,0 = 0, ∂w

∂ f 0,l = 0, ∂ w¯

∂ h 0,l = 0 ∂ w¯

and

∂ h l,0 = 0, ∂w

(6.45)

for l = 0, 1, . . . , and f 0,0 and h 0,0 are constant. Setting m = 0 in equation (6.43), we obtain the result that { f, h}0,0 = 0. Therefore, { f, h} mod J0 n ∞

(6.46)

∂ ∂ h n−k,0 − h k+1,0 f n−k,0 z n ∂ w¯ ∂w n=1 k=0 n ∞ 1 ∂ ∂ +2 h 0,n−k − h 0,k+1 f 0,n−k z¯ n . f 0,k+1 k!(n − k)! ∂w ∂ w¯

=2

1 k!(n − k)!

f k+1,0

n=1 k=0

It follows from equation (6.45) that ∂ ∂ [ f ]0 [h]0 ∂z ∂ w¯ ∞ 1 ∂ n n = f 0,0 + f n,0 (y)z + f 0,n (y)¯z ∂z n! n=1 ∞ ∂ 1 n n h 0,0 + h n,0 (y)z + h 0,n (y)¯z × ∂ w¯ n! n=1 ∞ ∞ 1 ∂ 1 n−1 n f n,0 (y)z h n,0 (y)z = (n − 1)! n! ∂ w¯ n=1 n=1 n ∞ 1 ∂ n z = f k+1,0 (y) h n−k,0 (y) . k!(n − k)! ∂ w¯ n=1

k=0

Similarly, n ∞ ∂[ f ]0 ∂[h]0 1 ∂h n−k,0 (y) n = f 0,k+1 (y) , z¯ ∂ z¯ ∂w k!(n − k)! ∂w n=1 k=0 n ∞ ∂[h]0 ∂[ f ]0 1 (y) ∂ f n−k,0 zn = h k+1,0 (y) , ∂z ∂ w¯ k!(n − k)! ∂ w¯ n=1 k=0 n ∞ 1 ∂ f n−k,0 (y) ∂[h]0 ∂[ f ]0 n = h 0,k+1 (y) . z¯ ∂ z¯ ∂w k!(n − k)! ∂w n=1

k=0

Taking equation (6.46) into account, we arrive at the following result.

6.9 Algebraic reduction

149

Proposition 6.9.7/7 The Poisson bracket of f mod J0 and h mod J0 in (C ∞ (P)/J0 )G is given by {[ f ]0 , [h]0 }J0 ∂[h]0 ∂[ f ]0 ∂[ f ]0 ∂[h]0 ∂[h]0 ∂[ f ]0 ∂[ f ]0 ∂[h]0 + + −2 . =2 ∂z ∂ w¯ ∂ z¯ ∂w ∂z ∂ w¯ ∂ z¯ ∂w This completes our discussion of Example 6.9.7.

7 Commutation of quantization and reduction

The theory of geometric quantization forms a bridge between quantum mechanics and the representation theory of Lie groups. In representation theory, geometric quantization is a geometric technique for obtaining a unitary representation of a connected Lie group from its action on a symplectic manifold. In quantum mechanics, geometric quantization provides a geometric method of quantization of a classical system. This dual role of geometric quantization enables us to use representation theory to test hypotheses in quantum mechanics and vice versa. Geometric quantization of a Hamiltonian action of a connected Lie group G on a connected symplectic manifold (P, ω) gives rise to a representation of the Lie algebra g of G by skew-adjoint operators on a Hilbert space H. We assume that this representation of g can be integrated to a unitary representation U of G on H F . This representation can be decomposed into a direct integral of irreducible unitary representations of U λ of G. Hence, Hλ dμ(λ), (7.1) HF =

where is the space of equivalence classes of irreducible unitary representations of G, Hλ is the representation space of U λ and dμ(λ) is a spectral measure on ; see Kirillov (1962). If G is compact, then every irreducible unitary representation U λ of G is finite-dimensional and can be obtained by geometric quantization of the corresponding co-adjoint orbit Oλ (Kostant, 1966). The term ‘commutation of quantization and reduction’ was coined by Guillemin and Sternberg (1982), who investigated a quantization representation corresponding to an action of a compact connected Lie group G on a compact Kähler manifold P. In this case, the quantization representation of G is finite-dimensional, the set is discrete and the spectral decomposition (7.1) reads

7.1 Review of geometric quantization HF =

m λ Hλ ,

151

(7.2)

λ∈

where the multiplicities m λ are positive integers. Assuming that the action of G on J −1 (Oλ ) is free, Guillemin and Sternberg showed that the multiplicity m λ of Uλ in equation (7.2) is equal to the dimension of the representation space obtained by quantization of the Marsden–Winstein reduction of J −1 (Oλ ). Sjamaar (1995) extended the results of Guillemin and Sternberg by removing the condition of freeness of the action of G on J −1 (Oλ ). Taken together, the results of Guillemin and Sternberg and those of Sjamaar show that for an action of a compact connected Lie group G on a compact Kähler manifold, all the multiplicities m λ occurring in equation (7.2) are completely determined by the quantization of the reduced spaces J −1 (Oλ )/G, where Oλ is the co-adjoint orbit corresponding to Uλ . We use the term ‘commutation of quantization and reduction’ to denote a programme of determining the spectral measure dμ(λ) in terms of the quantization of the symplectic reduction of the inverse images of co-adjoint orbits under the momentum map J : P → g∗ . We begin with a brief review of geometric quantization. Next, we discuss some partial results on the commutation of quantization and singular reduction of the zero level of the momentum map, which generalize the results of Guillemin, Sternberg and Sjamaar to noncompact symplectic manifolds that need not be Kähler, and to non-compact Lie groups. We continue with a discussion of the commutation of quantization and reduction of non-zero co-adjoint orbits. Next, we proceed to a discussion of the commutation of quantization and algebraic reduction. We conclude with an example of commutation of quantization and algebraic reduction at J = 0 for an improper action of a connected Lie group.

7.1 Review of geometric quantization Geometric quantization of a symplectic manifold consists of three stages: prequantization, polarization and unitarization. In the prequantization stage, we construct a faithful unitary representation of the Poisson algebra of the symplectic manifold, called a prequantization representation. The prequantization representation is reducible and fails to satisfy Heisenberg’s Uncertainty Principle. In the next stage, polarization, we restrict the prequantization representation to the kernel of a subalgebra of the Poisson algebra. In this way, we obtain a smaller representation, but may lose the scalar product. The last stage, unitarization, deals with the recovery of a scalar product in the representation space.

152

Commutation of quantization and reduction

7.1.1 Prequantization Let λ : L → P be a complex line bundle with a connection and a connectioninvariant Hermitian form · | · . A connection on L is given by a covariant derivative operator ∇, which associates to each section σ of L and each vector field X on P a section ∇ X σ of L such that for each f ∈ C ∞ (P), ∇ X ( f σ ) = X ( f )σ + f ∇ X σ and ∇ f X σ = f ∇ X σ . For every section σ of L, f ∈

C ∞ (P),

(7.3)

and X 1 , X 2 ∈ X(P),

(∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )( f σ ) = f (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ. Hence, there is a 2-form α on P such that (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = 2πiα(X 1 , X 2 )σ .

(7.4)

The form α is the pull-back by the section σ of the curvature form of the connection ∇. A Hermitian form · | · on L is connection-invariant if, for every pair of sections σ1 , σ2 of L and every vector field X on P, X ( σ1 | σ2 ) = ∇ X σ1 | σ2 + σ1 | ∇ X σ2 . The quantization of a mechanical system is defined in terms of an additional free parameter . In quantum mechanics, is the value of Planck’s constant divided by 2π . However, in the quasi-classical approximation, we consider limits of various expressions as → 0. The line bundle L over P with a connection ∇ and a connection-invariant Hermitian form on L is a prequantization line bundle of (P, ω) if the following prequantization condition is satisfied: i (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = − ω(X 1 , X 2 )σ, (7.5) for every X 1 , X 2 ∈ X(P) and each section σ of L. In other words, we require −1 that 2π ω is the curvature of ∇. The prequantization condition (7.5) requires that the de Rham cohomology class [(2π )−1 ω] on P is in H 2 (Z). If σ is a non-zero section of L, the covariant derivative ∇ X σ is proportional to σ , and there is a complex-valued 1-form θ on P such that ∇ X σ = −i−1 θ |X σ for every vector field X on P. Hence, ∇σ = −i−1 θ ⊗ σ. The 1-form θ is called the pull-back by σ of the connection form of ∇. Equation (7.5) implies that dθ = ω. C ∞ (P)

(7.6)

generates a local one-parameter group exp t X f of A function f ∈ local symplectomorphisms of (P, ω). The Hamiltonian vector field X f on f

7.1 Review of geometric quantization

153

can be lifted to a vector field X f on L such that exp t X f is a lift of exp t X f that preserves the connection ∇. For each p ∈ P and non-zero z ∈ L p , the horizontal component of X f (z) is the horizontal lift of X f at z, and the vertical X f acts component of X f (z) is proportional to f ( p). If X f ( p) = 0, then exp t on the fibre L p by multiplication by e2πi f ( p) . We denote by S ∞ (L) the space of smooth sections of L. For each σ ∈ S ∞ (L), we set d (exp t X f ◦ σ ◦ exp(−t X f ))|t=0 . dt

(7.7)

P f σ = (−i∇ X f + f )σ ;

(7.8)

P f σ = i Direct computation yields

´ see Sniatycki (1980). We refer to P f as the prequantization operator corresponding to f. The map P : C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → P f σ is called the prequantization map. Proposition 7.1.1 For each f 1 , f2 ∈ C ∞ (P) and σ ∈ S ∞ (L) [ P f1 , P f2 ] = i P { f1 , f 2 } .

(7.9)

Proof [ P f 1 , P f2 ]σ = ((−i∇ X f1 + f 1 )(−i∇ X f2 + f 2 ) + −(−i∇ X f2 + f 2 )(−i∇ X f1 + f 1 ))σ = (−2 (∇ X f1 ∇ X f 2 − ∇ X f2 ∇ X f1 ) − i(X f1 ( f 2 ) − X f2 ( f 1 )))σ i = −2 ∇[X f1 ,X f2 ] − ω(X f1 , X f2 ) −i(X f1 ( f 2 ) − X f2 ( f 1 )) σ = (2 ∇ X { f1 , f2 } + iω(X f1 , X f2 ) − i(X f1 ( f 2 ) − X f2 ( f 1 )))σ = (i(−i∇ X { f1 , f2 } + { f 1 , f 2 } − { f 1 , f 2 }) + +iω(X f1 , X f2 ) − iX f1 ( f 2 ) + iX f2 ( f 1 ))σ = i P { f1 , f 2 } σ, because −{ f 1 , f 2 } + ω(X f1 , X f2 ) −X f1 ( f 2 ) +X f2 ( f 1 ) = 0 by equation (6.7).

Corollary 7.1.2 The map C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → − i P f σ is a representation of the Lie algebra structure of C ∞ (P) on S ∞ (L).

154

Commutation of quantization and reduction

Proof For f 1 , f 2 ∈ C ∞ (P) and σ ∈ S ∞ (L), equation (7.9) yields i2 i i2 i i − P f1 , − P f2 σ = 2 [ P f1 , P f2 ]σ = 2 (i) P { f1 , f 2 } σ = − P { f1 , f 2 } σ, as required. The space S0∞ (L) of compactly supported smooth sections of L has a Hermitian scalar product (7.10) (σ1 | σ2 ) = σ1 | σ2 ωn , P

C ∞ (P),

where n = dim P. For each f ∈ the prequantization operator P f is symmetric with respect to the scalar product (7.10). If the Hamiltonian vector field X f of f is complete, then P f is self-adjoint on the Hilbert space H obtained by completion of S0∞ (L) with respect to the norm given by equation (7.10). Equation (7.9) gives the usual commutation relations imposed in quantum mechanics. However, prequantization does not correspond to a quantum theory, because an interpretation of (σ | σ )( p) as the probability density of localizing the state σ at a point p ∈ P fails to satisfy Heisenberg’s Uncertainty Principle.1 Now consider a Hamiltonian action of a connected Lie group G on (P ω) with an equivariant momentum map J :P → g∗ . Proposition 6.1.6 in Chapter 4 states that the map ξ → Jξ is a homomorphism of g of G to the Poisson algebra C ∞ (P). Hence, the map ξ → (−i/) P Jξ is a linear representation of the Lie algebra g on the space S ∞ (L), which we call the prequantization representation of g. Since the Hamiltonian vector fields X Jξ are complete, each operator (−i/)PJξ is skew-adjoint on the Hilbert space H obtained by the completion of S0∞ (L) with respect to the norm given by equation (7.10). Recall that the action of g on L is given by vector fields X Jξ on L; see equation (7.7). We assume that this action integrates to an action of G on L that covers the action of G on P. We refer to this action as the prequantization action of G on P. This assumption implies that the prequantization representation of g described above integrates to a representation of G. That is, we have a linear representation 1 2

R : G × S ∞ (L) → S ∞ (L) : (g, σ ) → R g σ

(7.11)

d (Rexp tξ σ )|t=0 = (−i/) P Jξ σ dt

(7.12)

such that

1 Heisenberg’s Uncertainty Principle states that for every quantum state of a particle, the accu-

racies %x and %p with which one can determine the position x and momentum p satisfy the inequality %x %p ≥ .

7.1 Review of geometric quantization

155

for each ξ ∈ g. The linear representation R induces a unitary representation U : G × H → S ∞ (L) : (g, σ ) → U g σ

(7.13)

such that U g σ = R g σ for each σ ∈ S ∞ (L) ∩ H. We refer to R and U as prequantization representations of G. In general, the unitary prequantization representation U fails to be irreducible.

7.1.2 Polarization A complex distribution F ⊂ T C P = C⊗T P on a symplectic manifold (P, ω) is Lagrangian if for each p ∈ P, the restriction of the symplectic form ω to the subspace F p ⊂ T pC P vanishes identically and rankC F = 12 dim P. If F is a complex distribution on P, we denote its complex conjugate by F. Let D = F ∩ F ∩ T P and E = (F + F) ∩ T P. A polarization of (P, ω) is an involutive complex Lagrangian distribution F such that D and E are involutive distributions on P. Let C ∞ (P)0F be the space of smooth complex-valued functions on P that are constant along F; that is, C ∞ (P)0F = { f ∈ C ∞ (P) ⊗ C | u f = 0 for all u ∈ F}.

(7.14)

The polarization F is said to be strongly admissible if the spaces P/D and P/E of integral manifolds of D and P, respectively, are quotient manifolds of P and the natural projection P/D → P/E is a sumbersion. A strongly admissible polarization F is locally spanned by Hamiltonian vector fields of ¯ ≥ 0 for every functions on C ∞ (P)0F . A polarization F is positive if iω(w, w) w ∈ F. A positive polarization F is semi-definite if ω(w, w) ¯ = 0 for w ∈ F implies that w ∈ D C . Let C F∞ (P) denote the space of functions on P whose Hamiltonian vector fields preserve F. In other words, f ∈ C F∞ (P) if, for every h ∈ C ∞ (P)0F , the Poisson bracket { f, h} ∈ C ∞ (P)0F . If f 1 , f 2 ∈ C F∞ (P) and h ∈ C ∞ (P)0F , then the Jacobi identity implies that {{ f 1 , f 2 }, h} = −{ f 2 , { f 1 , h}} + { f 1 , { f 2 , h}} ∈ C ∞ (P)0F . Hence, the ring C F∞ (P) is a Poisson subalgebra of C ∞ (P). Let S F∞ (L) denote the space of smooth sections of L that are covariantly constant along F, namely, S F∞ (L) = {σ ∈ S ∞ (L) | ∇u σ = 0 for all u ∈ F}.

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We shall refer to S F∞ (L) as the space of polarized sections. For each h ∈ C ∞ (P)0F , f ∈ C F∞ (P) and σ ∈ S F∞ (L), we have ∇ X h ( Q f σ ) = 0. Thus, for every f ∈ C F∞ (P), the prequantization operator P f maps S F∞ (L) to itself. Definition 7.1.3 The quantization map Q relative to a polarization F is the restriction of the prequantization map P : C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → P f σ = (−i∇ X f + f )σ to the domain C F∞ (P) × S F∞ (L) ⊂ C ∞ (P) × S ∞ (L) and codomain S F∞ (L) ⊂ S ∞ (L). In other words, Q : C F∞ (P) × S F∞ (L) → S F∞ (L) : ( f, σ ) → Q f σ = (−i∇ X f + f )σ . (7.15) We assume that the action : G × P → P preserves the polarization F. Hence, for each ξ ∈ g, the momentum Jξ is in C F∞ (P). Restricting the prequantization representation to the Poisson algebra spanned by Jξ , for ξ ∈ g, we obtain a representation ξ → (i)−1 Q Jξ of g on S F∞ (L). If the action of G on P lifts to an action of G on L by connection-preserving automorphisms, then this representation of g integrates to a linear representation R : G × S F∞ (L) → S F∞ (L) : (g, σ ) → R g σ

(7.16)

of G on S F∞ (L). For each g ∈ G, f ∈ C ∞ (P)0F and σ ∈ S F∞ (L), R g ( f σ ) = (∗g−1 f )R g σ. We refer to R : G × S F∞ (L) → S F∞ (L) as the quantization representation of G. Note that the quantization representation of G introduced in equation (7.16) is the restriction of the prequantization representation R introduced in equation (7.11) to the domain G × S F∞ (L) and codomain S F∞ (L). Therefore, using the same symbol R for both representations should not lead to any contradiction.

7.1.3 Examples of unitarization S ∞ (L)

The space of smooth sections of L is endowed with a scalar product given by equation (7.10). In general, polarized sections in S F∞ (L) need not have a finite norm corresponding to this scalar product. Unitarization deals with the choice of the scalar product and a modification of the quantization representation R leading to a unitary representation. Here, we consider two special cases for which unitarization is straightforward.

7.1 Review of geometric quantization

157

Kähler polarization A Kähler polarization of (P, ω) is a strongly admissible polarization F such that F ⊕ F¯ = T C P and iω(w, w) ¯ > 0 for all non-zero w ∈ F. These assumptions imply that there is a complex structure J on P such that F is the space of antiholomorphic directions. Moreover, P is a Kähler manifold such that −ω is the Kähler form on P. For a Kähler polarization F on (P, ω), the prequantization line bundle L over P is holomorphic and the space S F∞ (L) of polarized sections coincides with the space of holomorphic sections. Moreover, the holomorphic sections of L which are normalizable with respect to the scalar product (7.10) form a Hilbert space H F . In other words, H F = H ∩ S F∞ (L). Hence, the linear representation R of G on S F∞ (L) gives rise to a unitary representation U of G on H. Thus, a Kähler polarization does not require additional unitarization. Proposition 7.1.4 A co-adjoint orbit (O, ) of a compact connected Lie group G admits a Kähler polarization. Proof Since G is compact, its Lie algebra g admits a positive definite AdG invariant metric k, which allows an identification of g with g∗ . Under this identification, co-adjoint orbits go to adjoint orbits. Hence, we can treat O as an adjoint orbit. For each ξ ∈ O, the tangent space Tξ O is the quotient of g by the Lie algebra hξ of the isotropy group Hξ = {g ∈ G | Adg ξ = ξ }. The map adξ : g → g : ζ → [ξ, ζ ] preserves hξ and induces a map Aξ of Tξ O onto itself. The map Aξ is skew-symmetric with respect to k. Hence, the eigenvalues of Aξ are purely imaginary, and half of them lie on the positive imaginary axis. Let Fξ ⊂ Tξ O ⊗ C be the space spanned by these positive eigenvalues. It can be shown that the set F = ∪ξ ∈O Fξ ⊂ T C O is a Kähler polarization of the symplectic manifold (O, ). Theorem 7.1.5 Let O be a quantizable co-adjoint orbit. The unitary representation U of G on the Hilbert space H F obtained by the quantization of (O, ) with respect to the Kähler polarization F described in Proposition 7.1.4 is irreducible. Moreover, the map O → U O is a bijection of the space of quantizable co-adjoint orbits of G onto the space of irreducible representations of G. Theorem 7.1.5 is the Borel–Weil Theorem in the formulation due to Kostant (1966).

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Commutation of quantization and reduction

Cotangent polarization We assume here that P = T ∗ Q is the cotangent bundle of a manifold Q, ω is the canonical symplectic form of T ∗ Q, and the polarization F is the complexification of ker T ϑ, where ϑ : T ∗ Q → Q is the cotangent bundle projection. The canonical 1-form θ of the cotangent bundle T ∗ Q is defined as follows. For each p ∈ T ∗ Q and every u ∈ T p T ∗ Q, θ (u) = p | T ϑ(u) . The canonical symplectic form of T ∗ Q is ω = dθ . Since ω is exact, the prequantization line bundle is trivial; that is, L = C × P. We denote by σ0 : P → L : p → (1, p) the trivializing section of L. We choose the covariant derivative operator ∇ such that ∇σ0 = −i−1 θ ⊗ σ0 . Moreover, we normalize the Hermitian form σ1 , σ2 appearing in equation (7.10) so that σ0 , σ0 = 1. The space C ∞ (P)0F consists of complex-valued functions on P = T ∗ Q that are constant along the fibres of the cotangent bundle projection. In other words, C ∞ (P)0F = {ϑ ∗ f | f ∈ (C ⊗ C ∞ (Q))}. The space S F∞ (L) of polarized sections of L is given by S F∞ (L) = {ϑ ∗ (ψ)σ0 | ψ ∈ C ⊗ C ∞ (Q)}. For each σ = ϑ ∗ (ψ)σ0 , we have ¯ ¯ ϑ ∗ (ψ)σ0 , ϑ ∗ (ψ)σ0 = ϑ ∗ (ψψ) = (ψψ) ◦ ϑ. Since the fibres of the cotangent bundle projection ϑ are not compact, it follows that ¯ σ, σ ωn = (ψψ) ◦ ϑ ωn = ∞ T∗Q

T∗Q

unless σ = 0. Let D = {ϑ ∗ (ψ)σ0 ∈ S F∞ (L) | ψ ∈ C ⊗ C0∞ (Q)}, where C ⊗ C0∞ (Q) is the space of compactly supported complex-valued smooth functions on Q. We endow D with a topology of uniform convergence with all derivatives on compact sets. We may introduce an alternative scalar product on D by setting ∗ ∗ ψ¯ 1 (q)ψ2 (q) dμ(q), (7.17) (σ1 | σ2 ) Q = (ϑ (ψ1 )σ0 | ϑ (ψ2 )σ0 ) Q = Q

7.1 Review of geometric quantization

159

where dμ(q) is a Lebesgue measure on Q. We denote the completion of D with respect to the scalar product (7.17) by H F , and the topological dual of D by D . Then, D ⊂ H F ⊂ D . We consider a Hamiltonian action : G ×T ∗ Q → T ∗ Q of a connected Lie group such that the momentum map J : T ∗ Q → g∗ is constant along the fibres of the cotangent bundle projection. Hence, there exists a map j : Q → g∗ such that J = ϑ ∗ j = j ◦ ϑ. For each ξ ∈ g, the differential d Jξ annihilates ker T ϑ, which implies that X Jξ has values in ker T ϑ. Therefore, the action of G on T ∗ Q preserves each fibre of the cotangent bundle projection. Since the fibres of the cotangent bundle projection are Lagrangian submanifolds of T ∗ Q, it follows that the action of G on T ∗ Q is abelian. We may assume that G is an abelian group. Examples of such an action may be found in Examples 6.9.5 and 6.9.7. In both of these examples, the action of G on P was improper on the zero level set of the momentum map. Half-densities and half-forms Linear frames of the tangent bundle T M of a manifold M form a principal G L(n, R)-fibre bundle over M, where n = dim M; see Kobayashi and Nomizu (1963). Similarly, the complexified tangent bundle T C M = T M ⊗ C is a principal G L(n, C)-fibre bundle F T C M over M. If (v1 , . . . , vn ) ∈ F T C M and A = (ai j ) ∈ G L(n, C), then n n vi ai1 , . . . , vi ain (v1 , . . . , vn )A = i=1

i=1

is also a linear frame on T C M. A density of weight s on a manifold M, of dimension n, is a function d on the bundle F T C M of linear frames of T C M such that, for every linear frame (v1 , . . . , vn ) ∈ F T C M and A ∈ G L(n, C), d((v1 , . . . , vn )A) = |det A|s d(v1 , . . . , vn ).

(7.18)

We can generalize the notion of densities to frame bundles of arbitrary vector bundles on a manifold. In particular, a half-density on a polarization F of a symplectic manifold (P, ω) is a density of weight 12 on the bundle of linear frames F F of F. In other words, a half-density ν on F associates to each frame (v1 , . . . , vn ) of F a number ν(v1 , . . . , vn ) such that, for every A ∈ G L(n, C), 1

ν(v1 , . . . , vn ) = |det A| 2 ν(v1 , . . . , vn ).

(7.19)

The half-densities on a polarization F of (P, ω) form a complex line bundle √ over P, which we denote by |∧n F|. This bundle has a flat partial connection

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Commutation of quantization and reduction

√ over F. Thus, covariant derivatives of sections of |∧n F| are well defined. There is a canonical pairing of sections of the bundle of half-densities that √ associates to sections ν1 and ν2 of |∧n F| a density ν1 | ν2 on F, defined by ν1 | ν2 (v1 , . . . , vn ) = (v1 (v1 , . . . , vn ))ν2 (v1 , . . . , vn ). Unitarization of the quantization representation in terms of half-densities is obtained by tensor multiplication of the prequantization line bundle L on √ (P, ω) by the bundle |∧n F|. The representation space of the quantization √ representation is modified from S F∞ (L) to S F∞ (L ⊗ |∧n F|). If F = D ⊗ C is a strongly admissible real polarization, and σ1 , σ2 ∈ S F∞ (L) and ν1 , ν2 are √ sections of |∧n F| that are covariantly constant along F, then a pairing of σ1 ⊗ ν1 and σ2 ⊗ ν2 gives a density σ1 ⊗ ν1 | σ2 ⊗ ν2 of weight 1 on P/D defined as follows. Recall that the assumption that F is strongly admissible implies that P/D is a quotient manifold of P. Let ϑ : P → P/D denote the canonical projection. Consider a linear frame (u 1 , . . . , u n ) in Tq (P/D). Let p ∈ ϑ −1 ( p), and let u˜ i be the lift of u i to T p P. Choose a linear frame (v1 , . . . , vn ) in D p so that (v1 , . . . , vn ; u˜ 1 , . . . , u˜ n ) is a symplectic frame in T p P. Therefore, ω(vi , u˜ j ) = δi j and ω(vi , v j ) = ω(u i , u j ) = 0, where δi j is the Kronecker symbol. We set σ1 ⊗ ν1 | σ2 ⊗ ν2 (u 1 , . . . , u n ) = σ1 | σ2 (q)ν1 (v1 , . . . , vn )ν2 (v1 , . . . , vn ). We can show that σ1 ⊗ ν1 | σ2 ⊗ ν2 is a well-defined density of weight √ 1 on P/D. Hence, we can define a scalar product on S F∞ (L ⊗ |∧n F|) by integration of σ1 ⊗ ν1 | σ2 ⊗ ν2 over P/D. If P = T ∗ Q and F is the cotangent polarization defined in the preceding section, then the scalar product (7.17) is equivalent to the scalar product defined here. A half-form μ on F associates to each frame (v1 , . . . , vn ) of F a number μ(v1 , . . . , vn ) such that, for every A ∈ G L(n, C), 1

ν(v1 , . . . , vn ) = (det A) 2 ν(v1 , . . . , vn ).

(7.20)

Note that the difference between half-densities and half-forms is the replacement of the square root of the absolute value of det A in equation (7.19) by the square root of the determinant of A in equation (7.20). In order to 1 make (det A) 2 well defined, we need to introduce a double covering of the ´ symplectic frame bundle of (P, ω). For details, see Sniatycki (1980). The use of half-forms is essential for obtaining the metaplectic representation corresponding to the action of the symplectic group Sp(n) on R2n .

7.2 Commutation of quantization and singular reduction at J = 0 161

7.2 Commutation of quantization and singular reduction at J = 0 In this section, we consider a quantization representation of a connected Lie group G corresponding to a proper Hamiltonian action of G on a symplectic manifold (P, ω). Let J : P → g∗ be the equivariant momentum map corresponding to this action. The quantization structure on (P, ω) considered here consists of a polarization F, and a prequantization line bundle L over P with a connection ∇ that satisfies the prequantization condition (7.5). We assume that the action of G on P preserves the polarization F and lifts to a connection-preserving action on L. Geometric quantization gives rise to a linear representation R of G on the space S F∞ (L) of smooth sections of L that are covariantly constant along F. A G-invariant scalar product on S F∞ (L) leads from the linear representation R to a unitary representation U of G on a Hilbert space H. The unitary representation U can be decomposed into a direct sum or integral of irreducible unitary representations of G. If an irreducible unitary representation U λ of G can be obtained by the geometric quantization of a co-adjoint orbit Oλ , we would like to describe the contribution of U λ to U in terms of a quantization of J −1 (Oλ )/G. If the action of G on P is not free and proper, then the orbit space J −1 (Oλ )/G is not a symplectic manifold, and we have first to define what we mean by quantization of J −1 (Oλ )/G. First, we shall concentrate on the definition of the quantization of J −1 (0)/G, and discuss the commutation of quantization and reduction at the zero level of the momentum map. For non-zero co-adjoint orbits, we shall use the shifting trick discussed in Section 6.6. Consider the diagram below:

F

→

L ↓ P

=⇒ F|J −1 (0)

L|J −1 (0) ↓ −1 → J (0) ⇓

F|J −1 (0) /G

→

(7.21) L|J −1 (0) /G , ↓ J −1 (0)/G

where L is the prequantization line bundle and F is a G-invariant polarization of (P, ω). The horizontal double arrow symbolizes the restriction to

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J −1 (0), and the vertical double arrow symbolizes the projection to the space of G-orbits in J −1 (0). Our aim in this chapter is to describe the geometric structure of the spaces and maps occurring in the diagram (7.21) in the case when the action of G on P is proper, without assuming that G and P are compact. The case when P and G are compact, the action of G on J −1 (0) is free, and the polarization F is Kähler was the object of the work of Guillemin and Sternberg (1982) referred to at the beginning of this chapter. These authors showed that all of the spaces in the diagram (7.21) are manifolds and all of the maps are complex line bundle projections. In particular, J −1 (0)/G is symplectic. Moreover, the restriction F|J −1 (0) of F to J −1 (0) projects to a Kähler polarization (F|J −1 (0) )/G of J −1 (0)/G. Thus, geometric quantization of J −1 (0)/G is a quantization of a symplectic manifold with respect to a Kähler polarization. Furthermore, the restriction to J −1 (0) gives rise to a linear isomorphism from the space S F∞ (L)G of G-invariant polarized sections of L → P onto the space S F∞ (L|J −1 (0) )G of G-invariant sections of L|J −1 (0) → J −1 (0) that are covariantly constant along F|J −1 (0) . Similarly, we might expect that the orbit map projection J −1 (0) → J −1 (0)/G will give rise to a linear isomorphism from the space S F∞ (L|J −1 (0) )G to the space S F∞ (L|J −1 (0) /G) of polarized sections of L|J −1 (0) /G → J −1 (0)/G. In the general case considered here, when the action of G on P is proper but not free and neither G nor P need be compact, we cannot hope to obtain results as strong as those of Guillemin and Sternberg. Nevertheless, we can use the tools developed in Part I to investigate the singularities of the various spaces and mappings occurring in the diagram (7.21). Moreover, we can try to determine relationships between the spaces S F∞ (L)G , S F∞ (L|J −1 (0) )G and S F∞ (L|J −1 (0) /G). We are now going to discuss the structure on J −1 (0) induced by the quantization data on (P, ω). First, we need a few additional results, which were not given in Chapter 6 because they are specific to the zero level of the momentum map. Recall that, for each compact subgroup H of G, we have introduced the sets PH = { p ∈ P | G p = H }, P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G}, where G p = {g ∈ G | gp = p} is the isotropy group of p ∈ P. Connected components of ρ(J −1 (0) ∩ P(H ) ), where ρ : P → R = P/G is the orbit map, give rise to a stratification of ρ(J −1 (0)) = J −1 (0)/G. Each stratum Q of the stratification of J −1 (0)/G is the orbit of a family of Poisson derivations of

7.2 Commutation of quantization and singular reduction at J = 0 163 C ∞ (R), and it is the projection to R of a connected component K of J −1 (0) ∩ PH for some subgroup H of G. In other words, Q = ρ(K ), where K is a submanifold of P. Moreover, Q is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K ,

(7.22)

where ρ K : K → Q is the restriction of the orbit map ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Let L be a connected component of PH . Then the pull-back ω L of ω to L by the inclusion map L → P is symplectic. Moreover, the group N L = {g ∈ G | gp ∈ L ∀ p ∈ L} is a closed subgroup of G containing H as a normal subgroup. Hence, G L = N L /H is a Lie group. For each g ∈ N L , we denote the class of g in G L by [g]. Since H acts trivially on L, the action of N L on L induces an action L of G L on L, given by L : G L × L → L : ([g], p) −→ [g] p = gp. The action L is Hamiltonian provided L ∩ J −1 (0) = ∅. Proposition 7.2.1 (i) For each connected component L of PH , the action of the group G L on L preserves the symplectic form ω L on L given by the pullback to L of the symplectic form ω on P. If L ∩ J −1 (0) = ∅, then the action of G L on L is Hamiltonian with an Ad ∗ -equivariant momentum map JL : L → g∗L . (ii) J −1 (0) ∩ L = JL−1 (0). (iii) The action of G L on L induces on K the structure of a left principal fibre bundle with structure group G L , base space Q and principal bundle projection ρ K : K → Q. Proof (i) Let L be a connected component of PH . Recall that G L = N L /H , where N L = {g ∈ G | p ∈ L =⇒ gp ∈ L}.

(7.23)

We have shown that the action of G L on L, given by G L × L → L : ([g], p) → gp, is free and proper. Since the action of N L on P preserves ω and L, it follows that it preserves the pull-back ω L of ω to L. Let h and n L denote the Lie algebras of H and N L , respectively. The Lie algebra of G L is the quotient n L /h. For ξ ∈ n L , we denote the equivalence class of ξ in G L by [ξ ]. The action of the one-parameter group exp t[ξ ] of G L is given by the action of exp tξ on L. If we denote the restriction of X Jξ to L by X Jξ |L , we have

164

Commutation of quantization and reduction X Jξ |L ω L = −d Jξ |L .

Each point p ∈ L ⊆ PH is a fixed point of H . Hence, for every η ∈ h, the Hamiltonian vector field X Jη ( p) vanishes. This implies that d Jη ( p) = 0. Since L is a connected component of PH , it follows that Jη is constant on L. Moreover, L ∩ J −1 (0) = ∅ implies that Jη|L = 0. Hence, for every η ∈ h, X Jξ +η |L ω L = −d Jξ +η|L = −d Jξ |L − d Jη|L = −d Jξ |L . Therefore, we can define a momentum map JL : L → g∗L such that for each [ξ ] ∈ g L , JL[ξ ] = Jξ |L .

(7.24)

Since Jη|L = 0 for all η ∈ h, the map JL : L → g∗L is AdG∗ L -equivariant. (ii) Clearly, J −1 (0) ∩ L ⊆ JL−1 (0). Suppose that there is a ξ ∈ g such that Jξ |J −1 (0) is not identically zero. Then d Jξ does not vanish on JL−1 (0), and X Jξ L

does not vanish on JL−1 (0). We can average X Jξ over H , obtaining a vector field (X Jξ ) H ( p) = T g ◦ X Jξ (g −1 ( p)) dμ(g). H

The momentum map J : P →

g∗

is AdG∗ -equivariant. That is,

J (gp) = Adg∗ J ( p) for all g ∈ G. Therefore, for every ξ ∈ g, ∗g Jξ ( p) = Jξ (gp) = J (gp) | ξ = Adg∗ J ( p) | ξ = J ( p) | Adg −1 ξ . If η = H Adg −1 ξ is the average of ξ over G, then (X Jξ ) H is the Hamiltonian vector field of Jη . In other words, (X Jξ ) H = X Jη . But η is Ad H -invariant, which implies that η is in the Lie algebra of n L . Therefore, we have a decomposition X Jξ = X Jη +X J(ξ −η) . By part (iii) of Proposition 4.2.6, X Jη ( p) ∈ T p L for each p ∈ L. (iii) Every connected component of PH ∩ P0 has to be contained in a connected component of PH . Let K be a connected component of PH ∩ P0 contained in L. Then, K = L ∩ P0 . The action of N L on P preserves P0 , because N L ⊆ G and P0 = J −1 (0) is G-invariant. Hence, K = L ∩ P0 is N L -invariant, which implies that the action of G L = N L /H on L preserves K . Therefore, G L acts on K . Since the action of G L on L is free and proper, it follows that the action of G L on K is free and proper. Therefore, K is a

7.2 Commutation of quantization and singular reduction at J = 0 165

left principal fibre bundle with structure group G L . Its base space is the space K /G L of G L -orbits in K . By Proposition 4.2.6, for each p ∈ K , Gp ∩ L = G L p. Hence, K /G L = ρ(K ) = Q, and the principal bundle projection is ρ K :K → Q, obtained by restricting ρ : P → R to the domain K and codomain Q. We begin with a discussion of the structure induced on J −1 (0)/G by the G-invariant polarization F of (P, ω). The intersection F ∩ T C (J −1 (0)) is a linear subset of the complexified tangent bundle space of J −1 (0). Hence, it is a complex distribution on J −1 (0) in the sense of Definition 3.4.6. Moreover, F ∩ T C (J −1 (0)) is invariant under the action of G on J −1 (0). Recall that, in general, only the regular component of a differential space has its tangent bundle spanned by global derivations; see Proposition 3.3.15. Therefore, F ∩ T C (J −1 (0)) need not be spanned by global derivations in neighbourhoods of singular points of J −1 (0). In other words, F ∩T C (J −1 (0)) need not be smooth. Nevertheless, we shall refer to F0 = F ∩ T C (J −1 (0))

(7.25)

as the polarization of J −1 (0). The orbit type stratification M of P enables us to discuss F stratum by stratum. For each M ∈ M, consider the restriction F|M of F to points in M. Recall that M is a connected component of M(H ) for a compact subgroup H of G. Let L be a connected component of PH contained in M. Then M = G L is the union of G-orbits through L. The G-invariance of F ensures that F|M is uniquely determined by the restriction F|L of F to L. Recall that, for every p ∈ L, we have a decomposition of the tangent space to P at p, given by T p P = T p L ⊕ T p⊥ L , where T p L = T p PH , because L is an open subset of PH , and T p⊥ L consists of vectors v ∈ T p P such that the H -average of v vanishes; see Proposition 4.2.6. Proposition 7.2.2 The space T p⊥ L is the symplectic complement of T p L. In other words, T p⊥ L = {v ∈ T p P | ω(u, v) = 0 for all u ∈ T p L}. Proof

Since ω is G-invariant, for every u, v ∈ T p P and g ∈ H , we have ω(T g (u), T g (v)) = ω(u, v).

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Commutation of quantization and reduction

If u ∈ T p L, then u is H -invariant, and T g (u), T g (v) dμ(g) = ω u, T g (v) dμ(g) ω(u, v) = ω H

H

= ω(u, v H ), where v H = H T g (v) dμ(g) is the H -average of v. Since v H ∈ T p L, we have ω(u, v H ) = ω L (u, v H ). Therefore, ω(u, v) = 0 for all u ∈ T p L if and only if ω L (u, v H ) = 0 for all u ∈ T p L. Since ω L is symplectic, it follows that ω(u, v) = 0 for all u ∈ T p L if and only if v H = 0. Thus, ω(u, v) = 0 for all u ∈ T p L if and only if v ∈ T p⊥ L. The restriction ω|L of ω to points in L decomposes as the sum ω|L = ω L + ω⊥ L, where ω L is the pull-back of ω by the inclusion map L → P, and ω⊥ L = ω|L − ω L is the restriction of ω to vectors in T ⊥ L. Moreover, (T ⊥ L , ω⊥ ) L is a symplectic vector bundle over L. Let F|L be the restriction of F to points in L. We introduce the notation FL = F|L ∩ T C L and FL⊥ = F ∩ T ⊥C L . Proposition 7.2.3 If F is a G-invariant polarization of (P, ω), then: (i) F|L = FL ⊕ FL⊥ ; (ii) FL is a G L -invariant polarization of (L , ω L ); (iii) FL⊥ is a G L -invariant complex Lagrangian subbundle of the complexification of (T ⊥ L , ω⊥ L ). Proof (i) Since F is G-invariant, it follows that F|L is H -invariant and G L -invariant. For p ∈ L, let (w1 , . . . , wn ) be a basis in F|L . For each i = 1, . . . , n, we denote by wi H the H -average of wi . The H -invariance of ⊥ = FL implies that the vectors w1H , . . . , wn H are in FL and the vectors w1H ⊥ ⊥ w1 − w1H , . . . , wn H = wn − wn H are in FL . Since the basis (w1 , . . . , wn ) in FL is arbitrary, it follows that FL and FL⊥ span F|L . But FL ∩ FL⊥ = 0, and therefore F|L = FL ⊕ FL⊥ . (ii) For every w1 , w2 ∈ FL ⊆ F, ω L (w1 , w2 ) = ω(w1 , w2 ) = 0, which implies that FL is an isotropic complex distribution on (L , ω L ). Moreover, if w ∈ T pC L is such that ω L (w, u) = 0 for all u ∈ FL attached at p, then ω(w, u) = 0 for all u ∈ F p . Since F is a maximal isotropic complex distribution on P, it follows that w ∈ F. Hence, w ∈ FL , which ensures that FL is a maximal isotropic complex distribution on (PL , ω L ). In other words, FL is Lagrangian.

7.2 Commutation of quantization and singular reduction at J = 0 167

By definition, F is an involutive distribution. Since L is a manifold, it follows that FL = F ∩ T C L is involutive. Thus, FL is an involutive complex Lagrangian distribution on (L , ω L ), which means that FL is a polarization of (L , ω L ). Moreover, FL is G L -invariant because L and F|L are G L -invariant. (iii) We can show, as above, that FL⊥ is a maximal isotropic subbundle of ⊥ ⊥ (T ⊥ L , ω⊥ L ). Moreover, FL is G L -invariant because T L is G L -invariant. Let K be a connected component of J −1 (0) ∩ L = JL−1 (0), where JL : L → g L is the momentum map for a free and proper action of G L on L; see Proposition 7.2.1. Since the action of G L on L is free and proper, K is a submanifold of L, and the orbit space K /G L is a quotient manifold of K . We can identify K /G L with the projection ρ(K ) of K by the orbit map ρ : P → R. We know that Q = ρ(K ) = K /G L is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K ,

(7.26)

where ρ K : K → Q is the restriction of ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Since K ⊆ L ⊆ P, we can say that ω K is the pull-back of ω L by the inclusion map K → L. Definition 7.2.4 A G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0) if, for every compact subgroup H of G and each connected component K of J −1 (0) ∩ PH , the intersection FK = F ∩ T C K has constant rank. Let T C ρ K : T C K → T C Q be the complexification of the derived map Tρ K : T K → T Q of ρ K : K → Q. Proposition 7.2.5 Suppose a G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0). Then, for every compact subgroup H of G and each connected component K of J −1 (0) ∩ PH , Fˆ Q = T C ρ K (FK )

(7.27)

is a polarization of (Q, ω Q ), where Q = ρ K (K ) and ρ K∗ ω Q = ω K . Proof Since FK has constant rank, it follows that FK is an involutive complex distribution on K . If the projection T C ρ K (FK ) has constant rank on Q, then

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Commutation of quantization and reduction

T C ρ K (FK ) is a complex distribution on Q. However, T C ρ K (FK ) has constant rank if the rank of FK ∩ ker T C ρ K is constant. Let FL|K denote the restriction of FL to points of K . Since FL|K ⊇ FK ⊇ (FK ∩ ker T C ρ K ), we obtain rankC FL|K = rankC (FL|K /FK ) + rankC FK = rankC (FL|K /FK ) + rankC FK /(FK ∩ ker T C ρ K ) + rankC (FK ∩ ker T C ρ K ) = rankC FL|K − rankC FK + rankC FK /(FK ∩ ker T C ρ K ) + rankC (FK ∩ ker T C ρ K ). Therefore, rankC (FK ∩ ker T C ρ K ) = rankC FK − rankC FK /(FK ∩ ker T C ρ K ). Moreover, FL is a maximal isotropic distribution on L. On the other hand, equation (7.26) implies that T C K / ker T C ρ K is symplectic. Since FL is a maximal isotropic distribution on L, it follows that the projection FK /(FK ∩ ker T C ρ K ) is a maximal isotropic subbundle in T C K / ker T C ρ K . In other words, rank FK /(FK ∩ ker T C ρ K ) = dim Q. Therefore, rankC (FK ∩ ker T C ρ K ) = rankC FK − dim Q. This implies that rankC (FK ∩ ker T C ρ K ) is constant provided rankC FK is constant. Thus, F has a clean intersection with J −1 (0), and T C ρ K (FK ) is a maximal isotropic complex distribution on Q. The involutivity of FK implies that T C ρ K (FK ) is involutive. Hence, C T ρ K (FK ) is a polarization of (Q, ω Q ). We shall refer to Fˆ = T C ρ(F ∩ T C (J −1 (0)))

(7.28)

as the polarization of J −1 (0)/G. Let Fˆ|Q be the restriction of Fˆ to points in Q. Since FK = F ∩ T K and ρ K : K → Q is the restriction of ρ to the domain K and codomain Q, we have T C ρ K (FK ) = T C ρ(F ∩ T C (J −1 (0)))|Q ∩ T C Q = Fˆ|Q ∩ T C Q.

7.2 Commutation of quantization and singular reduction at J = 0 169 Proposition 7.2.5 states that the part of Fˆ|Q that is tangential to Q is a polarization Fˆ Q of (Q, ω Q ). It is of interest to interpret the quotient Fˆ|Q /FQ , which describes directions in Fˆ|Q that are not tangent to Q. Since Q is a stratum of the stratification of J −1 (0)/G, directions in Fˆ|Q that are not tangent to Q describe how the polarization Fˆ varies from stratum to stratum. Directions in Fˆ|Q that are not tangent to Q can be characterized by quotients FL|K /FK ⊥ / ker T C ρ, where F ⊥ is the restriction of F ⊥ to points of K , and FL|K |K L|K L and ker|K T C ρ is the restriction of ker T C ρ to points of K . The quotient T C ρ(FL|K )/ Fˆ Q corresponds to the directions in TQ R which are tangent to the stratum N of the orbit type stratification N of R = P/G that contains ⊥ ) corresponds to directions in T R that are not Q. The projection T C ρ(FL|K Q tangent to the stratum N . We now proceed to a discussion of the reduction of the prequantization line bundle λ : L → P of (P, ω) under the following simplifying assumption. Assumption 7.2.6 For every connected component L of PH such that L ∩ J −1 (0) = ∅, the action of G on the prequantization line bundle L induces a trivial action of H on the restriction L L of L to L. Consider next the restriction of the prequantization line bundle λ : L → P to P0 = J −1 (0). In order to simplify the notation, we shall write L0 for the restriction of L to J −1 (0). In other words, L0 = λ−1 (J −1 (0)).

(7.29)

The continuity of λ implies that L0 is a closed subset of L. Hence, smooth functions on L0 are restrictions to L0 of smooth functions on L. Let λ0 : L0 → J −1 (0) be the restriction of λ to the domain L0 and codomain J −1 (0). If f |J −1 (0) is the restriction of f ∈ C ∞ (P) to J −1 (0), then λ∗0 f |J −1 (0) = f |J −1 (0) ◦ λ0 = ( f ◦ λ)|L0 ∈ C ∞ (L0 ). Therefore, λ0 is a smooth map. We shall refer to L0 as a complex line bundle with base space J −1 (0) and projection map λ0 . A map σ0 : J −1 (0) → L0 is a section of L0 if λ0 ◦ σ0 is the identity on −1 J (0). By definition, a section σ0 of L0 is smooth if, for every f ∈ C ∞ (L), the pull-back σ0∗ f |L0 = f ◦ σ0 is in C ∞ (J −1 (0)). We denote the space of smooth sections of L0 by S(L0 ). In other words, S(L0 ) = {σ0 : P0 → L0 | λ0 ◦ σ0 = id P0 and σ0∗ f |L0 ∈ C ∞ (P0 ) ∀ f ∈ C ∞ (L).

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Commutation of quantization and reduction

Proposition 7.2.7 A section of L0 is smooth if and only if it is a restriction to P0 of a smooth section of L. Proof Let σ0 be a section of L0 . Suppose σ0 = σ|J −1 (0) for some section of L. Then, for every f ∈ C ∞ (L) in C ∞ (J −1 (0)), σ0∗ f |L0 = f ◦ σ0 = f ◦ σ|J −1 (0) = ( f ◦ σ )|J −1 (0) . Since f ◦ σ ∈ C ∞ (P), it follows that ( f ◦ σ )|P0 ∈ C ∞ (J −1 (0)). Therefore, σ0 is smooth. Conversely, suppose that σ0 is smooth. For each p ∈ J −1 (0), a local trivialization of L in an open neighbourhood U of p in P is given by a smooth map τ : λ−1 (U ) → U × C such that λ|λ−1 (U ) = π1 ◦ τ , where π1 : U × C → U is the projection on the first factor. The restriction of σ0 to J −1 (0) ∩ U gives a smooth function π2 ◦ τ ◦ σ0|U ∩P0 on J −1 (0) ∩ U with values in C. Therefore, there is a neighbourhood V of p0 in U and a complex-valued function ϕ : V → C such that π2 ◦ τ ◦ σ0|U ∩J −1 (0) = ϕ|V ∩J −1 (0) . The map σV : V → L : q → σV (q) = τ −1 (q, ϕ(q)) is a smooth section of the restriction of L to V . Moreover, if q ∈ J −1 (0) ∩ V , then σV (q) = τ −1 (q, π2 ◦ τ ◦ σ0|U ∩J −1 (0) ) = σ0|U ∩J −1 (0) (q). Hence, σV |V ∩J −1 (0) = σ0|V ∩J −1 (0) . This implies that, for every p ∈ J −1 (0), there is an extension σV of σ0 to a local section σV of L defined in a neighbourhood V of p in P. Using a partition-of-identity argument, we can extend σ0 to a smooth section σ of L. The prequantization action of G on L leaves L0 invariant. Hence, it induces an action of G on L0 such that every G-invariant section of L restricts to a G-invariant section of L0 . Proposition 7.2.8 Every smooth G-invariant section of L0 extends to a smooth G-invariant section of L. Proof Let σ0 be a smooth G-invariant section of L0 . By Proposition 7.2.7, there exists a smooth section σ of L that extends σ0 . For each p ∈ P, let S p be a slice through p for the action of G on P. We denote by σ p the restriction of σ to S p . Averaging σ p over G p , we obtain a G p -invariant section σ˜ p of L|S p such that for each q ∈ S p , σ˜ p (q) = gσ (g −1 q) dμ(g). Gp

7.2 Commutation of quantization and singular reduction at J = 0 171 Let h p ∈ C ∞ (S p ) be a compactly supported G p -invariant non-negative function such that h p = 1 in a neighbourhood U p of p in S p and f = 0 in the complement of an open neighbourhood V p of p that contains the closure of U p . The product gives a new section h p σ˜ p of L|S p , which coincides with σ p in U p and vanishes in the complement of V p . The set G S p is a G-invariant neighbourhood of p in P. We extend h p σ˜ p to a G-invariant section σˇ p of L|G S p as follows. For each q ∈ S p and g ∈ G, we set σˇ p (gq) = h p (q)g σ˜ p (q). If g1 , g2 ∈ G and q1 , q2 ∈ S p are such that g1 q1 −1 g2 g1 q1 , which implies that g2−1 g1 ∈ G p . Since h p and

(7.30) = g2 q2 , then q2 = σ˜ p are G p -invariant,

it follows that h p (q2 )g2 σ˜ p (q2 ) = h p (g2−1 g1 q1 )g2 σ˜ p (g2−1 g1 q1 ) = h p (q1 )g2 (g2−1 g1 σ˜ p (q1 )) = h p (q1 )g1 σ˜ p (q1 ). Hence, σˇ p is well defined by equation (7.30). The collection {GU p | p ∈ P} is a covering of P by G-invariant open sets. Using a G-invariant partition of unity subordinate to this covering, we obtain a G-invariant section σˇ of L. We want to show that σˇ |J −1 (0) = σ0 . Suppose that p ∈ J −1 (0). Then, σ p|S p ∩J −1 (0) = σ |S p ∩J −1 (0) = σ0|S p ∩J −1 (0) . The G-invariance of σ0 implies that σ˜ p|S p ∩J −1 (0) = σ0|S p ∩J −1 (0) . Since h p = 1 on U p ⊆ S p , we have h p σ˜ p|U p ∩J −1 (0) = σ0|U p ∩J −1 (0) , and σˇ p|GU p ∩J −1 (0) = σ0|GU p ∩J −1 (0) . Therefore, σˇ |GU p ∩J −1 (0) = σ0|GU p ∩J −1 (0) . The connection ∇ on L induces a connection ∇ 0 on L0 . Since J −1 (0) has singularities, we have to define what we mean by a connection on a singular space. We interpret ∇ 0 as the covariant differential of sections of L0 . In other words, for each σ0 ∈ S(L0 ), we have a linear map ∇ 0 σ0 : T (J −1 (0)) → L0 : u → ∇u0 σ0 such that ∇ 0 ( f 1 σ1 + f 2 σ2 ) = f 1 ∇ 0 σ1 + d f 1 ⊗ σ1 + f 2 ∇ 0 σ2 + d f 2 ⊗ σ2 for all f 1 , f 2 ∈ C ∞ (J −1 (0)) and σ1 , σ2 ∈ S(L0 ). With this interpretation, the connection ∇ 0 on L0 induced by the connection ∇ on L is given by ∇u0 σ0 = ∇u σ, where σ is an extension of σ0 to a section in S(L).

(7.31)

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Commutation of quantization and reduction

With this definition, the connection ∇ 0 can be described in terms of a Zariski form on J −1 (0); see Chapter 5. Hence, the curvature of ∇ 0 is not defined globally on J −1 (0), but only on the regular part J −1 (0)reg of J −1 (0), because T (J −1 (0)reg ) is locally spanned by global derivations; see Proposition 3.3.15. By assumption, the connection ∇ satisfies the prequantization condition (7.5). Therefore, for every pair of global derivations X 0 , Y0 of C ∞ (J −1 (0)) and each section σ0 of L0 , we have i 0 )σ0 = − ω0K (X 0 , Y0 )σ0 , (∇ X0 0 ∇Y00 − ∇Y00 ∇ X0 0 − ∇[X 0 ,Y0 ]

(7.32)

where ω0K is the Koszul form on Der C ∞ (P0 ), defined by evaluation of ω on vector fields X and Y on P that extend X 0 and Y0 , respectively. In other words, ω0K (X 0 , X 0 ) = ω(X, Y )|J −1 (0) . The ∇-invariant Hermitian form · | · on L restricts to a ∇ 0 -invariant Hermitian form · | · 0 on L0 . We may consider the line bundle λ0 : L0 → P0 with the connection ∇ 0 and the ∇ 0 -invariant Hermitian form · | · 0 as the prequantization structure on (P0 , ω0K ) induced by the prequantization of (P, ω). Since J −1 (0) is G-invariant, the prequantization action of G on L induces an action of G on L0 = L|J −1 (0) , which preserves ∇ 0 and · | · 0 . Let σ0 be a section of L0 . We say that σ0 is covariantly constant along F0 = F ∩ T C (J −1 (0)) if ∇w0 σ0 = 0 for every w ∈ F0 . We denote by S F0 (L0 ) the space of sections of L0 that are covariantly constant along F0 . In other words, S F0 (L0 ) = {σ0 ∈ S(L0 ) | ∇w0 σ0 = 0 for all w ∈ F0 = F ∩ T C (J −1 (0))}. For simplicity, we shall refer to sections in S F0 (L0 ) as polarized sections of L0 . Similarly, we denote by S F (L0 )G the space of G-invariant polarized sections of L0 . Let σ be a polarized section of L; that is, σ is a section of L that is covariantly constant along F. In other words, ∇ X σ = 0 for every vector field X on P with values in F. By the definition of the connection ∇ 0 on L0 , it follows that the restriction σ|J −1 (0) of σ to J −1 (0) is a section of L0 which is covariantly constant along F0 . Therefore, the restriction of sections of L to J −1 (0) gives rise to a linear map F : S F (L) → S F0 (L0 ) : σ → σ|J −1 (0) .

(7.33)

Moreover, if σ is G-invariant, then σ|J −1 (0) is G-invariant. Hence, the restriction of F to G-invariant sections gives a linear map FG : S F (L)G → S F0 (L0 )G : σ → σ|J −1 (0) .

(7.34)

7.2 Commutation of quantization and singular reduction at J = 0 173

Remark 7.2.9 The horizontal double arrow in the diagram (7.21) corresponds to a study of the following questions: (i) Under what conditions is FG : S F (L)G → S F0 (L0 )G one-to-one? (ii) Under what conditions is FG : S F (L)G → S F0 (L0 )G onto? Our next task is to describe the vertical double arrow in the diagram (7.21). We have already described the polarization Fˆ = T C ρ(F ∩ T C (J −1 (0))) of J −1 (0)/G. Now, we need to consider the projection to J −1 (0)/G of the complex line bundle L0 on J −1 (0). We begin with the space Lˆ = L0 /G

(7.35)

ˆ Since L0 is the of G-orbits in L0 , and denote the orbit map by κ : L0 → L. restriction of L to J −1 (0), which is a locally compact subcartesian space, it follows that L0 is a locally compact subcartesian space. Theorem 4.4.7 and Proposition 4.4.8 ensure that Lˆ is a locally compact differential space with the quotient topology. On the other hand, L0 is a subset of L, which is a manifold. Therefore, Lˆ is a subset of a stratified space L/G, which implies that Lˆ is a subcartesian space. Since the projection map λ0 : L0 → J −1 (0) is G-equivariant, there exists a map λˆ : Lˆ → J −1 (0)/G such that the following diagram commutes: L0

κ

λ0

J −1 (0)

/ Lˆ λˆ

ρ0

(7.36)

/ J −1 (0)/G.

Proposition 7.2.10 The map λˆ : Lˆ → J −1 (0)/G is smooth. Proof We need to show that for each f ∈ C ∞ (J −1 (0)/G), the pull-back λˆ ∗ f ˆ But ρ ∗ f ∈ C ∞ (J −1 (0))G , and λ∗ ρ ∗ f ∈ C ∞ (L0 )G . Moreover, is in C ∞ (L). 0 0 0 the commutativity of the diagram (7.36) yields κ ∗ λˆ ∗ f = λ∗0 ρ0∗ f ∈ C ∞ (L0 )G , ˆ which implies that λˆ ∗ f ∈ C ∞ (L). Given p ∈ J −1 (0), we denote the fibre of L0 over p by L0| p ; that is, L0| p =

λ−1 0 ( p).

Proposition 7.2.11 For each p ∈ P, the isotropy group G p of p acts on L0| p as a finite group of C-linear transformations.

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Proof Recall that the action of G on L is given by translations along the integral curves of lifts to L of Hamiltonian vector fields X Jξ , where ξ ∈ g. The infinitesimal action on L p of the Lie algebra g p of Gp is given by lifts Xˆ Jξ of Hamiltonian vector fields X Jξ such that X Jξ ( p) = 0. Hence, the horizontal component of Xˆ Jξ vanishes. The vertical component of X Jξ is proportional to Jξ ( p), which vanishes because p ∈ J −1 (0). Hence, the action of g p on L0| p is trivial. Therefore, G p acts on L0| p as a discrete group. To be more precise, let K p be the maximal subgroup of G p that acts on L0| p by the identity transformation. Then, K p is a normal subgroup of G p and we have an action G p /K p × L0| p → L0| p : ([g], z) → gz of the quotient group G p /K p on L0| p . Since the action of g p on L0| p is trivial, the Lie algebra k p of K p coincides with the Lie algebra g p of G p . Hence, G p /K p is a discrete group, and it is finite because G p is compact. Finally, the action of G p /K p on L0| p is C-linear because G acts on L by complex line bundle isomorphisms. Quotients of Cn by finite groups of C-linear transformations are called Vmanifolds, in the sense of Satake (1957). It follows from Proposition 7.2.11 that the fibres of λˆ : Lˆ → J −1 (0)/G have the structure of V-manifolds. However, Assumption 7.2.6 implies that for each p ∈ J −1 (0), the isotropy group G p of p acts trivially on the fibre L0| p of L0 . Hence, for each orbit Gp in J −1 (0), the fibre of Lˆ over Gp is a complex line C. Corollary 7.2.12 Assumption 7.2.6 ensures that λˆ : Lˆ → J −1 (0)/G is a complex line bundle with a singular base. Consider the restriction L L of the prequantization line bundle L to a symplectic manifold (L , ω L ), where L is a connected component of PH contained in J −1 (0). We denote by λ L : L L → L the restriction to L of the projection map λ : L → P, and by ∇ L the restriction to L L of the connection ∇ on L. It is easy to see that the complex line bundle L L with the connection ∇ L is a prequantization line bundle for (L , ω L ). By construction, prequantization gives an action of the Lie algebra g of G on L. We have assumed that this action integrates to an action of G on L that covers the action of G on P. Let H be a compact subgroup of G, and let L be a connected component of PH . The action on P of the subgroup N L of G given by equation (7.23) preserves L, by definition. Since the action of G on L is a lift of the action of G on P, it follows that the action of N L on L preserves L L . Therefore, we have an action of N L on L L that covers the action of N L on L.

7.2 Commutation of quantization and singular reduction at J = 0 175 Proposition 7.2.13 If L is a connected component of PH contained in J −1 (0), then the action of N L on L L induces a free and proper action of G L = N L /H on L L . Proof

The action of G L = N L /H on L L is given by G L × L L → L L : ([g], z) → gz

(7.37)

for every g ∈ N L . For every g ∈ N L such that g −1 g ∈ H , Assumption 7.2.6 implies that g −1 g z = z for all z ∈ L L . Hence, g z = gz, and [g]z is well defined by equation (7.37). For g1 , g2 ∈ N L and z ∈ L L , we have [g1 ]([g2 ]z) = g1 (g2 z) = (g1 g2 )z = ([g1 ][g2 ])z. Together with Assumption 7.2.6, this implies that equation (7.37) defines an action of G L on L L . Let λ L : L L → L be the restriction of the complex line bundle projection λ : L → P to the domain L and codomain L L . Since λ intertwines the action of G on L and the action of G on P, it follows that λ L intertwines the action of N L on L L and the action of N L of L. Therefore, for every g ∈ N L and z ∈ L L , we have λ L ([g]z) = λ L (gz) = gλ L (z) = [g]λ L (z). This implies that λ L intertwines the action of G L on L L and the action of G L on L. In other words, the action of G L on L L is a lift of the action of G L on L. Since the action of G L on L is proper, it follows that the action of G L on L L is proper. Moreover, suppose that [g]z = z for some [g] ∈ G L , and that z ∈ L L . Then, [g]λ L (z) = λ L ([g]z) = λ L (z). But the action of G L on L is free. Hence, [g] is the identity in G L , which implies that the action of G L on L L is free. Let K be a connected component of L ∩ J −1 (0) ⊆ PH , and let Q = ρ0 (K ) be the corresponding stratum of J −1 (0)/G. We denote the restriction of the prequantization line bundle L to K by L K , and the restriction of λ0 to the domain L K and codomain K by λ K : L K → K . In other words, −1 L K = L0|K = λ−1 0 (K ) = λ K (K ).

We denote the restriction of Lˆ to Q by Lˆ Q ; that is, Lˆ Q = λˆ −1 (Q). Similarly, we denote the restriction of λˆ : Lˆ → J −1 (0)/G by λˆ Q : Lˆ Q → Q. Since λˆ ◦ κ = ρ0 ◦ λ0 , it follows that λˆ ◦ κ(L K ) = ρ0 ◦ λ0 (L0|K ) = ρ0 (K ) = Q.

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Therefore, κ(L K ) ⊆ λˆ −1 (Q) = Lˆ Q , and the restriction of κ to L K has its range in Lˆ Q . We denote the restriction of κ : L0 → Lˆ to the domain L K and codomain Lˆ Q by κ K : L K → Lˆ Q . Proposition 7.2.14 If the action of G on the prequantization line bundle L induces a trivial action of H on the restriction L L of L to L, then: (i) Lˆ Q is a complex line bundle over Q with projection map λˆ Q : Lˆ Q → Q. (ii) Every smooth section σˆ of Lˆ Q has a unique lift to a G L -invariant section σ : K → L K such that σˆ ◦ ρ K = κ K ◦ σ . (iii) The G-invariant connection ∇ on L induces a connection ∇ Q on Lˆ Q . (iv) The Hermitian form · | · on L induces a ∇ Q -invariant Hermitian form · | · Q on L|Q . Proof (i) Consider a point q0 ∈ Q. By Proposition 7.2.1, the action of G L on L induces on K the structure of a left principal fibre bundle with structure group G L , base space Q and principal bundle projection ρ K : K → Q. Hence, there is a neighbourhood U of q in Q and a section τ : U → K of ρ K : K → Q. We denote by G L τ (U ) the collection of points in K that lie on orbits of G L intersecting τ (U ). In other words, G L τ (U ) = {gτ (q) ∈ K | g ∈ G L and q ∈ U }. G L U is an open G-invariant neighbourhood of τ (q0 ) in K . Moreover, the map G L τ (U ) → G L × U : gτ (q) → (g, q)

(7.38)

is a trivialization of G L U = ρ K−1 (U ). The restriction L K of L to K is a complex line bundle over K . Hence, there exist a neighbourhood V of τ (q0 ) in K and a trivializing section σ0 : V → L K of λ K : L K → K . Without loss of generality, we may assume that τ (U ) ⊆ V . We can construct a G L -invariant section σU : G L τ (U ) → L K that extends the restriction of σ0 to τ (U ). For each p = gτ (q) ∈ G L τ (U ), σU ( p) = σU (gτ (q)) = gσ0 (τ (q)). The smoothness of the trivialization map (7.38) implies that σU is smooth. By construction, Lˆ Q = L K /G L is the space of G L -orbits in L K . Hence, the G L -invariant section σU : G L τ (U ) → L K gives rise to a section σˆ U : U → Lˆ Q defined as follows. Points in Lˆ Q are G L -orbits in L K . Similarly, each q ∈ U is the G L -orbit in K through τ (q). In other words, q = G L τ (q). We set σˆ U : U → Lˆ Q : q = G L τ (q) → σˆ U (q) = G L σU (τ (q)).

(7.39)

7.2 Commutation of quantization and singular reduction at J = 0 177 Clearly, λˆ Q ◦ σˆ U is the identity on U . Moreover, σˆ U ◦ ρ K = κ K ◦ σU . Furthermore, if fˆ ∈ C ∞ (Lˆ K |U ) and κU is the restriction of κ K to L K |τ (U ) , then f = fˆ ◦ κU is in C ∞ (L K |τ (U ) ). Hence, for each q ∈ U , fˆ ◦ σˆ U (q) = fˆ ◦ σˆ (ρ K (τ (q))) = fˆ ◦ σˆ U ◦ ρ K (τ (q)) = fˆ(κ K (σU (τ (q))) = fˆ ◦ κU ◦ σU ◦ τ (q). Therefore, σˆ U∗ fˆ = fˆ ◦ σˆ U ∈ C ∞ (U ) because κU , σU and τ are smooth. This implies that the section σˆ U : U → Lˆ Q is smooth. Since G L acts on L K by complex line bundle automorphisms, it follows that the fibres of Lˆ Q are complex lines. By assumption, σ0 : V → L K is nowhere zero. Hence, σU : G L τ (U ) → L K is nowhere zero, which implies that σˆ U : U → Lˆ Q is nowhere zero. Thus, the bundle Lˆ Q is a locally trivial complex line bundle. (ii) Consider the following commutative diagram: LK

κK

/ Lˆ Q Lˆ Q

λK

K

ρK

/ Q,

where λ K : L K → K is the restriction of λ : L → P to the domain L K and codomain K . Here, the vertical arrows denote complex line bundle projections and the horizontal arrows denote principal G L -bundle projections. Let σˆ : Q → Lˆ Q be a section of Lˆ Q . We define its G-invariant lift σ : K → L K to a section of L K as follows. For each p ∈ K , ρ K ( p) = G L p is the orbit through p of the action of G L on L K . Moreover, σˆ (ρ K ( p)) is the orbit of the action of G L on L |K that covers G L ( p). Since the action of G L on L K is the lift of a free and proper action of G L on K , it follows that the fibre λ−1 K ( p) and the orbit σˆ (ρ K ( p)) in L K intersect at a single point. We denote by σ ( p) the unique point of intersection of the fibre λ−1 ˆ (ρ K ( p)) K ( p) and the orbit σ in L K . Clearly, λ K (σ ( p)) = p and κ K (σ ( p)) = σˆ (ρ K ( p)). Repeating this construction at every p ∈ K , we obtain a section σ : K → L K such that κ K ◦ σ = σˆ ◦ ρ K . It remains to show that σ is smooth. We have shown in part (i) of this proof that for each q0 ∈ Q, there exist a neighbourhood U of q0 in Q and trivializing sections σˆ U : U → Lˆ Q and σU : G L τ (U ) → L K such that σˆ U ◦ρ K = κ K ◦σU .

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Since σˆ is smooth, there exists a smooth complex-valued function fˇ on U such that the restriction σˆ |U of σˆ to U can be expressed in the form σˆ |U = fˇσˆ U . We can extend the function fˇ on U to a G L -invariant function on G L τ (U ) by setting f (gq) = fˇ(q) for every gq ∈ G L τ (U ). By construction, for every p ∈ K , σ ( p) is the unique point of intersection of the fibre λ−1 ˆ (ρ K ( p)) in L K . On the other hand, equation K ( p) and the orbit σ (7.39) implies that σU ( p) is the unique intersection of the orbit σˆ U (q) and the fibre λ−1 K ( p) for every p ∈ τ (U ). Since the action of G L on L K is linear, it follows that for every p ∈ τ (U ), σ ( p) = σ|U ( p) = f (ρ K ( p))σU (q). Moreover, the sections σ : K → L K and σU : G L τ (U ) → L K are G L invariant. Hence, σ restricted to G L τ (U ) coincides with a smooth section (ρ K∗ f )σU . This holds for a neighbourhood U of every q ∈ Q. Therefore, σ is smooth. (iii) Since K is a submanifold of P, the G-invariant connection ∇ on L restricts to a G L -invariant connection ∇ K on L K . We define a connection ∇ Q on L|Q as follows. Let σ be a G L -invariant section of L K . We denote by ρ K ∗ σ the section of L|Q such that ρ K ∗ σ ◦ ρ K = κ K ◦ σ . Conversely, if σˆ is a section of L|Q , then we denote by lift σˆ the G L -invariant section σ constructed above. This satisfies the condition σˆ ◦ ρ K = κ K ◦ lift σˆ . A G L -invariant vector field X on K pushes forward to a unique vector field Xˆ = ρ K ∗ X on Q. Conversely, every vector field Xˆ on Q lifts to a G L -invariant vector field X K on K , but this lift is not unique. Given a section σˆ of L |Q and a vector field Xˆ on Q, we set ∇ Qˆ σˆ = ρ K ∗ (∇ XK (lift σˆ )). X

(7.40)

Since ∇ K , X and lift σˆ are G L -invariant, it follows that ∇ XK (lift σˆ ) pushes forward to a section of L|Q . We need to show that the right-hand side of equation (7.40) does not depend on the choice of the G L -invariant vector field X such that Xˆ = ρ K ∗ X . Suppose that X is another G L -invariant vector field on K satisfying Xˆ = ρ K ∗ X . Then Y = X − X is a vertical G L -invariant vector field on K . Since the vertical directions are spanned by the restrictions to K of Hamiltonian vector fields of J L , equation (7.24) gives Y =

k i=1

f i X JL[ξi ] |K =

k i=1

f i X Jξi |L |K ,

7.2 Commutation of quantization and singular reduction at J = 0 179 where ξ1 , . . . , ξk ∈ n L . The G-invariance of lift σˆ implies that (P Jξ )|K lift σˆ = 0 for every ξ ∈ n L , where (P Jξ )|K is the restriction of the partial differential operator P Jξ to points of K . Taking equation (7.8) into account, we obtain (−i∇ XKJ + Jξ |K )lift σˆ = 0 ξ

for all ξ ∈ n L . But K ⊆ J −1 (0), which implies that Jξ |K = 0. Therefore, ∇ XKJ (lift σˆ ) = 0 for every ξ ∈ n L , which ensures that ∇YK (lift σˆ ) = 0. Hence, ξ

∇ XK (lift σˆ ) is independent of the choice of the G L -invariant vector field X such Q that Xˆ = ρ K ∗ X . Hence, ∇ ˆ σˆ is well defined by equation (7.40). X For f ∈ C ∞ (Q), ρ K∗ f is a G L -invariant function on K , and lift f σˆ = ∗ (ρ K f )lift σˆ , so that K ∇ Q ˆ σˆ = ρ K ∗ (∇(ρ ∗ fX

K

ˆ )) f )X (lift σ

= ρ K ∗ ((ρ K∗ f )∇ XK (lift σˆ ))

= f ∇ Qˆ σˆ , X

and ∇ Qˆ ( f σˆ ) = ρ K ∗ (∇ XK ((ρ K∗ f )lift σˆ )) X

= ρ K ∗ (X (ρ K∗ f )) lift σˆ + (ρ K∗ f )∇ XK (lift σˆ ) Q = Xˆ ( f )σˆ + f ∇ ˆ σˆ . X

Hence, ∇ Q gives a connection on L Q . (iv) The Hermitian form · | · on L is invariant under the parallel transport defined by the connection ∇. Since the action of G L on L K is given by parallel transport along integral curves of the Hamiltonian vector fields of components of JL , it follows that · | · induces a Hermitian form · | · Q on L|Q such that ρ K∗ σˆ 1 | σˆ 2 Q = lift σˆ 1 | lift σˆ 2 . We have shown that for every symplectic stratum (Q, ω Q ) of J −1 (0)/G, the restriction Lˆ Q of Lˆ to Q is a prequantization line bundle of (Q, ω Q ). In Proposition 7.2.5, we have shown that if our G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0), then F induces a polarization Fˆ Q of (Q, ω). Thus, J −1 (0)/G is stratified by prequantized, polarized symplectic manifolds.2 It is tempting to define quantum states of the quantization of J −1 (0)/G to be sections σˆ of Lˆ such that for each symplectic stratum (Q, ω Q ), the restriction of σˆ to Q gives a section of Lˆ Q that is covariantly constant along FQ . 2 This result was obtained for a Kähler polarization by Huebschmann (2006).

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However, such a definition is not sufficiently restrictive, because it ignores the components of Fˆ = T C ρ(F ∩ T C (J −1 (0))) that are not tangential to strata of J −1 (0)/G. In order to define a quantization of J −1 (0)/G that might satisfy the principle of commutation of quantization and reduction, we need to extend some of the results of Proposition 7.2.14 to the complex line bundle Lˆ over the singular space J −1 (0). Proposition 7.2.15 Each G-invariant section σ0 of L0 pushes forward to a smooth section σˆ of Lˆ such that κ ◦ σ0 = σˆ ◦ ρ0 . Proof The projection of a point p ∈ J −1 (0) to J −1 (0)/G is the G-orbit through p. In other words, ρ0 ( p) = Gp. If z ∈ L0 , then κ(z) = Gz. Therefore, κ(gz) = Ggz = Gz for every g ∈ G. Define σˆ : J −1 (0)/G → Lˆ by σˆ (ρ0 ( p)) = κ(σ0 ( p)). For every g ∈ G, we have σˆ (ρ0 (gp)) = κ(σ0 (gp)) = κ(gσ0 ( p)) = κ(σ0 ( p)) = σˆ (ρ0 ( p)). Hence, σˆ : Pˆ → Lˆ is well defined, and κ ◦ σ0 = σˆ ◦ ρ0 . In order to show that σˆ is smooth, consider the pull-back σˆ ∗ f of a function ˆ We need to verify that σˆ ∗ f ∈ C ∞ (J −1 (0)/G). By the definif ∈ C ∞ (L). tion of the quotient differential structure, it suffices to show that ρ0∗ σˆ ∗ f ∈ C ∞ (J −1 (0)). However, κ ◦ σ0 = σˆ ◦ ρ0 implies that ρ0∗ σˆ ∗ f = f ◦ σˆ ◦ ρ0 = f ◦ κ ◦ σ0 = σ0∗ κ ∗ f . But κ ∗ f ∈ C ∞ (L0 ) and σ0∗ κ ∗ f ∈ C ∞ (J −1 (0)), because κ and σ0 are smooth. Proposition 7.2.16 Every smooth section σˆ of Lˆ can be lifted to a unique G-invariant section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . Proof

Recall that we have the following commutative diagram: L0

κ

λ0

J −1 (0)

/ Lˆ λˆ

ρ0

/ J −1 (0)/G.

For each p ∈ J −1 (0), the projection of p to J −1 (0)/G is the G-orbit through p; that is, ρ0 ( p) = Gp. Similarly, for each z ∈ L0 , we have κ(z 0 ) = Gz and λˆ (Gz) = G λˆ (z).

7.2 Commutation of quantization and singular reduction at J = 0 181 ˆ For each p ∈ J −1 (0), σˆ (Gp) Let σˆ : J −1 (0)/G → Lˆ be a section of L. −1 is the orbit in L0 covering the orbit Gp in J (0). Hence, there exists a point z ∈ σˆ (Gp) such that p = λ0 (z). Therefore, σˆ (Gp) = Gz. We are going to show that z is uniquely defined by the conditions z ∈ σˆ (Gp) and p = λ0 (z). Suppose that there exists z ∈ σˆ (Gp) such that p = λ0 (z ). Since z and z are in Gz, there exists g ∈ G such that z = gz. Moreover, λ0 (z) = λ0 (z ) = p implies that g is in the isotropy group H = {g ∈ G | gp = p} of p. Let L be the connected component of PH containing p. Assumption 7.2.6 ensures that the action of g on the restriction L L of L to L is trivial. Hence, z = z. Since z is uniquely defined by the conditions z ∈ σˆ (Gp) and p = λ0 (z), there is a well-defined map σ0 : J −1 (0) → L0 such that σ0 ( p) = z. The condition p = λ0 (z) implies that σ0 is a section of L0 , which need not be smooth. For every g ∈ G, σ0 (gp) is the unique element of σˆ (Ggp) = σˆ (Gp) = Gz such that λ0 (σ0 (gp)) = gp. Since gz ∈ Gz and λ0 (gz) = gλ0 (z) = gp, it follows that σ0 (gp) = gσ0 ( p). Hence, σ0 is G-invariant. In Proposition 7.2.16, we have shown that every smooth section σˆ of Lˆ can be lifted to a unique G-invariant section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . However, we have not proved that the section σ0 is smooth. Throughout the remainder of this section, we make the following assumption. Assumption 7.2.17 Each section σˆ of Lˆ lifts to a unique G-invariant smooth section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . Our next step is to discuss the notion of a connection ∇ˆ on the complex line bundle Lˆ over a singular space J −1 (0)/G induced by the connection ∇ 0 on L0 ; see equation (7.31). The main difficulty is that the map Tρ0 : T (J −1 (0)) → T (J −1 (0)/G) need not be onto. However, in geometric quantization, we are interested mainly in a partial connection on the complex prequantization line ´ bundle covering the polarization (Sniatycki, 1980). Definition 7.2.18 Let L be a complex line bundle over a singular space S, and let F ⊆ T C S be a complex distribution on S. A partial connection on L over F is given by a covariant derivative operator ∇ which associates to each section σ of L a C-linear map ∇σ : F → L : w → ∇w σ that commutes with the projection to S. Moreover, we assume that ∇(σ1 + σ2 ) = ∇σ1 + ∇σ2 , and for every section σ of L and every f ∈ C ∞ (S), ∇w ( f σ ) = w( f )σ + f (x)∇w σ, for all w ∈ Fx .

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Commutation of quantization and reduction

Clearly, the restriction of the connection ∇ on the prequantization line bundle L of (P, ω) to L0 = L |J −1 (0) and F0 = F ∩ T C J −1 (0) gives rise to a connection ∇ 0 on L0 over F0 in the sense of Definition 7.2.18. This connection induces a connection ∇ˆ on Lˆ over Fˆ such that for every w ∈ F0 , ∇ˆ T C ρ0 (w) σˆ = κ(∇w0 σ0 ),

(7.41)

where σ0 is the unique G-invariant section of L 0 such that κ ◦ σ0 = σˆ ◦ ρ0 , which exists by Assumption 7.2.17. Proposition 7.2.19 The connection ∇ˆ is well defined by equation (7.41). Proof Let w1 and w2 be two vectors in F0 attached at the same point p and such that T C ρ0 (w1 ) = T C ρ0 (w2 ). Then, T C ρ0 (w1 − w2 ) = 0 and w1 − w2 ∈ T pC (Gp). But T p (Gp) is spanned by Hamiltonian vector fields X Jξ of momenta Jξ corresponding to elements ξ ∈ g. By assumption, σ0 is a G-invariant section of L0 = L|J −1 (0) . This implies that (i∇ X Jξ + Jξ )σ0 = 0 for every ξ ∈ g; see

equation (7.8). But Jξ ( p) = 0 for every p ∈ J −1 (0). Hence, ∇ X Jξ σ0 = 0 for all ξ ∈ g. Therefore, ∇w1 −w2 σ0 = 0, which implies that ∇w1 σ0 = ∇w2 σ0 . Thus, the right-hand side of equation (7.41) is independent of the choice of the vector w ∈ F0 attached at p that projects to T C ρ0 (w). Suppose now that w1 and w2 are vectors in F0 , attached at p1 and p2 , respectively, such that T C ρ0 (w1 ) = T C ρ0 (w2 ). Then, there exists g ∈ G such that gp1 = p2 . Moreover, the G-invariance of F0 implies that T C g (w1 ) is also in F0 , that it is attached at p2 and that T C g (w1 ) − w2 ∈ ker T C ρ0 . The G-invariance of σ0 and ∇ 0 implies that g(∇w0 1 σ0 ) = ∇T0 C Since ∇T0 C

g (w1 )−w2

g (w1 )

σ0 = ∇T0 C

g (w1 )−w2

σ0 + ∇w0 2 σ0 .

σ0 = 0, it follows that κ(∇w0 1 σ0 ) = κg((∇w0 2 σ0 )).

Hence, ∇ˆ is well defined by equation (7.41). Proposition 7.2.20 A smooth section σˆ of Lˆ is covariantly constant along Fˆ if and only if the lift of σˆ to a unique G-invariant section σ0 of L 0 is covariantly constant along F0 . Proof The proof is a straightforward consequence of the definition of the ˆ partial connection ∇ˆ over F.

7.2 Commutation of quantization and singular reduction at J = 0 183 Let S(L0 )G denote the space of smooth G-invariant sections of L0 , and let ˆ be the space of smooth sections of L. ˆ Proposition 7.2.15 ensures that S(L) there is a linear map ˆ : σ0 → (σ0 ) such that κ ◦ σ0 = (σ0 ) ◦ ρ0 . (7.42) : S(L0 )G → S(L) Clearly, the map is one-to-one. In Assumption 7.2.17, we assumed that ˆ is onto. This assumption enables us to define a partial : S(L0 )G → S(L) ˆ connection ∇ˆ on Lˆ that covers F.

Let S F0 (L0 )G denote the space of smooth G-invariant sections of L0 that are ˆ be the space of smooth sections covariantly constant along F0 , and let S Fˆ (L) ˆ Proposition 7.2.20 ensures that the of Lˆ that are covariantly constant along F. G restriction of to S(L0 ) gives rise to a linear isomorphism

ˆ : σ0 → F (σ0 ) such that κ ◦ σ0 = (σ0 ) ◦ ρ0 . F0 : S F0 (L0 )G → S Fˆ (L) 0 (7.43) Composing F0 with the restriction map FG : S F (L)G → S F0 (L0 )G : σ → σ|J −1 (0) given in equation (7.34), we obtain a linear map ˆ : σ → F0 (σ|J −1 (0) ). F0 ◦ FG : S F (L)G → S Fˆ (L) If F0 ◦ FG is an isomorphism, then the space S F (L)G that corresponds to ˆ of a trivial representation of G on S F (L) is isomorphic to the space S Fˆ (L) −1 quantization data on J (0)/G. Recall that we have defined the term ‘commutation of quantization and reduction’ as a programme of determining the spectral measure dμ(λ) in terms of the quantization of the symplectic reduction of the inverse images of co-adjoint orbits under the momentum map J : P → g∗ . Here, we have considered the trivial representation corresponding to λ = 0. If F0 ◦ FG is an isomorphism, then the space S F (L)G that corresponds to a trivial representaˆ of quantization data on tion of G on S F (L) is isomorphic to the space S Fˆ (L) −1 J (0)/G. We would like to determine the contribution of the trivial representation to the spectral measure dμ(λ) at λ = 0. If F0 ◦ FG is a monomorphism but not an isomorphism, then we might still be able to determine the space ˆ of quantization data on J −1 (0)/G. However, S F (L)G from the space S Fˆ (L) in this case we would have to identify the range of F0 ◦ FG . If F0 ◦ FG has a non-zero kernel, there would be a loss of information in passing from S F (L)G ˆ to S Fˆ (L). The discussion above was given under Assumption 7.2.17, which requires ˆ be a vector space epimorphism. By Proposithat : S(L0 )G → S(L) ˆ is one-to-one. If is not an tion 7.2.15, the map : S(L0 )G → S(L)

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epimorphism, we cannot use equation (7.41) to define a connection ∇ˆ on Lˆ ˆ of sections of Lˆ that ˆ and we have no notion of the space S ˆ (L) over F, F ˆ are covariantly constant along F. In this case, we can use the restriction map FG : S F (L)G → S F0 (L0 )G to relate the space S F (L)G of G-invariant sections of L that are covariantly constant along F to the space S F0 (L0 )G of their restrictions to J −1 (0). If FG is a monomorphism, then we might still be able to determine the space S F (L)G from the a knowledge of the range of FG in S F0 (L0 )G .

7.3 Special cases 7.3.1 The results of Guillemin and Sternberg We discuss first the case investigated by Guillemin and Sternberg (1982). These authors considered a Kähler polarization F on a compact symplectic manifold (P, ω), and a Hamiltonian action of a compact connected Lie group G that is free on the zero level of the momentum map J . Hence, P is a Kähler manifold, the prequantization line bundle L of (P, ω) is holomorphic and the space S F (L) consists of holomorphic sections of L. Moreover, J −1 (0) is a submanifold of P, and J −1 (0)/G is a quotient manifold of J −1 (0). This means that the stratification of J −1 (0)/G has only one stratum, which we denote by Q. In other words, Q = J −1 (0)/G. With this notation, Lˆ is a line bundle over Q. By Proposition 7.2.5, the polarization F of (P, ω) gives rise to a polarization FQ of (Q, ω Q ). It is easy to show that the assumption that F is Kähler implies that FQ is Kähler. Hence, Q has the structure of a Kähler manifold. By Proposition 7.2.14, the prequantization structure of L, consisting of a connection ∇ and a connection-invariant Hermitian form · | · , induces ˆ With this structure, Lˆ is a a prequantization structure ∇ Q and · | · Q on L. ˆ of polarized sections holomorphic line bundle over Q, and the space S FQ (L) ˆ of L consists of holomorphic sections. ˆ Proposition 7.2.14 ensures that the monomorphism : S(L0 )G → S(L) defined by equation (7.42) is an isomorphism of vector spaces. Moreover, the results of Guillemin and Sternberg show that the restriction of to polarized ˆ sections gives a vector space isomorphism F : S F (L0 )G → S FQ (L). G G G Now consider the restriction map F : S F (L) → S F (L0 ) : σ → σ|J −1 (0) ; see equation (7.33). Guillemin and Sternberg showed that FG is a vector space isomorphism. Their approach makes use of the action on P of the complex Lie group G C such that G is the real form of G C . It should be noted that the compactness of P and G implies that the spaces of polarized sections considered here are finite-dimensional. Hence, the results of

7.3 Special cases

185

Guillemin and Sternberg are formulated in terms of equality of the dimensions of the spaces under consideration. Sjamaar (1995) extended the results of Guillemin and Sternberg to the case when the action of the compact Lie group G on J −1 (0) need not be free. He retained the assumption that P is a compact Kähler manifold. For a sufficiently ‘positive’ polarization, the multiplicity m α can be obtained from the Riemann–Roch formula; see Guillemin and Sternberg (1982). This result has been studied by several authors. We shall not pursue this line of research here, however, because we are interested in cases in which the multiplicities may be infinite.

7.3.2 Kähler polarization without compactness assumptions We consider here the case when F is a Kähler polarization of a symplectic manifold (P, ω) and G is a connected Lie group with a proper Hamiltonian action on (P, ω) that preserves F. We do not assume that P or G is compact. In this case, we have no information about whether the projection : S(L0 )G → ˆ is an epimorphism. Similarly, we do not know whether G : S F (L)G → S(L) F S F (L0 )G : σ → σ|J −1 (0) is an epimorphism. However, we have the following result. Proposition 7.3.1 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a Kähler polarization F, and that J −1 (0) contains a Lagrangian submanifold of (P, ω). Then the restriction map F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism. Similarly, FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. Proof We show first that the complexified tangent space TC P of P along the Lagrangian manifold is the direct sum of T C and F| . Consider a point p ∈ . Suppose that w ∈ F p ∩ T pC . Since F is Kähler, iω(w, w) ¯ ≥ 0, and iω(w, w) ¯ = 0 implies that w = 0. On the other hand, from the assumption that is a Lagrangian submanifold, it follows that if w ∈ T C , then w¯ ∈ T C and ω(w, w) ¯ = 0. Hence, F p ∩ T pC = 0. Since dimC F = dimC T C = 1 2 dim P, it follows that TC P = F| ⊕ T C . Since F is a Kähler polarization of (P, ω), it follows that P has the structure of a Kähler manifold, that the prequantization line bundle L is holomorphic and that polarized sections of L are holomorphic. Let σ ∈ S F∞ (L) be in the kernel of . This means that σ is a holomorphic section of L that vanishes on J −1 (0). In particular, σ is identically zero on . Hence, all derivatives of σ in

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Commutation of quantization and reduction

the directions in T C are zero. On the other hand, all derivatives of σ in the directions in F are zero because σ is holomorphic. Hence, all derivatives of σ vanish on ⊂ P. Since σ is holomorphic, it follows that σ = 0. It follows that ker F = 0 and that F is a monomorphism. Since F (σ ) is G-invariant if σ is G-invariant, it follows that FG : ∞ S F (L)G → S F∞0 (L0 )G is a monomorphism. If the Hamiltonian action of G on (P, ω) is free and proper, then it is easy to show that the zero level of the momentum map contains a Lagrangian submanifold of (P, ω). Hence, Proposition 7.3.1 implies that FG is a monomorphism whenever the action of G is free and proper. On the other hand, for a proper non-free Hamiltonian action of a compact group G on a symplectic manifold (P, ω), the zero level of the momentum map need not contain a Lagrangian submanifold of (P, ω). Hence, we cannot use Proposition 7.3.1 to obtain Sjamaar’s results (Sjamaar, 1995). If P and G are not compact, then the quantization representation R of G on S F∞ (L) may be infinite-dimensional, and the space S F∞ (L)G of G-invariant sections in S F∞ (L) may also have infinite dimension. Unitarization leads from R to a unitary representation U on a Hilbert space H F . With some abuse of terminology, we can say that a section σ in S F∞ (L) is in the Hilbert space H F if unitarization leads to a vector in H F . In other words, we ignore the details of the unitarization process and continue the discussion as if the Hilbert space H F were the completion of a dense open subset of S F∞ (L) with respect to some scalar product (· | ·). The aim of the programme of commutation of quantization and reduction is to discuss the decomposition of the unitary representation U into irreducible unitary representations U α of G; see equation (7.1). Here, we concentrate on the trivial representation of G occurring in U. Let HG F denote the closed subspace of H F consisting of G-invariant vectors in H F . If we denote the trivial representation of G by U 0 , the space HG F corresponds to the atomic part of the spectral measure dμ(λ) at λ = 0. Suppose that quantization commutes with reduction on the zero level of the momentum map. In other words, we assume that the restriction map FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. This assumption means that all sections in S F∞ (L)G can be uniquely determined in terms of sections in range FG ⊆ S F∞0 (L0 )G . The scalar product (· | ·) on S F∞ (L), restricted to S F∞ (L)G , can be pushed forward by to a scalar product (· | ·)0 on range FG . For each normalizable σ1 and σ2 in S F∞ (L)G , we have (σ1 | σ2 ) = ( FG (σ1 ) | FG (σ2 ))0 .

7.3 Special cases

187

It would be of great help if we could describe ( FG (σ1 ) | FG (σ2 ))0 directly in terms of sections FG (σ1 ) and FG (σ2 ) in range FG . This would allow us to G G determine HG F in terms of the reduced data in range F . Sections in range F that are not normalizable in terms of (· | ·)0 correspond to the continuous part of the spectral measure dμ(λ) at λ = 0. Using unitarization in terms of half-forms, Hall and Kirwin obtained an explicit expression for ( FG (σ1 ) | FG (σ2 ))0 in terms of FG (σ1 ) and FG (σ2 ) in the case when F is a Kähler polarization of a compact symplectic manifold (P, ω) and G is a compact group with a free action on J −1 (0) (Hall and Kirwin, 2007). The case when the action of G on J −1 (0) is not free was investigated by Li (2008), who found estimates for (· | ·)0 . For compact G and P, all representations considered here are finite-dimensional and the spectral measure dμ(λ) is atomic. This implies that equation (7.1) can be rewritten as a direct sum. Nevertheless, the results of Hall, Kirwin and Li may serve as a starting point for investigations of the case when neither G nor P is compact. It is not necessary to know the scalar product (· | ·)0 on range FG in order G to determine HG F in terms of the reduced data in range F . It would suffice if we knew any other scalar product (· | ·)1 such that m(σ0 | σ0 )1 ≤ (σ0 | σ0 )0 ≤ M(σ0 | σ0 )1

(7.44)

for some positive numbers m and M, and all σ0 ∈ range . The inequalities (7.44) imply that σ0 is normalizable with respect to (· | ·)1 if and only if it is normalizable in terms of (· | ·)0 .

7.3.3 Real polarization We say that a polarization F of (P, ω) is real if F = D ⊗ C, where D is an involutive Lagrangian distribution on (P, ω). A real polarization F is strongly admissible if the space P/D of integral manifolds of D is a quotient manifold of P. In this case, D is spanned by Hamiltonian vector fields. Moreover, for each integral manifold of D, the restriction ∇| of ∇ to is a flat connection on L | . Proposition 7.3.2 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a strongly admissible real polarization F = D ⊗ C. If every integral manifold of D intersects J −1 (0), then the restriction maps F : S F∞ (L) → S F∞0 (L0 ) and FG : S F∞ (L)G → S F∞0 (L0 )G are monomorphisms. If the integral manifolds of D are simply connected, then F and FG are epimorphisms.

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Commutation of quantization and reduction

Proof Each section σ ∈ S F∞ (L)G is covariantly constant along F = D ⊗ C. Hence, σ is covariantly constant along D. This means that, for every integral manifold of D, the restriction σ| is covariantly constant on . Suppose that σ ∈ ker ; that is, σ|J −1 (0) = 0. Let be an integral manifold of D. By assumption, the intersection ∩ J −1 (0) is not empty, and there exists p ∈ ∩ J −1 (0). Then, σ ( p) = 0. Since σ is covariantly constant on , it follows that σ| = 0. This holds for every integral manifold of D. Therefore, σ = 0. Hence, F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism, which implies that FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. For each integral manifold of D, the connection ∇| on L| is flat. The assumption that is simply connected implies that the holonomy group of this connection is trivial. This means that for every point z ∈ L, we can use parallel transport along the integral manifold through the point p = λ(z) to obtain a covariantly constant section σ p of L| such that σ p ( p) = z. Now consider a section σ0 ∈ S F∞0 (L0 )G . For each p ∈ J −1 (0), we denote by p the integral manifold p of D through p, and by σ p the unique section of L| p such that σ p ( p) = σ0 ( p). Since σ0 is covariantly constant along D|J −1 (0) , it follows that σ p is independent of the choice of the point p in p ∩ J −1 (0). By assumption, all integral manifolds of D intersect J −1 (0). Hence, we have a map σ : P → L such that for every integral manifold of D, the restriction σ| of σ to coincides with σ p for any p ∈ ∩ J −1 (0). By construction, σ is invariant under parallel transport along D. The smoothness of the parallel transport and the assumed smoothness of σ0 imply that σ is smooth. Hence, F : S F∞ (L) → S F∞0 (L0 ) is an epimorphism. The G-invariance of D and ∇ implies that the action of G commutes with parallel transport along integral manifolds of D. Hence, the assumed G-invariance of σ0 implies that σ is G-invariant. Therefore, FG : S F∞ (L)G → S F∞0 (L0 )G is an epimorphism. Example 7.3.3 Consider the case when P = T ∗ R3 is the cotangent bundle R3 , ω is the canonical symplectic form of T ∗ R3 and the polarization F is the complexification of ker T ϑ, where ϑ : T ∗ R3 → R3 is the cotangent bundle projection. If x, y, z are the canonical coordinates on R3 and px , p y . pz are the corresponding conjugate momenta, then the canonical 1-form θ of T ∗ R3 is θ = px d x + p y dy + pz dz, and ω = dθ = dpx ∧ d x + dp y ∧ dy + dpz ∧ dz.

7.3 Special cases

189

The polarization F = D C , where D = span

∂ ∂ ∂ , , ∂ p x ∂ p y ∂ pz

,

and integral manifolds of D are fibres of the cotangent bundle projection ϑ : T ∗ R5 → R3 : ( p, x) = ( px , p y . pz , x, y, z) → x = (x, y, z). Consider the action of G = SU (2) on P = T ∗ R3 given by (eiϕ , ( px , p y . pz , x, y, z)) → (cos ϕpx + sin ϕp y , − sin ϕpx + cos ϕp y , pz , cos ϕx + sin ϕy, − sin ϕx + cos ϕy, z). This action is Hamiltonian, and its momentum map is the z-component of the angular momentum for the action of the rotation group on T R3 ; that is, J = px y − p y x. Moreover, this action preserves the cotangent bundle polarization. For x ∈ R3 , the cotangent space Tx∗ R3 is the integral manifold of D projecting to x. The intersection of Tx∗ R3 with J −1 (0) depends on x as follows. If x = (0, 0, z), then J −1 (0) ∩ Tx∗ R3 = {( px , p y , pz , 0, 0, z) ∈ Tx∗ R3 | ( px , p y , pz ) ∈ R3 }. If x = (0, y, z), where y = 0, then J −1 (0) ∩ Tx∗ R3 = {(0, p y , pz , 0, y, z) ∈ Tx∗ R3 | ( p y , pz ) ∈ R2 }. Similarly, if x = (x, 0, z), where x = 0, then J −1 (0) ∩ Tx∗ R3 = {( px , 0, pz , 0, y, z) ∈ Tx∗ R3 | ( px , pz ) ∈ R2 }. Finally, if x = (x, y, z), where x = 0 and y = 0, then J −1 (0) ∩ Tx∗ R3 = {(xt, yt, pz , x, y, z) ∈ Tx∗ R3 | (t, pz ) ∈ R2 }. Thus, every integral manifold of D is simply connected and has a nonempty intersection with J −1 (0). By Proposition 7.3.2, the restriction map FG : S F∞ (L)G → S F∞0 (L0 )G is an isomorphism.

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Commutation of quantization and reduction

7.4 Non-zero co-adjoint orbits In this section, we use the shifting trick discussed in Section 6.6 to extend the results for J −1 (0)/G obtained in the previous section to quantization of J −1 (O)/G, where O is a quantizable co-adjoint orbit. Let O ⊆ g∗ be a co-adjoint orbit of G, and let be the Kirillov–Kostant– Souriau form on O; see Section 6.1.1. We assume that O is an embedded submanifold of g∗ . Moreover, we assume that is quantizable, which means that it admits a prequantization line bundle λ O : L O → O with a connec−1 tion ∇ O and a connection-invariant Hermitian form · | · O such that 2π ˜ ω), ˜ where is the curvature of L O . We construct a symplectic manifold ( P, P˜ = P × O, with projections π1 : P˜ → P and π2 : P˜ → Oμ , and ω˜ = π1∗ ω ⊕ (−π2∗ ). ˜ given by The action of G on P, ˜ : G × P˜ → P˜ : (g, ( p, λ)) → ˜ g ( p, λ) = (g ( p), Adg∗ λ), is Hamiltonian with an AdG∗ -equivariant momentum map J˜ = π1∗ J − π2∗ I , where I : O → g ∗ is the inclusion map. We denote the space of G-orbits in ˜ ˜ P˜ by R˜ = P/G and the corresponding orbit map by ρ˜ : P˜ → R. According to Theorem 6.6.3, J −1 (O)/G and J˜−1 (0) are isomorphic Poisson differential spaces. We use this result to define the geometric quantization of J −1 (O)/G as being given by the geometric quantization of J˜−1 (0)/G discussed in the preceding sections. Our aim in this section is to express the quantization data for J˜−1 (0)/G in terms of data obtained by the geometric quantization of (P, ω) and (O, ). Let us consider geometric quantization of (O, ) in terms of the prequantization line bundle λ O : L O → O and a positive polarization FO . We denote by C F∞ (O) the space of functions in C ∞ (O) such that their Hamiltonian vector fields preserve the polarization FO . Similarly, we denote by S F∞ (L O ) the space of smooth sections of L O that are covariantly constant along FO . Quantization associates to each f ∈ C F∞ (O) a linear operator Q Of on the space S F∞ (L O ) given by Q Of σ O = (−i(∇ O ) X f + f )σ O ,

(7.45)

for every σ O ∈ S F∞ (L O ). Here, X f denotes the Hamiltonian vector field on O defined in terms of the symplectic form , and ∇ O is the covariant derivative operator on sections of L O . We denote by R O the representation of G on

7.4 Non-zero co-adjoint orbits

191

S F∞ (L O ) obtained by integration of the representation ξ → (i)−1 Q O Iξ of g. Note that Iξ = I | ξ is a smooth function on O. Complex conjugation z → z¯ in L O is an automorphism of L O considered as a real vector bundle over O, but it conjugates the complex structure. We denote by L O the complex line bundle over O with the conjugated complex structure. Note that for every section σ O of L O , the complex conjugate σ¯ O is O a section of L O . We denote by P the prequantization map given by the line O bundle L O with connection ∇ . For every ξ ∈ g∗ , the co-adjoint action of exp tξ on O is given by translations along integral curves of the Hamiltonian vector field of X Iξ relative to the symplectic form , where I : O → g∗ is the inclusion map. We denote by X − Iξ the Hamiltonian vector field of Iξ relative to the symplec-

−ξ . This means tic form − on O. We have X − Iξ = −X Jξ = X J−ξ = X

that X − Iξ corresponds to the action of exp(−tξ ) on O. On the other hand, − − ξ X− I−ξ = X −Iξ = −X Iξ = X Iξ = X , which means that the momentum map for the action of exp tξ on (O, −) is −Iξ .

Proposition 7.4.1 The connection ∇ O on L O induces a connection ∇ O with curvature 2π1 such that, for all σ O ∈ S ∞ (L O ), O

∇ σ¯ O = ∇ O σ O .

(7.46)

For each f ∈ C ∞ (O) and σ O ∈ S ∞ (L O ), O

P f σ¯ O = P Of σ O .

(7.47)

Moreover, the ∇ O -invariant Hermitian form · | · O on L O induces a ∇ O invariant Hermitian form · | · − O on L O such that O O σ¯ 1O | σ¯ 2O − O = σ2 | σ1 O

(7.48)

for all σ1O , σ2O ∈ S ∞ (L O ). Proof For every real f ∈ C ∞ (O) and X ∈ X(O), and each σ O ∈ S ∞ (L O ), equation (7.46) gives O

O

∇ f X σ¯ O = ∇ OfX σ O = f ∇ XO σ O = f ∇ XO σ O = f ∇ X σ¯ O , and O

∇ X ( f σ¯ O ) = ∇ XO ( f σ O ) = (X f )σ O + f ∇ XO σ O = (X f )σ¯ O + f ∇ XO σ O O

= (X f )σ¯ O + f ∇ X σ¯ O . Hence, equation (7.46) defines a connection ∇

O

on L O .

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Commutation of quantization and reduction

−1 O Since 2π is the curvature form of ∇ , equation (7.5) reads (∇ X 1 ∇ X 2 − −i O ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = (X 1 , X 2 )σ O . Taking into account equation (7.46), we get (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇ [X 1 ,X 2 ] )σ¯ O = i (X 1 , X 2 )σ¯ O , which implies that 2π1 is the curvature of ∇. For each f ∈ C ∞ (O), let X −f be the Hamiltonian vector field of f with respect to the symplectic form −. Equation (7.8) implies that O

P f σ¯ O = (−i∇ X − + f )σ¯ O = (i∇ X f + f )σ¯ O f

= i∇ X f σ¯ O + f σ¯ O = i∇ X f σ O + f σ¯ O

because X −f = −X f by equation (7.46)

= −i∇ X f σ O + f σ O = (−i∇ X f + f )σ O = P Of σ O . Here, we follow Dirac’s convention Dirac (1950), and our Hermitian forms are antilinear in the first argument and linear in the second argument. Therefore, σ2O | σ1O O is antilinear in σ2O and linear in σ1O , which implies that σ2O | σ1O O is linear in σ¯ 2O and antilinear in σ¯ 1O . Hence, equation (7.48) defines a Hermitian form on L O . Denoting by d O the differential on O, we have O O O O O O O O d O σ¯ 1O | σ¯ 2O − O = d O σ2 | σ1 O = ∇ σ2 | σ1 O + σ2 | ∇ σ1 O O O ¯ O − = σ = σ¯ 1O | ∇ O σ2O − ¯ 1O | ∇ σ¯ 2O − ¯ 1O | σ¯ 2O − 2 O O + ∇ σ1 | σ O + ∇ σ O. O

O

This implies that the Hermitian form · | · − O on L O is ∇ -invariant. O

Proposition 7.4.2 (i) If FO is a positive polarization of (O, ), then F O is a positive polarization of (O, −). (ii) If a function f ∈ C ∞ (O) is quantizable with respect to a polarization FO , then it is quantizable with respect to the polarization F O . In other words, ∞ O C F∞ (O) = C F∞ ¯ O ∈ S F∞ ¯ (P). Similarly, if σ ∈ S F (L O ), then σ ¯ (L O ). ∞ ∞ O (iii) For each f ∈ C F (O) and σ ∈ S F (L O ), O

Q Of σ O = Q f σ¯ O .

(7.49)

Proof (i) FO is a positive polarization of (O, ) if i(w, w) ¯ ≥ 0 for all w ∈ FO . However, for each w ∈ FO , its complex conjugate w¯ is in F O , and i(−)(w, ¯ w) ≥ 0, which implies that F O is positive. ∞ (ii) C F (O) denotes functions in C ∞ (O) whose Hamiltonian vector fields preserve F. If X f preserves F O , then X −f = −X f preserves F O . Therefore, (O). C F∞ (O) = C ∞ F Recall that if σ O ∈ S F∞ (L O ), then ∇w σ O = 0 for all w ∈ FO . Hence, ∇ w¯ σ¯ O = ∇w σ O = 0 for all w¯ ∈ F O .

7.4 Non-zero co-adjoint orbits

193

O

(iii) By Definition 7.1.3, the quantization map Q is the restriction of the O ∞ prequantization map P to the domain C F∞ ¯ (O) × S F¯ (L O ) and codomain S F∞ ¯ O (L O ). Equation (7.49) is a consequence of the results in part (ii) above and equation (7.47). ˜ ω) The next step is to relate the quantization structure on ( P, ˜ to the quantization structures on (P, ω) and (O, ). We take a prequantization line bundle to be the tensor product of bundles λ : L → P and λ¯ O : L O → O. More precisely, we first have to pull back the bundles λ : L → P and λ¯ O : L O → O by the projection maps π1 : P × O → P and π2 : P × O → O, respectively. We obtain bundles π1∗ L and π2∗ L O such that the following diagrams commute: π1∗ L

λ∗ π1

π1∗ λ

P×O

/L λ

π1

/P

π2∗ L O and

∗

λ0 π2

π2∗ λ0

P×O

/ L0 λ0

π2

/ O.

Here, π1∗ L is the complex line bundle over P × O such that its fibre over ( p, μ) ∈ P × O is the fibre of L over p = π1 ( p, μ). If z ∈ π1∗ L is a point in the fibre over ( p, μ) ∈ P × O, then the projection map π1∗ λ(z) is equal to ( p, μ). By definition, a point z in the fibre of π1∗ L over ( p, μ) ∈ P × O is a point in the fibre of L over p, which we denote by λ∗ π1 (z). Similarly, π2∗ L O is the complex line bundle over P × O such that its fibre over ( p, μ) ∈ P × O is the fibre of L O over μ = π2 ( p, μ). The maps π1∗ λ¯ O : π2∗ L O → P × O and λ¯ O∗ π1 : π2∗ L O → L O are defined in the same way as for π1∗ L. We define the prequantization line bundle L˜ over P˜ = P × O to be L˜ = π1∗ L ⊗ π2∗ L O . ˜ This is a complex line bundle over P˜ = (P × O) with projection map λ˜ : L˜ → P, ˜ such that for each ( p, μ) ∈ P, λ˜ −1 ( p, μ) = λ−1 ( p) ⊗ (λ¯ O )−1 (μ), where the tensor product of the complex lines λ−1 ( p) and (λ¯ O )−1 (μ) is taken over the field C of complex numbers. If z 1 ∈ λ−1 ( p) and z 2 ∈ (λ¯ O )−1 (μ), we can identify z 1 ⊗ z 2 ∈ λ˜ −1 ( p, μ) with the product z 1 z 2 of complex numbers. Let σ : P → L be a section. The pull-back of σ by π1 is the π1∗ σ : P ×O → ∗ π1 L that associates to ( p, μ) ∈ P × O the element σ ( p) ∈ L, considered as a point in the fibre of π1∗ L over ( p, μ). Similarly, we define the pull-back of a section σ¯ O : O → L O by π2 : P × O → O. If σ is a section of L and σ¯ O is

194

Commutation of quantization and reduction

a section of L O , we may define a section π1∗ σ ⊗ π2∗ σ¯ O : P × O → L˜

(7.50)

= π1∗ L ⊗ π2∗ L O : ( p, μ) → σ ( p) ⊗ σ¯ O (μ) = σ ( p)σ¯ O (μ) ˜ Moreover, a local section σ˜ of L˜ can be expressed in the form of L. σ˜ = f˜ π1∗ σ ⊗ π2∗ σ¯ O ,

(7.51)

where f˜ ∈ C ∞ (P × O) ⊗ C, and σ and σ¯ iO are local non-zero sections of L ˜ the space of smooth sections of L. ˜ and L O , respectively. We denote by S ∞ (L) O

Proposition 7.4.3 (i) The connections ∇ on L and ∇ on L O give rise to a ˜ a section σ˜ = connection ∇˜ on L˜ such that for every vector field X on P, ∗ ∗ O ˜ we have f˜π σ ⊗ π σ¯ of L˜ and a point ( p, μ) ∈ P, 1

2

∇˜ X ( p,μ) σ˜ = ∇˜ X ( p,μ) ( f˜π1∗ σ ⊗ π2∗ σ¯ O ) = X ( f )( p, μ)σ ( p)σ¯ O (μ) + f ( p, μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O

+ f ( p, μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O .

(7.52)

−1 ∗ ∗ The curvature of ∇˜ is 2π {π1 ω ⊕ (−π2 )}. (ii) The Hermitian forms · | · and · | · − O on L and L O , respectively, give rise to a Hermitian form · | · P˜ on L˜ such that, if σ˜ 1 = f˜1 π1∗ σ ⊗ π2∗ σ¯ O and σ˜ 2 = f˜2 π1∗ σ ⊗ π2∗ σ¯ O , then

σ˜ 1 | σ˜ 2 P˜ = f˜1 f˜2 π1∗ σ1 | σ2 π2∗ σ¯ 1O | σ¯ 2O − O ,

(7.53)

where f˜2 denotes the complex conjugate of f˜2 . The Hermitian form · | · P˜ on ˜ L˜ is ∇-invariant. (iii) If F and F¯ O are positive polarizations of (P, ω) and (O, −), respectively, then F˜ = π1∗ F ⊕ π2∗ F¯ O ˜ ω). is a positive polarization of ( P, ˜ For every section σ of L and section σ¯ O of ∗ L O , the tensor product σ˜ = π1 σ ⊗ π2∗ σ¯ O is covariantly constant along F˜ if σ is covariantly constant along F and if σ¯ O is covariantly constant along F¯ O . Proof that

(i) First, we show that ∇˜ σ˜ is well defined by equation (7.52). Suppose σ˜ = f˜π1∗ σ ⊗ π2∗ σ¯ O = f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O .

7.4 Non-zero co-adjoint orbits

195

Then, σ1 = hσ and σ1O = h O σ O for some h ∈ C ∞ (P) and h O = C ∞ (O), and f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O = f˜1 π1∗ (hσ ) ⊗ π2∗ (h O σ¯ O ) = f˜1 (π1∗ h)(π2∗ h O )π1∗ σ ⊗ π2∗ σ¯ O , which implies that f˜ = f˜1 (π1∗ h)(π2∗ h O ). Moreover, (X ( f˜))( p, μ) = X ( f˜1 (π1∗ h)(π2∗ h O ))( p, μ) = (X ( f˜1 )( p, μ))h( p)h O (μ) + f˜1 ( p, μ)(X (π1∗ h)( p, μ))h O (μ) + f˜1 ( p, μ)h( p)(X (π2∗ h O )( p, μ)) = (X ( f˜1 )( p, μ))h( p)h O (μ) + f˜1 ( p, μ)(T π1 (X ( p, μ))(h))h O (μ) + f˜1 ( p, μ)h( p)(T π2 (X ( p, μ))(h O )). Therefore, ∇˜ X ( p,μ) σ˜ = ∇ X ( p μ) ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = (X ( f˜)( p, μ))σ ( p)σ¯ O (μ) + f˜( p, μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O + f˜( p, μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O (μ)

= (X ( f˜1 )( p, μ))h( p)h O (μ)σ ( p)σ O (μ) + f˜1 ( p, μ)(T π1 (X ( p, μ))(h))h O (μ)σ ( p)σ¯ O (μ) + f˜1 ( p, μ)h( p)(T π2 (X ( p, μ))(h O ))σ ( p)σ¯ O (μ) + f˜1 ( p, μ)h( p)h O (μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O + f˜1 ( p, μ)h( p)h O (μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O (μ).

Since ∇ and ∇

O

satisfy equation (7.3), we have

(T π1 (X ( p, μ))(h))σ ( p) + h( p)(∇T π1 (X ( p,μ)) σ ) = ∇T π1 (X ( p,μ)) (hσ ) and (T π2 (X ( p, μ))(h O ))σ O (μ) + h O (μ)(∇T π2 (X ( p,μ)) σ ) = ∇T π2 (X ( p,μ)) (h O σ O ).

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Commutation of quantization and reduction

Hence, ∇˜ X ( p,μ) σ˜ = (X ( f˜1 )( p, μ))(hσ )( p)(h O σ O )(μ) + f˜1 ( p, μ)(∇T π1 (X ( p,μ)) (hσ ))(h O σ O )(μ) + f˜1 ( p, μ)(∇T π2 (X ( p,μ)) (h O σ O ))(hσ )( p) = (X ( f˜1 )( p, μ))(σ1 ( p))(σ1O (μ)) + f˜1 ( p, μ)(∇T π1 (X ( p,μ)) (σ1 ))(σ1O )(μ) + f˜1 ( p, μ)(∇T π2 (X ( p,μ)) (σ1O ))σ1 ( p) = ∇˜ X ( p,μ) ( f˜1 (π1∗ σ1 ) ⊗ (π2∗ σ O )), which implies that ∇˜ X ( p,μ) σ˜ is well defined by equation (7.52). ˜ For every h˜ ∈ C ∞ ( P), ˜ p, μ)∇˜ X ( p,μ) σ˜ , ∇˜ (h˜ X )( p,μ) σ˜ = h( ˜ σ˜ + h( ˜ p, μ)∇˜ X ( p,μ) σ˜ . ∇˜ X ( p,μ) h˜ σ˜ = (X ( p, μ)(h)) ˜ Therefore, ∇˜ is a connection on L. −1 −1 We know that 2π ω is the curvature of ∇ and 2π is the curvaO ture of ∇ . This leads by a straightforward computation to the result that −1 ∗ ∗ ˜ 2π {π1 ω ⊕ (−π2 )} is the curvature of ∇. (ii) As before, we have to show that the form σ˜ 1 | σ˜ 2 P˜ is well defined by equation (7.53). Suppose that σ˜ 1 = f˜1 π1∗ σ ⊗ π2∗ σ¯ O = f˜11 π1∗ σ1 ⊗ π2∗ σ¯ 1O , σ˜ 2 = f˜2 π1∗ σ ⊗ π2∗ σ¯ O = f˜21 π1∗ σ1 ⊗ π2∗ σ¯ 1O , where σ1 = hσ and σ1O = h O σ O , for some h ∈ C ∞ (P) and h O = C ∞ (O). Then f˜1 = (π1∗ h)(π2∗ h O ) f˜11 and f˜2 = (π1∗ h)(π2∗ h O ) f˜21 . Hence, σ˜ 1 | σ˜ 2 P˜ = f˜1 f˜2 π1∗ σ | σ π2∗ σ¯ O | σ¯ O − O = (π1∗ h)(π2∗ h O ) f˜11 (π1∗ h)(π2∗ h O ) f˜21 π1∗ σ | σ π2∗ σ¯ O | σ¯ O − O = f˜11 f˜21 π1∗ hσ | hσ π2∗ h O σ¯ O | h O σ¯ O − O = f˜11 f˜21 π1∗ σ1 | σ1 π2∗ σ¯ 1O | σ¯ 1O − O , which implies that σ˜ 1 | σ˜ 2 P˜ is well defined. Since the forms · | · and · | · − O ˜ are Hermitian, it follows that · | · P˜ is Hermitian. The ∇-invariance of · | · P˜ follows by straightforward but tedious computations from the ∇-invariance of O · | · and the ∇ -invariance of · | · − O.

7.4 Non-zero co-adjoint orbits

197

(iii) Let 2n = dim P and 2k = dim O. Then dimC F˜ = n + k. Consider ˜ Without loss of generality, we may assume that a basis (w1 , . . . , wn+k ) in F. (w1 , . . . , wn ) is a basis in π1∗ F, and (wn+1 , . . . , wn+k ) is a basis in π2∗ F¯ O . For i, j = 1, . . . , n and r, s = 1, . . . , k, ω(w ˜ i , wn+r ) = (π1∗ ω ⊕ (−π2∗ ))(wi , wn+r ) = (π1∗ ω)(wi , wn+r ) − (π2∗ )(wi , wn+r ) = ω(T π1 (wi ), T π1 (wn+r )) − (T π2 (wi ), T π2 (wn+r )) = 0 because T π1 (wn+r ) = 0 and T π2 (wi ) = 0. Moreover, ω(w ˜ i , w j ) = ω(T π1 (wi ), T π1 (w j )) = 0, because T π1 (wi ) and T π1 (w j ) are in F. Similarly, T π2 (wr ) and T π2 (ws ) are in F¯ O , which implies that ω(w ˜ r , ws ) = −(T π2 (wr ), T π2 (ws )) = 0. ˜ The positivity of the polarization Hence, F˜ is a Lagrangian distribution on P. ¯ F of (P, ω) and the polarization FO of (O, −) ensures that F˜ is a positive ˜ ω). polarization of ( P, ˜ It remains to show that F˜ is involutive. Let X 1 , . . . , X n be vector fields on P locally spanning F. Similarly, we consider vector fields X 1O , . . . , X kO on O which locally span F¯ O . We can lift these vector fields to vector fields on P˜ as follows. For each i = 1, . . . , n, we define π1∗ X i to be the unique vector field on P˜ such that π1∗ X i (π1∗ f ) = π1∗ (X i ( f )) for all f ∈ C ∞ (P), π1∗ X i (π2∗ f O ) = 0 for all f O ∈ C ∞ (O). Similarly, for every r = 1, . . . , k, we define π2∗ X rO to be the unique vector field on P˜ such that π2∗ X rO (π1∗ f ) = 0 for all f ∈ C ∞ (P), π2∗ X rO (π2∗ f O ) = π2∗ (X rO ( f O )) for all f O ∈ C ∞ (O). The vector fields π1∗ X 1 , . . . , π1∗ X n locally span π1∗ F, and the vector fields π2∗ X 1O , . . . , π2∗ X kO locally span π2∗ F¯ O . For every i = 1, . . . , n and r = 1, . . . , k, the vector fields π1∗ X i and π2∗ X rO commute because second partial derivatives of smooth functions commute. On the other hand,

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[π1∗ X i , π1∗ X j ](π1∗ f ) = π1∗ X i ((π1∗ X j )(π1∗ f )) − π1∗ X j ((π1∗ X i )(π1∗ f )) = π1∗ X i (π1∗ (X j ( f ))) − π1∗ X j (π1∗ (X i ( f ))) = π1∗ (X i (X j ( f )) − X j (X i ( f ))) = π1∗ ([X i , X j ]( f )) = (π1∗ [X i , X j ])(π1∗ f ), and [π1∗ X i , π1∗ X j ](π1∗ f O ) = 0. Since F is involutive, the commutator [X i , X j ] is a linear combination of the vector fields X 1 , . . . , X n , which implies that [π1∗ X i , π1∗ X j ] is a linear combination of the vector fields π1∗ X 1 , . . . , π1∗ X n . Hence, π1∗ F is involutive. In a similar way, we can show that π2∗ F¯O is involutive. Therefore, F˜ = π1∗ F ⊕ π2∗ F¯ O is involutive. Thus, we ˜ ω). have shown that F˜ is a positive polarization of ( P, ˜ Consider σ˜ = π1∗ σ ⊗ π2∗ σ¯ O . By equation (7.52), ∇˜ X ( p,μ) σ˜ = ∇˜ X ( p,μ) (π1∗ σ ⊗ π2∗ σ¯ O ) O

= (∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) + σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O . ˜ then T π1 (X ( p, μ)) ∈ F and T π2 (X ( p, μ)) ∈ F¯ O . ThereIf X ( p, μ) ∈ F, fore, σ˜ π1∗ σ ⊗π2∗ σ¯ O is covariantly constant along F˜ if σ is covariantly constant along f and if σ O is covariantly constant along F¯ O . ˜ of smooth sections of It follows from equation (7.51) that the space S ∞ (L) ˜ L can be presented locally as follows: ˜ = {σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O | f˜ ∈ C ∞ ( P), ˜ σ¯ O ∈ S ∞ (L˜ O )}. ˜ σ ∈ S ∞ (L), S ∞ (L) 1 2 Similarly, ˜ ˜ σ¯ O ∈ S ∞ (L˜ O )}. ˜ ˜ 0 , σ ∈ S F∞ (L), S F∞ ¯ O | f˜ ∈ C ∞ ( P) ˜ (L) = { f σ ⊗ σ F¯ F˜ Proposition 7.4.4 ˜ is given by For each ξ ∈ g∗ , the prequantization operator P˜ J˜ξ on S ∞ (L) P˜ J˜ξ ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( P O Iξ σ )).

(7.54) ˜ Similarly, the quantization operator on S ∞ ˜ (L) is given by F

Q˜ J˜ξ ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( Q Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( Q O Iξ σ )).

(7.55)

7.4 Non-zero co-adjoint orbits

199

Proof Since P˜ = P × O, ω˜ = π1∗ ω ⊕ (−π2∗ ), and J˜ = π1∗ J − π2∗ I , for each ξ ∈ g∗ , the Hamiltonian vector field X J˜ξ of P˜ξ = π1∗ Jξ −π2∗ Iξ is given by X J˜ξ = X π1∗ Jξ − X π2∗ Iξ = π1∗ X Jξ + π2∗ X Iξ = π1∗ X Jξ − π2∗ X − Iξ , where X − Iξ is the Hamiltonian field of Iξ on (O, −). Hence, the prequantization operator P˜ ˜ is given by Jξ

P˜ J˜ξ σ˜ = (−i∇˜ X J˜ + J˜ξ )σ˜ = (−i∇˜ X J˜ + J˜ξ )( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) ξ

ξ

= −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + f˜ (−i∇˜ X J˜ + J˜ξ )(π1∗ σ ⊗ π2∗ σ¯ O ) ξ

= −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + − f˜ (i∇˜ X π ∗ J − i∇˜ X −∗ )(π1∗ σ ⊗ π2∗ σ¯ O ) 1 ξ

π2 I ξ

+ f˜ (π1∗ Jξ − π2∗ Iξ )(π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + +(π1∗ (−i∇ X Jξ + Jξ )σ ) ⊗ π2∗ (σ¯ O ) + +π1∗ σ ⊗ π2∗ ((i∇ X − − Iξ )σ¯ O ) Iξ

=

−i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O O − f˜π1∗ σ ⊗ π2∗ ( P Iξ σ¯ O ).

+ f˜π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O

Taking into account Proposition 7.4.1, we get P˜ J˜ξ σ˜ = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( P O Iξ σ )).

Equation (7.55) follows from the fact that Q˜ is the restriction of P˜ to the ∞ ˜ ∞ ˜ ∞ ˜ ∞ ˜ ∞ ˜ ˜ domain C ∞ ˜ ( P)× S ˜ (L) ⊂ C ( P)× S (L) and codomain S ˜ (L) ⊂ S (L). F

F

F

By assumption, the prequantization actions of g on L and L O lift to actions of G on L and L O , respectively. Hence, the prequantization action of G on ˜ For each z ∈ λ−1 ( p) ⊆ L and L˜ = π1∗ L ⊗ π2∗ L O lifts to an action of G on L. O −1 z O ∈ (λ¯ ) (μ) ⊆ L O , the product z ⊗ z O satisfies z ⊗ z O ∈ λ˜ −1 ( p, μ) ⊆ π1∗ L ⊗ π2∗ L O . For every g ∈ G, we have g(z ⊗ z O ) = gz ⊗ gz O ∈ λ˜ −1 (gp, gμ). In order to continue, we need a scalar product on the space S F∞ (L O ). For simplicity, we assume that the polarization FO of (O, ) is Kähler. This

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Commutation of quantization and reduction

implies that L O is a holomorphic line bundle and that sections in S F∞O (L O ) are holomorphic. The scalar product in S F∞O (L O ) is given by (σ1O | σ2O ) O = σ1O | σ2O O k , (7.56) O

where k = 12 dim O; see equation (7.10). We denote by H FO the Hilbert space of sections in S F∞ (L O ) for which the scalar product (7.56) exists. ˜ where σ ∈ S ∞ (L), Theorem 7.4.5 Consider σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O ∈ S ∞ (L), 1

F˜

2

F

σ O ∈ H FO and f˜|π −1 ( p) σ O ∈ H FO for every p ∈ P.

(i) For each σ0O ∈ H FO , the map 'σ˜ (σ0O ) : P → L : p → σ O | f˜|π −1 ( p) σ0O O k σ ( p) = (σ

(7.57)

1

O O

| f˜|π −1 ( p) σ0O ) O σ ( p), 1

is in S F∞ (L). ˜ as the product (L) (ii) 'σ˜ (σ0 ) is independent of the presentation of σ˜ ∈ S ∞ F˜ f˜ π ∗ σ ⊗ π ∗ σ¯ O , where σ ∈ S ∞ (L) and f˜|π −1 ( p) σ O ∈ H FO for every 1

F

2

p ∈ P. ˜ (iii) 'σ˜ : σ0O → 'σ˜ (σ0 ) is a linear homomorphism of H FO to S ∞ ˜ (L). F

(iv) If σ˜ is G-invariant, then 'σ˜ : σ0O → 'σ˜ (σ0 ) intertwines the actions of ˜ G on H FO and S ∞ ˜ (L). F

Proof (i) Since σ0O ∈ H FO , and f˜|π −1 ( p) σ O ∈ H FO for every p ∈ P, it follows that the function p → (σ O | f˜|π −1 ( p) σ0O ) O is in C ∞ (P). The 1 ˜ implies that ∇˜ u˜ σ˜ = 0 for assumption that σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O ∈ S ∞ (L) 1

2

F˜

˜ f˜) = 0 for all u˜ ∈ π1∗ F. Therefore, the funceach u˜ ∈ π1∗ F. Hence, u( O ˜ tion p → (σ | f |π −1 ( p) σ0O ) O is constant along F. The assumption that 1

σ ∈ S F∞ (L) ensures that 'σ˜ (σ0O ) ∈ S F∞ (L). (ii) Suppose that

f˜π1∗ σ ⊗ π2∗ σ¯ O = f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O , where σ and σ1 are in S F∞ (L), and σ O and σ1O are in S F∞ (L O ). Then there exist h ∈ C F∞ (P) and h O ∈ C F∞O (O) such that σ1 = f σ and σ1O = h O σ O . Therefore, f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O = ( f˜1 π1∗ hπ1∗ h¯ O )π1∗ σ ⊗ π2∗ σ¯ O = f˜π1∗ σ ⊗ π2∗ σ¯ O ,

7.4 Non-zero co-adjoint orbits

201

and f˜ = ( f˜1 π1∗ h π1∗ h¯ O ). Hence, (σ1O | f˜1|π −1 ( p) σ0O ) O σ1 ( p) = (h O σ O | f˜1|π −1 ( p) σ0O ) O h( p)σ ( p) 1

1

= (σ O | h¯ O f˜1|π −1 ( p) h( p)σ0O ) O σ ( p) 1

= (σ O | ( f˜1 π1∗ h π1∗ h¯ O )|π −1 ( p) σ0O ) O σ ( p) 1

= (σ O | f˜|π −1 ( p) σ0O ) O σ ( p), 1

which shows that 'σ˜ (σ0O ) is well defined by equation (7.57). (iii) Since 'σ˜ (σ0O ) is linear in σ0O ∈ H FO , it follows that 'σ˜ : σ0O → ˜ 'σ˜ (σ0 ) is a linear homomorphism of H FO to S ∞ (L). F˜ ∗ ∗ O ˜ (iv) Suppose that σ˜ = f π σ ⊗π σ¯ is G-invariant. Then, for every g ∈ G, 1

2

p ∈ P and μ ∈ O, σ˜ (gp, Adg∗ μ) = g(σ˜ ( p, μ)). Hence,

f˜(gp, Adg∗ μ)σ (gp)(σ¯ O (Adg∗ μ)) = g( f˜( p, μ)σ ( p)σ¯ O (μ))

(7.58)

= f˜( p, μ)g(σ ( p))g(σ¯ (μ)). O

Recall that R O denotes the quantization representation of G on S F∞O (L O ). For σ0O ∈ S F∞O (L O ), and each g ∈ G and μ ∈ O, (R O σ0O )(μ) = g −1 (σ0O (Adg∗ μ)). Moreover, R gO ( f O σ0O ) = ((gO−1 )∗ f O )R gO σ0O . If σ0O ∈ H O , then U O σ0O = R O σ0O defines a unitary transformation on H FO , so that (σ O | U gO σ0O ) O = (U gO−1 σ O | σ0O ) O . Therefore, (σ O | f˜| π −1 ( p) U gO σ0O ) O = ( f˜ | π −1 ( p) σ O | U gO σ0O ) O 1

1

=

(U gO−1 ( f˜ | π −1 ( p) σ O ) | σ0O ) O 1

= (((gO−1 )∗ f˜ | π −1 ( p) )U gO−1 σ O | σ0O ) O 1 f˜ −1 (Adg −1 (μ)) gσ O (Ad ∗−1 μ) | σ O O k = O

= O

| π1 ( p)

g

0

f˜| π −1 ( p) (Adg∗−1 μ) gσ O(Adg∗−1 μ) | σ0O (μ) Ok . 1

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Commutation of quantization and reduction

Equation (7.58) implies that f˜(g −1 p, Adg∗−1 μ)σ (g −1 p)(gσ O (Adg∗−1 μ)) = f˜( p, μ)g −1 (σ ( p))g −1 (gσ O (μ)) = f˜( p, μ)g −1 (σ ( p)σ¯ O (μ)).

Replacing p by gp, we obtain f˜( p, Adg∗−1 μ)σ ( p)(gσ O (Adg∗−1 μ)) = f˜(gp, μ)g −1 (σ (gp)σ¯ O (μ)). Hence, 'σ˜ (U gO σ0O )( p) = (σ O | f˜|π −1 ( p) U g σ0O ) O σ ( p) 1 ˜ = f |π −1 ( p) (Adg∗−1 μ) gσ O (Adg∗−1 μ) | σ0O (μ) O k σ ( p) 1 O f˜|π −1 (gp) (μ) σ O (μ) | σ0O (μ) O k g −1 σ (gp) = =g

O −1

1

'σ˜ (σ0O )(gp) = (R g 'σ˜ (σ0O ))( p).

Therefore, 'σ˜ ◦ U gO = R g ◦ 'σ˜ , as required. Suppose that we have a scalar product on S F∞ (L) leading to a Hilbert space H F of square-integrable polarized sections of L. Then, the quantization representation G can be decomposed into a direct integral of irreducible unitary representations of G. In particular, we have the following decomposition of H F : Hλ dμ(λ), (7.59) HF =

where is the space of equivalence classes of irreducible unitary representations of G, Hλ is the representation space of a representation U λ corresponding to the class λ, and dμ(λ) is a spectral measure on . Suppose that U O is the irreducible unitary representation corresponding to a quantizable coadjoint orbit O, and λ O is the equivalence class of U O . If 'σ˜ (σ O ) = 0, then the spectral measure dμ(λ) does not vanish at λ = λ O . If, in addition, 'σ˜ (σ O ) ∈ Hλ O , then the spectral measure dμ(λ) contains an atomic part of the form m λ δ(λ − λ O ), where m λ O is the multiplicity of λ O , possibly infinite, in the decomposition of U F . If m λ O is finite, there exists a closed subspace m λ O Hλ O of H F on which U F is equivalent to the direct sum of m α copies of U λ O . For infinite m λ O , m λ O Hλ O is still a closed invariant subspace of H F , and U F restricted to m λ O Hλ O is equivalent to an appropriately defined direct sum of copies of U λ O . Thus, sections 'σ˜ (σ O ) in S F∞ (L) that are not normalizable correspond to the continuous part of the spectrum of dμ(λ) in the decomposition (7.59).

7.5 Commutation of quantization and algebraic reduction

203

7.5 Commutation of quantization and algebraic reduction In order to discuss commutation of quantization and algebraic reduction, we first have to define what we mean by quantization of algebraic reduction. As before, we consider only the zero level set of the momentum map.

7.5.1 Quantization of algebraic reduction Recall that algebraic reduction at the zero level of the momentum map is defined in terms of the ideal

k ∞ J = Jξi f i | f 1 , . . . , f k ∈ C (P) , i=1

where (ξ1 , . . . , ξk ) is a basis in g. The reduced Poisson algebra (C ∞ (P)/J )G consists of G-invariant equivalence classes [ f ] of functions f ∈ C ∞ (P). Since G is connected, [ f ] ∈ C ∞ (P)/J ⇐⇒ X Jξ ( f ) ∈ J ∀ ξ ∈ g. The prequantization of (P, ω) gives rise to a representation f → i1 P f of the Poisson algebra C ∞ (P) on the space S ∞ (L) of smooth sections of L. Let J S ∞ (L) = span { f σ | f ∈ J and σ ∈ S ∞ (L)}. S ∞ (L),

For each σ ∈ S ∞ (L)/J S ∞ L) .

(7.60)

we denote the equivalence class of σ by [σ ] ∈

Proposition 7.5.1 For each ξ ∈ g and σ ∈ S ∞ (L), the class [ P Jξ σ ] ∈ S ∞ (L)/J S ∞ (L) is independent of the choice of the representative σ of the class [σ ] ∈ S ∞ (L)/J S ∞ (L). Proof If [σ ] = [σ ] then σ = σ + i f i Jξi σi . Taking into account equations (7.8) and (6.6), we get P Jξ = −i∇ X Jξ + Jξ , which implies that P Jξ Jζ = Jζ P Jξ − iX Jξ (Jζ ) = Jζ P Jξ − iJ[ζ,ξ ] . Hence, P Jξ σ = P Jξ σ + P Jξ ( = P Jξ σ +

f i Jξi σi ) = P Jξ σ +

i

Jξi P Jξ ( f i σi ) − i

i

= P Jξ σ +

P Jξ (Jξi f i σi )

i

X Jξ (Jξi ) f i σi

i

Jξi P Jξ ( f i σi ) − i

i

Therefore, [ P Jξ σ ] = [ P Jξ σ ].

i

J[ξi ,ξ ] f i σi .

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Commutation of quantization and reduction

We have assumed that the action of g on S ∞ (L), given by (ξ, σ ) → i1 P Jξ σ integrates to a representation of G on S ∞ (L). Hence, the action of g on S ∞ (L)/J S ∞ (L), given by (ξ, [σ ]) → i1 [ P Jξ σ ], integrates to a representation of G on S ∞ (L)/J S ∞ (L). We denote (S ∞ (L)/J S ∞ (L))G the space of G-invariant elements in S ∞ (L)/J S ∞ (L). Since G is connected, it follows that [σ ] ∈ (S ∞ (L)/J S ∞ (L))G ⇐⇒ P Jξ σ ∈ J S ∞ (L) for all ξ ∈ g. Proposition 7.5.2 The map P : (C ∞ (P)/J )G × (S ∞ (L)/J S ∞ (L))G → (S ∞ (L)/J S ∞ (L))G : ([ f ], [σ ]) → P [ f ] [σ ] = [ P f σ ] is well defined. Proof For [ f ] ∈ (C ∞ (P)/J )G , we have X Jξ ( f ) ∈ J . Hence, for each Jξ σ ∈ J S ∞ (L), P f (Jξ σ ) = (−i∇ X f + f )(Jξ σ ) = Jξ (−i∇ X f + f )σ − i(X f (Jξ ))σ = Jξ (−i∇ X f + f )σ − i(X Jξ ( f ))σ ∈ J S ∞ (L). This implies that, for [ f ] ∈ (C ∞ (P)/J )G , the operator P f maps J S ∞ (L) to itself. Hence, [ P f σ ] does not depend on the representative σ of [σ ]. For k Jξ ∈ J and [σ ] ∈ (S ∞ (L)/J S ∞ (L))G , we have X k Jξ = k X Jξ + Jξ X k . Therefore, P k Jξ σ = (−i∇ X k Jξ + k Jξ )σ

= (−ik∇ X k Jξ − iJξ ∇ X k + k Jξ )σ

= −iJξ ∇ X k σ + ik P Jξ σ ∈ J S ∞ (L).

Hence, [ P f σ ] does not depend on the representative f of [ f ]. Combining these results, we obtain that an equivalence class [ P f σ ] ∈ S ∞ (L)/J S ∞ (L) depends only on [σ ] ∈ (S ∞ (L)/J S ∞ (L))G and [ f ] ∈ (C ∞ (P)/J )G . It remains to show that [ P f σ ] is G-invariant. For [ f ] ∈ (C ∞ (P)/J )G , [σ ] ∈ (S ∞ (L)/J S ∞ (L))G and ξ ∈ g, P Jξ P f σ = P f P Jξ σ + [ P Jξ , P f ]σ. But [σ ] ∈ (S ∞ (L)/J S ∞ (L))G implies P Jξ σ ∈ J S ∞ (L) so that P f P Jξ σ ∈ J S ∞ (L) by the first part of the proof. On the other hand [ P Jξ , P f ]σ = i P {Jξ , f } σ = i(−i∇ X {Jξ , f } + {Jξ , f })σ.

7.5 Commutation of quantization and algebraic reduction

205

But, {Jξ , f } = −X Jξ f ∈ J because [ f ] is G-invariant. Hence, {Jξ , f } = ∞ j f j Jζ j for some f j ∈ C (P) and ζ j ∈ g. Moreover, X f 1 f 2 = f 1 X f 2 + f 2 X f1 implies that ∇ X f j Jζ = ( f j ∇ X Jζ + Jζ j ∇ X f j ). j

j

Therefore, (−i∇ X {Jξ , f } + {Jξ , f })σ =

−i( f j ∇ X Jζ + Jζ j ∇ X f j ) + f j Jζ j σ j

=

j

j

j

f j P Jζ j − iJζ j ∇ X f j σ ∈ J S ∞ (L)

j

because [σ ] is G-invariant. Therefore, [ P f σ ] ∈ (S ∞ (L)/J S ∞ (L))G . Definition 7.5.3 The map that associates to each [ f ] ∈ (C ∞ (P)/J )G an operator P [ f ] on the space (S ∞ (L)/J S ∞ (L))G is a prequantization of the reduced Poisson algebra. Let F be a strongly admissible polarization of (P, ω), and let C F∞ (P) be a subalgebra of C ∞ (P) consisting of functions f such that X f preserves F. Quantization in terms of the polarization F assigns to each f ∈ C F∞ (P) an operator Q f on the space S F∞ (L) = {σ ∈ S ∞ (L) | ∇u σ = 0 for all u ∈ F}. Moreover, Q f σ = P f σ for each f ∈ C F∞ (P) and σ ∈ S F∞ (L). By analogy, we expect quantization after reduction to give a representation of the restriction of (C ∞ (P)/J )G to the set (C F∞ (P)/J )G consisting of G-invariant J -equivalence classes of functions in C F∞ (P). The representation space will be the space (S F∞ (L)/J S ∞ (L))G consisting of G-invariant J S ∞ (L)-equivalence classes of sections in S F∞ (L). We shall use the following identifications: (C F∞ (P)/J )G = (C F∞ (P)/(C F∞ (P) ∩ J ))G , (S F∞ (L)/J

∞

S (L)) = G

(S F∞ (L)/(S F∞ (L) ∩ J

∞

(7.61)

S (L)) . G

(7.62)

Proposition 7.5.4 The map Q : (C F∞ (P)/J )G × (S F∞ (L)/J S ∞ (L))G → (S F∞ (L)/J S ∞ (L))G : ([ f ], [σ ]) → Q [ f ] [σ ] = [ P f σ ] is well defined. Proof We know that if f ∈ C F∞ (P) and σ ∈ S F∞ (L), then P f σ ∈ S F∞ (L). In Proposition 7.5.2, we proved that [ P f σ ] ∈ (S ∞ (L)/J S ∞ (L))G is independent of the representatives f of [ f ] ∈ (C ∞ (P)/J )G and σ of

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[σ ] ∈ (S ∞ (L)/J S ∞ (L))G . Therefore, [ P f σ ] ∈ (S F∞ (L)/J S ∞ (L))G . Moreover, [ P f σ ] is independent of the representative f ∈ C F∞ (P) of [ f ] ∈ (C F∞ (P)/J )G . Similarly, [ P f σ ] is independent of the representative σ ∈ S F∞ (L) of [σ ] ∈ (S F∞ (L)/J S ∞ (L))G . Hence, the map Q : ([ f ], [σ ]) → Q [ f ] [σ ] = [ P f σ ] is well defined. Definition 7.5.5 A map that associates to each [ f ] ∈ (C F∞ (P)/J )G an operator Q [ f ] on the space (S F∞ (L)/J S ∞ (L))G is a quantization of the subalgebra (C F∞ (P)/J )G of the reduced Poisson algebra.

7.5.2 Kähler polarization Now consider the case when F is a Kähler polarization of (P, ω). We denote by F : S F∞ (L) → S F∞ (L)/J S ∞ (L) the projection map that associates to each σ in S F∞ (L) its class [σ ] in S F∞ (L) → S F∞ (L)/J S ∞ (L). If σ is G-invariant, then its class [σ ] is G-invariant. Hence, the restriction of F to S F∞ (L)G gives a projection map ∞ G ∞ ∞ G G F : S F (L) → (S F (L)/J S (L)) : σ → [σ ].

(7.63)

Proposition 7.5.6 Assume that the Hamiltonian action of G on (P, ω) preserves a Kähler polarization F of (P, ω). If J −1 (0) contains a Lagrangian submanifold of (P, ω), then the projection G F is a vector space isomorphism such that G G F ◦ Q f = Q[ f ] ◦ F

for every f ∈ C F∞ (P)G . Proof Consider σ ∈ S F∞ (L) ∩ J S ∞ (L). Since σ ∈ J S ∞ (L), it follows that σ|J −1 (0) = 0. By Proposition 7.3.1, σ = 0, because J −1 (0) contains a Lagrangian submanifold of (P, ω). Hence, S F∞ (L) ∩ J S ∞ (L) = 0. Using the identification (7.62), we obtain S F∞ (L)/J S ∞ (L) = S F∞ (L)/(S F∞ (L) ∩ J S ∞ (L)) = S F∞ (L)/(0) = S F∞ (L). Hence, (S F∞ (L)/J S ∞ (L))G = S F∞ (L)G , ∞ ∞ G ∞ G which implies that the projection map G F : S F (L) → (S F (L)/J S (L)) is an isomorphism. For every G-invariant function f ∈ C F∞ (P), the class [ f ] is in f ∈ ∞ C F (P)G . Moreover, for each σ ∈ S F∞ (L)G , we have Q f σ ∈ S F∞ (L)G and ∞ ∞ G [σ ] = G F σ ∈ (S F (L)/J S (L)) . Hence,

7.5 Commutation of quantization and algebraic reduction

207

G Q [ f ] ◦ G F σ = Q [ f ] [σ ] = [ Q f σ ] = F ◦ Q f σ,

which completes the proof.

7.5.3 Real polarization We consider here a Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) that preserves a real strongly admissible polarization F = D ⊗ C, as in Section 7.3.3. Proposition 7.5.7 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a strongly admissible real polarization F = D⊗C. If every integral manifold of D intersects J −1 (0), then the projection map ∞ G ∞ ∞ G G F : S F (L) → (S F (L)/J S (L)) : σ → [σ ]

is an isomorphism such that G G F ◦ Q f = Q[ f ] ◦ F

for every f ∈ C F∞ (P)G . Proof Consider σ ∈ S F∞ (L) ∩ J S ∞ (L). Since σ ∈ J S ∞ (L), it follows that σ|J −1 (0) = 0. By Proposition 7.3.2, the assumptions made here ensure that the restriction map F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism. Hence, σ|J −1 (0) = 0 implies that σ = 0. Therefore, S F∞ (L) ∩ J S ∞ (L) = 0. We continue as in the proof of Proposition 7.5.4. The identification (7.62) yields S F∞ (L)/J S ∞ (L) = S F∞ (L)/(S F∞ (L) ∩ J S ∞ (L)) = S F∞ (L)/(0) = S F∞ (L). Hence, (SF∞ (L)/J S ∞ (L))G = S F∞ (L)G , ∞ ∞ G ∞ G which implies that the projection map G F : S F (L) → (S F (L)/J S (L)) is an isomorphism. For every G-invariant function f ∈ C F∞ (P), the class [ f ] is in f ∈ ∞ C F (P)G . Moreover, for each σ ∈ S F∞ (L)G , we have Q f σ ∈ S F∞ (L)G and ∞ ∞ G [σ ] = G F σ ∈ (S F (L)/J S (L)) . Hence, G Q [ f ] ◦ G F σ = Q [ f ] [σ ] = [ Q f σ ] = F ◦ Q f σ,

which completes the proof.

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Commutation of quantization and reduction

We see in the cases considered here that the same assumptions lead to commutation of quantization and algebraic reduction and to commutation of quantization and singular reduction. In general, for a proper group action, algebraic reduction may give a larger Poisson algebra than singular reduction; see Example 6.9.4. However, in that example also, quantization of singular reduction and quantization of algebraic reduction give equivalent results (Bates et al., 2009).

7.5.4 Improper action Unlike singular reduction, algebraic reduction does not require properness of the action of G on (P, ω). The action of G on the cotangent bundle space endowed with the canonical symplectic form, which we have described in Section 7.1.3, is improper. In this section, we discuss a slight generalization of the special case described in Section 7.1.3. We assume that the polarization F is real and that the momentum map J : P → g∗ is constant along D. In this case, for each ξ ∈ g and σ ∈ S F∞ (L), we have Q Jξ σ = Jξ σ .

(7.64)

Hence, a section σ ∈ S F∞ (L) is G-invariant only if J σ = 0. Thus, the support of σ is contained in J −1 (0). We assume that J −1 (0) is nowhere dense in P. This implies that the equation J σ = 0 has only weak solutions. In order to discuss generalized sections of S F∞ (L), we need to describe the topology on S F∞ (L). We assume that the space Q of integral manifolds of the distribution D is a quotient manifold of P and that the leaves of D are complete affine spaces. Let ϑ : P → Q be the map that associates to each p ∈ P the maximal integral manifold of D through p. We assume that ϑ is a submersion. Moreover, we assume that the prequantization bundle L is trivial, that is, L = P × C, and that there exists a section σ0 ∈ S F∞ (L) such that σ0 ( p) = ( p, 1). All of these assumptions are satisfied by the system described in Section 7.1.3. Under the assumptions made here, S F∞ (L) = {ϑ ∗ (ψ)σ0 | ψ ∈ C ⊗ C ∞ (Q)}.

(7.65)

Let D = {ϑ ∗ (ψ)σ0 ∈ S F∞ (L) | ψ ∈ C ⊗ C0∞ (Q)}, where C⊗C0∞ (Q) is the space of compactly supported complex-valued smooth functions on Q. We endow D with a topology of uniform convergence on

7.5 Commutation of quantization and algebraic reduction

209

compact sets of all derivatives of functions ψ ∈ C ⊗ C0∞ (Q). We define a scalar product on D as follows: ∗ ∗ (σ1 | σ2 ) Q = (ϑ (ψ1 )σ0 | ϑ (ψ2 )σ0 ) Q = ψ¯ 1 (q)ψ2 (q) dμ(q). (7.66) Q

Let H F be the completion of D with respect to the norm given by the scalar product on D, and let D be the topological dual of D. Clearly, D ⊂ H F ⊂ D .3 For each ξ ∈ g, the quantum operator Q Jξ on S F∞ (L) preserves D. Hence, it extends to a self-adjoint operator on H and gives rise to a dual operator Q Jξ on D such that, for every ξ ∈ g, ϕ ∈ D and σ ∈ D, we have Q Jξ ϕ | σ = ϕ | Q Jξ σ , where · | · denotes the evaluation map. The space of generalized invariant vectors is ker Q J = {ϕ ∈ D | Q Jξ ϕ = 0 for all ξ ∈ g}. On the other hand, the range of Q J in D is range Q J = { Q Jξ σ1 + Q Jξ σ2 + . . . + Q Jξ σk | σ1 , . . . , σk ∈ D}, 1

2

k

where (ξ1 , . . . , ξk ) form a basis of g. There is a duality between ker Q J and D/range Q J such that for every ϕ ∈ ker Q J and every class [σ ] ∈ D/range Q J , we have ϕ | [σ ] = ϕ | σ . Since each Q Jξ is a multiplication operator, it follows that range Q J = {Jξ1 σ1 + Jξ2 σ2 + . . . + Jξk σk | σ1 , . . . , σk ∈ D} = J D. In the following discussion, we look for conditions under which D/range Q J = D/J D and (S F∞ (L)/J S ∞ (L))G are isomorphic. We begin with a simple lemma. Lemma 7.5.8 The class [σ ] ∈ S ∞ (L)/J S ∞ (L) of σ ∈ S ∞ (L) is uniquely determined by the restriction of σ to any open set containing J −1 (0). Proof

For σ ∈ S ∞ (L), if (support σ ) ∩ J −1 (0) = ∅, then [σ ] = 0.

Theorem 7.5.9 If Q is locally compact and ϑ(J −1 (0)) is compact, then D/J D and (S F∞ (L)/J S ∞ (L))G are isomorphic. 3 The triplet of spaces D ⊂ H ⊂ D obtained here is usually called a Gelfand triplet.

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Commutation of quantization and reduction

Proof For every σ ∈ S F∞ (L), there exists an open set Vσ ⊂ Q = P/D such that Vσ ⊇ ϑ(J −1 (0)), [σ ] is uniquely determined by ϑ −1 (Vσ ), and V¯σ is compact. Hence, there exists σ ∈ D = S F∞ (L) ∩ H F such that σ|ϑ −1 (Vσ ) = σ|ϑ −1 (V ) and [σ ] = [σ ]. Therefore, σ

S F∞ (L)/J

S ∞ (L) = S F∞ (L)/(J S ∞ (L) ∩ S F∞ (L)) = D/(J S ∞ (L) ∩ D).

For σ ∈ D, the class [σ ] ∈ D/(J S ∞ (L) ∩ D) is given by

[σ ] = σ + f i Jξi σi | f i Jξi σi ∈ D . i

But f i Jξi σi ∈ D implies that fi Jξi σi = ϑ ∗ (ψi )σ0 , where ψi has compact support in Q. There exists a function χi ∈ C ⊗ C0∞ (Q) such that χi (q) = 1 for every q in the support of ψi . Then, ψi = ψi χi and f i Jξi σi = ϑ ∗ (ψi )σ0 = ϑ ∗ (ψi χi )σ0 = ϑ ∗ (ψi )ϑ ∗ (χi )σ0 = f i Jξi ϑ ∗ (χi )σi ∈ J D. This implies [σ ] = [σ ] ∈ D/J D, so that S F∞ (L)/J S ∞ (L) = D/J D.

(7.67)

By definition, (S F∞ (L)/J S ∞ (L))G = {[σ ] ∈ S F∞ (L)/J S ∞ (L) | [ Q Jξ σ ] = 0 for all ξ ∈ g}. But Q Jξ σ = Jξ σ . Hence, [ Q Jξ σ ] = 0 for all σ ∈ S F∞ (L) and ξ ∈ g. Therefore, (S F∞ (L)/J S ∞ (L))G = S F∞ (L)/J S ∞ (L).

(7.68)

Equations (7.67) and (7.68) yield (S F∞ (L)/J S ∞ (L))G = D/J D, which completes the proof. We have shown that in the case under consideration, the representation space (S F∞ (L)/J S ∞ (L))G of the quantization of the singular reduction at J = 0 is naturally isomorphic to the space D/J D of generalized invariant vectors of the geometric quantization of the original phase space (P, ω).

8 Further examples of reduction

8.1 Non-holonomic reduction ‘Non-holonomic reduction’ is the term used for the reduction of symmetries of non-holonomically constrained Hamiltonian systems. The dynamics of such a system can be described in terms of a symplectic distribution (D, ) on a manifold P. Here, D is a distribution on P and is a symplectic form on D. In other words, associates to each p ∈ P a linear symplectic form p on D p ⊆ T p P. We assume that is smooth in the following sense. For each pair of smooth vector fields X 1 and X 2 with values in D, (X 1 , X 2 ) ∈ C ∞ (P). The symplectic form on D associates to each f ∈ C ∞ (P) a unique vector field Y f on P, with values in D, such that (Y f , X ) = X ( f )

(8.1)

for each vector field X on P with values in D. We call Y f the distributional Hamiltonian vector field of f . There is a bracket operation [·, ·] on the space C ∞ (P) of smooth functions on P, called the almost-Poisson bracket, such that [ f 1 , f 2 ] = Y f1 ( f2 )

(8.2)

for every f 1 , f 2 ∈ C ∞ (P). The almost-Poisson bracket need not satisfy Jacobi’s identity. The ring C ∞ (P) endowed with the almost-Poisson bracket [·, ·] is called an almost-Poisson algebra. Consider a proper action of a Lie group G on P which preserves the distribution D, the symplectic form on D and the Hamiltonian h. Non-holonomic reduction is concerned with the description of the reduced dynamics on the space P/G of orbits of G in P. Since the action of G on P preserves , it follows that for each g ∈ G, the pull-back ∗g : C ∞ (P) → C ∞ (P) preserves the almost-Poisson bracket [·, ·]. In other words, for each f 1 , f 2 ∈ C ∞ (D), ∗g [ f 1 , f 2 ] = [∗g f 1 , ∗g f 2 ].

(8.3)

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Further examples of reduction

The orbit space P/G, endowed with a differential structure C ∞ (P/G) = { f¯ ∈ C 0 (P/G) | ρ ∗ f ∈ C ∞ (P)},

(8.4)

where ρ : P → P/G is the orbit map, is a subcartesian differential space with the quotient topology. Since the action of G on P is proper, the orbit space P/G is stratified by orbit type. As before, we denote this stratification by N. Each stratum N ∈ N is an orbit of the family X(P/G) of all vector fields on P/G. Let C ∞ (P)G denote the ring of G-invariant functions in C ∞ (P). Equation (8.3) implies that C ∞ (P)G is closed under the almost-Poisson bracket. Since ρ ∗ gives an isomorphism of C ∞ (P/G) onto C ∞ (P)G , it follows that C ∞ (P/G) inherits an almost-Poisson bracket [·, ·] P/G such that [ρ ∗ f¯1 , ρ ∗ f¯2 ] = ρ ∗ [ f¯1 , f¯2 ] P/G .

(8.5)

Thus, C ∞ (P/G) is an almost-Poisson algebra. We can proceed as in Section 6.2, even though the almost-Poisson algebra C ∞ (P/G) does not satisfy the Jacobi identity and the action of G on P fails to have a momentum map. In a sense, the non-holonomic reduction discussed here consists of picking out the results of symplectic reduction which do not depend on the Jacobi identity and the existence of the momentum map. This is the reason why this section is so short. A comprehensive presentation of the current state of the geometry of non-holonomically constrained Hamiltonian systems can be found in a recent book Cushman et al. (2010), which contains several examples. Following the convention adopted in Cushman et al. (2010), we use a bar above a symbol in this chapter to denote that the symbol refers to an object on the orbit space. This should not lead to confusion, because we have no operation of complex conjugation in this chapter. For each f¯ ∈ C ∞ (P/G), we denote by Y f¯ the derivation of C ∞ (P/G) such that Y ¯ ( f¯ ) = [ f¯, f¯ ] P/G (8.6) f

for each

f¯

∈

C ∞ (P/G).

Proposition 8.1.1 The derivation Y f¯ is the push-forward of the distributional Hamiltonian vector field Yρ ∗ f¯ on P by the orbit map ρ : P → P/G. Moreover, Y f¯ is a vector field on P/G; that is, exp tY f¯ is a local one-parameter local group of diffeomorphisms of P/G. Proof For f¯ ∈ C ∞ (P/G), its pull-back ρ ∗ f¯ by ρ is in C ∞ (P)G and Yρ ∗ f0 is a G-invariant distributional Hamiltonian vector field on P. Moreover, for each f¯ ∈ C ∞ (P/G), equation (8.2) implies that

8.1 Non-holonomic reduction

213

Yρ ∗ f¯ (ρ ∗ f¯ ) = [ρ ∗ f¯, ρ ∗ f¯ ] = ρ ∗ [ f¯, f¯ ] P/G = ρ ∗ (Y f¯ ( f¯ )). Thus, Y f¯ is the push-forward of Yρ ∗ f¯ by the orbit map ρ. Since Yρ ∗ f¯ generates a local one-parameter local group exp tYρ ∗ f¯ of diffeomorphisms of P, it follows that translations along integral curves of Y f¯ give rise to a local one-parameter local group exp tY f¯ of diffeomorphisms of P/G such that ρ ◦ (exp t Tρ ∗ f¯ ) = (exp tY f¯ ) ◦ ρ.

(8.7)

Hence, Y f¯ is a vector field on P/G. For each f¯ ∈ C ∞ (P/G), we refer to Y f¯ as the almost-Poisson vector field of f¯ and denote by P(P/G) = {Y f¯ | f¯ ∈ C ∞ (P/G)}

(8.8)

the family of all almost-Poisson vector fields on P/G. Since P(P/G) ⊆ X(P/G), it follows that for each stratum N ⊆ P/G and every x ∈ N , the value at x of the almost-Poisson bracket [ f¯1 , f¯2 ] P/G of functions in C ∞ (P/G) depends only on the restrictions f¯1|N and f¯2|N of f¯1 and f¯2 , respectively, to N . Hence, the space R(N ) = { f¯|N | f¯ ∈ C ∞ (P/G)} of the restrictions to N of smooth functions on P/G inherits the structure of an almost-Poisson algebra from C ∞ (P/G). The almost-Poisson bracket on R(N ) is given by [ f¯1|N , f¯2|N ] N = ([ f¯1 , f¯2 ] P/G )|N for every f¯1 , f¯2 ∈ C ∞ (P/G). By the definition of a stratification, strata N ∈ N are locally closed connected submanifolds N of P/G. Proposition 2.1.8 ensures that every f¯ ∈ C ∞ (N ) coincides locally with a function in R(N ). Hence, the almost-Poisson algebra structure of R(N ) extends to C ∞ (N ). Thus, we have proved the following result. Proposition 8.1.2 Each stratum N of the orbit type stratification N of D/G is an almost-Poisson manifold. Now consider the family P(P/G) of almost-Poisson vector fields on P/G, given by equation (8.8). By Theorem 3.4.5, orbits of P(P/G) are smooth manifolds. Since P(P/G) ⊆ X(P/G), for each x ∈ P/G, the orbit Q of P(P/G) through x is a manifold immersed in the stratum N of P/G that contains x. Moreover, for each f¯ ∈ C ∞ (D/G), the restriction Y f¯|Q of the almost-Poisson vector field of f¯ to Q is a vector field on Q. The set D Q = {Y f¯ (x) | x ∈ Q, f¯ ∈ C ∞ (D/G)}

(8.9)

214

Further examples of reduction

is a generalized distribution on Q locally spanned by vector fields. For each x ∈ Q, consider a form Q on D Q defined by Q (Y f¯ (x), Y f¯ (x)) = (Y f¯ ( f¯))(x).

(8.10)

Proposition 8.1.3 Equation (8.10) gives a well-defined symplectic form Q on the generalized distribution D Q . Proof Suppose that f¯ and f¯ are functions in C ∞ (P/G) such that Y f¯ (x) = Y f¯ (x). Then (Y f¯ ( f¯))(x) = (Y f¯ (x))( f¯) = (Y f¯ (x))( f¯) = (Y f¯ ( f¯))(x), which implies that Q (Y f¯ (x), Y f¯ (x)) does not depend on the choice of f¯ in equation (8.10). Equation (8.6) implies that Q is skew-symmetric. Hence, Q (Y f¯ (x), Y f¯ (x)) depends on Y f¯ (x) but not on the choice of f¯ defining Y f¯ . Therefore, Q is well defined. It remains to show that Q is non-singular. Suppose Y f¯ (x) is a vector in D Q such that Q (Y f¯ (x), Y f¯ (x)) = 0 for all vectors Y f¯ (x) ∈ D Q ∩ Tx Q. Then, (Y f¯ (x))( f¯) = 0 for all f¯ ∈ C ∞ (P/G). But Q is an orbit of the family P(P/G) of almost-Poisson vector fields. Hence, for each function f˜ ∈ C ∞ (Q), there exist a neighbourhood U of x in Q and a function f¯ ∈ C ∞ (P/G) such that f˜|U = f¯|U . Thus, (Y f¯ (x))( f˜) = 0 for all f˜ ∈ C ∞ (Q). Therefore, Y f¯ (x) = 0 ∈ Tx Q, and Q is non-singular. A distributional Hamiltonian system on a manifold P is a symplectic distribution (D, ) on P and a Hamiltonian h ∈ C ∞ (P). The evolution of the system is given by integral curves of the distributional Hamiltonian vector field Yh of the Hamiltonian h. The action of G on P is a symmetry of the distributional Hamiltonian system (D, , h) on P if preserves H , and h. ¯ where h¯ ∈ C ∞ (P/G) is called In this case h ∈ C ∞ (P)G , so that h = ρ ∗ h, the reduced Hamiltonian. The vector field Yh pushes forward to a vector field Yh¯ on P/G. Consider the maximal integral curve c : I → P of Yh through p = c(0) ∈ P. Let Q be the orbit of P(P/G) through ρ( p), and let h¯ |Q be the restriction of h¯ to Q. The reduced motion c¯ = ρ ◦ c : I → Q is the maximal integral curve of the distributional Hamiltonian vector field Yh¯ |Q on Q defined in terms of the symplectic distribution (D Q , Q ).

8.2 Dirac structures The Pontryagin bundle of a manifold Q is the direct sum P of the tangent and cotangent bundles of Q. A Dirac structure is an isotropic subbundle D; it describes a class of dynamical systems. For example, Hamiltonian systems on

8.2 Dirac structures

215

Q corresponding to a given symplectic form ω on Q are elements of the Dirac structure {(u, p) ∈ T Q ⊕ T ∗ Q | p = u ω}. Similarly, distributional Hamiltonian systems corresponding to a distribution H and a symplectic form on H are elements of the Dirac structure {(u, p) ∈ H ⊕ T ∗ Q | p|H = u }.

8.2.1 Symmetries of the Pontryagin bundle The Pontryagin bundle of a manifold Q is the direct sum T Q ⊕ T ∗ Q of the tangent and cotangent bundles of Q. It is naturally isomorphic to the fibre product P = T Q × Q T ∗ Q. Let τ : T Q → Q and ϑ : T ∗ Q → Q be the tangent and the cotangent bundle projections, respectively, and let π : P = T Q × Q T ∗ Q → Q : (u, p) → π(u, p) = (τ (u), ϑ( p)). The Pontryagin bundle carries a symmetric form ·, ·

defined as follows. For each (u 1 , p1 ) and (u 2 , p2 ) in the same fibre of π , (u 1 , p1 ), (u 2 , p2 )

= p1 | u 2 + p2 | u 1 .

(8.11)

The form ·, ·

is indefinite, with signature (dim Q, dim Q). Moreover, the space (P) of smooth sections of the Pontryagin bundle carries a bilinear skew-symmetric bracket [(X, α), (Y, β)] = ([X, Y ], £ X β − £Y α + 12 d(α(Y ) − β(X ))),

(8.12)

called the Courant bracket. A Dirac structure on Q is a maximal isotropic subbundle D of (P, ·, ·

); that is, D is a subbundle of P = T Q × Q T ∗ Q, the form ·, ·

vanishes on all pairs of elements of D, and rank D = dim Q. We denote the inclusion map by ι : D → P and the projection of D onto Q by δ = π ◦ ι : D → Q. The space (D) of smooth sections of D consists of sections (X, α) of P = T Q × Q T ∗ Q with values in D. We assume that D is locally spanned by smooth sections (X 1 , α1 ), . . . , (X m , αm ). A Dirac structure is said to be closed (or integrable) if, for every pair (X, α) and (Y, β) of sections of D, the bracket [(X, α), (Y, β)] is a section of D.1 Let : G × Q → Q : (g, x) → g (x) = gx be an action of a connected Lie group G on the manifold Q. This induces an action T : G × T Q → T Q : (g, u) → T g (u) 1 There is an alternative terminology also used in the literature, in which the ‘Dirac structure’

defined here is called an ‘almost-Dirac structure’, whereas a ‘closed Dirac structure’ is called simply a ‘Dirac structure’.

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Further examples of reduction

of G on the tangent bundle T Q of Q. The push-forward of a vector field X on Q by g is given by (g )∗ X = T g ◦ X ◦ g −1 , where we treat X as a section of the tangent bundle projection τ : T Q → Q. A vector field X is G-invariant if (g )∗ X = X for each g ∈ G. Similarly, we have an induced action T ∗ : G × T ∗ Q → T ∗ Q : (g, p) → T ∗ g ( p), where T ∗ g ( p) | u = p | T g−1 (u)

for every pair (u, p) ∈ P = T Q × Q T ∗ Q. This definition implies that the action of G on P preserves the evaluation. In other words, T ∗ g ( p) | T g (u) = p | u

(8.13)

for all g ∈ G. If α is a 1-form on Q, considered as a section of the cotangent bundle ϑ : T ∗ Q → Q, then (g )∗ α = T ∗ g ◦ α ◦ g −1 is a section of ϑ, which we shall also call the push-forward of α by g . A form α is G-invariant if (g )∗ α = α for every g ∈ G. For every vector field X and a 1-form α on Q, (g )∗ α | (g )∗ X (x) = ((g )∗ α)(x) | ((g )∗ X )(x)

= T ∗ g (α(g −1 x)) | T g (X (g −1 x))

= α(g −1 x) | X (g −1 x) = α | X (g −1 (x)) = ∗g −1 α | X (x). Therefore, (g )∗ α = ∗g−1 α. The product of T and T ∗ gives rise to the action : G × P → P : (g, (u, p)) → g (u, p) = (T g (u), T ∗ g ( p)). (8.14) For a section σ = (X, α) of π : P → Q, we denote by (g )∗ σ the section of π given by (g )∗ σ = g ◦ σ ◦ g −1 = ((g )∗ X, (g )∗ α) = ((g )∗ X, ∗g−1 α). (8.15) A section σ of π is G-invariant if (g )∗ σ = σ for each g ∈ G. We consider here a Dirac structure D ⊂ P that is invariant under the action of G on P.

8.2 Dirac structures

217

8.2.2 Free and proper action If the action of G on Q is free and proper, the action of G on the Pontryagin bundle is also free and proper. Therefore, Q is a left principal fibre bundle with structure group G, base manifold Q/G and projection map ρ Q : Q → Q/G.2 Similarly, P is a left principal G-bundle with base manifold P/G and projection map ρ P : P → P/G. Since the Pontryagin bundle projection π : P → Q intertwines the actions of G on P and on Q, that is, for each g ∈ G, π ◦ g = g ◦ π , it follows that there exists a map π : P/G → Q/G such that the following diagram commutes: P π

Q

ρP

/ P/G π

ρQ

/ Q/G.

Moreover, the action on P is linear on fibres of the projection π . This implies, as in Proposition 7.2.14, that π : P/G → Q/G is a vector bundle (in the category of manifolds). In general, we have the following result. Remark 8.2.1 Let ε : E → M be a vector bundle, and let G be a Lie group acting freely and properly on E by vector bundle automorphisms. This means that the action of G on E covers a free and proper action of G on M, and the action of G on fibres of ε is linear. Then, the space E/G of G-orbits in E has the structure of a vector bundle over the space M/G of the space of G-orbits in M. Proof The proof of this remark is analogous to the proof of part (i) of Proposition 7.2.14. If σ = (X, α) : Q → P is a G-invariant section of π , then there exists a section σ = (X , α) : Q/G → P/G of π such that the following diagram commutes: PO

ρP

σ

/ P/G O σ

Q

ρQ

/ Q/G.

Here, X : Q/G → T (Q/G) is the section induced by the G-invariant vector field X , and α : Q/G → T ∗ (Q/G) is the section induced by the G-invariant 2 In order to keep the notation simple, since there are several spaces involved here, we do not

assign separate symbols to the quotients.

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Further examples of reduction

1-form α. In order to give an interpretation of the sections X and α, we introduce a connection on Q. A connection on the principal bundle Q is a G-invariant distribution hor T Q that is complementary to the vertical distribution ver T Q = ker T π . This implies that we have a direct-sum decomposition T Q = ver T Q ⊕ hor T Q.

(8.16)

Every vector u ∈ Tq Q can be decomposed into a vertical part ver u and a horizontal part hor u; that is, u = ver u + hor u. Similarly, every covector p ∈ Tq∗ Q can be decomposed into a vertical part ver p and a horizontal part hor p such that p | u = ver p + hor p | ver u + hor u = ver p | ver u + hor p | hor u . (8.17) This gives rise to a decomposition of the cotangent bundle T ∗ Q = ver T ∗ Q ⊕ hor T ∗ Q.

(8.18)

The decompositions (8.16) and (8.18) lead to a decomposition of the Pontryagin bundle P = ver P ⊕ hor P, where the vertical Pontryagin bundle ver P and the horizontal Pontryagin bundle hor P are given by ver P = ver T Q ⊕ ver T ∗ Q

and

hor P = hor T Q ⊕ hor T ∗ Q.

Equations (8.11) and (8.17) imply that the bilinear form on P decomposes into its vertical and horizontal components (u 1 , p1 ), (u 2 , p2 )

= (ver u 1 , ver p1 ), (ver u 2 , ver p2 )

+ (hor u 1 , hor p1 ), (hor u 2 , hor p2 )

.

(8.19) (8.20)

We denote the restrictions of ·, ·

to ver P and hor P by ver ·, ·

and hor ·, ·

, respectively. Equation (8.19) implies that ·, ·

= ver ·, ·

+ hor ·, ·

.

(8.21)

The Courant bracket (8.12) on the space of sections of P does not decompose into horizontal and vertical parts, because the bracket of a horizontal section of P with a vertical section of P need not vanish. The orbit spaces (ver P)/G and (hor P)/G are vector bundles over P/G. We call (ver P)/G the reduced vertical Pontryagin bundle and (hor P)/G the reduced horizontal Pontryagin bundle. The adjoint bundle of a principal fibre bundle Q is Q[g] = (Q × g)/G. Similarly, the co-adjoint bundle of Q is Q[g∗ ] = (Q × g∗ )/G.3 3 A comprehensive discussion of the associated bundles of a principal bundle can be found in

Kobayashi and Nomizu (1963).

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219

Proposition 8.2.2 (i) The reduced vertical Pontryagin bundle (ver P)/G is isomorphic to the direct sum Q[g] ⊕ Q[g∗ ] of the adjoint and co-adjoint bundles of Q. (ii) The reduced horizontal Pontryagin bundle is isomorphic to the Pontryagin bundle of the orbit space Q/G. Proof (i) For each ξ ∈ g, we denote by X ξ the vector field on Q generating the action on Q of the one-parameter subgroup exp tξ of G. We shall refer to X ξ as the fundamental vector field corresponding to ξ . For each g ∈ G, we have g(exp tξ )g −1 = exp t Adg ξ . Hence, d ξ ξ −1 −1 exp tξ (g q) ((g )∗ X )(q) = T g (X (g (q))) = T g dt |t=0 d d = g (exp tξ (g −1 q))|t=0 = exp t Adg ξ (q)|t=0 dt dt = X Adg ξ (q). The vertical distribution ver T Q consists of values of vertical vector fields; that is, ver T Q = {X ξ (q) | ξ ∈ g, q ∈ g}. The connection 1-form on Q is a g-valued 1-form θ such that θ (X ξ ) = ξ for every ξ ∈ g. For each g ∈ G, θ (T g ◦ X ξ ◦ g −1 ) = θ ((g )∗ X ξ ) = θ (X Adg ξ ) = Adg ξ . Hence, θ ◦ T g = Adg ◦ θ ; that is, θ intertwines the action of G on ver T Q and the adjoint action of G on g. Consider the map (τ, θ ) : ver T Q → Q × g : u → (τ (u), θ (u)). This is an isomorphism of the vector bundles ver T Q and Q × g over Q. For each g ∈ G and u ∈ ver T Q, we have (τ, θ )(T g (u)) = (τ (T g (u)), θ (T g (u))) = (gτ (u), Adg θ (u)). Hence, (τ, θ ) : ver T Q → Q × g intertwines the action of G on ver T Q with the action G × (Q × g) → Q × g : (g, (q, ξ )) → (gq, Adg ξ ). Therefore, (τ, θ ) induces an isomorphism (τ, θ ) : (ver T Q)/G → (Q × g)/G of the corresponding orbit spaces. In other words, (ver T Q)/G is isomorphic to the adjoint bundle Q[g∗ ]. Now consider the vertical subbundle of T ∗ Q given by ver T ∗ Q = {ver p | p ∈ T ∗ Q}. Let ϕ : ver T ∗ Q → g∗ be a map such that for each p ∈ ver Tq∗ Q and ξ ∈ g∗ ,

220

Further examples of reduction ϕ( p) | ξ = p | X ξ (q) .

For each g ∈ G, ϕ(T ∗ g ( p)) | ξ = T ∗ g ( p) | X ξ (gq) = p | T g −1 (X ξ (gq))

= p | X

Adg−1 ξ

(q) = ϕ( p) | Adg−1 ξ = Adg∗ ϕ( p) | ξ .

Hence, ϕ intertwines the action of G on ver T ∗ Q and the co-adjoint action of G on g∗ . Hence, the map (τ, ϕ) : ver T ∗ Q → Q × g∗ : p → (τ ( p), ϕ( p)) is a G-equivariant isomorphism of vector bundles. Therefore, (τ, ϕ) induces an isomorphism of the orbit spaces (ver T ∗ Q)/G and Q × g∗ )/G, which we denote by (τ, ϕ) : (ver T ∗ Q)/G → (Q × g∗ )/G. The orbit space (Q × g∗ )/G is a vector bundle over Q/G, called the co-adjoint bundle of Q and denoted by Q[g∗ ]. This is the dual bundle of the adjoint bundle Q[g]. The preceding discussion shows that (τ, θ, ϕ) : ver P → Q × g ⊕ g∗ : p → (τ ( p), θ ( p), ϕ( p)) is a isomorphism of vector bundles which preserves the direct-sum structure of the fibres and intertwines the actions of G on ver P and on Q × g ⊕ g∗ . Hence, (τ, θ, ϕ) induces an isomorphism (τ, θ, ϕ) : (ver P)/G → (Q × g ⊕ g∗ )/G = Q[g] ⊕ Q[g∗ ] of bundles over Q/G. (ii) Consider next the horizontal distribution hor T Q on Q. The tangent map Tρ Q maps hor T Q onto T (Q/G). For each g ∈ G, Tρ Q ◦ T g = Tρ Q . Moreover, Tρ Q (u) = 0 if and only if u ∈ ver T Q. Hence, Tρ Q induces a vector bundle isomorphism η : (hor T Q)/G → T (Q/G). −1 (x), we denote by liftq w the For each w ∈ Tx (Q/G) and q ∈ ρ Q unique vector in hor Tq Q that projects on w; that is, Tρ Q (liftq w) = w. The vector liftq w is called the horizontal lift of w to q. The dual map η ∗ : (hor T ∗ Q)/G → T ∗ (Q/G) of η is defined as follows. For each p ∈ Tq∗ Q and w ∈ Tρ Q (q) Q, we have η∗ ( p) | w = p | liftq w . The map (η⊕η∗ ) : (hor P)/G = (hor T Q)/G⊕(hor T ∗ Q)/G → T (Q/G)⊕T ∗ (Q/G) is an isomorphism of bundles over Q/G. The bundle T (Q/G) ⊕ T ∗ (Q/G) is the Pontryagin bundle of the orbit space Q/G. Corollary 8.2.3 The reduced Pontryagin bundle P/G is isomorphic to the direct sum of Q[g] ⊕ Q[g∗ ] and T (Q/G) ⊕ T ∗ (Q/G).

8.2 Dirac structures

221

The bilinear form ·, ·

on P is G-invariant and gives rise to a bilinear form ·, ·

P/G on the reduced Pontryagin bundle such that p1 , p2

= ρ P ( p1 ), ρ P ( p2 )

P/G

(8.22)

for every p1 , p2 in the same fibre of P. Similarly, ver ·, ·

defines a bilinear form ver ·, ·

P/G in (ver P)/G, and hor ·, ·

defines a bilinear form hor ·, ·

P/G in (hor P)/G. Equations (8.21) and (8.22) imply that ·, ·

P/G = ver ·, ·

P/G ⊕ hor ·, ·

P/G .

(8.23)

The Courant bracket (8.12) evaluated on G-invariant sections of P → Q gives a G-invariant section of P → G. Hence, there is a bracket [·, ·] P/G on the space (P/G) of sections of π : P/G → Q/G such that if σ1 and σ2 are G-invariant sections of P → G, then [σ 1 , σ 2 ] P/G = [σ1 , σ2 ].

(8.24)

The bracket [σ1 , σ2 ] of two sections of P, σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ), can be decomposed into a vertical component ver [σ1 , σ2 ] and a horizontal component hor [σ1 , σ2 ]. If σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ) are G-invariant sections of ver P, then ver [σ1 , σ2 ] is a G-invariant section of ver P. Therefore, ver [σ1 , σ2 ] projects to a section of (ver P)/G → Q/G, which we denote by ver [σ 1 , σ 2 ] P/G ; this depends only on the sections σ 1 and σ 2 . Hence, the bracket ver [·, ·] gives rise to a bracket ver [·, ·] P/G on the space of sections (ver P)/G. Similarly, if σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ) are G-invariant sections of hor P, then hor [σ1 , σ2 ] is a G-invariant section of hor P. Therefore, hor [σ1 , σ2 ] projects to a section of (hor P)/G → Q/G, which we denote by hor [σ 1 , σ 2 ] P/G ; this depends only on the sections σ 1 and σ 2 . Hence, the bracket hor [·, ·] gives rise to a bracket hor [·, ·] P/G on the space of sections (hor P)/G. Lemma 8.2.4 We assume that the action of G on Q is free and proper. A G-invariant Dirac structure D ⊂ P = T Q ⊕ T ∗ Q is locally spanned by G-invariant sections. Proof Consider q0 ∈ Q, and let m = dim Dq0 = dim Q. Since D is smooth, there exists an m-tuple ((X 1 , α 1 ), . . . , (X m , α m )) of smooth sections of D that span D in a neighbourhood W of q0 . Since Q is a principal bundle over Q/G, there exists a local section σ : U → Q of ρ Q : Q → Q/G passing through q0 . Let S = σ (U ). Without loss of generality, we may assume that S is contained in W. Let m 1 1 m m ((X 1S , α S1 ), . . . , (X m S , α S )) be the restrictions of ((X , α ), . . . , (X , α )) to

222

Further examples of reduction

points of S. These are sections of TS Q ⊕ TS∗ Q over S. Moreover, they span the restriction D S of D to S. Let f : S → R be a bump function with compact support equal to 1 in a neighbourhood S0 of q0 in S. Then, m (( f X 1S , f α S1 ), . . . , ( f X m S , f α S )) span D S in S0 . Since Q is a principal G-bundle over Q/M and σ : U → M is a section of −1 (U ) → G × S intertwining ρ Q : Q → Q/G, there is a diffeomorphism δ : ρ Q

−1 the action of G on ρ Q (U ) and multiplication on the left in G. Without loss of generality, we may assume that δ maps each point s ∈ S to (e, s), where e is −1 (U ) the identity in G. For each i = 1, . . . , m, let Y˜ i be a vector field on ρ Q −1 i i defined as follows. For each q ∈ ρ (U ), Y˜ (q) = T g ( f (s)X (s)), where Q

−1 q ∈ δ −1 (g, s) ∈ (G, S). Clearly, Y˜ i is a G-invariant vector field on ρ Q (U ) −1 i i ˜ extending X S . Similarly, we define a G-invariant 1-form β on ρ Q (U ) by setting β˜ i (q) | u = f (s)α iS (s) | T g −1 u ∀ u ∈ Tq Q.

Since f has compact support in S, the vector fields Y˜ i and the forms β˜ i vanish −1 (U ) in Q. Hence, they can be extended by zero to near the boundary of ρ Q G-invariant globally defined vector fields Y i and 1-forms β i on Q. In other words,

−1 (U ) Y˜ i (q) for q ∈ ρ Q i Y (q) = −1 0 for q ∈ / ρ Q (U ),

−1 β˜ i (q) for q ∈ ρ Q (U ) β i (q) = −1 0 for q ∈ / ρ Q (U ). m Since (( f X 1S , f α S1 ), . . . , ( f X m S , f α S )) span D S0 and D is G-invariant, it −1 1 1 m m follows that ((Y , β ), . . . , (Y , β )) span D in ρ Q (ρ Q (S0 )).

Proposition 8.2.5 Let G be a Lie group acting freely and properly on a manifold Q, and let D be a G-invariant Dirac structure on Q. (i) The space D/G of G-orbits in D is a maximal isotropic subbundle of (P/G, ·, ·

P/G ). (ii) If the Dirac structure D is closed in the sense that, for each pair σ1 and σ2 of G-invariant sections of D → Q, the bracket [σ1 , σ2 ] has values in D, then [σ 1 , σ 2 ] P/G has values in D/G for every pair of sections σ 1 , σ 2 of (D/G) → Q/G. Proof By Corollary 8.2.3, Q/G is a manifold and P/G is a vector bundle over Q/G isomorphic to the direct sum of Q[g] ⊕ Q[g∗ ] and the Pontryagin bundle of Q/G. By assumption, the Dirac structure D is a G-invariant

8.2 Dirac structures

223

subbundle of the Pontryagin bundle P. Moreover, G acts freely and properly on Q, and it acts freely and properly on P by vector bundle automorphisms. By Lemma 8.2.4, D is locally spanned by G-invariant sections. Following an argument analogous to that in the proof of part (i) of Proposition 7.2.14, we conclude that the orbit space D/G is a locally trivial vector bundle over Q/G; see Remark 8.2.1. Equation (8.22) ensures that D/G is isotropic with respect to the bilinear form ·, ·

P/G . Consider a G-invariant section σ of P such that for some q ∈ Q, σ (ρ Q (q)), σ 1 (ρ Q (q))

P/G = 0 for all G-invariant sections σ1 of D. Equation (8.22) implies that σ (q), σ1 (q)

= 0 for all G-invariant sections σ1 of D. Since D is maximal, it follows that σ (q) ∈ D. Hence, there exists a G-invariant section σ of D such that σ (q) = σ (q). Therefore, σ (ρ Q (q)) = σ (ρ Q (q)). This ensures that D/G is a maximal isotropic subbundle of (P/G, ·, ·

P/G ). This completes the proof of part (i). Part (ii) follows from equation (8.24).

8.2.3 Proper non-free action In this subsection, we drop the assumption that the action of G on Q is free. We begin by showing that the freeness of the action of G on Q is not a necessary condition for the conclusion of Lemma 8.2.4. Proposition 8.2.6 A Dirac structure D ⊂ P = T Q ⊕ T ∗ Q, invariant under a proper action of a connected Lie group G on Q, is locally spanned by G-invariant sections. Proof Consider q0 ∈ Q, and let m = dim Dq0 = dim Q. Since D is smooth, there exists an m-tuple (σ 1 , . . . , σ m ) of smooth sections of D that span D in a neighbourhood W of q0 . Let H be the isotropy group of q0 , and let S be the slice at q0 for the action of G on Q. Without loss of generality, we may assume that S is contained in W . The restrictions σ S1 , . . . , σ Sm of σ 1 , . . . , σ m to S span D in S. Let f : S → R be a bump function with compact support equal to 1 in a neighbourhood V0 of q0 in S. The sections f σ1 , . . . , f σn have compact support in S and span D in V0 . Since D is G-invariant, the restriction of D to S is H -invariant. Averaging the sections f σ S1 , . . . , f σ Sm over H , we obtain H -invariant sections σ˜ S1 , . . . , σ˜ Sm of D, given by

224

Further examples of reduction σ˜ Si =

H

(g )∗ ( f σ Si ) dμ(g)

for each i = 1, . . . , m. The sections σ˜ S1 , . . . , σ˜ Sm have compact support in S and span D in a neighbourhood of q0 in S. The sections σ˜ S1 , . . . , σ˜ Sm can be extended to G-invariant sections σ˜ U1 , . . . , σ˜ Um of D in a G-invariant neighbourhood U = G S of q0 in Q, which are defined as follows. Each q ∈ U is of the form q = gs, where g ∈ G and s ∈ S. For each i = 1, . . . , m, and q ∈ U , we set σ˜ Ui (q) = σ˜ Ui (gs) = g (σ Si (s)). If g1 s1 = g2 s2 for g1 , g2 ∈ G and s1 , s2 ∈ S, then s1 = g1−1 g2 s2 , which implies that h = g1−1 g2 ∈ H . Hence, g1 (σ Si (s1 )) = g1 (σ Si (hs2 )) = g1 (h (σ Si (s2 ))) because σ Si is H -invariant. Moreover, g1 (h (σ Si (s2 ))) = g1 h (σ Si (s2 )) = g1 g−1 g2 (σ Si (s2 )) = g2 (σ Si (s2 )), 1

which implies that σ˜ Ui is well defined. Also, for each q = gs ∈ Q and g¯ ∈ G, (( g¯ )∗ σ˜ Ui )(gs) = g¯ ◦ σ˜ Ui ◦ g¯ −1 (gs) = g¯ (σ˜ Ui (g¯ −1 gs)) = g¯ (g¯ −1 g (σ Si (s))) = g¯ g¯ −1 g (σ Si (s)) = g (σ Si (s)) = σ˜ Ui (gs). Therefore, σ˜ Ui is G-invariant. We can extend each section σ˜ Ui to a global section σ˜ i of π : P → Q by setting σ˜ i (q) = 0 for every q in the complement of U = G S in P. Since U is G-invariant and σ˜ Ui is G-invariant, it follows that σ˜ i is G-invariant. Moreover, the support of σ˜ Ui is the G-orbit of the support of σ˜ Si , which is compact in S. Hence, σ˜ i is a smooth global section. Finally, each section σ˜ i has values in D because D is G-invariant. We have obtained smooth, G-invariant, global sections σ˜ 1 , . . . , σ˜ m of π : P → Q, with values in D, which span D in a neighbourhood of q0 . This completes the proof. Lemma 8.2.7 Let σ = (X, α) be a G-invariant section of the Pontryagin bundle P of Q. Given q ∈ Q, let H be the isotropy group of q and let Q H = {q ∈ Q | G q = H }. Let k be a G-invariant Riemannian metric on Q,, and let Tq Q ⊥ H be the k-orthogonal complement of Tq Q H in Tq Q. Then, X (q) ∈ Tq Q H and α(q) annihilates Tq Q ⊥ H.

8.2 Dirac structures

225

Proof For q ∈ Q, let H be the isotropy group of q, and let Q H ⊆ Q be the set of fixed points of the action of H in Q. According to Proposition 4.2.5, Q H is a local submanifold of Q, and Tq Q H = (Tq Q) H is the space of H -invariant vectors in Tq Q. It follows from the proof of Proposition 4.2.6 that Q H is an open subset of Q H . Hence, Tq Q H = Tq Q H = (Tq Q) H . Therefore, if X is a G-invariant vector field on Q, then X (q) is invariant under the action of H on Tq Q, which implies that X (q) ∈ Tq Q H . A G-invariant Riemannian metric k on Q gives rise to a unique vector bundle isomorphism k$ : T ∗ Q → T Q such that if p ∈ Tq∗ Q, then k$ ( p) ∈ Tq Q and k(k$ ( p), u) = p | u . Since α and k are G-invariant, Y = k$ ◦ α is a G-invariant vector field on Q. Hence, Y (q) ∈ Tq Q H . Therefore, for each u ∈ Tq Q ⊥ H, α(q) | u = k(k$ (α( p)), u) = k(Y ( p), u) = 0, which completes the proof. Theorem 8.2.8 Let D ⊂ P = T Q ⊕ T ∗ Q be a Dirac structure. We assume that D is invariant under a proper action of a connected Lie group G on Q. Let H be a compact subgroup of G, and let L be a connected component L of Q H . (i) If (X, α) and (Y, β) are G-invariant sections of P, then their restrictions to points of L give rise to sections (X L , α L ) and (Y L , β L ) of the Pontryagin bundle PL of L that are invariant under the action of G L on L, where G L is defined by equation (4.7). (ii) The restriction to L of the Courant bracket of the G-invariant sections (X, α) and (Y, β) gives the Courant bracket of the sections (X L , α L ) and (Y L , β L ) of PL ; that is, [(X, α), (Y, β)] L = [(X L , α L ), (Y L , β L )]. (iii) The Dirac structure D on Q restricts to a Dirac structure D L on L. (iv) The Dirac structure D is uniquely determined by the collection of all Dirac structures D L as L varies over connected components of Q H and H varies over compact subgroups of G. Proof (i) Let L be a connected component of Q H , where H is a compact subgroup of G. By Proposition 4.2.6, L is a submanifold of Q. By Proposition 8.2.6, every p ∈ D is in the image of a smooth G-invariant section σ = (X, α) of D. In other words, p = (X (q), α(q)), where q = π( p). Proposition 4.2.6 ensures that X (q) ∈ Tq L. Let α L be the pull-back of α to L, and let X L be the

226

Further examples of reduction

restriction of X to points in L. Then, (X L , α L ) is a section of the Pontryagin bundle PL = T L ⊕ T ∗ L. (ii) If (X, α) and (Y, β) are G-invariant, then [X, Y ] is G-invariant, and [X, Y ] restricted to points in L is a vector field [X, Y ] L tangent to L. Moreover, [X, Y ] L = [X L , Y L ]. Similarly, all forms appearing in the expression £ X β − £Y α + 12 d(α(Y ) − β(X )) are G-invariant. Hence, 1 £ X β − £Y α + d(α(Y ) − β(X )) 2 L 1 = (£ X β) L − (£Y α) L + (d(α(Y ) − β(X )) L ) 2 1 = £ X L β L − £YL α L + d(α L (Y L ) − β L (X L )). 2 Therefore, [(X, α), (Y, β)] L = [(X L , α L ), (Y L , β L )]. (iii) We denote by D L the subset of PL spanned by the sections (X L , α L ), which are the restrictions to L of G-invariant sections of D. Let ·, ·

L be the bilinear form of the Pontryagin bundle PL of L. If (X, α) and (X , α ) are G-invariant sections of D, then for each q ∈ L, (X L , α L ), (X L , α L )

L (q) = α L (q) | X L (q) + α L (q) | X L (q)

= α(q) | X (q) + α (q) | X (q)

= (X, α), (X , α )

(q) = 0. This implies that D L is isotropic. Suppose now that (u L , a L ) ∈ Tq L ⊕Tq∗ L satisfies (u L , a L ), (X L (q), α L (q))

L = 0 for every G-invariant section (X, α) of D. For each q ∈ Q, let Tq L ⊥ be the k-orthogonal complement of Tq L in Tq Q, where k is a G-invariant Riemannian metric on Q. We can extend a L ∈ Tq∗ L to a covector a ∈ Tq∗ Q by setting a | v = 0 for each v ∈ Tq L ⊥ . Since Tq L ⊆ Tq Q, the vector u L ∈ Tq L is in Tq Q, and (u L , a) ∈ P. For each G-invariant section (X, α) of D, (X (q), α(q)), (u L , a)

= α(q) | u L + a | X (q)

= α L (q) | u L + a L | X L (q)

= (X L (q), α L (q)), (u L , a L )

L = 0. Hence, (u L , a) ∈ D because D is a maximal isotropic subbundle of (P, ·, ·

). Therefore, (u L , a L ) ∈ D L , which implies that D L is a maximal isotropic subbundle of (PL , ·, ·

L ). Thus, D L is a Dirac structure on L. (iv) We still need to prove that D is uniquely determined by the collection of all Dirac structures D L as L varies over connected components of

8.2 Dirac structures

227

Q H and H varies over compact subgroups of G. It suffices to show that for each q ∈ L ⊆ Q H and a G-invariant section (X, α) of D, the value of α at q is uniquely determined by α L (q), where α L is the pull-back of α to L. Lemma 8.2.7 implies that α(q) | v = 0 for each v ∈ Tq L ⊥ . Hence, α(q) is determined by its restriction α L (q) to Tq L. In Proposition 4.2.6, we showed that the action of G L = N L /H on L is free and proper. Here, N L is the closed subgroup of G consisting of elements g ∈ G that preserve the manifold L. Since the Dirac structure D is G-invariant, it follows that the Dirac structure D L is G L -invariant. Hence, we can apply the results obtained in the preceding subsection to the Dirac structure D L on L. Corollary 8.2.9 For each compact subgroup H of G and each connected component L of Q H , we have the following result. (i) The reduced Pontryagin bundle PL /G L decomposes into a direct sum of the reduced vertical Pontryagin bundle (ver PL )/G L and the reduced horizontal Pontryagin bundle (hor PL )/G L . (ii) (ver PL )/G L is isomorphic to L[g L ]⊕ L[g∗L ], where g L is the Lie algebra of G L , and (hor PL )/G L is isomorphic to T (L/G L ) ⊕ T ∗ (L/G L ). (iii) The reduced Dirac structure D L /G L is a maximal isotropic subbundle (PL /G L , ·, ·

PL /G L ). (iv) If the Dirac structure D L is closed, then the Dirac structure D L /G L is closed. Recall that the connectedness of G implies that M = G L is the stratum of the orbit type stratification M of Q. Hence, the stratum N = ρ(M) of the orbit type stratification of Q/G can be identified with L/G L . Therefore, the restriction D M of D to M can be identified with the G-orbit G D L of D L in P, which implies that D M /G = (G D L )/G = D L /G L . Thus, for each stratum N of the orbit type stratification of Q/G, we have a complete description of D/G restricted to N .

References

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Ortega, J.-P. and T.S. Ratiu (2002), ‘The optimal momentum map’. In Geometry, Mechanics and Dynamics: Volume in Honor of the 60th Birthday of J.E. Marsden, P. Newton, P. Holmes and A. Weinstein (eds.), Springer, Berlin, pp. 329–362. Ortega, J.-P. and T.S. Ratiu (2004), Momentum Maps and Hamiltonian Reduction, Birkhäuser, Boston. Palais, R. (1961), ‘On the existence of slices for actions of noncompact Lie groups’, Ann. of Math. 73 295–323. Pasternak-Winiarski, Z. (1984), ‘On some differential structures defined by actions of groups’. In Proceedings of the Conference on Differential Geometry and Its Applications, Part 1 (Nove Mesto na Morave, 1983), Charles University, Prague, pp. 105–115. Pflaum, M.J. (2001), Analytic and Geometric Study of Stratified Spaces, Springer, Berlin. Pukanszky, L. (1971), ‘Unitary representations of solvable groups’, Ann. Sci. Ecole Normale Sup. 4 457–608. Sasin, W. (1986), ‘On some exterior algebra of differential forms over a differential space’, Demonstratio Math. 19 1063–1075. Satake, I. (1957), ‘The Gauss–Bonnet theorem for V-manifolds’, J. Math. Soc. Japan 9 464–492. van der Schaft, A.J. and B.M. Maschke (1994), ‘On the Hamiltonian formulation of nonholonomic mechanical systems’, Rep. Math. Phys. 34 225–233. Schwarz, G. (1975), ‘Smooth functions invariant under the action of compact Lie groups’, Topology 14 63–68. Sikorski, R. (1967), ‘Abstract covariant derivative’, Colloq. Math. 18 252–272. Sikorski, R. (1972), Wst˛ep do Geometrii Ró˙zniczkowej, PWN, Warsaw. Sjamaar, R. (1995), ‘Holomorphic slices, symplectic reduction and multiplicities of representations’, Ann. of Math. 141 87–129. ´ Sniatycki, J. (1980), Geometric Quantization and Quantum Mechanics, Applied Mathematical Science, vol. 30, Springer, New York. ´ Sniatycki, J. (1983), ‘Constraints and quantization’. In Non-Linear Partial Differential Operators and Quantization Procedures: Proceedings of a Workshop Held at Clausthal, Federal Republic of Germany, 1981, S. Andersson and H. Doebner (eds.), Lecture Notes in Mathematics, vol. 1037, Springer, Berlin, pp. 301–334. ´ Sniatycki, J. (2003a), ‘Integral curves of derivations on locally semi-algebraic differential spaces’. In Dynamical Systems and Differential Equations (Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24–27, 2002, Wilmington, NC, USA), W. Feng, S. Hu and X. Lu (eds.), American Institute of Mathematical Sciences Press, Springfield, MO, pp. 825–831. ´ Sniatycki, J. (2003b), ‘Orbits of families of vector fields on subcartesian spaces’, Ann. Inst. Fourier (Grenoble) 53 2257–2296. ´ Sniatycki, J. (2005), ‘Poisson algebras in reduction of symmetries’, Rep. Math. Phys. 56 53–73. ´ Sniatycki, J. (2011), ‘Reduction of symmetries of Dirac structures’, J. Fixed Point Theory Appl. 10 339–358.

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Index

adjoint action, 105 algebraic reduction, 138 almost-Poisson algebra, 211 almost-Poisson bracket, 211 Arms, J., 6, 8, 142 Aronszajn, N., 2 Bates, L., 4, 6–10 Bierstone, E., 5 Blankenstein, G., 11 Bleuler, K., 9 Bochner, S., 5 Borel Extension Lemma, 145 Buchner, K., 2 Bursztyn, H., 11 Cavalcanti, G.R., 11 Cegiełka, K., 3 co-adjoint action, 106 co-adjoint orbit, 106 commutation of quantization and reduction, 151 connection, 152 on principal bundle, 218 cotangent polarization, 158 Courant, T., 11 covariant derivative, 152 Cushman, R., 4–7, 9, 10, 105 density, 159 derivation, 25 at a point, 26 derived map, 39 diffeomorphism, 16 differential space, 15 differential structure, 15 generated by family of functions, 17 differential subspace, 18 Dirac, P.A.M., 9 Dirac structure, 214

distribution, 51 Duistermaat, J.J., 4, 5, 10 Duval, C., 9 Elhadad, M.J., 9 exponential map, 57, 59 frontier condition, 52 global derivation, 26 Gotay, M., 6, 8, 9 Gualtieri, M., 11 Guillemin, V., 7, 9, 10, 130, 150, 162, 184 Gupta, S.N., 9 H -slice, 82 half-density, 159 half-form, 160 Hall, B., 187 Hamilton, M., 7, 9 Hamiltonian action, 109 Hamiltonian vector field, 108 Heller, M., 2 Hilbert, D., 5 horizontal Pontryagin bundle, 218 Huebschmann, J., 9 immersed manifold, 18 integral curve, 31 integral manifold, 51 isotropy group, 56 isotropy type, 56 Jacobi identity, 110 Jennings, G., 8 Jotz, M., 11 Kähler polarization, 157 Kimura, T., 8 Kirillov, A.A., 6, 9 Kirillov–Kostant–Souriau form, 106 Kirwin, W., 187

234

Koch curve, 41 Kolk, J.A.C., 4 Koon, W.S., 10 Kostant, B., 6, 9, 157 Koszul, J.L., 5 Koszul form, 91 Lagrangian distribution, 155 Leibniz’s rule, 25 Li, H., 187 Libermann, P., 7 local extension, 54 local one-parameter group, 34 local submanifold, 60 locally complete, 46 locally trivial stratification, 53 Lusala, T., 4 Marle, C.-M., 7 Marsden, J.E., 6, 7, 10, 11, 105 Marsden–Weinstein reduction, 134 Marshall, C.D., 3, 5, 102 Marshall form, 100 Maschke, B.M., 10 Meyer, K.R., 6, 105 minimal stratification, 53 Misiołek, G., 7 momentum, 109 momentum map, 109 Moncrief, V., 6 Multarzynski, P., 2 Noether, E., 8 non-holonomic reduction, 211 normalizer, 61 orbit of group action, 56 of vector fields, 45 orbit type, 56 orbit type stratification, 67, 75 Ortega, J.-P., 6, 7 Palais, R., 5, 81 partition of unity, 23 Pasternak-Winiarski, Z., 3 Perlmutter, M., 7 Pflaum, M.J., 4 Poisson algebra, 110 bracket, 110 derivation, 111 differential space, 131

Index

ideal, 132 reduction, 111 vector field, 111 polarization, 155 Pontryagin bundle, 214 positive polarization, 155 prequantization action, 154 condition, 152 map, 153 operator, 153 representation, 154, 155 principal fibre bundle, 69 proper action, 56 push-forward, 27 quantization map, 156 representation, 156 Ratiu, T., 6, 7, 11 refinement, 53 regular component, 41 regular point, 41 regular reduction, 134 Sasin, W., 2, 102 Satake, I., 174 Schwarz, G., 5 second countable, 22 section, 38 Seidenberg, A., 5 semi-definite polarization, 155 shifting maps, 130 shifting trick, 130 Sikorski, R., 1, 2, 15, 16 singular point, 41 singular reduction, 105 singular set, 41 Sjamaar, R., 9, 151, 185 slice, 57 smooth functions, 15 smooth map, 16 ´ Sniatycki, J., 3–12 Souriau, J.-M., 6, 9 space of derivations, 25 Stefan, P., 4 Sternberg, S., 7, 9, 10, 130, 150, 162, 184 strata, 52 stratification, 52 strongly admissible polarization, 155 structural dimension, 41

Index

subcartesian differential space, 21 submanifold, 18 Sussmann, H., 4 symplectic complement, 165 symplectic form, 108 symplectomorphism, 108 tangent bundle, 37 projection, 37 tangent map, 40 tangent space, 26 tangent vector, 26 Tarski, A., 5 Thom, R., 4 Tuynman, G.M., 9 unitarization, 156

van der Schaft, A.J., 10, 11 vector field, 34 vertical Pontryagin bundle, 218 Walczak, P., 2 Watts, J., 3, 5, 102 Wazewski, T., 20 Weinstein, A., 6–8, 10, 11, 105 Weyl, H., 5 Whitney, H., 4 Wilbour, D.C., 8 Woodhouse, N., 9 Yoshimura, H., 11 Zariski, O., 5 Zariski form, 94

235

Differential Geometry of Singular Spaces and Reduction of Symmetry In this book, the author illustrates the power of the theory of subcartesian differential spaces for investigating spaces with singularities. Part I gives a detailed and comprehensive presentation of the theory of differential spaces, including integration of distributions on subcartesian spaces and the structure of stratified spaces. Part II presents an effective approach to the reduction of symmetries. Concrete applications covered in the text include the reduction of symmetries of Hamiltonian systems, non-holonomically constrained systems, Dirac structures and the commutation of quantization with reduction for a proper action of the symmetry group. With each application, the author provides an introduction to the field in which relevant problems occur. This book will appeal to researchers and graduate students in mathematics and engineering.

J. S´ NIATYCKI is a Professor in the Department of Mathematics and Statistics at the University of Calgary.

N E W M AT H E M AT I C A L M O N O G R A P H S Editorial Board Béla Bollobás, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry Simon, Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit www.cambridge.org/mathematics. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups J. B. Garnett and D. E. Marshall Harmonic Measure P. Cohn Free Ideal Rings and Localization in General Rings E. Bombieri and W. Gubler Heights in Diophantine Geometry Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. Wüstholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry

Differential Geometry of Singular Spaces and Reduction of Symmetry J . S´ N I AT Y C K I Department of Mathematics and Statistics University of Calgary Calgary, Alberta, Canada

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107022713 ´ c J. Sniatycki 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Sniatycki, Jedrzej. Differential geometry of singular spaces and reduction of symmetry / Jedrzej Sniatycki. pages cm. – (New mathematical monographs ; 23) ISBN 978-1-107-02271-3 (hardback) 1. Geometry, Differential. 2. Function spaces. 3. Symmetry (Mathematics) I. Title. QA641.S55 2013 516.3 6–dc23 2012047532 ISBN 978-1-107-02271-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface List of selected symbols 1 Introduction

page vii ix 1

PART I D I F F E R E N T I A L G E O M E T RY O F SI N G U L A R SPAC ES 2 Differential structures 2.1 Differential spaces 2.2 Partitions of unity

15 15 21

3 Derivations 3.1 Basic properties 3.2 Integration of derivations 3.3 The tangent bundle 3.4 Orbits of families of vector fields

25 25 31 37 44

4 Stratified spaces 4.1 Stratified subcartesian spaces 4.2 Action of a Lie group on a manifold 4.3 Orbit space 4.4 Action of a Lie group on a subcartesian space

52 52 56 67 81

5 Differential forms 5.1 Koszul forms 5.2 Zariski forms 5.3 Marshall forms

91 91 94 99

vi

Contents

PART II R E D U C T I O N O F S Y M M E T R I E S 6 Symplectic reduction 6.1 Symplectic manifolds with symmetry 6.1.1 Co-adjoint orbits 6.1.2 Symplectic manifolds 6.1.3 Poisson algebra 6.2 Poisson reduction 6.3 Level sets of the momentum map 6.4 Pre-images of co-adjoint orbits 6.5 Reduction by stages for proper actions 6.6 Shifting 6.7 When the action is free 6.8 When the action is improper 6.9 Algebraic reduction

105 105 105 108 110 111 114 125 126 129 134 135 136

7 Commutation of quantization and reduction 7.1 Review of geometric quantization 7.1.1 Prequantization 7.1.2 Polarization 7.1.3 Examples of unitarization 7.2 Commutation of quantization and singular reduction at J = 0 7.3 Special cases 7.3.1 The results of Guillemin and Sternberg 7.3.2 Kähler polarization without compactness assumptions 7.3.3 Real polarization 7.4 Non-zero co-adjoint orbits 7.5 Commutation of quantization and algebraic reduction 7.5.1 Quantization of algebraic reduction 7.5.2 Kähler polarization 7.5.3 Real polarization 7.5.4 Improper action

150 151 152 155 156 161 184 184 185 187 190 203 203 206 207 208

8 Further examples of reduction 8.1 Non-holonomic reduction 8.2 Dirac structures 8.2.1 Symmetries of the Pontryagin bundle 8.2.2 Free and proper action 8.2.3 Proper non-free action

211 211 214 215 217 223

References Index

228 233

Preface

My first encounter with differential spaces was in the mid 1980s. At a conference in Toru´n, I presented the notion of algebraic reduction of symmetries of a Hamiltonian system. After the lecture, Constantin Piron asked me if my reduced spaces were the differential spaces of Sikorski. I had to admit that I did not know what Sikorski’s differential spaces were. To this Piron replied something like ‘You should be ashamed of yourself! You are a Pole and you do not know what are differential spaces of Sikorski!’ During the lunch break I went to the library to consult Sikorski’s work. In the afternoon session, I told Piron that the spaces we were dealing with were not the differential spaces of Sikorski. At that time I did not realize that they were differential schemes. Around the same time, Richard Cushman was working out his examples of singular reduction. I was fascinated by his pictures of reduced spaces with singularities. However, I had not the faintest idea what he was really doing. Since Richard was spending a lot of time in Calgary working on his book with Larry Bates, I had a chance to ask him to explain singular reduction to me. It took me a long time to realize that he was talking the language of differential spaces without being aware of it. From conversations with Richard, it became clear that differential spaces provided a convenient language for the description of the reduction of symmetries for proper actions of symmetry groups. The next push in the direction of serious investigations of differential spaces came from Ryer Sjamaar and Eugene Lerman. In their Annals of Mathematics paper on reduction of symmetries of Hamiltonian systems, they proved a theorem using techniques that are natural to the theory of differential spaces. Studying their proof, I realized that it was very simple and that I could not think of an equally simple proof that would not utilize their techniques. It convinced me that the language of differential spaces facilitated obtaining new results, and I decided to investigate if reduction of symmetries could be completely formulated and analysed within the category of differential spaces.

viii

Preface

The theory of differential spaces is essentially differential geometry not restricted to smooth manifolds. Roman Sikorski, who is considered the father of the theory, called his book (in Polish) Wst˛ep do Geometrii Ró˙zniczkowej. This translates as ‘Introduction to Differential Geometry’. Originally, differential geometry meant the description, in terms of differentiable functions, of curves and surfaces in Rn . Singularities of curves or surfaces under consideration could also be described in terms of smooth functions. Differential geometry evolved in two different directions: the theory of manifolds and singularity theory. Manifolds are smooth spaces not presented as subsets of Rn . Singularity theory is the study of the failure of the manifold structure. Differential geometry in the sense of Sikorski is a reunification of the two theories. It contains the theory of manifolds and also allows the investigation of singularities. It is the investigation of geometry in terms of differentiable functions. Differential geometry, understood in this way, is analogous to algebraic geometry, which is the investigation of geometry in terms of polynomials. The difference between the two theories is in the choice of the space of functions. I am grateful to Constantin Piron for drawing my attention to Sikorski’s book. I greatly appreciate the support and encouragement of Hans Duistermaat. I would like to thank Larry Bates for his support and for bringing Richard Cushman to Calgary, and to thank Jordan Watts for his interest in my work. Above all, I want to thank Richard Cushman for his patience in explaining to me the foundations of his theory of singular reduction and his subsequent collaboration, encouragement and criticism. I also want to thank Cathy Beveridge and Leslie McNab for their help in editing the manuscript. Both Cathy and Leslie have worked hard to make sure that this book is written in proper English. However, I am sure that, in spite of their vigilance, I will have managed to slip in some phrases that go against the proper use of English. Last but not least, I want to thank my wife, Pamela Plummer, without whose support this book would not have been possible. Partial support from the National Science and Engineering Research Council of Canada is gratefully acknowledged.

List of selected symbols

Upper case Latin alphabet Ad ∗ B C C 0 (S) C ∞ (S) C ∞ (S)G D D D Der C ∞ (S) E Exp : T p P → P F F F FF FTCM G H H I J J0 K L L M

co-adjoint action open ball complex numbers continuous functions on S smooth functions on S G-invariant functions on S distribution; real part of polarization in Chapter 7 space of compactly supported sections dual of D space of derivations of C ∞ (S) family of Hamiltonian vector fields exponential map defined by connection function; polarization in Chapter 7 family of functions family of vector fields bundle of linear frames of F bundle of linear frames of T C M Lie group Lie group Hilbert space interval; inclusion map of co-adjoint orbit into g∗ in Chapter 7 momentum map ideal generated by components of J manifold manifold prequantization line bundle manifold; stratum

x M N NH N (S) N O O P P PH PH P(H ) P(R) Q Q R R R R(S) S Sp S ∞ (L) S F∞ (L) S ∞ (L)G T CM T S, T ∗ S Tϕ T p⊥ L U, V, W U X, Y, Z X( f ) X(S)

List of selected symbols

stratification manifold, stratum normalizer of H space of functions with vanishing restrictions to S stratification orbit partition by orbits manifold, differential space prequantization map set of points in P fixed by action of H set of points in P of symmetry type H set of points in P of orbit type H Poisson vector fields on R manifold quantization map manifold; differential space; orbit space real numbers linear representation space of restrictions to S of functions defined on a larger space manifold; differential space slice at p space of smooth sections of L space of smooth polarized sections of L space of G-invariant smooth sections of L complexified tangent bundle of M tangent and cotangent bundle spaces of S derived map of ϕ symplectic complement of T p L open subsets unitary representation global derivations; vector fields; sections of tangent bundle evaluation of X on f family of all vector fields on S

Lower case Latin alphabet a, b c c:I →S d

real numbers complex number curve in S differential

List of selected symbols

e exp : g → G exp(t X ) exp(t X )(x) f f −1 (I ) g g g∗ h h h∗ hor T P k m n p q s supp f t u, v ver T P w x, y z

group identity exponential map local one-parameter local group of diffeomorphisms defined by X point on maximal integral curve of X through x function inverse image of I under f element of group G Lie algebra of G dual of g function Planck’s constant divided by 2π Lie algebra of H dual of h horizontal distribution on P Riemannian metric Lie algebra Lie algebra of N point point parameter support of f parameter derivation at a point; vector vertical distribution on P derivation at a point; vector point point of L

Lower case Greek alphabet α β δi j ζ, η θ ϑ λ, μ, ν λ:L→ P ξ

1-form 1-form Kronecker δ elements of Lie algebra 1-form cotangent bundle projection elements of co-adjoint orbit complex line bundle projection element of Lie algebra

xi

xii π ρ ρ∗ ρ∗ σ σ∗ σ∗ τ τ∗ ϕ ϕ∗ ϕ∗ ω

List of selected symbols

projection map form; distributional symplectic form in Chapter 8 map pull-back by ρ push-forward by ρ section pull-back by σ push-forward by σ map; tangent bundle projection pull-back by τ map pull-back by ϕ push-forward by ϕ symplectic form

Upper case Greek alphabet kK (S) kM (S) kZ (S)

Lagrangian submanifold projection of G-invariant section restriction of section action action symplectic form of co-adjoint orbit space of Koszul k-forms on S space of Marshall k-forms on S space of Zariski k-forms on S

Non-alphabetic symbols ∇ · | ·

√ n |∧ F| [·, ·] {·, ·} |

(· | ·)

covariant derivative evaluation; sesquilinear form on a line bundle half-densities on F left interior product Lie bracket Poisson bracket restriction scalar product on a Hilbert space

1 Introduction

This book is written for researchers and graduate students in the field of geometric mechanics, especially the theory of systems with symmetries. A wider audience might include differential geometers, algebraic geometers and singularity theorists. The aim of the book is to show that differential geometry in the sense of Sikorski is a powerful tool for the study of the geometry of spaces with singularities. We show that this understanding of differential geometry gives a complete description of the stratification structure of the space of orbits of a proper action of a connected Lie group G on a manifold P. We also show that the same approach can handle intersection singularities; see Section 8.2. We assume here that the reader has a working knowledge of differential geometry and the topology of manifolds, and we use theorems in these fields freely without giving proofs or references. On the other hand, the material on differential spaces is developed from scratch. The results on differential spaces are proved in detail. This should make the book accessible to graduate students. The book is split into two parts. In Part I, we introduce the reader to the differential geometry of singular spaces and prove some results, which are used in Part II to investigate concrete systems. The technique of differential geometry presented here is fairly straightforward, and the reader might get a false impression that the scope of the theory does not differ much from that of the geometry of manifolds. However, the examples given in Part I will serve as warnings that such an impression is false. Part II is devoted to applications of the general theory. Each chapter in this part may be considered as an extensive example of the use of differential geometry to deal with singularities in concrete problems. Since these problems occur in various theories, each chapter begins with a section introducing elements of the underlying theory, in order to show the reader the relevance of the problem under consideration.

2

Introduction

The book contains no exercises, because the actual techniques involved are very simple. In addition to the standard techniques of the differential geometry of manifolds, we use techniques of algebraic geometry for rings of smooth functions. The fact that algebraically defined derivations of smooth functions admit integral curves is the main difference between differential and algebraic geometry. The technical details of the presentation are based on the TEX style file chosen for the preparation of this book. Displayed results are labelled by the number of the chapter, the number of the section in the chapter and the number of the result within the section. For example, ‘Lemma 2.1.3’ stands for Lemma 1.3 in Chapter 2; it can also be read as the third lemma in Section 2.1. Displayed equations are referenced by the number of the chapter and the number of the equation within the chapter. For example, ‘equation (3.21)’ stands for equation 21 in Chapter 3. This book is based on several years of research. Some of the results presented here were obtained by the author. Some other results have been taken directly from the work of other researchers. The remainder corresponds to an adaptation and reformulation of the work of other authors so that it fits into the theory presented here. In order to keep the flow of the presentation in the subsequent chapters free from obstructions, we give below a detailed description of the content of the book and the references to the literature. Part I is devoted to a comprehensive presentation of the current status of the differential geometry of singular spaces. A comprehensive bibliography of the literature on differential spaces during the period 1965–1992 was published in 1993 by Buchner, Heller, Multarzy´nski and Sasin (Buchner et al., 1993). According to these authors, the first paper on differential spaces was Sikorski (1967). In the same year, at a meeting of the American Mathematical Society, Aronszajn presented an extensive programme of differential-geometric study of subcartesian spaces in terms of singular charts. Aronszajn’s subcartesian spaces included arbitrary subspaces of Rn (see Aronszajn, 1967). In 1973, Walczak showed that subcartesian spaces are special cases of differential spaces (see Walczak, 1973). In Section 2.1, we describe the basic definitions and constructions of Sikorski’s theory following his book (see Sikorski, 1972). The fundamental notion of this theory is the differential structure C ∞ (S) of a space S, consisting of functions on S deemed to be smooth. The differential structure of a space carries all information about the geometry of the space. In particular, a map ϕ : S → T is smooth if it pulls back smooth functions to smooth functions. A diffeomorphism is an invertible smooth map with a smooth inverse. As in topology, subsets, products and quotients of differential spaces are differential

Introduction

3

spaces. However, the quotient differential space need not have the quotient topology. Proposition 2.1.11, which gives conditions for equivalence of the quotient differential-space topology and the quotient topology, is taken from the work of Pasternak-Winiarski (1984) . A differential space S is subcartesian if every point of S has a neighbourhood diffeomorphic to a subset of some Cartesian space Rn . The category of subcartesian differential spaces is the main object of our study. Manifolds are subcartesian spaces that are locally diffeomorphic to open subsets of Rn . If M is a manifold, the collection of all local diffeomorphisms to open subsets of Rn forms the maximal atlas on M. Differential geometry, understood as the study of the geometry of a space in terms of the ring of smooth functions on that space, naturally extends from manifolds to subcartesian spaces. We do not go beyond subcartesian spaces, because a differential space which is not subcartesian need not have a locally finite dimension. In Section 2.2, we show that subcartesian spaces admit partitions of unity. The importance of partitions of unity stems from the fact that they enable us to globalize collections of local data. The existence of partitions of unity on locally compact and paracompact differential spaces was first proved by Cegiełka (1974). Here, we follow the proof of Marshall (1975a). In Chapter 3, we discuss vector fields on subcartesian spaces. A vector field on a manifold M can be described either as a derivation of a ring C ∞ (M) of smooth functions on M or as a generator of a local one-parameter group of local diffeomorphisms of M. These two notions are equivalent if M is a manifold. However, they may be inequivalent on a subcartesian space S that is not a manifold. In Section 3.1, we study the basic properties of derivations of the differential structure C ∞ (S) of a subcartesian space S. We show that every derivation X of C ∞ (S) can be locally extended to a derivation of C ∞ (R n ). This result allows the study of ordinary differential equations on subcartesian spaces, which we discuss in Section 3.2. The existence and uniqueness theorem for integral ´ curves of derivations on a subcartesian space was first proved by Sniatycki (2003a). In Section 3.3, we discuss the tangent bundle space T S of S, defined as the space of derivations of C ∞ (S) at points of S. In the literature, T S is also called the tangent pseudobundle or the Zariski tangent bundle. Following Watts (2006), we define the regular component Sreg of S as the set of all points p of S at which dim T p S is locally constant, and prove that Sreg is open and dense in S and that the restriction T Sreg of T S to Sreg is locally spanned by global derivations; see Lusala et al. (2010). Example 3.3.12, taken from Epstein and

4

Introduction

´ Sniatycki (2006), shows that a differential space that is regular everywhere need not be a manifold. In Section 3.4, we study global derivations of S that generate local oneparameter groups of local diffeomorphisms. We call such global derivations vector fields. We show that the orbits of any family of vector fields on a subcartesian space S are smooth manifolds immersed in S. This result, first ´ proved by Sniatycki (2003b), is a generalization of some theorems of Sussmann (1973) and Stefan (1974). In particular, it implies that orbits of the family X(S) of all vector fields on S give a partition of S by smooth manifolds. Therefore, every subcartesian space S has a minimal partition by smooth manifolds. This result gives us an alternative interpretation of the strata of a minimal stratification of a subcartesian space, which we study in Chapter 4. In Chapter 4, we discuss stratified spaces, first investigated by Whitney (1955), who called them ‘manifold collections’. The term ‘stratification’ is due to Thom (1955–56). A stratified space is usually described as a topological space partitioned in a special way by smooth manifolds. Here, we restrict our considerations to stratified spaces that are also subcartesian differential spaces. In Section 4.1, we discuss stratified subcartesian spaces following the work ´ ´ of Sniatycki (2003b) and Lusala and Sniatycki (2011). A stratified space is, by definition, partitioned by smooth manifolds. The results of Chapter 3 show that a subcartesian space is also partitioned by smooth manifolds, which are orbits of the family of all vector fields. We show that if a stratified space S is subcartesian and the stratification of S is locally trivial, then the partition of S by orbits of the family of all vector fields is also a stratification of S. Moreover, this second stratification of S is coarser than the original stratification. If the original stratification is minimal, then it is the same as the stratification given by the orbits of the family of all vector fields. In other words, a minimal locally trivial stratification of a subcartesian space is completely determined by its differential structure. In Section 4.2, we describe the orbit type stratification M of a manifold P given by a proper action on P of a connected Lie group G. This stratification is not minimal, because the union of all the strata is the manifold P. The presentation adopted here borrows from the presentations of the same topic in the books by Cushman and Bates (1997), Duistermaat and Kolk (2000), and Pflaum (2001). Section 4.3 is devoted to a discussion of the structure of the orbit space R = P/G. We show that the projection to the orbit space R of the strata of M is a locally trivial and minimal stratification of R. This is called the orbit type stratification of the orbit space R. We also show that R is a subcartesian space.

Introduction

5

The material presented in Section 4.3 is based on the results of many authors. In particular, results of Bierstone (1975; 1980), Bochner’s Linearization Theorem (Duistermaat and Kolk, 2000), the Hilbert–Weyl Theorem (Weyl, 1946), Palais’s Slice Theorem (Palais, 1961), a theorem by Schwarz (1975) and the Tarski–Seidenberg Theorem (Abraham and Robbin, 1967). The form of pre´ sentation adopted here follows that of Cushman, Duistermaat and Sniatycki (2010). By combining the results of Sections 4.1 and 4.3, we conclude that the strata of the orbit type stratification of the orbit space R are orbits of the family of all vector fields on R. This result is the basis for the singular reduction of symmetries discussed in subsequent chapters. In Section 4.4, we study a proper action of a Lie group on a locally compact subcartesian space. Palais’s Slice Theorem applies to this case, and we prove that the space of orbits of the action is a locally compact differential space. We have no extension of Bochner’s Linearization Theorem to subcartesian spaces, and we can prove neither that the orbit space is subcartesian nor that it is stratified. Nevertheless, the result obtained here suffices to prove singular reduction by stages in Section 6.5. Chapter 5 is devoted to a discussion of differential forms on subcartesian spaces. We are led to three inequivalent notions of differential forms. Zariski differential forms on S are defined as alternating multilinear maps from spaces of pointwise derivations of C ∞ (S) to real numbers. Zariski differential forms can be pulled back by smooth maps. If S is not a manifold, then exterior differentials of Zariski differential forms are not defined. The second possibility is Koszul differential forms, defined as alternating multilinear maps from spaces of global derivations of C ∞ (S) to C ∞ (S). We can take exterior differentials of Koszul forms, but we cannot define their pull-backs by differential maps. The third possibility is Marshall forms, which agree with Zariski forms and Koszul forms on the regular component Sreg of S. Marshall forms allow pull-backs, as well as exterior differentials. The presentation adopted here follows a paper by Marshall (1975a), Watts’ theses (Watts, 2006; 2012) and his unpublished notes. In Part II, we apply the general theory introduced in Part I to the problem of reduction of the symmetries of various systems. In most cases, we make an assumption that the action of the symmetry group G on the phase space P of the system is proper. This assumption implies that the orbit space P/G is stratified, and the study of reduction involves an investigation of the interplay between the stratification structure of P/G and the geometric structure characterizing the system under consideration. There is no satisfactory theory of the structure of the space of orbits of an improper action of a Lie group on a manifold. However, if P is a symplectic

6

Introduction

manifold and the improper action of G on P is Hamiltonian, we can show that algebraic reduction, in terms of differential schemes, encodes a lot of information about the action of G on P. We also show that the information obtained by algebraic reduction may survive the process of quantization and may be decoded on the quantum level. The objective of symplectic reduction, discussed in Chapter 6, is to describe the structure of the space of orbits of a Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω). For a proper action, we know that the orbit space R = P/G is stratified, and we investigate the interaction between the stratification structure of R and the Poisson structure of R induced by the symplectic structure of P. We also discuss the case when the action of G on P fails to be proper. In Section 6.1, we give a brief review of Hamiltonian actions of a Lie group G on a symplectic manifold (P, ω), the properties of the momentum map J : P → g∗ , and the Poisson algebra structure of C ∞ (P) induced by the symplectic form ω on P. We begin with a discussion of the co-adjoint action of G on co-adjoint orbits in g∗ and describe the Kirillov–Kostant–Souriau symplectic form of a co-adjoint orbit (Kirillov, 1962; Kostant, 1966; Souriau, 1966). Moreover, we show that the momentum map for a co-adjoint orbit is the inclusion of the orbit in g∗ . This introductory material is included here in order to establish the notation and to introduce the problem to readers who might be unfamiliar with symplectic geometry. Symplectic reduction for a free and proper action was introduced by Meyer (1973) and Marsden and Weinstein (1974). It is known as regular reduction or Marsden–Weinstein reduction. The first study of the structure of the orbit space for a proper non-free Hamiltonian action of the symmetry group was the paper of Arms, Marsden and Moncrief (Arms et al., 1981), who showed that the zero level of the momentum map is stratified. The technique of singular reduction in terms of the Poisson algebra structure was initiated by Cushman (1983), and later formalized by Arms, Cushman and Gotay (Arms et al., 1991). The role of Sikorski’s theory of differential spaces in ´ singular reduction was first described by Cushman and Sniatycki (2001). Comprehensive presentations of singular reduction have been given in the books by Cushman and Bates (1997) and Ortega and Ratiu (2004). Our discussion of singular reduction is contained in Sections 6.2–6.6. Our presentation differs from the presentations in Cushman and Bates (1997) and Ortega and Ratiu (2004) because we have the general theory developed in Part I at our disposal. Nevertheless, it has many points in common with earlier approaches. In Section 6.2, we describe the structure of the orbit space R = P/G in terms of the structure of the ring C ∞ (R) of smooth functions on R. Using

Introduction

7

the results of Chapter 4, we describe strata of the orbit type stratifications of P/G as orbits of the family of all vector fields X(R) on R. For each stratum of R, the Poisson structure on C ∞ (R) induces the structure of a Poisson manifold. Since Poisson derivations of C ∞ (R) are vector fields on R, orbits of the family P(R) of all Poisson derivations of C ∞ (R) give foliations of strata of R by symplectic leaves. A proof that a Poisson manifold is singularly foliated by symplectic leaves was given in the book by Libermann and Marle (1987). In Section 6.3, we show that for each μ ∈ g∗ , the projection to R of the level set J −1 (μ) is a stratified space with symplectic strata, which are symplectomorphic to the corresponding symplectic leaves of strata of R. In Section 6.4, we obtain similar results for projections to R of J −1 (O), provided that the co-adjoint orbit O is locally closed.1 The main results obtained in Sections 6.3 and 6.4 are not new. However, the proofs of these results are new. In Section 6.5, we apply the results of Section 4.4 to the case when the symmetry group G of (P, ω) has a normal subgroup H . In this case, we can first reduce the action of H . The result is a stratified Poisson space P/H ´ with symmetry group G/H . Following Lusala and Sniatycki (to appear), we prove that the structure of the orbit space (P/H )/(G/H ) is isomorphic to that of P/G. This result is called ‘reduction by stages’ in the literature; see the book by Marsden, Misiołek, Ortega, Perlmutter and Ratiu (Marsden et al., 2007). In Section 6.6, we discuss the process of shifting, which gives rise to an equivalence between the reduction of J −1 (O) and the reduction at zero for a shifted momentum map on P × O, where O is a co-adjoint orbit. This is essential for the extension to non-zero co-adjoint orbits of the results on the commutation of quantization and reduction of J −1 (0) discussed in the next chapter. Shifting was introduced for a free and proper action by Guillemin and Sternberg (1984). For a proper non-free action, shifting was first proved by ´ Bates, Cushman, Hamilton and Sniatycki (Bates et al., 2009). In Section 6.7, we restrict singular reduction to the case when the action of G on P is free and proper. As a corollary, we obtain the results of the Marsden–Weinstein reduction (Marsden and Weinstein, 1974). In Section 6.8, we discuss the case when the action of G on P is not proper. In this case, the ring of G-invariant functions on P need not separate the orbits, and singular reduction is not applicable. At present, there is no satisfactory theory of the structure of the space of orbits of an improper 1 An example of a co-adjoint orbit which is not locally closed was first given by Pukanszky

(1971). Here, we do not study such co-adjoint orbits; however, they were discussed by Ortega and Ratiu (2004).

8

Introduction

action of a Lie group on a manifold. However, in our case, P is a symplectic manifold and the improper action of G on P is Hamiltonian, which allows algebraic reduction as discussed in Section 6.9. Algebraic reduction gives rise to a Poisson algebra defined in terms of differential schemes, which are differential-geometry analogues of schemes in algebraic geometry. The Poisson algebra of algebraic reduction encodes a lot of information about the action of G on P. The problem arises as to how to decode the information encoded in algebraic reduction and use it in applications. We return to this question in Chapter 7. Algebraic reduction of the zero level of the momentum map was introduced ´ by Sniatycki and Weinstein (1983). Algebraic reduction at non-zero co-adjoint orbits was introduced independently by Wilbour (1993) and Kimura (1993). Theorem 6.9.6 (the shifting theorem) was proved by Arms (1996). Example 6.9.4 was first investigated in the context of algebraic reduction by Arms, Gotay and Jennings (Arms et al., 1990). Example 6.9.7 was first outlined in ´ Sniatycki and Weinstein (1983); a full analysis of this example was given in ´Sniatycki (2005). Lemma 3.8.1 was proved by Bates (2007). Chapter 7 is devoted to the problem of commutation of geometric quantization and reduction. The term ‘geometric quantization’ is used in mechanics and in representation theory. In both cases, it describes essentially the same mathematical procedure, but its starting points and aims are different in the two cases. In representation theory, quantization is a technique for obtaining a unitary representation of a connected Lie group from its action on a symplectic manifold. In quantum mechanics, geometric quantization provides a geometric way to transition from the classical to the quantum description of a physical system. In physics, we often study a quantum subsystem of a classical system. This is usually done by starting with a classical description of the whole system and then imposing constraints to single out the subsystem, followed by reduction of spurious degrees of freedom and subsequent quantization. We expect that the physical results obtained will be the same as the results of a study of the subsystem in terms of quantization of the whole system. This expectation can be rephrased as the principle that quantization commutes with reduction. The importance of commutation of quantization and reduction was realized in the study of the quantization of gauge theories and general relativity. According to Noether’s Second Theorem (Noether, 1918), the presence of a gauge symmetry leads to a constraint in the theory, given by J = 0, where J is the momentum map for the gauge group action (Binz et al., 2006). In

Introduction

9

the studies by Bleuler (1950) and Gupta (1950) of the quantization of electrodynamics, these authors quantized the full space of the Cauchy data for the electromagnetic field and imposed an appropriate constraint on the space of quantum states. On the other hand, Dirac’s study of the quantization of gravity led to a distinction between first-class and second-class constraints (Dirac, 1964). First-class constraints could be imposed on the quantum level, whereas second-class constraints had to be imposed on the classical level. It is rather difficult to give a definite answer in the framework of quantum field theory to the question of whether quantization and reduction commute. Guillemin and Sternberg (1982) proved that geometric quantization commutes with reduction provided that some strong technical assumptions are satisfied. Their approach was formulated in the framework of the representation theory of Lie groups. Geometric quantization has its roots in the work of Kirillov (1962), Auslander and Kostant (1971), Kostant (1966; 1970) and Souriau (1966). A comprehensive bibliography was given in a book by Woodhouse (1992). We begin with a discussion of the significance of commutation of quantization and reduction in the framework of representation theory. In Section 7.1, ´ we give a review of geometric quantization following Sniatycki (1980). In Section 7.2, we discuss in general terms the problem of commutation of geometric quantization and singular reduction. This problem has been stud´ ied by Bates, Cushman, Hamilton and Sniatycki (Bates et al., 2009), using ´ an algebraic approach based on Sniatycki’s earlier results on commutation of ´ quantization and algebraic reduction (Sniatycki, 2012). The approach to the problem of commutation of geometric quantization and singular reduction, as well as many of the results presented in this section, is new. In Section 7.3, we discuss various special cases. We begin with the case of a Kähler quantization of a compact symplectic manifold (P, ω) with a Hamiltonian action of a compact connected Lie group G, investigated by Guillemin and Sternberg (1982) and by Sjamaar (1995). We discuss which of the results of Guillemin and Sternberg and of Sjamaar follow from our general approach, and which of these results are specific to the approach that they used. Our results also hold when the symplectic manifold P and the Lie group G are not compact, and agree with the results of Huebschmann (2006). Next, we discuss conditions for commutation of singular reduction and quantization with respect to a real polarization. For a free and proper action of G on P, these conditions ´ were first introduced by Sniatycki (1983), and subsequently studied by Duval, ´ Elhadad, Gotay, Sniatycki and Tuynman; see Duval et al. (1990; 1991) and the references therein.

10

Introduction

In Section 7.4, we discuss commutation of quantization and reduction at non-zero quantizable co-adjoint orbits using the shifting trick described in ´ Section 6.6. The approach adopted here follows Sniatycki (2012). In Section 7.5, we discuss the problem of commutation of geometric quantization and algebraic reduction. In fact, algebraic reduction was invented for this problem. In 1980, at a conference in Banff, Guillemin presented some unpublished results from his work with Sternberg. This lecture motivated the present author to investigate possible ways to generalize the results of Guillemin and Sternberg to singular momentum maps. In 1981, the author presented at a conference in Clausthal a paper discussing some examples in quantum mechanics which could be interpreted as quantum reduction of sin´ gular constraints (Sniatycki, 1983). Weinstein’s reaction to this lecture led to ´ a collaboration, which culminated in publication of a joint paper (Sniatycki and Weinstein, 1983). We discuss some special cases when the polarization is Kähler or real, and obtain results similar to the results for singular reduction. We conclude with some partial results on commutation of quantization and reduction for an improper action of the symmetry group. Chapter 8 contains two more examples of reduction of symmetry. In Section 8.1, we discuss reduction of symmetry for a proper action of the symmetry group G of a non-holonomically constrained Hamiltonian system. We begin with a description of the distributional Hamiltonian formulation of con´ strained dynamics, following Bates and Sniatycki (1993). Next, we reformulate the distributional Hamiltonian formulation in terms of the almost-Poisson formulation of van der Schaft and Maschke (1994). This encodes the distributional Hamiltonian structure of the theory in the structure of C ∞ (P). The space C ∞ (P)G of G-invariant functions is an almost-Poisson subalgebra of C ∞ (P). Since the differential structure C ∞ (P/G) of the orbit space P/G is isomorphic to C ∞ (P)G , it inherits an almost-Poisson algebra structure, which was first used to discuss reduction by Koon and Marsden (1998). The almost-Poisson bracket is a derivation and gives rise to a family P(P/G) of almost-Poisson vector fields on P/G. The orbits of this family are manifolds. Each orbit Q carries a generalized distribution D Q spanned by the restriction of P(P/G) to Q. Moreover, D Q carries a symplectic form Q defined by the almost-Poisson structure of C ∞ (Q). A comprehensive presentation of the current state of the geometry of non-holonomically constrained Hamiltonian systems can be found in a recent book by Cushman, Duistermaat ´ and Sniatycki (Cushman et al., 2010). In Section 8.2, we discuss reduction of symmetries for a proper action of the symmetry group G of a Dirac structure. A Dirac structure on a manifold P is a maximal isotropic subbundle D of the Pontryagin bundle P = T Q × Q T ∗ Q

Introduction

11

of a manifold Q. The notion of a Dirac structure was introduced by Courant and Weinstein (1988); see also Courant (1990) and Dorfman (1993). A proper action of a Lie group G on Q is a symmetry of D if the action of G lifted to P preserves D. Since the action of G on Q is proper, it follows that the action of G on P is proper. Moreover, the action of G on P preserves D and induces a proper action of G on D. Hence, the orbit space D/G is a stratified subcartesian space. The main problem of the reduction is to relate the stratification of P/G to stratifications of P/G and Q/G. We introduce a G-invariant Riemannian metric k on Q, and decompose T Q into its vertical component ver T Q, which is tangent to orbits of G, and its horizontal component hor T Q, which is k-orthogonal to ver T Q. We decompose T ∗ Q and P in a similar way. Even for a free and proper action of G, we may encounter intersection singularities because the intersection of D with T Q need not be clean. Therefore, we begin with a study of regular reduction for a free and proper action of G on Q. In this case Q is a left principal fibre bundle with structure group G. We show that the reduced Pontryagin bundle P/G is isomorphic to the direct sum of T (Q/G) ⊕ T ∗ (Q/G) and Q[g] ⊕ Q[g∗ ], where Q[g] and Q[g∗ ] are the adjoint and the co-adjoint bundle, respectively, of Q. The regular reduction gives rise to the quotient D/G in the form of the direct sum of T (Q/G) ⊕ T ∗ (Q/G) and Q[g] ⊕ Q[g∗ ], which can be interpreted as a generalized Dirac structure on Q/G. Next, we consider a proper non-free action of G on Q. For each stratum N of the orbit type stratification of Q, the quotient D/G defines a generalized Dirac structure on N . Moreover, D/G is uniquely determined by the collection of generalized Dirac structures on all strata of the orbit type stratification of Q. Reduction of the symmetries of Dirac structures was carried out for a free and proper Dirac action by Blankenstein and van der Schaft (2001) and Blankenstein (2000) in the context of generalized Poisson structures, and by Bursztyn, Cavalcanti and Gualtieri (Bursztyn et al., 2007) in the setting of Courant algebroids. Blankenstein and Ratiu (2004) treated a Dirac structure with symmetries as a generalized Poisson structure with a momentum map, and performed singular reduction at singular values of the momentum map. Jotz, ´ Ratiu and Sniatycki (Jotz et al., 2011) studied singular reduction completely within the Dirac category. In a recent paper, Jotz and Ratiu (2012) discussed the reduction of non-holonomic systems in terms of Dirac reduction. The authors of the references mentioned above were interested mainly in the horizontal reduced Dirac structure. The presence of a vertical Dirac structure was observed by Yoshimura and Marsden (2007) in an example in which the action of the symmetry group was free and proper and the horizontal

12

Introduction

reduced Dirac structure vanished identically. The presentation given here ´ follows that of Sniatycki (2011). The interdependence of the chapters is shown in the diagram below. Ch. 2

→

Ch. 3 ↓ Ch. 5

→

Ch. 4

→

Ch. 6 ↓ Ch. 8

→

Ch. 7

PART I Differential geometry of singular spaces

2 Differential structures

2.1 Differential spaces In this section, we describe the category of differential spaces, which includes finite-dimensional manifolds as a subcategory. Definition 2.1.1 A differential structure on a topological space S is a family C ∞ (S) of real-valued functions on S satisfying the following conditions: 1. The family { f −1 (I ) | f ∈ C ∞ (S) and I is an open interval in R} is a subbasis for the topology of S. 2. If f 1 , . . . , f n ∈ C ∞ (S) and F ∈ C ∞ (Rn ), then F( f 1 , . . . , f n ) ∈ C ∞ (S). 3. If f : S → R is a function such that, for every x ∈ S, there exist an open neighbourhood U of x and a function f x ∈ C ∞ (S) satisfying f x|U = f |U , then f ∈ C ∞ (S). Here, the subscript vertical bar | denotes a restriction. Functions f ∈ C ∞ (S) are called smooth functions on S. It follows from Condition 1 above that smooth functions on S are continuous. Condition 2 with F( f 1 , f 2 ) = a f 1 + b f 2 , where a, b ∈ R, implies that C ∞ (S) is a vector space. Similarly, taking F( f1 , f 2 ) = f 1 f 2 , we conclude that C ∞ (S) is closed under multiplication of functions. A topological space S endowed with a differential structure is called a differential space. In his original definition, Sikorski (1972) defined C ∞ (S) to be a family of functions satisfying Condition 2. Then, he used Condition 1 to define a topology on S. Finally, he imposed Condition 3 as a consistency condition. An example of a differential space is the Euclidean space Rn with the standard topology and the standard differential structure C ∞ (Rn ) as defined in

16

Differential structures

calculus. Another example is a smooth manifold M with the differential structure given by the ring C ∞ (M) of smooth functions on M. We also have some more exotic examples below. Example 2.1.2 S is an arbitrary set endowed with the trivial topology (the empty set and S are the only open sets), and its differential structure C ∞ (S) is the set of all constant functions on S. Example 2.1.3 S is an arbitrary set endowed with the discrete topology (every subset of S is open), and C ∞ (S) is the set of all functions on S. Let (R, C ∞ (R)) and (S, C ∞ (S)) be differential spaces. Definition 2.1.4 A map ϕ : R → S is smooth if ϕ ∗ f = f ◦ ϕ ∈ C ∞ (R) for every f ∈ C ∞ (S). A smooth map ϕ between differential spaces is a diffeomorphism if it is invertible and its inverse is smooth. Proposition 2.1.5 A smooth map between differential spaces is continuous. Proof Let ϕ : R → S be smooth, and let U be an open set in S. We need to show that ϕ −1 (U ) is open in R. Let x ∈ ϕ −1 (U ). By Condition 1, there exist functions f 1 , . . . , f n ∈ C ∞ (S) and open intervals I1 , . . . , In in R such that f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In ) is an open neighbourhood of ϕ(x) contained in U . Then x ∈ ϕ −1 ( f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In )) ⊆ ϕ −1 (U ). But ϕ −1 ( f 1−1 (I1 ) ∩ . . . ∩ f n−1 (In )) = ( f 1 ◦ ϕ)−1 (I1 ) ∩ . . . ∩ ( f n ◦ ϕ)−1 (In ) = (ϕ ∗ f 1 )−1 (I1 ) ∩ . . . ∩ (ϕ ∗ f n )−1 (In ).

Since the functions ϕ ∗ f 1 , . . . , ϕ ∗ f n ∈ C ∞ (R), Condition 1 applied to C ∞ (R) implies that (ϕ ∗ f 1 )−1 (I1 ) ∩ . . . ∩ (ϕ ∗ f n )−1 (In ) is open in R. Thus, every point x ∈ ϕ −1 (U ) has an open neighbourhood contained in ϕ −1 (U ). Therefore ϕ −1 (U ) is open in R. Hence, ϕ : R → S is continuous. Corollary 2.1.6 A diffeomorphism of differential spaces is a homeomorphism of the underlying topological spaces. An alternative way of constructing a differential structure on a set S, also used by Sikorski (1972), goes as follows. Let F be a family of real-valued functions on S. Endow S with the topology generated by a subbasis { f −1 (I ) | f ∈ F and I is an open interval in R}.

(2.1)

2.1 Differential spaces

17

Define C ∞ (S) by the requirement that h ∈ C ∞ (S) if, for each x ∈ S, there exist an open subset U of S, functions f 1 , . . . , f n ∈ F, and F ∈ C ∞ (Rn ) such that h |U = F( f 1 , . . . , f n )|U .

(2.2)

Clearly, F ⊆ C ∞ (S). In Theorem 2.1.7 below, we show that C ∞ (S) defined here is a differential structure on S. We refer to it as the differential structure on S generated by F. Theorem 2.1.7 The family C ∞ (S) defined above is a differential structure on S. Proof Condition 1 of Definition 2.1.1 is satisfied by the choice of topology on S. To show that Condition 2 is satisfied, let h 1 , . . . , h n ∈ C ∞ (S) and F ∈ C ∞ (Rn ). By definition, for each x ∈ S, there exist an open neighbourhood U of x, functions Fi ∈ C ∞ (Rn i ) for i = 1, . . . , n, and functions fi ji ∈ F, where ji = 1, . . . , n i , such that h i|U = Fi ( f i1 , . . . , f ini )|U for every i = 1, . . . , n. Then, F(h 1 , . . . , h n )|U = F(h 1|U , . . . , h n|U ) = F(F1 ( f 11 , . . . , f 1n 1 )|U , . . . , Fn ( f n1 , . . . , f nn n )|U ) = F(F1 ( f 11 , . . . , f 1n 1 ), . . . , Fn ( f n1 , . . . , f nnn ))|U = F(F1 , . . . , Fn )(( f 11 , . . . , f 1n 1 ), . . . , ( fn1 , . . . , f nn n ))|U . Since F(F1 , . . . , Fn ) ∈ C ∞ (Rm ), where m = n 1 + . . . + n n and f i ji ∈ F, it follows that F(h 1 , . . . , h n ) ∈ C ∞ (S). Hence, Condition 2 is satisfied. To verify Condition 3, suppose that h : S → R is a function satisfying the assumption of Condition 3. In other words, for every x ∈ S, there exists an open neighbourhood U of x and h x ∈ C ∞ (S) such that h |U = h x|U . By the construction of C ∞ (S), there exist a neighbourhood Ux of x in S, functions f x1 , . . . , f xn ∈ F and a function Fx ∈ C ∞ (Rn ) such that h x|Ux = Fx ( f x1 , . . . , f xn )|Ux . Hence, h |U ∩Ux = h x|U ∩Ux = Fx ( f x1 , . . . , f xn )|U ∩Ux , which implies that h ∈ C ∞ (S). Hence, C ∞ (S) is a differential structure on S.

18

Differential structures

Let R be a differential space with a differential structure C ∞ (R), and let S be an arbitrary subset of R endowed with the subspace topology (open sets in S are of the form S ∩ U , where U is an open subset of R). Let R(S) = { f |S | f ∈ C ∞ (R)}.

(2.3)

In other words, R(S) is the space of restrictions to S of smooth functions on R. Proposition 2.1.8 The space R(S) of restrictions to S ⊆ R of smooth functions on R generates a differential structure C ∞ (S) on S such that the differential-space topology of S coincides with its subspace topology. In this differential structure, the inclusion map ι : S → R is smooth. Proof Theorem 2.1.7 ensures that C ∞ (S) is a differential structure on S. By assumption, the family { f −1 (I ) | f ∈ C ∞ (S) and I is an open interval in R} is a subbasis for the topology of R. Hence, { f −1 (I ) ∩ S | f ∈ C ∞ (R) and I is an open interval in R} = { f |S−1 (I ) | f ∈ C ∞ (R) and I is an open interval in R} = { f |S−1 (I ) | f |S ∈ R(S) and I is an open interval in R} is a subbasis for the subspace topology of S. Therefore, the differential-space topology of S coincides with its subspace topology. For each f ∈ C ∞ (R), the pull-back ι∗ f of f by the inclusion map is the restriction of f to S. Hence, ι∗ f ∈ R(S) ⊆ C ∞ (S) and the inclusion map ι : S → R is smooth. In the following, for every subset S of a differential space R, we use the differential structure C ∞ (S) on S described above. If we want to emphasize that S has the subspace topology and C ∞ (S) is generated by restrictions to S of functions in C ∞ (R), we say that S is a differential subspace of R. If S with the differential structure C ∞ (S) is a manifold, we say that S is a submanifold of the differential space R. We shall also encounter the situation in which S is a subset of a differential space R endowed with a topology T that is finer than the subspace topology. In this case, we consider the space C ∞ (S) of functions on S obtained from C ∞ (R) as follows. A function f ∈ C ∞ (S) if, for each set U ⊆ S that is open in the topology T , there exists a function h ∈ C ∞ (R) such that f |U = h |U . If C ∞ (S) satisfies Condition 1 of Definition 2.1.1, then we say that S is an immersed differential space. If, in addition, S with the differential structure C ∞ (S) is a manifold, then we say that S is an immersed manifold.

2.1 Differential spaces

19

differential structures C ∞ (S1 ) and S1 and π2 : S1 × S2 → S2 be the canonical projections on the first and the second factor, respectively. Consider the family π1∗ (C ∞ (S1 )) ∪ π2∗ (C ∞ (S2 )) of functions on S1 × S2 consisting of pull-backs to S1 × S2 of functions in C ∞ (S1 ) and functions in C ∞ (S2 ). Let S1 and S2 be differential spaces with C ∞ (S2 ), respectively. Let π1 : S1 × S2 →

Proposition 2.1.9 The family π1∗ (C ∞ (S1 )) ∪ π2∗ (C ∞ (S2 )) of functions on S1 × S2 generates a differential structure C ∞ (S1 × S2 ) on S1 × S2 such that the differential-space topology on S1 × S2 coincides with its product topology. In this differential structure, the projections π1 and π2 are smooth. Proof Theorem 2.1.7 ensures that C ∞ (S1 × S2 ) is a differential structure on S1 × S2 . By assumption, the families of sets { f i−1 (I ) | f i ∈ C ∞ (Si ) and I is an open interval in R}, where i = 1, 2, are subbases for the topologies of S1 and S2 , respectively. Hence, the family of sets { f 1−1 (I1 ) × S2 , S1 × f 2−1 (I2 )}, where f i ∈ C ∞ (Si ) and I1 and I2 are open intervals in R, is a subbasis for the product topology in S1 × S2 . On the other hand, f 1−1 (I1 ) × S2 = {(x1 , x2 ) ∈ S1 × S2 | f 1 (x1 ) ∈ I1 }

= {(x 1 , x2 ) ∈ S1 × S2 | π1∗ f 1 ((x1 , x2 )) ∈ I1 }.

Similarly, S1 × f 2−1 (I2 ) = {(x1 , x2 ) ∈ S1 × S2 | f 2 (x2 ) ∈ I2 }

= {(x 1 , x2 ) ∈ S1 × S2 | π2∗ f 2 ((x1 , x2 )) ∈ I2 }.

Therefore, the family of sets { f 1−1 (I1 )× S2 , S1 × f 2−1 (I2 )}, where fi ∈ C ∞ (Si ) and I1 and I2 are open intervals in R, is contained in the family { f −1 (I ) | f i ∈ C ∞ (S1 × S2 ) and I is an open interval in R}. Therefore, the differential-space topology on S1 ×S2 coincides with the product topology. By construction, for each i = 1, 2 and f i ∈ C ∞ (Si ), the pull-back πi∗ f i ∈ ∞ C (S1 × S2 ). This implies that the projections π1 and π2 are smooth. Now consider an equivalence relation ∼ on a differential space S with differential structure C ∞ (S). Let R = S/ ∼ be the set of equivalence classes of ∼, and let ρ : S → R be the map assigning to each x ∈ S its equivalence class ρ(s).

20

Differential structures

Theorem 2.1.10 The space of functions on R, given by C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (S)}, is a differential structure on R. In this differential structure, the projection map ρ : S → R is smooth. Proof The differential-space topology of R defined by C ∞ (R) has a subbasis consisting of sets f −1 (I ), where f ∈ C ∞ (R) and I is an open interval. Since ρ −1 ( f −1 (I )) = (ρ ∗ f )−1 (I ) and ρ ∗ f ∈ C ∞ (S), it follows that ρ −1 ( f −1 (I )) is open in S. Hence, f −1 (I ) is open in the quotient topology of S for every open interval I and each f ∈ C ∞ (R). Therefore, the quotient topology of S is finer than the topology defined by C ∞ (R). This implies that the projection map ρ : S → R is continuous. For f 1 , . . . , f n ∈ C ∞ (R) and F ∈ C ∞ (Rn ), ρ ∗ F( f 1 , . . . , f n ) = F(ρ ∗ f 1 , . . . , ρ ∗ f n ) ∈ C ∞ (S). This shows that C ∞ (S) satisfies Condition 2 of Definition 2.1.1. To verify Condition 3, suppose that we have a function f : R → R such that, for every y ∈ R, there exist a neighbourhood U y of y in R and a function f y ∈ C ∞ (R) satisfying f |U y = f y|U y . Hence, for every x ∈ S, there exist a neighbourhood ρ −1 (Uρ(x) ) in S and a function ρ ∗ f ρ(x) ∈ C ∞ (S) such that ρ ∗ f |ρ −1 (Uρ(x) ) = ρ ∗ f ρ(x)|ρ −1 (Uρ(x) ) . This implies that ρ ∗ f ∈ C ∞ (S). Hence, f ∈ C ∞ (R). Thus, C ∞ (R) is a differential structure on R. By definition, f ∈ C ∞ (R) implies that ρ ∗ f ∈ C ∞ (S). Hence, the projection map ρ : S → R is smooth. It should be emphasized that, in general, the quotient topology of R = S/ ∼ is finer than the differential-space topology defined by C ∞ (R). Waz˙ ewski (1934) gave an example of a free improper action of R1 on R2 such that the only invariant smooth functions on R2 are constant functions. Let R denote the space of orbits of this action, and let ρ : R2 → R denote the orbit map. In this case, the differential structure C ∞ (R) consists of constant functions and the differential-space topology of R is trivial. A condition for the differential-space topology to coincide with the quotient topology is given below. Proposition 2.1.11 We use the notation of Theorem 2.1.10. The topology of R induced by C ∞ (R) coincides with the quotient topology of R if, for each set U in R which is open in the quotient topology, and each y ∈ U , there exists a function f ∈ C ∞ (R) such that f (y) = 1 and f |R\U = 0, where R\U denotes the complement of U in R.

2.2 Partitions of unity

21

Proof Let U be a set in R which is open in the quotient topology of R. For each y ∈ U , there exists f ∈ C ∞ (R) such that f (y) = 1 and f |R\U = 0. Hence, y ∈ f −1 (0, 2) ⊆ U . Moreover, f ∈ C ∞ (R) implies that f −1 (0, 2) is open in the differential-space topology of R. Therefore, U is open in the differential-space topology of R. Thus, the differential-space topology of R is finer than its quotient topology. By Theorem 2.1.10, the quotient topology of R is finer than its differential-space topology. Hence, the quotient topology of R coincides with its differential-space topology. We can characterize smooth Hausdorff manifolds of dimension n as Hausdorff differential spaces S such that every point x ∈ S has a neighbourhood U diffeomorphic to an open subset V of Rn . Here, the differential structures on U and V are generated by restrictions of smooth functions of S and Rn , respectively. We can weaken this definition by not requiring that V is open in Rn , and allowing n to be an arbitrary non-negative integer. Definition 2.1.12 A differential space S is subcartesian if it is Hausdorff and every point x ∈ S has a neighbourhood U diffeomorphic to a subset V of Rn . It should be noted that V in Definition 2.1.12 may be an arbitrary subset of Rn , and n may depend on x ∈ S. As in the theory of manifolds, diffeomorphisms of open subsets of S onto subsets of Rn are called charts on S. The family of all charts is the complete atlas on S.1

2.2 Partitions of unity The aim of this section is to establish the existence of a partition of unity for locally compact, second countable Hausdorff differential spaces. Lemma 2.2.1 For every open subset U of a differential space S and every x ∈ U , there exists f ∈ C ∞ (S) satisfying f |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. Proof Let U be open in S, and let x ∈ U. It follows from Condition 1 of Definition 2.1.1 that there exist a map ϕ = ( f1 , . . . , f n ) : S → Rn , with f 1 , . . . , f n ∈ C ∞ (S), and an open set U˜ ⊆ Rn such that x ∈ ϕ −1 (U˜ ) ⊆ U. Since ϕ(x) ∈ U˜ ⊆ Rn , there exists F ∈ C ∞ (Rn ) such that F|V˜ = 1 for some neighbourhood V˜ of ϕ(x) in Rn contained in U˜ , and F ˜ = 0 for some open |W

1 The original definition of a subcartesian space, due to Aronszajn (1967), was formulated in

terms of charts.

22

Differential structures

set W˜ in Rn such that U˜ ∪ W˜ = Rn . Since ϕ is continuous, V = ϕ −1 (V˜ ) and W = ϕ −1 (W˜ ) are open in V . Moreover, ϕ −1 (U˜ ) ⊆ U and U˜ ∪ W˜ = Rn imply that U ∪ W = S. By Condition 2, f = F( f 1 , . . . , f n ) ∈ C ∞ (S). Furthermore, f |V = (F ◦ ϕ)|V = F|ϕ(V ) = F|V˜ = 1. Similarly, f |W = F|W˜ = 0, which completes the proof. It follows from Lemma 2.2.1 that, for a Hausdorff differential space S, functions in C ∞ (S) separate points. In other words, for every pair (x, y) of points in S, there exist disjoint open neighbourhoods V and W of x and y, respectively, and a function f ∈ C ∞ (S) such that f |V = 1 and f |W = 0. Lemma 2.2.2 Let S be locally compact, Hausdorff and second countable. Then every open cover {Uα } of S has a countable, locally finite refinement consisting of open sets with compact closures. Proof Since S is second countable, there exists a countable family {Un } of open sets in S, which forms a basis for the topology of S. Let {Un k } be a subcollection consisting of sets with compact closures. The assumption that S is Hausdorff and locally compact implies that {Un k } is a basis for the topology of S. This can be seen as follows. Since S is locally compact, given a point x ∈ S there exists a compact set C in S containing an open neighbourhood U of x. That is, x ∈ U ⊆ C. Since C is a compact subset of the Hausdorff space S, it is closed. Hence, U ⊆ C, which implies that U is compact. The assumption that {Un } is a basis implies that there exists n x such that x ∈ Un x ⊆ U . Hence, U n x is compact. Therefore, Un x ∈ {Un k }. We set V−1 = V0 = ∅ and take V1 = Un 1 . There exists a smallest integer k1 such that V 1 ⊆ Un 1 ∪. . .∪Un k1 . We now set V2 = V1 ∪ Un 2 ∪. . .∪Un k1 . Continuing in this way, we obtain a sequence of open sets V j = Un 1 ∪. . .∪Un k j for every j ∈ N, where n k j is the smallest integer such that V j−1 ⊆ V j−1 ∪ Un k j−1 +1 ∪. . .∪Un k j . For each j, the closure V j of V j is contained in V j+1 and ∪∞ j=1 V j = S. For each j ∈ N, the set V j \V j−1 is compact and is contained in the open set V j+1 \V j−2 . Let {Uα }α∈A be an arbitrary open cover of S. Hence, {Uα ∩ (V j+1 \V j−2 )}α∈A is an open cover of the compact set V j \V j−1 and it admits j j a finite subcover. We denote by {W1 , . . . , Wm j } the finite collection of sets in j

{Uα ∩ (V j+1 \V j−2 )}α∈A which cover V j \V j−1 . Each Wi is contained in a j Wi j Wi

j

compact set V j+1 . Hence, is compact. Moreover, for some α ∈ A, Wi = j Uα ∩(V j+1 \V j−2 ), so that ⊆ Uα . Moreover, Wi ∩ Wlk = ∅ if | j −k |> 4. mj j i Finally, ∪∞ j=1 ∪i=1 W j = S. Hence, the collection {Wi | i = 1, . . . , m j , j ∈ N}

2.2 Partitions of unity

23

is a countable, locally finite refinement of {Uα } and consists of open sets with compact closures. Definition 2.2.3 A countable partition of unity on a differential space S is a countable family of functions { f i } ⊆ C ∞ (S) such that: (a) The collection of their supports is locally finite. (b) fi (x) ≥ 0 for each i and each x ∈ S. ∞ (c) i=1 f i (x) = 1 for each x ∈ S. Let {Uα } be an open cover of S. A partition of unity { f i } is subordinate to {Uα } if, for each i, there exists α such that the support of fi is contained in Uα . Theorem 2.2.4 Let S be a differential space with differential structure C ∞ (S), and let {Uα } be an open cover of S. If S is Hausdorff, locally compact and second countable, then there exists a countable partition of unity { fi } ⊆ C ∞ (S), subordinate to {Uα } and such that the support of each fi is compact. Proof Let {Uα } be an open cover of a differential space S. Since S is Hausdorff, locally compact and second countable, there exists a family Vi , i = 0, 1, . . . , of open sets with compact closures in S such that V0 = ∅, V j ⊆ V j+1 and ∪∞ j=1 V j = S. For x ∈ S, let i x be the largest integer such that x ∈ S\V i x . Choose αx such that x ∈ Uαx . By Conditions 1 and 2 of Definition 2.1.1, there exist functions f 1 , . . . , f n ∈ C ∞ (S) such that the smooth map ϕx = ( f 1 , . . . , f n ) : S → Rn takes x to ϕx (x) = (0, . . . , 0) ∈ Rn and ϕx−1 ((−3, 3)n ) ⊆ Uαx ∩ (Vi x +2 \V i x ). Let F ∈ C ∞ (Rn ) be a non-negative function such that F|[−1,1]n = 1 and F|Rn \(−2,2)n = 0. Then f x = F( f 1 , . . . , f n ) is in C ∞ (S) and has compact support contained in ϕx−1 ((−3, 3)n ), and f x has value 1 in ϕx−1 ((−1, 1)n ). For each i ≥ 1, choose a finite set of points xi1 , . . . , xiki in S whose ((−1, 1)n ) cover V i \Vi−1 . The functions f xi j ∈ C ∞ (S) neighbourhoods ϕx−1 ij are non-negative and their supports form a locally finite family of sets in S. Hence, f =

ki ∞

f xi j

i=1 j=1

is a well-defined positive function on S and Condition 3 implies that f ∈ C ∞ (S). Each function h i j = f −1 f xi j has compact support, and the family {h i j } forms a partition of unity on S subordinate to the cover {Uα }. Condition 2 ensures that the functions h i j are in C ∞ (S).

24

Differential structures

Many problems that arise in differential geometry are easy to solve locally. Partitions of unity are used to construct global solutions from such local solutions. In particular, we have the following corollary to Theorem 2.2.4. Corollary 2.2.5 Let S be a Hausdorff, locally compact and second countable differential space with differential structure C ∞ (S), and let R be a closed subset of S. The differential structure C ∞ (R) induced by the inclusion map ι : R → S consists of restrictions to R of functions in C ∞ (S). In the following chapters we restrict our attention to Hausdorff, locally compact, second countable subcartesian spaces. Hence, we shall be able to rely on the existence of partitions of unity. Note that a subcartesian space is Hausdorff; see Definition 2.1.12. Moreover, Definition 2.1.12 and Condition 1 of Definition 2.1.1 imply that a subcartesian space is locally compact. Thus, we make the following assumption. Assumption 2.2.6 All subcartesian differential spaces considered here are second countable.

3 Derivations

In this chapter, for a differential space S, we study the properties of derivations of C ∞ (S). We show that if S is subcartesian, then derivations of C ∞ (S) admit maximal integral curves. We define vector fields on a subcartesian space to be derivations that generate local one-parameter groups of local diffeomorphisms of the space. We conclude the chapter with a proof showing that orbits of a family of vector fields on a subcartesian space are immersed manifolds.

3.1 Basic properties Definition 3.1.1 A derivation of C ∞ (S) is a linear map X : C ∞ (S) → C ∞ (S) : f → X ( f ) satisfying Leibniz’s rule X ( f1 f 2 ) = X ( f1 ) f2 + f 1 X ( f2 )

(3.1)

for every f 1 , f 2 ∈ C ∞ (S). We denote by Der C ∞ (S) the space of derivations of C ∞ (S). This has the structure of a Lie algebra, with the Lie bracket [X 1 , X 2 ] defined by [X 1 , X 2 ]( f ) = X 1 (X 2 ( f )) − X 2 (X 1 ( f )) for every X 1 , X 2 ∈ Der C ∞ (S) and f ∈ C ∞ (S). Moreover, Der C ∞ (S) is a module over the ring C ∞ (S), and [ f 1 X 1 , f 2 X 2 ] = f 1 f 2 [X 1 , X 2 ] + f 1 X 1 ( f 2 )X 2 − f 2 X 2 ( f 1 )X 1 for every X 1 , X 2 ∈ Der C ∞ (S) and f 1 , f 2 ∈ C ∞ (S).

26

Derivations

Definition 3.1.2 A derivation of C ∞ (S) at x ∈ S is a linear map v : C ∞ (S) → R such that v( f 1 f 2 ) = v( f 1 ) f 2 (x) + f 1 (x)v( f 2 )

(3.2)

for every f 1 , f2 ∈ C ∞ (S). We interpret derivations of C ∞ (S) at x ∈ S as tangent vectors to S at x. The set of all derivations of C ∞ (S) at x is denoted by Tx S and is called the tangent space to S at x.1 If X is a derivation of C ∞ (S), then for every x ∈ S we have a derivation X (x) of C ∞ (S) at x given by X (x) : C ∞ (S) → R : f → X (x) f = (X f )(x).

(3.3)

The derivation (3.3) is called the value of X at x. Clearly, the derivation X is uniquely determined by the collection {X (x) | x ∈ S} of its values at all points in S. In order to avoid confusion between derivations of C ∞ (S) and derivations of C ∞ (S) at a point in S, we shall often refer to the former as global derivations of C ∞ (S), or simply as global derivations if C ∞ (S) is understood. Lemma 3.1.3 If f ∈ C ∞ (S) is a constant function, then X ( f ) = 0 for all X ∈ Der C ∞ (S). Proof If f ∈ C ∞ (S) is identically zero, then f 2 = f = 0, and Leibniz’s rule implies that X ( f ) = X ( f 2 ) = 2 f X ( f ) = 0 for every X ∈ Der C ∞ (S). Similarly, if f is a non-zero constant function, that is, f (x) = c = 0 for all x ∈ S, then f 2 = c f , and the linearity of derivations implies that X ( f 2 ) = X (c f ) = cX ( f ). On the other hand, Leibniz’s rule implies that X ( f 2 ) = 2 f X ( f ) = 2cX ( f ). Hence, cX ( f ) = 2cX ( f ). Since c = 0, it follows that X ( f ) = 0. Lemma 3.1.4 If f ∈ C ∞ (S) vanishes identically in an open set U ⊆ S, then X ( f )|U = 0 for all X ∈ Der C ∞ (S). Proof If f ∈ C ∞ (S) vanishes identically in an open set U ⊆ S, then for each x ∈ U , by Lemma 2.2.1 there exist h ∈ C ∞ (S) satisfying h |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. This implies that h f = 0. Therefore, 0 = X (h f ) = h X ( f )+ f X (h) for every derivation X. Evaluating this identity at x, we obtain X ( f )(x) = 0 because f (x) = 0. Hence, X ( f )|U = 0. 1 The term ‘Zariski tangent space’ is also used.

3.1 Basic properties

27

Proposition 3.1.5 Let U be an open subset of a differential space S. A derivation X of C ∞ (S) defines a unique derivation X |U of C ∞ (U ), called the restriction of X to U , such that X |U ( f |U ) = (X ( f ))|U

(3.4)

for every f ∈ C ∞ (S). Conversely, if Y is a derivation of C ∞ (U ), then for each x ∈ U there exist an open neighbourhood V of x contained in U , and X ∈ Der C ∞ (S) such that Y|V = X |V . Proof Given a derivation X of C ∞ (S), we need to define its restriction X |U . The action of X |U on the restrictions to U of functions in C ∞ (S) is defined in equation (3.4). In general, given h ∈ C ∞ (U ), for every x ∈ U there exist f x ∈ C ∞ (S) and an open neighbourhood Vx of x in U such that h |Vx = f x|Vx . We define X |U (h) to be the function in C ∞ (S) such that (X |U (h))|V = (X ( f ))|V . First, we show that X |U (h) is well defined. Suppose that for each x ∈ U , there is another choice Vx and f x such that h |Vx = f x|V . Then, h |Vx ∩Vx = x f x|Vx ∩Vx = f x|Vx ∩V . Therefore, ( f x − f x )|Vx ∩Vx = 0 and Lemma 3.1.3 implies x that X ( f x − f x )|Vx ∩Vx = 0. Hence, X ( f x ) is equal to X ( f x ) on Vx ∩ Vx . Since {Vx ∩ Vx }x∈U covers U , it follows that X |U is well defined. Suppose now that Y is a derivation of C ∞ (U ). It follows from Lemma 2.2.1 that there exists f ∈ C ∞ (S) satisfying f |V = 1 for some neighbourhood V of x contained in U , and f |W = 0 for some open subset W of S such that U ∪ W = S. Then f |U Y is a derivation of C ∞ (U ), which vanishes on U \W . Hence, it extends to a smooth derivation X of C ∞ (S) such that X |V = Y|V . Let S and R be differential spaces with differential structures C ∞ (S) and respectively, and let ϕ be a diffeomorphism of S onto R. For each X ∈ Der C ∞ (S), the map

C ∞ (R),

ϕ∗ X : C ∞ (R) → C ∞ (R) : f → ϕ∗ X ( f ) = (ϕ −1 )∗ (X (ϕ ∗ f ))

(3.5)

is a derivation of C ∞ (R), called the push-forward of X by the diffeomorphism ϕ. Moreover, ϕ∗ : Der C ∞ (S) → Der C ∞ (R) : X → ϕ∗ X is a Lie algebra isomorphism. Suppose now that ϕ : S → R is a smooth map. Derivations X in Der C ∞ (S) and Y in Der C ∞ (R) are said to be ϕ-related if ϕ ∗ (Y ( f )) = X (ϕ ∗ f ) for every f ∈ C ∞ (R). In this case, we say that X pushes forward to a derivation Y on R and write Y = ϕ∗ X .

(3.6)

28

Derivations

Subcartesian spaces are differential spaces that are locally diffeomorphic to subsets of Rn . Therefore, local information about subcartesian spaces can be obtained by investigating differential structures of subsets of Rn . Taking Proposition 3.1.5 into account, we see that local information about derivations on subcartesian spaces can be completely determined by a study of derivations on subsets of Rn . Theorem 3.1.6 Let S be a differential subspace of Rn , and let X be a derivation of C ∞ (S). For each x ∈ S ⊆ Rn , there exist a neighbourhood U of x in Rn and a vector field Y on Rn such that X (F|S )|U ∩S = (Y (F))|U ∩S for every F ∈ C ∞ (Rn ). Proof Let u be a derivation of C ∞ (S) at x ∈ S ⊆ Rn . For each F ∈ C ∞ (Rn ), the restriction F|S of F to S is in C ∞ (S). It is easy to see that the map C ∞ (Rn ) → R : F → u(F|S ) is a derivation at x of C ∞ (Rn ). We denote the natural coordinate functions on Rn by x 1 , . . . , x n : Rn → R. Every derivation Y of C ∞ (Rn ) is of the form n

Fi

i=1

∂ , ∂xi

where F i = Y (x i ) for i = 1, . . . , n. Let X be a derivation of C ∞ (S), and let F ∈ C ∞ (Rn ). For each x ∈ S, the derivation X (x) of C ∞ (S) at x gives a derivation of C ∞ (Rn ) at x. Hence, X (F|S )(x) = X (x)(F|S ) =

n ∂F i (x)(X (x)(x|S )) ∂xi i=1

=

n i=1

∂F i (x)(X (x|S ))(x). ∂xi

This is valid for every x ∈ S. Hence, n ∂F i (X (x|S )). X (F|S ) = ∂xi i=1

i ) are in C ∞ (S). Since S is a differFor i = 1, . . . , n, the coefficients X (x|S n ential subspace of R , for each x ∈ S there exist a neighbourhood U of x in i ) i Rn and functions F 1 , . . . , F n ∈ C ∞ (Rn ) such that X (x |S |U ∩S = F|U ∩S for each i = 1, . . . , n. Hence,

3.1 Basic properties X (F|S )|U ∩S =

n i=1

∂F F ∂xi

29

i

. |U ∩S

Since the F 1 , . . . , F n are smooth functions on Rn , it follows that Y =

n

Fi

i=1

is a vector field on

∂ ∂xi

Rn .

We can rephrase Theorem 3.1.6 by saying that every derivation on a differential subspace S of Rn can be locally extended to a vector field on Rn . Suppose that S is closed. In this case, we can use a partition of unity on Rn to extend every derivation of C ∞ (S) to a global vector field on Rn . Conversely, suppose that Y =

n i=1

Fi

∂ ∂xi

Rn .

is a vector field on We want to know under what conditions Y restricts to a derivation of S. Since S is a differential subspace of Rn , its differential structure C ∞ (S) is generated by the ring R(S) = {F|S | F ∈ C ∞ (Rn )},

(3.7)

of smooth functions on Rn

in the sense of Propoconsisting of restrictions to S sition 3.1.8. Let N (S) denote the ideal of functions in C ∞ (Rn ) that vanish identically on S: N (S) = {F ∈ C ∞ (Rn ) | F|S = 0}.

(3.8)

We identify R(S) with the quotient C ∞ (Rn )/N (S). Proposition 3.1.7 Every derivation of R(S) at x extends to a unique derivation of C ∞ (S) at x. Proof Let w be a derivation of R(S) at x ∈ S. Consider f ∈ C ∞ (S). There exist an open neighbourhood U of x in Rn and a function Fx ∈ C ∞ (Rn ) such ( f ) = w(F x |S ). Let V be another open neighthat f |U ∩S = Fx |U ∩S . Set w bourhood of x in Rn , and let Hx ∈ C ∞ (Rn ) be a function such that f |V ∩S = Hx |V ∩S . Now U ∩ V ∩ S is an open subset of S, and Fx |U ∩V ∩S = Hx |U ∩V ∩S . Therefore (Fx − Hx )|U ∩V ∩S = 0, i.e. (Fx − Hx )|S ∈ R(S) ⊂ C ∞ (Rn ) vanishes identically on the open subset U ∩V ∩S of S. Hence, w(Fx |S −Hx |S ) = 0. This proves that the extension w is a well-defined derivation of C ∞ (S) that extends the derivation w of R(S) at x. Finally, it is clear that such an extension w of w is uniquely defined.

30

Derivations

Proposition 3.1.8 A derivation w of C ∞ (Rn ) at x ∈ S ⊆ Rn defines a derivation of C ∞ (S) at x if and only if w annihilates N (S); that is, w(F) = 0 for all F ∈ N (S). Proof It follows from Proposition 3.1.7 that derivations at x of C ∞ (S) are determined by derivations of R(S) at x. But R(S) = C ∞ (Rn )/N (S). Hence, a derivation w of C ∞ (Rn ) at x passes to the quotient if and only if w(F) = 0 for all F ∈ N (S). Corollary 3.1.9 A smooth vector field Y on Rn restricts to a derivation of C ∞ (S) if Y preserves the null ideal N (S) of S. In other words, Y (F) ∈ N (S) for every F ∈ N (S). Proof By Proposition 3.1.8, if Y restricts to a derivation of C ∞ (S), then for every x ∈ S and F ∈ N (S), we have Y (F)(x) = 0. Hence, Y (F)|S = 0, which implies that Y (F) ∈ N (S). Conversely, suppose that Y preserves the null ideal of N (S) of S. If F, H ∈ C ∞ (Rn ) are such that F − H ∈ N (S), then Y (F) − Y (H ) ∈ N (S) and Y (F)|S = Y (H )|S . Thus, the map F|S → Y (F)|S is a well-defined derivation of R(S). By Proposition 3.1.7, it extends to a derivation of C ∞ (S). Example 3.1.10 Let S = {(x, y) ∈ R2 | x y = 0}. In other words, S is the union of the x-axis and the y-axis in R2 . Since S is a closed subset of R2 , smooth functions on S are restrictions to S of smooth functions on R2 . In other words, C ∞ (S) = R(S). Corollary 3.1.9 implies that a vector field Y = Yx

∂ ∂ + Yy ∂x ∂y

on R2 restricts to a derivation of C ∞ (S) if and only if Y preserves the null ideal N (S) of S. Hence, Y (x y) = Yx (x, y)y + Y y (x, y)x = 0 if x y = 0, which implies Yx (0, y) = 0 and Y y (x, 0) = 0 for all x, y ∈ R.

(3.9)

Conversely, if F(x, y) ∈ N (S), then F(x, 0) = F(0, y) = 0 for all x, y ∈ R. Hence, Y (F)(x, y) = Yx (x, y)

∂F ∂F (x, y) + Y y (x, y) (x, y). ∂x ∂y

3.2 Integration of derivations

31

In particular, using equation (3.9), we obtain Y (F)(0, y) = Yx (0, y)

∂F ∂F (0, y) + Y y (0, y) (0, y) = 0, ∂x ∂y

(3.10)

Y (F)(x, 0) = Yx (x, 0)

∂F ∂F (x, 0) + Y y (x, 0) (x, 0) = 0 ∂x ∂y

(3.11)

for every x, y ∈ R. Equations (3.10) and (3.11) imply that Y (F) ∈ N (S). Thus, equation (3.9) ensures that Y restricts to a derivation of C ∞ (S). By Theorem 3.1.6, every derivation X of C ∞ (S) extends locally to a derivation of C ∞ (R2 ). Hence, X (0, 0) = 0 for every derivation X of C ∞ (S). On the other hand, if ∂ ∂ + vy v = vx ∂x ∂y is a non-zero derivation of C ∞ (R2 ) at the origin (0, 0), then for every F ∈ N (S), v(F) = vx

∂F ∂F (0, 0) + v y (0, 0) = 0, ∂x ∂y

because F(x, 0) = F(0, y) = 0 implies ∂∂ Fx (0, 0) = ∂∂Fy (0, 0) = 0. Therefore, v defines a derivation of C ∞ (S) at (0, 0) ∈ S, which is not a value of a global derivation X .

3.2 Integration of derivations We show here that derivations on a subcartesian space admit unique maximal integral curves. Let c : I → S be a smooth map of an interval I in R to a differential space S. If I has a non-empty interior, we say that c is an integral curve of a derivation X of C ∞ (R) if d f (c(t)) = X ( f )(c(t)) (3.12) dt for every f ∈ C ∞ (S) and every t ∈ I . We extend the notion of an integral curve to the case when the interior of I is empty by saying that if I is a single point in R, then c : I → S is an integral curve of every derivation of C ∞ (S). Integral curves of a given derivation X of C ∞ (S) can be ordered by inclusion of their domains. In other words, if c1 : I1 → S and c2 : I2 → S are two integral curves of X and I1 ⊆ I2 , then c1 c2 . An integral curve c1 : I → S of X is maximal if c1 c2 implies that c1 = c2 .

32

Derivations

Theorem 3.2.1 Let S be a subcartesian space, and let X be a derivation of C ∞ (S). For every x ∈ S, there exists a unique maximal integral curve c of X such that c(0) = x. Proof (i) Local existence. For x ∈ S, let ϕ be a diffeomorphism of a neighbourhood V of x in S onto a differential subspace R of Rn . Let Z = ϕ∗ X |V be a derivation of C ∞ (R) obtained by pushing forward the restriction of X to V by ϕ. In other words, Z ( f ) ◦ ϕ = X |V ( f ◦ ϕ) for all f ∈ C ∞ (R). Without loss of generality, we may assume that there is an extension of Z to a vector field Y on Rn . Let z = ϕ(x), and let c0 be an integral curve in Rn of the vector field Y such that c0 (0) = z. Let I x be the connected component of c0−1 (R) containing 0, and let c : I x → R be the curve in R obtained by the restriction of c0 to I x . Clearly, c(0) = z. For each t0 ∈ Ix and each f ∈ C ∞ (R), there exist a neighbourhood U of c(t0 ) in R and a function F ∈ C ∞ (Rn ) such that f |U = F|U . Therefore, d d f (c(t))|t=t0 = F(c(t))t=t0 = (Y (F))(c(t0 )) dt dt = (Y (F))|U (c(t0 )) = (Z ( f ))(c(t0 )), which implies that c : I x → R is an integral curve of Z through z. Since Ix is a connected subset of R containing 0, it is an interval (possibly the single point {0}). Then cx = ϕ −1 ◦ c : Ix → V ⊆ S satisfies cx (0) = ϕ −1 (c(0)) = ϕ −1 (z) = x. Moreover, for every t ∈ I x and h ∈ C ∞ (S), f = h ◦ ϕ −1 ∈ C ∞ (R) and d d d h(cx (t)) = h(ϕ −1 (c(t))) = (h ◦ ϕ −1 )(c(t)) dt dt dt d = ( f (c(t)) = Z ( f )(c(t)) dt = Z (h ◦ ϕ −1 )(ϕ ◦ cx (t)) = X (h)(cx (t)). Thus, cx : I x → S is an integral curve of X through x. (ii) Smoothness. It follows from the theory of differential equations that the integral curve c0 in Rn of a smooth vector field Y is smooth. Hence, c = c0|Ix is smooth. Since ϕ is a diffeomorphism of a neighbourhood of x in S to R, its inverse ϕ −1 is smooth, and the composition cx = ϕ −1 ◦ c is smooth. (iii) Local uniqueness. This follows from the local uniqueness of the solutions of first-order differential equations in Rn . (iv) Maximality. Suppose that p ≤ 0 ≤ q are the end points of the domain I of the integral curve c of X through x obtained in section (i) of the proof.

3.2 Integration of derivations

33

If q ∈ I , q = ∞ or limt→q − c(t) does not exist, then the curve c does not extend beyond q. If x 1 = limt→q − c(t) exists, then it is unique because S is Hausdorff and we can repeat the construction of section (i) beginning from the point x1 . In this way, we obtain an integral curve c1 : I1 → S of X with the initial condition c1 (0) = x 1 . Let I˜1 = I ∪ {t = q + s | s ∈ I1 ∩ [0, ∞)}, and let c˜1 : I˜1 → S be given by c˜1 (t) = c(t) if t ∈ I and c˜1 (t) = c1 (t − q) if t ∈ {q + s | s ∈ I1 ∩ [0, ∞)}. Clearly, c˜1 is continuous. Moreover, since x1 = limt→q − c(t), it follows that the lower end point p1 of I1 is strictly less than zero. Hence, the restriction of c to (max( p, p1 ) + q, q) differs from the restriction of c1 to (max( p, p1 ), 0) by the reparametrization t → t − q. Since c and c1 are smooth, it follows that c˜1 is smooth. Let q1 be the upper limit of I1 . If q1 ∈ I1 , q1 = ∞ or limt→q − c1 (t) does not exist, then the curve c1 does 1 not extend beyond q1 . Otherwise, we can extend c˜1 by an integral curve c2 of X through x2 = limt→q − c1 (t). Continuing the process, we obtain a maximal 1 extension for t ≥ 0. We can construct a maximal extension for t ≤ 0 in a similar way. (v) Global uniqueness. Let c : I → S and c : I → S be two maximal integral curves of X through x, and let T + = {t ∈ I ∩ I | t > 0 and c(t) = c (t)}. Suppose that T + = ∅. Since T + is bounded from below by 0, there exists a greatest lower bound l of T + . This implies that c(t) = c (t) for 0 ≤ t ≤ l and, for every ε > 0, there exists tε ∈ T + such that l < tε < l+ε and c(tε ) = c (tε ). Let xl = c(l) = c (l), and let cl : Il → S be an integral curve of X through xl constructed as in section (i). We denote by ql the upper end point of the interval Il . If ql > 0, the local uniqueness implies that c(t) = c (t) = cl (t − l) for all l ≤ t ≤ l + ql . Hence, we obtain a contradiction with the assumption that l is the greatest lower bound of T + . If ql = 0, then there is no extension of cl to t > 0. Let q and q be the upper end points of I and I , respectively. Since c and c are maximal integral curves of X , it follows that q = q = l. Hence, the set T + is empty. A similar argument shows that T − = {t ∈ I ∩ I | t < 0 and c(t) = c (t)} = ∅. Therefore, c(t) = c (t) for all t ∈ I ∩ I . If I = I , then we obtain a contradiction with the assumption that c and c are maximal. Hence, I = I and c = c . Let X be a derivation of C ∞ (S). We denote by (exp t X )(x) the point on the maximal integral curve of X through x corresponding to the value t of the parameter. Given x ∈ S, (exp t X )(x) is defined for t in an interval I x

34

Derivations

containing zero, and (exp 0X )(x) = x. If t, s and t + s are in Ix , if s ∈ I(exp t X )(x) , and if t ∈ I(exp s X )(x) , then (exp(t + s)X )(x) = (exp s X )((exp t X )(x)) = (exp t X )((exp s X )(x)). In the case when S is a manifold, the map exp t X is a local one-parameter group of local diffeomorphisms of S. For a subcartesian space S, the map exp t X might fail to be a local diffeomorphism. Definition 3.2.2 A vector field on a subcartesian space S is a derivation X of C ∞ (S) such that for every x ∈ S, there exist an open neighbourhood U of x in S and ε > 0 such that for every t ∈ (−ε, ε), the map exp t X is defined on U , and its restriction to U is a diffeomorphism from U onto an open subset of S. In other words, X is a vector field on S if exp t X is a local one-parameter group of local diffeomorphisms of S. Example 3.2.3 Consider S = [0, ∞) ⊆ R with the structure of a differential subspace of R. Let (X f ) = f (x) for every f ∈ C ∞ ([0, ∞)) and x ∈ [0, ∞). Note that the derivative at x = 0 is the right derivative; it is uniquely defined by f (x) for x ≥ 0. For this X , the map exp t X is given by (exp t X )(x) = x + t whenever x and x + t are in [0, ∞). In particular, for every neighbourhood U of 0 in [0, ∞), there exists δ > 0 such that [0, δ) ⊆ U . Moreover, exp t X maps [0, δ) onto [t, δ + t), which is not an open neighbourhood of t = (exp t X )(0) in [0, ∞). Hence, the derivation X is not a vector field on [0, ∞). On the other hand, for every f ∈ C ∞ [0, ∞) such that f (0) = 0, the derivation f X is a vector field, because 0 is a fixed point of f X . A subset A of a topological space T is locally closed if every point x ∈ A has a neighbourhood Ux in T such that A ∩ Ux is closed in Ux . The closure of A ∩ U x in Ux is the intersection A ∩ Ux , where A is the closure of A in T . Therefore, A ∩ Ux = A ∩ U x for every x ∈ A. Taking the union over all x ∈ A, we obtain A= A ∩ Ux = A ∩ Ux = A ∩ Ux . x∈A

x∈A

x∈A

Hence, A is an intersection of an open and a closed set or, equivalently, A is open in A. We extend the notion of local closedness to subcartesian spaces as follows. Definition 3.2.4 A subcartesian space S is locally closed if every point of S has a neighbourhood diffeomorphic to a locally closed subset of Rn . Next, we give a simple characterization of vector fields on locally closed subcartesian spaces.

3.2 Integration of derivations

35

Lemma 3.2.5 Let S be a differential subspace of Rn . If U and V are open subsets of Rn and ϕ : U → V is a diffeomorphism such that ϕ(U ∩ S) = V ∩ S, then the restriction of ϕ to U ∩ S is a diffeomorphism ψ of U ∩ S onto V ∩ S. Proof By assumption, S is a topological subspace of Rn , the mapping ϕ : U → V is a homeomorphism and ϕ(U ∩ S) = V ∩ S. Hence, for every open subset W of Rn , ϕ −1 (W ∩ (V ∩ S)) is open in U ∩ S and ϕ(W ∩ (U ∩ S)) is open in V ∩ S. Thus, ϕ induces a homeomorphism ψ : U ∩ S → V ∩ S. Moreover, ϕ induces a diffeomorphism of the open differential subspaces U and V of Rn . We want to show that f ∈ C ∞ (V ∩ S) implies that ψ ∗ f ∈ C ∞ (U ∩ S). Given x ∈ U ∩ S, let y = ψ(x) ∈ V ∩ S. Since S is a differential subspace of Rn and f ∈ C ∞ (V ∩ S), there exist a neighbourhood W of y in V and a function f W ∈ C ∞ (V ) such that f |W ∩S = f W |W ∩S . Moreover, ϕ −1 (W ) is a neighbourhood of x in U , ϕ ∗ f W is in C ∞ (U ), and (ψ ∗ f )|ϕ −1 (W )∩S = ( f ◦ ψ)|ϕ −1 (W )∩S = f ◦ ϕ|ϕ −1 (W )∩S = f |W ∩S = f W |W ∩S = f ◦ ϕ|ϕ −1 (W )∩S = f W ◦ ϕ|ϕ −1 (W )∩S = (ϕ ∗ f W )|ϕ −1 (W )∩S . Thus, for every x ∈ U ∩ S, there exist a neighbourhood ϕ −1 (W ) of x in U and a function ϕ ∗ f W in C ∞ (U ) such that (ψ ∗ f )|ϕ −1 (W )∩S = (ϕ ∗ f W )|ϕ −1 (W )∩S . This implies that ψ ∗ f ∈ C ∞ (U ∩ S). It follows that ψ is smooth. We can prove in a similar manner that ψ −1 is smooth. Hence, ψ is a diffeomorphism. Proposition 3.2.6 Let S be a locally closed subcartesian space. A derivation X of C ∞ (S) is a vector field on S if the domain of every maximal integral curve of X is open in R. Proof Consider first the case when S is a locally closed differential subspace of Rn . That is, S = O ∩ C, where O is open and C is closed in Rn . Let X be a derivation on S such that domains of all its integral curves are open in R. In other words, for each x ∈ S, the domain I x of the map t → (exp t X )(x) is an open interval in R. We need to show that the map x → (exp t X )(x) is a local diffeomorphism of S. Given x0 ∈ S, there exists an open neighbourhood W of x0 such that the restriction of X to W extends to a vector field Y on Rn . We show first that the restriction of X to W generates a local one-parameter group of local diffeomorphisms of W . Since open sets in S are intersections of open sets in Rn with S, we can write without loss of generality W = U ∩ C, where U is an open set in Rn contained in O. Let exp tY denote the local one-parameter group of local diffeomorphisms of U generated by Y . For each x ∈ W , we denote by Jx

36

Derivations

the maximum interval in R such that (exp t X )(x) = exp(tY )(x) for all t ∈ Jx . Note that Jx is the intersection of I x and K x = {t ∈ R | (exp tY )(x) ∈ U }. Since exp tY is a local one-parameter group of local diffeomorphisms of an open subset U in Rn , it follows that K x is open in R. Hence, the assumption that I x is an open interval implies that Jx is also an open interval. Given x ∈ W = U ∩ C, there exist ε > 0 and a neighbourhood U of x in U such that, for every t ∈ (−ε, ε), the map exp tY is defined on U , and its restriction to U is a diffeomorphism from U onto an open subset of U. In view of Lemma 3.2.5, it suffices to show that there exist δ ∈ (0, ε] and a neighbourhood U of x in U such that exp tY maps U ∩ C to ((exp tY )(U )) ∩ C for all t ∈ (−δ, δ). Suppose that there are no U and δ satisfying this condition. This means that for every neighbourhood U of x in U and every δ ∈ (0, ε], there exist a point y ∈ U ∩ C and s ∈ (−δ, δ) such that (exp sY )(y) ∈ / ((exp tY )(U )) ∩ C. Since (exp sY )(y) ∈ (exp tY )(U ) for every t ∈ (−ε, ε), it follows that (exp sY )(y) ∈ / C. Hence, s is not in the domain I y of the maximal integral curve of X through y. If s > 0, let u be the infimum of the set {t ∈ [0, s] | (exp tY )(y) ∈ / C}. Then, (exp t X )(y) ∈ C for all t ∈ [0, u). Since t → (exp t X )(y) is continuous and C is closed, it follows that (exp u X )(y) ∈ C. Moreover, for every v > u, there exists t ∈ (u, v) such that (exp t X )(y) ∈ / C. This implies that [0, ∞) ∩ Jy = [0, u]. Since Jy is open if the domain I y of the maximal integral curve of X through y is open, it follows that I y is not open in R, contrary to the assumption of the theorem. Hence, the case s > 0 is excluded. Similarly, we can show that the case s < 0 is inconsistent with the assumption that the domains of all maximal integral curves of X are open. We have shown that there exist δ ∈ (0, ε] and a neighbourhood U of x in U such that exp tY maps U ∩ C to ((exp tY )(U )) ∩ C for all t ∈ (−δ, δ). This implies that exp t X (z) = exp tY (z) is defined for every t ∈ (−δ, δ) and each z ∈ U . By Lemma 3.2.5, it follows that exp t X restricted to U ∩ W is a diffeomorphism onto (exp tY )(U ) ∩ W . Since this holds for every x ∈ W , we conclude that exp t X is a local one-parameter group of local diffeomorphisms of W . Hence, X |W is a vector field on W . Since for every x0 ∈ S there is an open neighbourhood W of x0 in S such that X |W is a vector field on W , it follows that X is a vector field on S. Now consider the case of a general locally closed subcartesian space S. Let X be a derivation on S such that the domains of all maximal integral curves of X are open. For each x ∈ S, the function I x → S : t → (exp t X )(x), where I x is an open interval in R, is continuous.

3.3 The tangent bundle

37

For every x ∈ S, there exist a neighbourhood W of x in S and a diffeomorphism χ of W onto a locally closed subspace U ∩ C of Rn . By the first part of the proof, the push-forward of X by the diffeomorphism χ is a vector field on U ∩C. Since χ is a diffeomorphism, it follows that there exist a neighbourhood W of x in W ⊆ S and ε > 0 such that, for every t ∈ (−ε, ε), the map exp t X is defined on W , and its restriction to W is a diffeomorphism from W onto an open subset of W ⊆ S. Hence, X is a vector field on S. The following example shows that the assumption that S is locally closed is essential in Proposition 3.2.6. Example 3.2.7 The set S = {(x1 , x2 ) ∈ R2 | x12 + (x2 − 1)2 < 1 or x2 = 0} is not locally closed at (0, 0). The vector field Y =

∂ ∂ x1

on R2 restricts to a derivation X of C ∞ (S). For every (x1 , x2 ) ∈ R2 , (exp tY )(x1 , x 2 ) = (x1 +t, x2 ) for all t ∈ R. All integral curves of X have open domains. Nevertheless, exp t X fails to be a local one-parameter local group of diffeomorphisms of S.

3.3 The tangent bundle Recall that a derivation of C ∞ (S) at x ∈ S is a linear map v : C ∞ (S) → R such that v( f 1 f 2 ) = v( f 1 ) f 2 (x) + f 1 (x)v( f 2 ) for every f 1 , f 2 ∈ C ∞ (S); see Definition 3.1.2. The existence of integral curves of global derivations established in the preceding section justifies the interpretation of derivations of C ∞ (S) at x ∈ S as tangent vectors to S at x. We denote by Tx S the set of all derivations of C ∞ (S) at x. This is a vector space, referred to as the tangent space to S at x. Definition 3.3.1 The tangent bundle of a differential space S is the union T S of all tangent spaces to S at all points in S. In other words, Tx S. TS = x∈S

The tangent bundle projection is the map τ : T S → S that assigns to each v ∈ T S the point x ∈ S such that v ∈ Tx S.

38

Derivations

We want to describe the differential structure of T S. First, consider the case when S is a differential subspace of Rn . We denote by q1 , . . . , qn the restrictions to S of the canonical coordinate functions (x 1 , . . . , xn ) on Rn . For every function f ∈ C ∞ (S) and x ∈ S, there exist a neighbourhood U of x in Rn and F ∈ C ∞ (Rn ) such that f |U ∩S = F(q1 , . . . , qn )|U ∩S .

(3.13)

Consider v ∈ Tx S, and let yi = v(qi ) for i = 1, . . . , n. Equation (3.13) yields v( f ) = (∂1 F)v(q1 ) + . . . + (∂n F)v(qn ) = y1 ∂1 F + . . . + yn ∂n F,

(3.14)

where ∂i f =

∂F . ∂qi

Equation (3.14) shows that v ∈ Tx S can be identified with a vector (y1 , . . . , yn ) ∈ Rn . Since Tx S has the structure of a vector space, the set Vx = {(v(q1 ), . . . , v(qn )) ∈ Rn | v ∈ Tx S}

(3.15)

is a vector subspace of Rn . The tangent bundle T S is the subset of R2n given by T S = {(x, y) = (x 1 , . . . , xn , y1 , . . . , yn ) ∈ R2n | x ∈ S and y ∈ Vx }. (3.16) We now consider the case of a general differential space S. For every f ∈ C ∞ (S), the differential of f is a function d f : T S → R given by d f (v) = v( f )

(3.17)

for every v ∈ T S. In the following discussion, we use the notation d f | v = d f (v)

(3.18)

to simplify complicated formulae. Definition 3.3.2 The differential structure C ∞ (T S) of T S is generated by the family of functions { f ◦ τ, d f | f ∈ C ∞ (S)}. Proposition 3.3.3 The tangent bundle projection τ : T S → S is smooth. Proof For every f ∈ C ∞ (S), the pull-back τ ∗ f = f ◦ τ ∈ C ∞ (T S), by the definition of C ∞ (T S). Hence, τ is smooth. Definition 3.3.4 A smooth section of τ : T S → S is a smooth map σ : S → T S such that τ ◦ σ = identity.

3.3 The tangent bundle

39

Proposition 3.3.5 If X is a derivation of C ∞ (S), then the map S → T S : x → X (x) is a smooth section of τ . Conversely, if σ : S → T S is a smooth section of the tangent bundle projection τ , then for each f ∈ C ∞ (S), the function X ( f ) such that X ( f )(x) = σ (x)( f ) for every x ∈ S is smooth, and the map X : C ∞ (S) → C ∞ (S) : f → X ( f ) is a derivation of C ∞ (S). Proof A section σ : S → T S of τ is smooth if, for every smooth function h ∈ C ∞ (T S), its pull-back σ ∗ h by σ is in C ∞ (S). It suffices to check this condition on the set {τ ∗ f, d f | f ∈ C ∞ (S)} of functions generating the differential structure of T S. For every f ∈ C ∞ (S), σ ∗ (τ ∗ f ) = σ ∗ ( f ◦ τ ) = f ◦ (τ ◦ σ ) = f ∈ C ∞ (S). Also, for every x ∈ S, σ ∗ (d f )(x) = σ (x)( f ) = X ( f )(x), which implies that σ ∗ (d f ) = X ( f ). Hence, σ ∗ (d f ) is smooth if and only if X ( f ) is smooth. Moreover, for every x ∈ S, X ( f 1 f 2 )(x) = σ (x)( f 1 f 2 ) = (σ (x)( f 1 )) f 2 (x) + f 1 (x)(σ (x)( f 2 (x)) = (X ( f 1 )(x)) f 2 (x) + f 1 (x)(X ( f 2 )(x)). Hence, X ( f 1 f 2 ) = X ( f 1 ) f 2 + f 1 X ( f 2 ). For every x ∈ S, σ (x) maps C ∞ (S) linearly to R. Hence, the map X : C ∞ (S) → C ∞ (S) is linear. Therefore, X is a derivation of C ∞ (S), which completes the proof. Definition 3.3.6 The derived map of a smooth map ϕ : S → R is the map T ϕ : T S → T R such that, for each x ∈ S and u ∈ Tx S, we have T ϕ(u) ∈ Tϕ(x) R and T ϕ(u)( f ) = u( f ◦ ϕ) for every f ∈ C ∞ (R). Proposition 3.3.7 If ϕ : S → R is smooth, then T ϕ : T S → T R is smooth. Proof Let τ S : T S → S, and let τ R : T R → R denote the tangent bundle projections. It follows from Definition 3.3.2 that it suffices to show that if f ∈ C ∞ (R), then (T ϕ)∗ (d f ) and (T ϕ)∗ (τ R∗ f ) are in C ∞ (T S). Since T ϕ maps Tx S to Tϕ(x) R, it follows that τ R ◦ T ϕ = ϕ ◦ τS .

40

Derivations

Hence, (T ϕ)∗ (τ R∗ f ) = (T ϕ)∗ ( f ◦ τ R ) = f ◦ τ R ◦ T ϕ = f ◦ (ϕ ◦ τ S ) = τ S∗ (ϕ ∗ f ) ∈ C ∞ (T S) because ϕ ∗ f ∈ C ∞ (S). Using equation (3.18), for every f ∈ C ∞ (R) and u ∈ T S, ((T ϕ)∗ (d f ))(u) = d f | T ϕ(u) = T ϕ(u)( f ) = u( f ◦ ϕ) = u(ϕ ∗ f ) = dϕ ∗ ( f ) | u . Hence, (T ϕ)∗ (d f ) = dϕ ∗ ( f ) ∈ C ∞ (S). It follows from Definition 3.3.6 that the following diagram commutes: TS

Tϕ

/ TR

τS

τR

S

ϕ

/ R.

Moreover, if ϕ is a diffeomorphism of S onto R, then T ϕ is a diffeomorphism of T S onto T R. In this case, if we identify derivations X and ϕ∗ X of C ∞ (S) and C ∞ (R), respectively, with the corresponding sections of τ S and τ R , we obtain ϕ∗ X = T ϕ ◦ X ◦ ϕ −1 . The derived map T ϕ is also referred to as the tangent map of ϕ. Proposition 3.3.8 For every derivation X of the differential structure C ∞ (S) of a subcartesian space and a diffeomorphism ϕ : S → R, exp(tϕ∗ X ) = ϕ ◦ exp(t X ) ◦ ϕ −1 . Proof

For each f ∈ C ∞ (R) and y = ϕ(x) ∈ R,

d d f ((ϕ ◦ exp(t X ) ◦ ϕ −1 )(y)) = f (ϕ ◦ exp(t X ))(x) dt dt d d = Tϕ exp(t X )(x) (f) = exp(t X )(x) (ϕ ∗ f ) dt dt ∗ by equation (3.12) = X (ϕ f )(exp(t X )(x)) = ϕ ∗ (ϕ∗ X ( f ))(exp(t X )(x))

by equation (3.5)

= (ϕ∗ X ( f ))(ϕ(exp(t X )(x))) = (ϕ∗ X ( f ))(ϕ(exp(t X )(ϕ −1 (y))) = (ϕ∗ X ( f ))(ϕ ◦ (exp(t X )) ◦ ϕ −1 )(y). Hence, t → (ϕ◦(exp(t X ))◦ϕ −1 )(y) is an integral curve of ϕ∗ X through y.

3.3 The tangent bundle

41

Definition 3.3.9 Let S be a subcartesian space. The structural dimension of S at a point x ∈ S is the smallest integer n such that for some open neighbourhood U ⊆ S of x, there is a diffeomorphism of U onto a subset V ⊆ Rn . Theorem 3.3.10 For a subcartesian space S, the structural dimension at x is equal to dim Tx S. Proof Let n be the structural dimension of S at x. There is a neighbourhood U of x in S and a diffeomorphism ϕ : U → V , where V is a differential subspace V of Rn . By Theorem 3.1.6, every derivation of C ∞ (S) can be extended locally to a derivation of C ∞ (Rn ). Hence, dim Tx S ≤ dim Rn = n. Now assume that dim Tx S < n. Then there exists a derivation u ∈ Tx Rn that is not the value of a local extension of a derivation of C ∞ (V ). This implies, by Corollary 3.1.9, that there is a function f ∈ N (V ) such that u( f ) = 0. In this case, if x 1 , . . . , x n are the canonical coordinate functions on Rn , then ∂f (ϕ(x)) = 0 ∂x j for some j ∈ {1, . . . , n}. Hence, there is a neighbourhood W ⊆ f −1 (0) of ϕ(x) that is a submanifold of Rn of dimension (n − 1). Therefore, there is a neighbourhood W˜ of ϕ(x) in W diffeomorphic to an open set in Rn−1 . Since f ∈ N (V ), it follows that V ⊆ f −1 (0). Therefore, there is a neighbourhood of ϕ(x) in V diffeomorphic to a subset of Rn−1 , which contradicts the assumption that n is the structural dimension of S at x. This completes the proof that dim Tx S is equal to the structural dimension of S at x. A point x ∈ S is a manifold point of S if there is an open neighbourhood U of x in S that is a manifold. In this case, dim Ty S = dim Tx S for all y ∈ U . It is convenient to weaken this condition as follows. Definition 3.3.11 A point x ∈ S is regular if there is a neighbourhood U of x in S such that dim Ty S = dim Tx S for all y ∈ U . The set of all regular points of S is called the regular component of S and is denoted by Sreg . The complement of Sreg in S is called the singular set of S. It is denoted Ssing , and its elements are called singular points of S. Example 3.3.12 The Koch curve is a subset K of R2 defined as follows. The set K 0 = {(0, 0), (1, 0)} consists of the end points of the line segment C0 = [0, 1] × {0} ∈ R2 . Construct a set C1 by removing the middle third from the segment C0 and replacing it with two equal segments that would form an equilateral triangle (pointing, say, upwards) with the removed piece. The resulting √ four-sided zigzag has vertices K 1 = {(0, 0), (0, 13 ), ( 12 , 63 ), ( 23 , 0), (1, 0)}.

42

Derivations

Next, construct a set C2 by applying the same construction to each line segment of the set C1 . We denote the set of vertices of C2 by K 2 . Continuing in this way, we obtain a sequence of piecewise linear sets Cn and the sets K n of their vertices. Let K ∞ be the union of all sets K n , i.e. K ∞ = ∪∞ n=0 K n . The Koch curve K is the topological closure of K ∞ . Since K is a closed subset of R2 , its differential structure C ∞ (K ) consists of the restrictions to K of smooth functions on R2 . We can show that dim Tx K = 2 for each x ∈ K . Hence, every point of K is regular in the sense of Definition 3.3.11. Lemma 3.3.13 Let n be the maximum of the structural dimensions of S at the points of an open subset V ⊂ S. If every open subset contained in V has a point at which the structural dimension is n, then V consists of regular points. Proof By Theorem 3.3.10, the structural dimension of S at x is equal to dim Tx S. Hence, the assumption implies that the subset W = {x ∈ V : dim Tx S = n} is dense in V . For each x ∈ V , let O x be an open neighbourhood of x in V diffeomorphic to a subset of Rn . Take y ∈ V \ W . Then dim Ty S < n (by Theorem 3.3.10 and the definition of the structural dimension). Let O y be an open neighbourhood of y in V diffeomorphic to a subset of Rn y . Since W is dense in V , there exists x ∈ W ∩ O y . So, Ox ∩ O y is diffeomorphic to a subset of Rn y . But n is the minimum of all m such that a neighbourhood of x is diffeomorphic to a subset of Rm . Since O x ∩O y is a neighbourhood of x diffeomorphic to a subset of Rn y , we have n ≤ n y . But n y < n, by assumption. Therefore, V \ W is empty; that is, the dimension of S at a point of the open subset V is n. This implies that every point in V is regular. Theorem 3.3.14 The regular component Sreg of a subcartesian space S is open and dense in S. Proof Let x ∈ Sreg . There exists an open neighbourhood U ⊆ S of x such that for every y ∈ U , dim Ty S = dim Tx S. Hence, U ⊆ Sreg . Therefore, Sreg is an open subset of S. Now, suppose that Sreg is not dense in S. In this case, there exists a nonempty open subset U ⊆ S such that U contains no regular points. Without loss of generality, we assume that U is locally diffeomorphic to a differential subspace of Rn for some n > 0. In other words, we assume that for x ∈ U , there is a neighbourhood V1 of x in U such that V1 is diffeomorphic to a subset of Rn . Suppose first that n = 0. This means that V1 is a non-empty set of isolated points. Since V1 is an open subset of U , and U is open in S, it follows that the points of V1 are isolated in S. Hence, dim Ty S = 0 for each y ∈ V1 . Therefore V1 ⊆ Sreg , which is a contradiction with the supposition that U has no regular points.

3.3 The tangent bundle

43

Define S i = {y ∈ S : dim Ty S ≤ i}. By construction, V1 ⊂ S n . Since V1 ⊆ U is open, it follows that V1 contains infinitely many points at which the structural dimensions are at least two different integers from the set {0, 1, . . . , n}. Let n 1 be the maximum of these structural dimensions at the points in V1 . By Lemma 3.3.13, there exists an open subset V2 ⊂ V1 such that the maximum of the structural dimensions of S at the points in V2 is n 2 < n 1 . Similarly, there exists an open subset V3 ⊂ V2 with a maximum of the structural dimensions at its points n 3 < n 2 . Thus, by continuing this process, we obtain the following decreasing sequence: n1 > n2 > n3 > · · · > ni , which stops at some n i ≥ 0. We reach some open subset Vi ⊂ U such that the structural dimension at all points of Vi is n i ≥ 0. Hence, all points of Vi are regular points. This contradicts the assumption that U contains no regular points. Therefore, the set Ssing of singular points of S has an empty interior, which means that Sreg is dense in S. Proposition 3.3.15 Let S be a subcartesian space. Then the restriction of the tangent bundle projection τ : T S −→ S to T Sreg is a locally trivial fibration over Sreg . For each x ∈ Sreg with structural dimension n, there are a neighbourhood W of x in S and a family X 1 , . . . , X n of global derivations of C ∞ (S) such that TW S = τ −1 (W ) is spanned by the restrictions X 1 , . . . , X n to V . Proof Let x ∈ Sreg , with dim Tx S = n. Since Sreg is open, there exists a neighbourhood U ⊂ Sreg of x such that dim Ty S = n for all y ∈ U . Since S is a subcartesian space and n is the structural dimension of S at x, we may assume without loss of generality that there is a diffeomorphism ϕ : U → V , where V is a subset of Rn . We first prove that T V , the set of all pointwise derivations of C ∞ (V ), is a trivial bundle. Let R(V ) consist of the restrictions to V of all smooth functions on Rn , and let N (V ) be the space of functions on Rn that vanish on V . We identify R(V ) with C ∞ (Rn ) modulo N (V ). Since n = dimx S, it follows that there are n linearly independent derivations of C ∞ (V ) at ϕ(x). Therefore, by Proposition 3.1.8, ∂i f |V = 0 for every i = 1, . . . , n and each f ∈ N (V ). This implies that the ∂1 , . . . , ∂n are derivations of C ∞ (V ). Hence, there are n sections X 1 , . . . , X n of the tangent bundle projection τV : T V −→ V such that X i (h mod N (V ))(y) = (∂i h)(y) for every i = 1, . . . , n, h ∈ R(V ) and y ∈ V . Now we need to prove that the sections X 1 , . . . , X n are smooth. Let q1 , . . . , qn be the restrictions to V of the

44

Derivations

coordinate functions on Rn . For i = 1, . . . , n, we denote by dqi the function on T V such that dqi (w) = w(qi ) for every w ∈ T V . The differential structure of T V is generated by the functions (τV∗ q1 , . . . , τV∗ qn , dq1 , . . . , dqn ) in the sense that every function f ∈ C ∞ (T V ) is of the form f = F(τV∗ q1 , . . . , τV∗ qn , dq1 , . . . , dqn ) for some F ∈ C ∞ (R2n ). In order to show that X i : V → T V is smooth, it suffices to show that for every f ∈ C ∞ (T V ), the pull-back X i∗ f is in C ∞ (V ). Since

1 if i = j dqi ◦ X j = δi j = 0 if i = j, it follows that X i∗ f = f ◦ X i = F(τV∗ q1 , . . . , τ V∗ qn , dq1 , . . . , dqn ) ◦ X i = F(τV∗ q1 ◦ X i , . . . , τV∗ qn ◦ X i , dq1 ◦ X i , . . . , dqn ◦ X i ) = F(q1 ◦ τV ◦ X i , . . . , qn ◦ τV ◦ X i , δ1i , . . . , δni ) = F(q1 , . . . , qn , δ1i , . . . , δni ). Hence, X i∗ f is in C ∞ (V ). This implies that the tangent bundle T V is globally spanned by n linearly independent smooth sections X 1 , . . . , X n . Thus, T V is a trivial bundle. We can choose an open neighbourhood W of y contained in V such that its closure W is also in V . Using a partition of unity, we can construct derivations of C ∞ (S) that extend restrictions of X 1 , . . . , X n to W . Hence, T W is spanned by the restrictions to W of global derivations of C ∞ (S). This completes the proof.

3.4 Orbits of families of vector fields In this section, we prove that orbits of families of vector fields are immersed manifolds. Let F be a family of vector fields on a subcartesian space S, and let x0 be a point in S. Let X 1 , . . . , X n be vector fields in F. Consider a piecewise smooth curve given by first following the integral curve of X 1 through x0 for a time τ1 , next following the integral curve of X 2 through x 1 = (exp τ1 X 1 )(x0 ) for a time τ2 , after that following the integral curve of X 3 through x2 = (exp τ2 X 2 )(x1 )

3.4 Orbits of families of vector fields

45

for some τ3 , and so on. For each i = 1, . . . , n, let Ji be the closed interval in R with end points 0 and τi . In other words, Ji = [0, τi ] if τi > 0 and Ji = [τi , 0] if τi < 0. Note that τi < 0 means that the integral curve of X i is followed in the negative time direction. Clearly, for every i, Ji is contained in the domain I xi−1 of the maximal integral curve of X i originating at xi−1 . The range of the curve is n {(exp ti X i )(xi−1 ) ∈ S | t ∈ Ji }. i=1

Definition 3.4.1 The orbit through x 0 of the family F is the union O x0 of the ranges of all the curves described above. In other words, Ox0 =

∞

n

{(exp ti X i )(xi−1 ) ∈ S | ti ∈ Ji },

(3.19)

n=1 X 1 ,...,X n J1 ,...,Jn i=1

where the vector fields X 1 , . . . , X n are in F and, for each i = 1, . . . , n, the interval Ji ⊂ I xi−1 is either [0, τi ] or [τi , 0], with xi = (exp τi X i )(xi−1 ). Proposition 3.4.2 Let Ox0 be the orbit through x0 of a family F of vector fields on a subcartesian space S. For each X ∈ F and f ∈ C ∞ (S), the integral curve of f X through x0 is contained in Ox0 . Similarly, if X, Y are in F, then the integral curve of (exp X )∗ Y through x0 is contained in O x0 . Proof For f ∈ C ∞ (S) and X ∈ F, the integral curves of X and f X differ by parametrization, provided f = 0. The integral curves of f X originating at the points for which f = 0 are constant. Hence, the integral curves of f X originating at x0 are contained in the orbit Ox0 of F. By Proposition 3.3.8, we have the equality exp(t (exp X )∗ Y )(x0 ) = exp(X ) ◦ exp(tY ) ◦ exp(−X )(x0 ) whenever both sides are defined. Let I0 ⊆ R1 be the domain of the maximal integral curve of (exp X )∗ Y through x 0 . Since exp(−X )(x 0 ) is defined, it follows that −1 is in the domain I1 of the maximal integral curve of X through x 0 . Let I2 be the domain of the maximal integral curve of Y through exp(−X )(x 0 ). There exists s ∈ I0 ∩ I2 such that 1 is contained in the domain of the maximal integral curve of X through exp(sY )[exp(−X )(x 0 )]. Hence, the curves c1 : [0, 1] → S : t → exp(−t X )(x0 ), c2 : [0, s] → S : t → exp(tY )[exp(X )(x0 )], c3 : [0, 1] → S : t → exp(t X )[exp(sY )[exp(−X )(x0 )]]

46

Derivations

are well defined and contained in Ox0 . Moreover, the point exp(s(exp X )∗ Y ))(x0 ) = {exp(X ) ◦ exp(sY ) ◦ exp(−X )}(x 0 ) can be obtained by first following c1 from x 0 to exp(−X )(x 0 ), next following c2 from exp(−X )(x 0 ) to [exp(sY )[exp(−X )(x0 )]], and finally following c3 from the point [exp(sY )[exp(−X )(x 0 )]] to the point exp((exp s X )∗ Y ))(x 0 ). Therefore, exp((exp s X )∗ Y ))(x 0 ) is contained in Ox0 . Definition 3.4.3 A family F of vector fields on S is locally complete if, for every X, Y ∈ F and x ∈ S, there exist an open neighbourhood U of x and Z ∈ F such that ((exp X )∗ Y )|U = Z |U . For example, a family consisting of a single vector field X is locally complete because (exp t X )∗ X (x) = X (x) at all points x ∈ S for which (exp t X )(x) is defined. Proposition 3.4.4 Every family F of vector fields on a subcartesian space S can be extended to a locally complete family F with the same orbits. Proof If F is not locally complete, we can find vector fields X and Y in F and an x0 ∈ S such that there do not exist a neighbourhood U of x0 and a Z ∈ F satisfying ((exp X )∗ Y )|U = Z |U . Since X is a vector field on S, there is a neighbourhood V of x0 in S such that exp X restricts to a diffeomorphism of V onto its image. Hence, (exp X )∗ Y is well defined on V . There exist f ∈ C ∞ (S) and open neighbourhoods U1 and U2 of x0 in S such that U 1 ⊂ U2 ⊂ V , f |U1 = 1 and f |S\U2 = 0. Define a vector field Z by Z |V = f (exp X )∗ Y and Z |S\U2 = 0. By Proposition 3.3.2, the integral curves of Z through x ∈ S are contained in the orbit O x of F. Hence, orbits of the family F1 = F ∪ {Z } are the same as orbits of F. Continuing this process, we obtain a locally complete family F of vector fields on S such that orbits of F coincide with orbits of F. Theorem 3.4.5 Each orbit O of a family F of vector fields on a subcartesian space S is a manifold. Moreover, in the manifold topology of O, the differential structure on O induced by its inclusion in S coincides with its manifold differential structure. Proof By Proposition 3.3.4, there exists a locally complete family of vector fields on S with the same set of orbits as F. Hence, without loss of generality, we may assume that the family F is locally complete. (i) Notation. In order to simplify the presentation, we introduce the following notation. For k > 0, let X = (X 1 , . . . , X k ) ∈ Fk , t = (t1 , . . . , tk ) and

3.4 Orbits of families of vector fields

47

exp(t X)(x) = exp(tk X k )◦ . . . ◦ exp(t1 X 1 ) (x). Given X, the expression for exp(t X)(x) is defined for all (t, x) in an open subset (X) of Rk × S. Let t (X) denote the set of all x ∈ S such that (t, x) ∈ (X). In other words, t (X) is the set of all x for which exp(t X)(x) is defined. In addition, we denote by xX ⊆ Rk the set of t ∈ Rk such that exp(t X)(x) is defined, and set expx X : xX → S : t → exp(t X)(x).

(3.20)

By construction, if x ∈ O, then expx X is smooth and its range is contained in O. (ii) Rank of a locally complete family of vector fields. For each x ∈ O, the rank of F at x, denoted by rank Fx , is the number of vector fields X 1 , . . . , X m in F such that X 1 (x), . . . , X m (x) form a basis of the subspace of Tx S spanned by the values at x of vector fields in F. Since linear independence is an open property, it follows that if X 1 , . . . , X m are linearly independent at x, then they are linearly independent in a neighbourhood of x. The assumption that the family F is locally complete ensures that the rank of F is constant on O. This can be seen as follows. Suppose that rank Fx0 = m. This implies that there is a basis X = (X 1 , . . . , X m ) of the linear span of F at x0 . For each i = 1, . . . , m, T exp(t X)(X i (x0 )) = T exp(tm X m )◦ . . . ◦ exp(t1 X 1 ) (X i (x0 )) = T (exp(tm X m )(. . . (T exp(t1 X 1 )(X i (x0 ))))) = exp(tm X m 1 )∗ (. . . (exp(t1 X 1 )∗ X i ))(exp(t X)(x0 )) = (exp(t X)∗ X i )((exp(t X)(x0 )). Since F is locally complete, for each i = 1, . . . , m there is a vector field Z i in F which is equal to exp(t X)∗ X i in a neighbourhood of exp(t X)(x 0 ). Each map exp(t X ) : x → exp(t X )(x) is a local diffeomorphism. Hence, the composition exp(t X 1 ) = exp(tm 1 X 1m 1 )◦ . . . ◦ exp(t1 X 11 ) is also a local diffeomorphism. This implies that T exp(t X) : Tx0 S → Texp(t X)(x0 ) S is a vector space isomorphism. Therefore, the linear independence of the vector fields X 11 , . . . , X 1m 1 at x 0 implies that the vector fields Z 1 , . . . , Z m 1 are linearly independent at exp(t X)(x0 ). Since m = rank Fx0 is the maximum number of vector fields X 01 , . . . , X 0m in F which are linearly independent at x0 , it follows that rank Fx0 ≤ rank Fexp(t X)(x0 ) . On the other hand, x0 = exp(−t X)(exp(t X)(x 0 )). Repeating the same argument as above, we obtain the result that rank Fexp(t X)(x0 ) ≤ rank Fx0 . Therefore,

48

Derivations

rank Fexp(t X)(x0 ) = rank Fx0 . Since the orbit O is connected, it follows that rank F is constant on O. (iii) Covering of the orbit by manifolds. Given x ∈ O, there exist X = (X 1 , . . . , X m ) ∈ Fm and a neighbourhood V of x in S such that {X 1 , . . . , X m } is a maximal linearly independent subset of F|V . For each i = 1, . . . , m, and u ∈ R, d u exp(t X i )(x) = u X i (exp(t X i )(x)). dt Hence, for each u = (u 1 , . . . , u m ) ∈ Rm , T expx X(u) = u 1 X 1 (x) + . . . + u m X m (x). The vectors X 1 (x), . . . , X m (x) are linearly independent, which implies that the derived map T expx X : Rm → Tx S is one-to-one. Since S is subcartesian, we may assume without loss of generality that there exists a smooth map ϕ : V → Rn that induces a diffeomorphism of V onto its image ϕ(V ) ⊆ Rn . By Theorem 3.1.6, for every i = 1, . . . , m, the vector field ϕ∗ X i on ϕ(V ) extends locally to a vector field Y i on Rn . Shrinking V if necessary, we may assume that all vector fields ϕ∗ X i are restrictions to ϕ(V ) of vector fields Y i on Rn . Let y = ϕ(x) and Y = (Y1 , . . . , Ym ). As before, we denote by exp y Y the map from the neighbourhood of 0 ∈ Rm to Rn given by exp y (Y )(t) = exp(tm Y m )◦ . . . ◦ exp(t1 Y 1 ) (y). The linear independence at x of the vector fields X 1 , . . . , X m implies that the vector fields Y1 , . . . , Yn are linearly independent at y. Hence, there exists a neighbourhood W of 0 in Rm such that exp y Y (W ) is a submanifold of Rn and that exp y Y , restricted to W , gives a diffeomorphism exp y Y |W : W → exp y Y (W ). Since y = ϕ(x) ∈ ϕ(V ) and, for i = 1, . . . , m, the restriction to ϕ(V ) of Y i gives the vector field ϕ∗ X i on ϕ(V ), the set exp y Y (W ) is contained in ϕ(V ), and it is the image of W ⊆ Rm under the map expϕ(x) (ϕ∗ X) : W → ϕ(V ) : t → exp(tm ϕ∗m X 1 )◦ . . . ◦ exp(t1 ϕ∗ X 1 ) (ϕ(x)). In other words, exp y Y (W ) = expϕ(x) (ϕ∗ X)(W ) ⊆ ϕ(V ).

(3.21)

The differential structure of exp y Y (W ) is generated by restrictions to exp y Y (W ) of smooth functions in C ∞ (Rm ). The differential structure of ϕ(V ) is also generated by restrictions to ϕ(V ) of smooth functions in C ∞ (Rm ). Hence, equation (3.21) implies that expϕ(x) (ϕ∗ X)(W ) is a manifold in the differential structure generated by restrictions of smooth functions on

3.4 Orbits of families of vector fields

49

ϕ(V ). We say that expϕ(x) (ϕ∗ X)(W ) is a submanifold of ϕ(V ). Moreover, expϕ(x) (ϕ∗ X)|W : W → expϕ(x) (ϕ∗ X)(W ) is a diffeomorphism. Since ϕ is a diffeomorphism of V on its image ϕ(V ) and V is open in S, it follows that expx (X)(W ) is a submanifold of S and expx (X)|W is a diffeomorphism of W onto expx (X)(W ). The construction above can be repeated for each point x in the orbit O, a finite collection X of vector fields in F and a neighbourhood W of 0 ∈ Rm , where m is the number of vector fields in X that satisfy the assumptions made above. In this way, we obtain a family of sets expx X(W ) in O covering O. In other words, O= expx X(W ). (3.22) x∈O

X

W

Each expx X(W ) is a submanifold of S diffeomorphic to W . (iv) Topology of the orbit. We have shown that the orbit O is covered by a family {expx X(W )} of subsets of O, where x ∈ O, X = (X 1 , . . . , X m ) ∈ Fm is a frame field for F in a neighbourhood of x, and W is a neighbourhood of 0 ∈ Rm such that expx X is a diffeomorphism of W on its image. We want to take this family of subsets of O to be a basis for the topology of O. For this definition to make sense, we must verify that if x0 ∈ expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ), then there exist a frame field X 0 for F in a neighbourhood of x 0 and an open neighbourhood W0 of 0 in Rm such that expx0 X 0 (W0 ) ⊆ expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ).

(3.23)

First, we show that for every point x0 ∈ expx1 X 1 (W1 ) and every X ∈ F, there is an open neighbourhood I of 0 ∈ R such that the integral curve of X through x 0 with domain I is contained in expx1 X 1 (W1 ). Let c : s → x(s) be an integral curve of X such that x(0) = x0 . It suffices to show that there is a curve s → t(s) = (t1 (s), . . . , tm (s)) in W1 such that x(s) = exp x X(t(s)) = exp(tm (s)X m )◦ . . . ◦ exp(t1 (s)X 1 )(x1 ) for all s in a neighbourhood I of 0 ∈ R. Differentiating this equation with respect to s at x(s), we obtain X (x(s)) = (T expx X 1 )(t(s))

d (t(s)) ds

dtm dtm−1 + exp(tm (s)X 1m )∗ X 1m−1 (x(s)) + ... ds ds dt1 . (3.24) + exp(tm (s)X 1m ) ◦ . . . ◦ exp(t2 (s)X 12 )∗ X 11 (x(s)) ds

= X 1m (x(s))

50

Derivations

Since expx X is a diffeomorphism of W onto its image expx X(W ), it follows that T (expx X)(t(s)) is a vector space isomorphism of Tt Rm onto Texpx X(t(s)) expx X(W ). In particular, the vectors X m (x(s)), . . . , (exp(tm (s)X m ) ◦ . . . ◦ exp(t2 (s)X 2 ))∗ X 1 (x(s)) are linearly independent. Hence, equation (3.24) is a system of linear differential equations of first order for t(s), and it has a unique smooth solution for s in a neighbourhood I of 0 ∈ R. Repeating this argument for X 11 , . . . , X 1m , we obtain the result that there is a neighbourhood W0 of 0 ∈ Rm such that expx0 X 1 is a diffeomorphism of W0 onto expx0 X 1 (W0 ) ⊂ expx1 X 1 (W1 ). By shrinking W0 if necessary, we can obtain expx0 X 1 (W0 ) ⊂ expx2 X 2 (W2 ), which proves the inclusion (3.23). Therefore, we can take the family {expx X(W )} of subsets of the orbit O, where x ∈ O, X = (X 1 , . . . , X m ) ∈ Fm is a frame field for F in a neighbourhood of x, and W is a neighbourhood of 0 ∈ Rm such that expx X is a diffeomorphism of W on its image, as a basis of a topology T on O. In this topology, O is a connected topological space locally homeomorphic to Rm . Note that the topology T of O may be finer than its subspace topology. (v) Differential structure of the orbit. The orbit O is covered by open sets {expx X(W )}, each of which is diffeomorphic to an open neighbourhood of 0 ∈ Rm . Moreover, if the intersection U12 = expx1 X 1 (W1 ) ∩ expx2 X 2 (W2 ) is not empty, then it is an open subset of O, and expx1 X 1 ◦ (expx2 X 2 )−1 is a diffeomorphism of expx2 X 2 (U12 ) onto expx1 X 1 (U12 ). Hence, O is a smooth manifold diffeomorphic to Rm . For each function f ∈ C ∞ (O) and each x ∈ O, the restriction of f to expx X(W ) is smooth. But expx X(W ) is a submanifold of S. This means that if f ∈ C ∞ (O), then for each x ∈ O there exist a neighbourhood U = expx X(W ) of x in O and a function h ∈ C ∞ (S) such that f |U = h |U . Conversely, let f : O → R be such that for each x ∈ O, there exist an open neighbourhood U of x in O and h ∈ C ∞ (S) such that f |U = h |U . Consider expx0 X 0 (W0 ) ⊆ O. By hypothesis, for each x ∈ expx0 X 0 (W0 ), there exist an open neighbourhood U of x in O and h ∈ C ∞ (S) such that f |U = h |U . In particular, the restrictions of f and h to the open neighbourhood U ∩ expx0 X 0 (W0 ) of x in expx0 X 0 (W0 ) coincide. Hence, the restriction of f to expx0 X 0 (W0 ) is smooth. This holds for every open set expx0 X 0 (W0 ) of our covering of O by manifolds. Hence, f is smooth in the manifold differential structure C ∞ (O) of the orbit. We have shown that the manifold differential structure of the orbit O coincides with the differential structure of O induced by its inclusion into S; see Proposition 2.1.8. This completes the proof.

3.4 Orbits of families of vector fields

51

We can restate the results obtained above in terms of distributions. Definition 3.4.6 (i) A distribution on a subcartesian space S is a subset D ⊆ T S such that for each x ∈ S, the intersection Dx = D ∩ Tx S is a vector subspace of T S. (ii) A distribution D on S is smooth if there is a family F of vector fields on S that spans D. In other words, for every x ∈ S, Dx = span {X (x) ∈ Tx S | X ∈ F}. We denote by X(S) the family of all vector fields on a subcartesian space S. Given a smooth distribution D on S, the family F D , defined by F D = {X ∈ X(S) | X (x) ∈ Dx for each x ∈ S}, is the maximal family of vector fields on S which spans D. An integral manifold of a distribution D is a connected manifold M in S such that Tx M = Dx for every x ∈ M. We say that M is a maximal integral manifold of D through x ∈ S if M is an integral manifold of D and contains x, and every other integral manifold of D containing x is an open submanifold of M. Definition 3.4.7 A distribution D on a subcartesian space S is integrable if, for every x ∈ S, there exists a maximal integral manifold of D containing x. Theorem 3.4.5 ensures that the orbits of every family F of vector fields on S are manifolds. However, the orbit of F passing through x ∈ S need not be an integral manifold through x of the distribution D spanned by F. According to Proposition 3.4.4, orbits of F are the same as orbits of its locally complete extension F. Therefore, in the proof of Theorem 3.4.5, we could assume that our family F of vector fields was locally complete. We are led to the following corollary. Corollary 3.4.8 If F is a locally complete family of vector fields on a subcartesian space S, then the orbits of F are integral manifolds of the distribution D on S spanned by F. Clearly, the family X(S) of all vector fields on a subcartesian space S is locally complete. Therefore, the distribution on S spanned by X(S) is integrable. It is the maximal integrable distribution on S. Its maximal integral manifolds, that is, orbits of X(S), give a partition O of S by smooth manifolds. This partition will be discussed further in Chapter 4.

4 Stratified spaces

Stratified spaces are examples of singular spaces and can be analysed in the framework of differential geometry. In this chapter, we describe stratified subcartesian spaces and develop a differential-geometric approach to their study. We shall use this approach in subsequent sections to describe the singular reduction of symmetries.

4.1 Stratified subcartesian spaces A stratification of a subcartesian space S is a partition of S by a locally finite family M of locally closed connected submanifolds M, called strata of M, which satisfy the following condition.1 Frontier Condition 4.1.1 For M, M ∈ M, if M ∩ M = ∅, then either M = M or M ⊂ M\M. Here, M denotes the topological closure of M in S and M\M the complement of M in M. The pair (S, M) is called a stratified space. Local finiteness of M means that, for each point x ∈ S, there exists a neighbourhood U of x in S that intersects only a finite number of manifolds M ∈ M. A subset M of S is locally closed if, for each x ∈ M, there exists a neighbourhood U of x in S such that M ∩ U is closed in U. If S is a manifold, an injectively immersed submanifold M of S is embedded if and only if M is locally closed in S. 1 There are a variety of definitions of the notion of ‘stratification’ in the literature. For example,

Mather (1973) defined a stratification of a topological space S as a map from S to a sheaf of germs of manifolds satisfying certain conditions. Our definition is equivalent to Mather’s in the case when S is a differential space. It is more convenient because it does not require the introduction of sheaves.

4.1 Stratified subcartesian spaces

53

In Chapter 3, we showed that every subcartesian space S admits a partition O by orbits of the family X(S) of all vector fields on S. It is of interest to see under what conditions this partition of S is a stratification. Proposition 4.1.2 The partition O of a subcartesian space S by orbits of the family X(S) of all vector fields on S satisfies Frontier Condition 4.1.1. Proof Let O and O be orbits of X(S). Suppose that x ∈ O ∩ O, with O = O. We first show that O ⊂ O. Note that the orbit O is invariant under the family of one-parameter local groups of local diffeomorphisms of S generated by vector fields. Since x ∈ O, it follows that, for every vector field X on S, exp(t X )(x) is in O if it is defined. But O is the orbit of X(S) through x. Hence, O ⊂ O. Corollary 4.1.3 The partition O of S by orbits of X(S) is a stratification of S if and only if it is locally finite and the orbits of X(S) are locally closed. Stratified subcartesian spaces form a category with morphisms ϕ : (S1 , M1 ) → (S2 , M2 ) given by smooth maps ϕ : S1 → S2 such that, for each M1 ∈ M1 , the range ϕ(M1 ) is contained in M2 ∈ M2 . If ϕ is a diffeomorphism of S1 onto S2 and M2 = {ϕ(M) | M ∈ M1 }, then ϕ is an isomorphism of (S1 , M1 ) onto (S2 , M2 ). Stratifications of S can be partially ordered by inclusion. If M1 and M2 are two stratifications of S, we say that M1 is a refinement of M2 and write M1 ≥ M2 if, for every M1 ∈ M1 , there exists M2 ∈ M2 such that M1 ⊆ M2 . We say that M is a minimal (coarsest) stratification of S if it is not a refinement of a different stratification of S. If S is a manifold, then the minimal stratification of S consists of a single manifold M = S. If (S, M) is a stratified subcartesian space and N is a manifold, the product S × N is stratified by the family M S×N = {M × N | M ∈ M}. If U is an open subset of a stratified space (S, M), we can consider a family MU = {M ∩ U | U ∈ M}. In general, MU need not be a stratification of U . Definition 4.1.4 A stratification M of a subcartesian space S is locally trivial if, for every M ∈ M and each x ∈ M: (i) there exists an open neighbourhood U of x in S such that MU is a stratification of U ; (ii) there exists a subcartesian stratified space (S , M ) with a distinguished point y ∈ S such that the singleton {y} ∈ M ; and (iii) there is an isomorphism ϕ : (U, MU ) → ((M ∩ U ) × S , M(M∩U )×S ) such that ϕ(x) = (x, y).

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Stratified spaces

Let M be a stratification of a subcartesian space S. We say that M admits local extensions of vector fields if, for each M ∈ M, each vector field X M on M and each point x ∈ M, there exist a neighbourhood V of x in M and a vector field X on S such that X |V = X M|V . In other words, the vector field X is an extension to S of the restriction of X M to V . Proposition 4.1.5 Every locally trivial stratification of a subcartesian space S admits local extensions of vector fields. Proof Let X M be a vector field on M ∈ M. Given x 0 ∈ M, by Definition 4.1.4, there exist a neighbourhood U of x0 in M, a stratified differential space (S , M ) with a distinguished point y ∈ S such that the singleton {y0 } ∈ M , and an isomorphism ϕ : U → (M ∩ U ) × S of stratified subcartesian spaces such that ϕ(x0 ) = (x0 , y0 ). Let exp(t X M ) be the local one-parameter group of local diffeomorphisms of M generated by X M , and let X (M∩U )×S be a derivation of C ∞ ((M ∩ U ) × S ) defined by (X (M∩U )×S h)(x, y) =

d h(exp(t X M )(x), y)|t=0 , dt

for every h ∈ C ∞ ((M ∩ U ) × S ) and each (x, y) ∈ (M ∩ U ) × S . Since X (M∩U )×S is defined in terms of a local one-parameter group (x, y) → (exp(t X M )(x), y) of diffeomorphisms, it is a vector field on (M ∩ U ) × S . We can use the inverse of the diffeomorphism ϕ : U → (M ∩U )×S to push forward X (M∩U )×S to a vector field X U = (ϕ −1 )∗ X (M∩U )×S on U . Choose a function f 0 ∈ C ∞ (S) with support in U and such that f (x) = 1 for x in some neighbourhood U0 of x0 contained in U . Let X be a derivation of C ∞ (S) extending f 0 X U by zero outside U . In other words, for every f ∈ C ∞ (S), if / U0 , then (X f )(x) = 0. x ∈ U , then (X f )(x) = f 0 (x)(X U f )(x), and if x ∈ Clearly, X is a vector field on S extending the restriction of X M to M ∩U0 . Theorem 4.1.6 Let M be a stratification of a subcartesian space S admitting local extensions of vector fields. The partition O of S by orbits of the family X(S) of all vector fields is a stratification of S, and M is a refinement of O. Moreover, if M is minimal, then M = O. Proof Let M be a stratification of S admitting local extensions of vector fields. Since every vector field X M on a manifold M ∈ M extends locally to a vector field on S and M is connected, it follows that M is contained in an orbit O ∈ O. Every orbit O ∈ O is a union of strata of M. Since M is locally finite, for each x ∈ O there exists a neighbourhood V of x in S which intersects only a finite number of strata M1 , . . . , Mk of M. Hence, V intersects only a finite

4.1 Stratified subcartesian spaces

55

number of orbits in O. Moreover, since strata of M form a partition of S, it follows that V =

k

Mi ∩ V.

(4.1)

i=1

Consider x ∈ M1 . Since M1 is locally closed, there exists a neighbourhood U of x contained in V , and such that M1 ∩ U is closed in U . We can relabel the manifolds M1 , . . . , Mk so that O ∩U =

l

Mi ∩ U

i=1

for some l ≤ k. Without loss of generality, we may assume that x ∈ M i for each i = 2, . . . , l. We want to find out whether O ∩ U is closed in U . Suppose we have a sequence (yk ) in O ∩U convergent to y ∈ U . Since O ∩U is a finite union of disjoint manifolds, there must be a subsequence of (yk ) contained in one of them. Without loss of generality, we may assume that each yk ∈ Mi for some i = 1, . . . , l. We want to show that the limit y = limk→∞ yk ∈ O ∩ U . If y ∈ Mi , then y ∈ Mi ∩ U ⊆ O ∩ U . If y ∈ M i \Mi , then y ∈ M j for some j = 1, . . . , k. By assumption, y ∈ U , and U intersects only the strata that have / O ∩ U implies x in their closure. If M j ⊆ O, then y ∈ O ∩ U . Therefore, y ∈ that M j is not contained in O. We have shown that O ∩ U is not closed in U only if there exist strata Mi and M j such that Mi ⊆ O, M j O, M j ⊆ M i and x ∈ M j ∩ M i . To prove that this is inconsistent with our assumptions, consider a neighbourhood of x in O of the form expx X(W ), where W is an open neighbourhood of 0 ∈ Rm , X = (X 1 , . . . , X m ) ∈ X(S)m is an m-tuple of vector fields on S, and m = dim O. By the construction in the proof of Theorem 3.4.5, expx X(W ) is an m-dimensional locally closed submanifold of S. Let U0 be an open neighbourhood of x in U such that U0 ∩ expx X(W ) is closed in U0 . As before, we consider a sequence (yk ) in Mi ∩ U0 ∩ expx X(W ) ⊆ O ∩ U0 which converges / U0 ∩expx X(W ) ⊆ U0 ∩ O. to y ∈ M j ∩U0 . Since M j O, it follows that y ∈ This contradicts the fact that U0 ∩expx X(W ) is closed in U0 . Therefore, O ∩U is closed in U . Since x is an arbitrary point of the orbit O, it follows that O is locally closed. We have shown that the partition O of S by orbits of the family X (S) of all vector fields on S is locally finite and that each orbit in O is locally closed. Corollary 4.1.3 asserts that O is a stratification of S. By construction, every stratum of the original stratification M is contained in a stratum of O. This implies that M ≥ O. If M is minimal, then M = O.

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4.2 Action of a Lie group on a manifold We are now going to apply the results of the preceding section to describe the structure of the space of orbits of a proper action of a connected Lie group G on a manifold P. We begin with a review of proper actions. An action : G × P → P : (g, p) → (g, p) ≡ g ( p) ≡ gp of a locally compact connected Lie group on a manifold P is proper if, for every convergent sequence ( pn ) in P and a sequence (gn ) in G such that the sequence (gn pn ) is convergent, the sequence (gn ) has a convergent subsequence (gn k ) and lim pn k . lim (gn k pn k ) = lim gn k k→∞

k→∞

k→∞

The isotropy group G p of a point p ∈ G is G p = {g ∈ G | gp = p}. Proposition 4.2.1 Isotropy groups of a proper action are compact. Proof Let (gn ) be a sequence in the isotropy group G p of p ∈ P. Then gn p = p for all n, and the sequence gn p converges to p. By the definition of a proper action, there exists a subsequence (gn k ) in G converging to g such that gp = limk→∞ (gn k p) = limk→∞ p = p. Hence, g ∈ G p , which implies that G p is compact. The orbit of G through p ∈ P is the set Gp = {gp | g ∈ G}. It is a manifold diffeomorphic to the quotient G/G p . Let H be a compact subgroup of G. We denote by P H the set of points in P fixed by the action of H . In other words, P H = { p ∈ P | gp = p ∀ g ∈ H } = { p ∈ P | H ⊆ G p }. We denote the set of points p ∈ P with isotropy group H by PH , and the set of points with an isotropy group conjugate to H by P(H ) . The set PH = { p ∈ P | G p = H } is usually called the subset of P of isotropy type H , and P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G} is called the subset of P of orbit type H .

4.2 Action of a Lie group on a manifold

57

Definition 4.2.2 A slice through p ∈ P for an action of G on P is a submanifold S p of P containing p such that: 1. S p is transverse and complementary to the orbit Gp of G through p. In other words, T p P = T p S p ⊕ T p (Gp). 2. For every p ∈ S p , the manifold S p is transverse to the orbit Gp ; that is, T p P = T p S p + T p (Gp ). 3. S p is G p -invariant. 4. Let p ∈ S p . If gp ∈ S p , then g ∈ G p . Given a G-invariant Riemannian metric k on P, we can construct slices for the action of G on P as follows. We denote by ver T P the generalized distribution on P consisting of vectors tangent to G-orbits in P, and by hor T P the k-orthogonal complement of ver T P. The tangent bundle space T P can be expressed as the product of ver T P and hor T P : T P = ver T P × hor T P,

(4.2)

where the product is taken in the category of differential spaces. The Ginvariance of k implies that this product structure of T P is invariant under the derived action of G on T P. Let p ∈ P and H = G p . We denote by H : H × P → P : (g, p) → H g ( p) = gp the restriction of to H . Since H leaves p fixed, for each g ∈ H , the derived map T gH : T P → T P preserves T p P, and it defines a linear action p : H × T p P → T p P : (g, v) → T gH (v) of H on T p P. Let Exp p be the exponential map from a neighbourhood of 0 in T p P to P corresponding to the metric k. In other words, Exp p (v) is the value at 1 of the geodesics of k originating from p in the direction v. It intertwines the linear action p of H on T p P and the action H of H on P. For each v in the domain of Exp p ∈ T p P and g ∈ H , g Exp p (v) = Exp p (T gH (v)).

(4.3)

Since T p P is a vector space, for each u ∈ T p P we have the natural identification of Tu (T p P) with T p P. With this identification, T0 Exp p : T p P → T p P : v → v. In other words, T0 Exp p is the identity on T p P.

(4.4)

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Proposition 4.2.3 2 (i) There is an open ball B in hor T p P centred at 0 such that S p = Exp p (B) is a slice through p for the action of G on P. (ii) The restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action p of H on hor T p S p and the action of H on S p . For each g ∈ H and q ∈ S p , β( p (g, v)) = g (β(v)) = g(β(v)). Proof The exponential map Exp p corresponding to the metric k is a diffeomorphism of a neighbourhood V of 0 in T p P onto a neighbourhood of p in P. Therefore, Exp p maps submanifolds of T p P contained in V onto submanifolds of P. In particular, if B is a ball in hor T p P centred at 0 and contained in V , then S p = Exp p (B) is a submanifold of P containing p. Let β : B → S p : u → β(u) = Exp p (u) be the diffeomorphism obtained by restricting Exp p to B. Equation (4.4) implies that T p S p = Tβ(T0 B) = Tβ(hor T p P) = hor T p P. Taking Equation (4.2) into account, we obtain T p P = ver T p P × hor T p P = T p (Gp) × T p S p = T p (Gp) ⊕ T p S p . Hence, Condition 1 of Definition 4.2.2 is satisfied. Moreover, this shows that the restriction |G×S p of : G × P → P to G × S p is a submersion at (e, p), where e is the identity in G. Hence, it is a submersion at (e, p ) for all p in a neighbourhood of p in S p . By shrinking B if necessary, we can redefine S p so that |G×S p is a submersion. This implies Condition 2 of Definition 4.2.2. Since k is G-invariant and p ∈ PH , it follows that Exp p intertwines the linear action p of H on T p P and the action of H on P. Moreover, the open ball B in hor T p P defined in terms of the metric k is invariant under the action of H in T p P. Therefore, S p is H -invariant and, for each g ∈ H and v ∈ B, β( p (g, v)) = g (β(v)). This proves the second part of Proposition 4.2.3 and shows that Condition 3 of Definition 4.2.2 is satisfied. It remains to show that we can choose the radius r of B so that Condition 4 of Definition 4.2.2 is met. We prove this by contradiction. Suppose that there is no r > 0 such that p ∈ S p and gp ∈ S p ; then g ∈ H . Taking r = n1 , we obtain a sequence ( pn ) of points in S p converging to p and a sequence (gn ) in G\H such that the sequence (gn pn ) in S p converges to p. Since the action of 2 This proposition is known as Bochner’s Linearization Lemma (see Duistermaat and Kolk

(2000)).

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59

G on P is proper, there exists a subsequence of (gn m ) converging to g ∈ G. Without loss of generality, we may assume that gn → g as n → ∞. Since lim pn = gp, p = lim gn pn = lim gn n→∞

n→∞

n→∞

it follows that g ∈ H . Moreover, g −1 gn → e as n → ∞. Let g = h ⊕ m be a decomposition of the Lie algebra g of G into the direct sum of the Lie algebra h of H and a subspace m of g complementary to h. Consider the map : h ⊕ m → G : (η, ξ ) → (exp η)(exp ξ ),

(4.5)

where exp denotes the exponential map from the Lie algebra g to G. The derived map T(0,0) of at (0, 0) ∈ h ⊕ m is the inclusion of h ⊕ m into g. Hence, there is a neighbourhood V × W of (0, 0) in h ⊕ m such that defines a diffeomorphism of V × W onto (V × W ) ⊆ G. Since g −1 gn → e as n → ∞, it follows that g −1 gn ∈ (V × W ) for sufficiently large n. Hence, there exist unique ηn ∈ V ⊆ h and ξn ∈ W ⊆ m such that g −1 gn = (exp ηn )(exp ξn ). Since g −1 gn → e, it follows that ξn → 0. Since Gp = G/G p = G/H , it follows that exp : g → G induces a diffeomorphism of a neighbourhood of 0 in m onto a neighbourhood of p in Gp. Consider a map m × S p → P : (ξ, q) → (exp ξ, q). Condition 1 of Definition 4.2.2 implies that it gives a diffeomorphism of a neighbourhood of (0, p) in m × S p onto its image. By shrinking W ⊆ m if necessary, we can find a neighbourhood U of p in S p such that ϕ : W × U → ϕ(W × U ) ⊆ G : ϕ(ξ, q) → (exp ξ, q)

(4.6)

is a diffeomorphism. For each n, qn = gn pn = g(exp ηn )(exp ξn ) pn ∈ S p . Since g(exp ηn ) ∈ H and S p is H -invariant, it follows that qn = (exp ξn ) pn = (exp(−ηn ))g −1 qn ∈ S p . Moreover, pn → p and ξn → 0 imply that, for sufficiently large n, we have pn ∈ U , ξn ∈ W and qn = (exp ξn ) pn ∈ ϕ(U × W ) ∩ U . Since pn , qn ∈ U and ξn ∈ W , we can write ϕ(ξn , pn ) = (exp ξn ) pn = qn = (exp 0)qn = ϕ(0, qn ). But ϕ is a diffeomorphism of the product W × U onto its image. Therefore, ϕ(ξn , pn ) = ϕ(0, qn ) implies that ξn = 0 and pn = qn . Hence, for sufficiently large n, gn = g(exp ηn )(exp ξn ) = g(exp ηn ) ∈ H , / H . This completes the proof. which contradicts the assumption that gn ∈

60

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Remark 4.2.4 (i) Let g = h ⊕ m be a decomposition of the Lie algebra g of G into the direct sum of the Lie algebra h of H and a subspace m of g complementary to h. For p ∈ PH , there exists a slice S p through p, a neighbourhood W of 0 in m and a neighbourhood U of p in P such that ϕ : W × S p → ϕ : U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. (ii) The set G S p = {gq | g ∈ G and q ∈ S p } is a G-invariant open neighbourhood of p in P. Proof (i) Since Gp = G/G p = G/H , it follows that the exponential map exp : g → G induces a diffeomorphism of a neighbourhood of 0 in m onto a neighbourhood of p in Gp. Let S p = Exp p (B p ) be a slice through p for an action of G on P. Condition 1 of Definition 4.2.2 implies that there are neighbourhoods of U and V of p in P and S p , respectively, such that ϕ : W × V → U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. If V = S p , we can find a ball B centred at 0 ∈ T p P, contained in B and such that S p = Exp p (B) ⊆ V . There exist W ⊆ W and U ⊆ U such that ϕ : W × S p → U : ϕ(q, ξ ) → (exp ξ, q) is a diffeomorphism. (ii) Clearly, G S p is G-invariant. Note that exp W is a subset of G diffeomorphic to a neighbourhood of p of the orbit Gp. It follows from part (i) that U = (exp W )S p = {gq | g ∈ exp W and q ∈ S p } ⊆ G S p is open in P. For each g ∈ G, the diffeomorphism g : P → P maps S p to a slice Sgp through gp. In other words, gS p = Sgp . Hence, gU = g(exp W )S p = g(exp W )g −1 Sgp ⊆ G S p is an open subset of P. Therefore, G Sp =

gU

g∈G

is open in P. We are now in a position to discuss the manifold properties of the subsets P H , PH and P(H ) of P. We say that a subset S of P is a local submanifold of P if each connected component of S is a submanifold. To verify that S is a

4.2 Action of a Lie group on a manifold

61

local submanifold of P, it suffices to show that, for each p ∈ S, there exists a neighbourhood U of p in P such that S ∩ U is a submanifold of P. Proposition 4.2.5 The set P H of fixed points of the action of a compact Lie group H on P is a local submanifold of P. Moreover, for each p ∈ P H , the tangent bundle space of P H at p consists of H -invariant vectors in T p P: T p P H = (T p P) H = {v ∈ T p P | p (g, v) = v ∀ g ∈ H }. Proof For p ∈ P H , let Exp p be the exponential map from a neighbourhood of 0 in T p P to P corresponding to the metric k. This defines a diffeomorphism of a neighbourhood V of 0 ∈ T p P onto a neighbourhood U of p in P. The vectors in T p P which are fixed by the action p form a subspace (T p P) H of T p P. Therefore, V ∩(T p P) H is a submanifold of T p P, and Exp p (V ∩(T p P) H ) is a submanifold of U . Since Exp p intertwines the linear action of H on V and the action of H on U , it follows that Exp p (V ∩ (T p P) H ) = U ∩ P H . Thus, every p ∈ P H has a neighbourhood U ∩ P H in P H which is a submanifold of P. Hence, P H is a local submanifold of P. Moreover, T Exp p restricted to T0 V = T0 (T p P) = T p P is the identity transformation. Hence, T p P H = (T p P) H . The normalizer N H of H in G is a closed subgroup of G given by N H = {g ∈ G | g H g −1 = H }. For each p ∈ P and g ∈ G, G gp = gG p g −1 . Hence, the action of g ∈ G on P preserves PH if and only if g H g −1 = H . In other words, g preserves PH if and only if g ∈ N H . Let L be a connected component of PH . The group N L = {g ∈ N H | gp ∈ L ∀ p ∈ L} is a closed subgroup of N H containing H as a normal subgroup. Hence, G L = N L /H

(4.7)

is a Lie group. For each g ∈ N L , we denote the class of g in G L by [g]. Since H acts trivially on L, the action of N L on L induces an action L of G L on L, given by L : G L × L → L : ([g], p) −→ [g] p = gp. Proposition 4.2.6 (i) The set PH = { p ∈ P | G p = H }

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of points with isotropy group H is a local submanifold of P. Moreover, connected components of PH are open in P H . (ii) For each p ∈ PH , the tangent space T p PH consists of H -invariant vectors in T p P. (iii) For every vector field X on P, let T g ◦ X ◦ g −1 dμ(g) XH = H

be the H -average of X . For any H -invariant Riemannian metric k on P, the restriction of X H to PH is k-orthogonal to the restriction of X ⊥ H = X − X H to PH . Thus, the restriction of X to PH has a unique decomposition into its component tangent to PH and its component normal to PH , which is independent of the choice of an H -invariant Riemannian metric on P. (iv) For each connected component L of PH , the action of G L = N L /H on L is free and proper. (v) For each p ∈ L, Gp ∩ L = G L p, where Gp is the orbit through p of the action of G on P, and G L p is the orbit through p of the action of G L on L. Proof (i) It follows from the definition that PH ⊆ P H . Let p ∈ PH , so that G p = H , and let S p be the slice through p for the action of G on P that satisfies the conditions of Proposition 4.2.3. Hence, there exist neighbourhoods W of 0 in m and U of p in P such that ϕ : W × S p → U : (ξ, q) → (exp ξ, q) is a diffeomorphism. Let q ∈ U ∩ P H . Then there exists (ξ, q ) ∈ m × S p such that q = (exp ξ, q ) = (exp ξ )q . Hence, G q = (exp ξ )G s (exp(−ξ )). However, q ∈ P H implies that G q ⊇ H . On the other hand, q ∈ S p implies that G q ⊆ H . Therefore, H ⊆ G q = (exp ξ )G q (exp(−ξ )) ⊆ (exp ξ )H (exp(−ξ )), so that (exp(−ξ ))H (exp ξ ) ⊆ H.

(4.8)

Hence, the Lie algebra of (exp(−ξ ))H (exp ξ ) is contained in h. Since the conjugation H → (exp(−ξ ))H (exp ξ ) is an isomorphism, it follows that the dimension of the Lie algebra of (exp(−ξ ))H (exp ξ ) is the same as the dimension of h. Therefore, the Lie algebra of (exp(−ξ ))H (exp ξ ) is equal to h. This

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63

implies that the connected component of the identity in (exp(−ξ ))H (exp ξ ) coincides with the connected component He of the identity in H . Thus, the connected component of g in (exp(−ξ ))H (exp ξ ) is g He , which is the same as the connected component of g in H . Hence, (exp(−ξ ))H (exp ξ ) = H . We have shown that for each q ∈ U ∩ P H , the isotropy group G q is equal to H , so that q ∈ PH . Therefore, U ∩ P H ⊆ U ∩ PH . But PH ⊆ P H . Hence, U ∩ P H = U ∩ PH . Therefore, PH is open in P H . Since P H is a local submanifold of P, it follows that PH is a local submanifold of P. (ii) If u ∈ T p PH ⊆ T p P, let t → c(t) be a curve in PH such that u = c(0). ˙ For every f ∈ C ∞ (P) and each g ∈ H , we have T g (u)( f ) = d f | T g (u) = d f ◦ T g | u = d( f ◦ g ) | u

d d = u(( f ◦ g )) = ( f (gc(t)))|t=0 = ( f (c(t)))|t=0 = u( f ). dt dt Hence, T g (u) = u for every g ∈ H . Conversely, suppose that u ∈ T p P is H -invariant. By Proposition 4.2.5, u ∈ T p P H . Since connected components of PH are open in P H , it follows that T p P H = T p PH . Therefore, every H -invariant vector in T p P is contained in T p PH . (iii) Let X H = H T g ◦ X ◦ g −1 dμ(g) be the H -average of a vector field X on P. Since X H is H -invariant, it follows from part (ii) that X H is tangent to PH . For p ∈ PH and u ∈ T p P, the average u H = H T g (u) dμ(g) of u over H is in T p PH . Suppose that k is an H -invariant Riemannian metric on P. For every u, v ∈ T p P, the H -invariance of k implies that k(T g (u), T g (v) = k(u, v) for all g ∈ H . Hence, if v ∈ T p PH , then v is H -invariant, and T g (u) dμ(g), v = k(T g (u), v) dμ(g) k(u H , v) = k H H k T g (u), T g (v) dμ(g) = k(u, v) = k(u, v). = H

H

If u is k-orthogonal to T p PH , then k(u H , v) = 0 for all v ∈ T p PH . Since u H is in T p PH , it follows that u H = 0. Conversely, suppose that u H = 0. Then, k(u, v) = 0 for all v ∈ T p PH , which implies that u is k-orthogonal to T p PH . If X is a vector field on P, then the H -average of X ⊥ H = X − X H van( p) is k-orthogonal to T p PH . Thus the ishes. Therefore, for every p ∈ PH , X ⊥ H

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decomposition X |PH = X H |PH + X ⊥ H |PH is k-orthogonal for every H -invariant Riemannian metric k on P. (iv) Let [g1 ] and [g2 ] be elements of G L such that [g1 ] p = [g2 ] p for some p ∈ L. This means that g1 and g2 are in N L and g1 p = g2 p. Hence, h = g1−1 g2 ∈ H , and g2 = g1 h. Thus, [g2 ] = [g1 ]. Therefore, the action of G L on L is free. We are now going to show that L is proper. Consider a convergent sequence ( pn ) in L with limit p ∈ L. Let ([g]n ) be a sequence in G L such that [g]n pn converges to p ∈ L. For each n ∈ N, let gn ∈ N L be such that [g]n = [gn ]. Then [g]n pn = gn pn converges to p ∈ L. Since the action of G on P is proper, there exists a subsequence (gn k ) of (gn ) converging to g ∈ G such that p = lim (gn k pn k ) = g lim pn k = gp. k→∞

k→∞

By assumption, p and p are in L, which implies that g ∈ N L . Since the quotient map N L → G L = N L /H is continuous, the sequence ([gn k ]) converges to [g]. Thus the action of G L on L is proper. (v) For p ∈ L, consider a point gp ∈ L ⊆ PH , and g H g −1 = H , which / NL . implies that g ∈ N H . We want to show that g ∈ N L . Suppose that g ∈ Then there exists p ∈ L such that gp ∈ / L. Let U = { p ∈ L | gp ∈ L} and V = { p ∈ L | gp ∈ L}. The map g|L : L → PH , obtained as a restriction of g to the domain L, with codomain PH , is continuous. Since L is a connected component of PH , −1 it follows that U = −1 g|L (L) and V = g|L (PH \L) are both open in L. By construction, U ∩ V = ∅ and U ∪ V = L. Hence, if U and V are not empty, then the sets U and V form a disconnection of L. But L is connected, and we have assumed that p ∈ U . Thus, V = ∅, which implies that g ∈ N L , so that [g] ∈ G L and gp = [g] p. Therefore, Gp ∩ L ⊆ G L p. Conversely, G L p = {[g] p | [g] ∈ G L } = {gp | g ∈ N L ⊆ G} ⊆ Gp, which completes the proof. Proposition 4.2.7 The set P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G} of points of orbit type H is a local submanifold of P.

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65

Proof Let p ∈ PH , and let S p be a slice at p for the action of G on P. As in Remark 4.2.4, let m be a subspace of g complementary to h, and let ϕ : W × S p → U : (ξ, q) → (exp ξ, q) be a diffeomorphism of an open neighbourhood W × S p of (0, p) in m × S p onto an open neighbourhood U of p in P. We can take U = Exp p (V ) for an open neighbourhood of 0 in T p P. We have shown in the proof of Proposition 4.2.6 that PH ∩ U = P H ∩ U . Observe that P(H ) = G PH . Hence, U ∩ P(H ) = {(exp ξ, q) | ξ ∈ W, q ∈ S p ∩ PH }. But S p ⊆ U , and so S p ∩ PH = S p ∩ P H = S pH is the set of H -invariant points in S p . By Proposition 4.2.5, S pH is a local submanifold of S p . Hence, W × S pH is a local submanifold of W × S p and U ∩ P(H ) = ϕ −1 (W × S pH ) is a local submanifold of U . Therefore, P(H ) is a local submanifold of P. In the remainder of this section, we are going to show that the family M of connected components of P(H ) as H varies over compact subgroups of G is a stratification of P. In order to prove this claim, we have to show that this family is locally finite and that it satisfies Frontier Condition 4.1.1. Connected components of P(H ) are locally closed, since they are submanifolds of P. Proposition 4.2.8 The family M of connected components of P(H ) as H varies over compact subgroups of G is locally finite. Proof We prove this result by induction on the dimension of P. If dim P = 0, then P is discrete. The assumption that the action of G on P is continuous and G is connected implies that every point of P is a fixed point of G. Hence, orbits of G in P are singletons. This implies that we have only one orbit type P(G) = PG = P. The assumption that the action is proper implies that G is compact. Since P has the discrete topology, for every p ∈ P the singleton { p} is open and intersects only one connected component of P(G) = P. Therefore, M is locally finite for dim P = 0. Suppose now that dim P = m and that we have proved local finiteness of M for all proper actions G × P → P such that dim P < m. Consider a point p ∈ P and a slice S p through p as in Proposition 4.2.3. That is, S p = Exp p (B), where Exp p is a G p -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a G p -invariant open ball in hor T p P centred at the origin. For sufficiently small r > 0, the sphere Sr = {v ∈ hor T p P | k(v, v) = r 2 }

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Stratified spaces

is contained in B. Since k is G p -invariant, it follows that the action p on T p P preserves Sr and induces a proper action of G p on Sr . By construction, dim Sr = dim(hor T p P) − 1 < m = dim P. Therefore, by the induction hypothesis, the family of connected components of (Sr )(H ) as H varies over compact subgroups of G p is locally finite. Here we have denoted by (Sr )(H ) the subset of Sr of orbit type (H ). Since the action p is linear, for every t ∈ R and g ∈ G p , we have (tv) = t(v), and the family of connected components of subsets (T p S p )(H ) of T p S p as H varies over compact subgroups of G p is locally finite. Therefore, the family of connected components of subsets (S p )(H ) of S p is locally finite. Since the group G is connected and G S p is a G-invariant neighbourhood of p in P, it follows that, for each compact subgroup H of G, the intersection with S p of every connected component of P(H ) is a connected component of (S p )(H ) . Hence, the family M of connected components of P(H ) as H varies over compact subgroups of G is locally finite. Proposition 4.2.9 The family M of connected components of P(H ) as H varies over compact subgroups of G satisfies Frontier Condition 4.1.1. Proof Let H and K be compact subgroups of G such that K ⊆ H . Suppose that P(H ) ∩ P (K ) = ∅, and let p ∈ PH ∩ P (K ) . Consider a slice S p at p for the action of G on P constructed as in Proposition 4.2.3. That is, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in hor T p P to a neighbourhood of p in P, and B is an H -invariant open ball in hor T p P centred at the origin. Since p ∈ PH , there is an action of H on S p . We have a linear action of H on hor T p P. For v ∈ hor T p P, we denote by Hv the isotropy group of v. Clearly, (hor T p P) H = {v ∈ hor T p P | Hv = H } = {v ∈ hor T p P | gv = v for all g ∈ H } = (hor T p P) H . Let (hor T p P) K = {v ∈ hor T p P | Hv = K }. Since the action of H on hor T p P is linear, for every g ∈ K ⊆ H , v ∈ (hor T p P) K and w ∈ (hor T p ) H , we have g(v + w) = gv + gw = v + w. Hence, K ⊆ Hv+w . On the other hand, if g ∈ Hv+w , then we also have g(v + w) = gv + gw = v + w, so that gv = v because w ∈ (T p S p ) H implies that gw = w. Therefore Hv+w = K , and (hor T p P) K = (hor T p P) K + (hor T p P) H . This implies that (hor T p H ) H = (hor T p P) H ⊆ (hor T p P) K .

4.3 Orbit space

67

Now, B is an H -invariant open ball in hor T p P centred at 0, B H = B ∩ (hor T p H ) H and B K = B ∩ (hor T p P) K . Therefore, B ∩ B K = B ∩ (hor T p P) K ⊇ B ∩ (hor T p P) H = B H . Since Exp p : B → S p is a diffeomorphism intertwining the linear action of H on T p S p and the action of H on S p , it follows that (S p ) K = Exp p (B K ), (S p ) H = Exp p (B H ) and (S p ) H ⊆ S p ∩ (S p ) K . Furthermore, G S p is a G-invariant neighbourhood of p in P, P(H ) = G PH and P(K ) = G PK . Hence, (G S p ) ∩ P(H ) = (G S p ) ∩ (G PH ) = G(S p ∩ PH ) = G(S p ) H is a G-invariant neighbourhood of p in P(H ) . Similarly, (G S p ) ∩ P(K ) = (G S p ) ∩ (G PK ) = G(S p ∩ PK ) = G(S p ) K is a G-invariant open subset of P(K ) such that (G S p ) ∩ P(K ) = G(G S p ∩ P(K ) ) = G(G S p ∩ (G S p ∩ PK )) = G(G S p ∩ (G S p ) K ) ⊇ G(G S p ) H = G(G S p ∩ PH ) = G S p ∩ P(H ) . Hence, the component M of P(H ) containing p is contained in the closure N of the component N of P(K ) such that p ∈ N . We have shown that the family M of connected components of P(H ) as H varies over compact subgroups of G gives a stratification of P. This stratification is called the orbit type stratification of P.3

4.3 Orbit space In this section, we are going to show that the projection to the orbit space R = P/G of the orbit type stratification M of P is a minimal stratification N of R. Hence, strata of N coincide with orbits of the family of all vector fields on R. 3 It should be noted that the orbit type stratification M of P is not minimal, because P is a

smooth manifold and the union of all strata of M.

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We endow the orbit space R with the quotient topology. In other words, a subset V of R is open if U = ρ −1 (V ) is open in P, where ρ : P → R is the canonical projection (the orbit map). Let C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}.

(4.9)

By Theorem 2.1.10, C ∞ (R) is a differential structure on R. We denote by C ∞ (P)G the ring of G-invariant smooth functions on P. For each f ∈ C ∞ (R), its pull-back ρ ∗ f is G-invariant. Conversely, if f˜ is a G-invariant smooth function on P, then it pushes forward to a function f = ρ∗ f˜ in C ∞ (R). Since ρ ∗ ◦ ρ∗ = identity, it follows that f˜ = ρ ∗ f . This means that the pull-back ρ ∗ : C ∞ (R) → C ∞ (P) induces a ring isomorphism between C ∞ (R) and the ring C ∞ (P)G of G-invariant functions in C ∞ (P). Proposition 4.3.1 The topology of R induced by C ∞ (R) coincides with the quotient topology. Proof In view of Proposition 2.1.11, it suffices to show that, for each set V in R which is open in the quotient topology, and each y ∈ R, there exists f ∈ C ∞ (R) such that f (y) = 1 and f |R\V = 0. Let y ∈ V ⊆ R, where V is open in the quotient topology. Consider a slice S p through a point p ∈ ρ −1 (V ) ⊆ P such that ρ( p) = y. The intersection W = ρ −1 (V ) ∩ S p is a neighbourhood of p in S p . Moreover, it is G p -invariant because ρ −1 (V ) and S p are invariant. There exists a compactly supported nonnegative function h ∈ C ∞ (S p ) such that h( p) = 1 and the support of h is contained in W . Since G p is compact, we may average h over G p , obtaining a G p -invariant function ˜h = ∗g h dμ(g), Gp

where dμ(g) is the Haar measure on G p normalized so that vol G p = 1. Since G p and the support of h are compact, W is G p -invariant, and the support of ˜ denoted by supp h, ˜ is compact and h ⊆ W , it follows that the support of h, contained in W . The set G S p is a G-invariant open neighbourhood of p in P. We can define a G-invariant function f˜1 on G S p as follows. For each p ∈ G S p , there exists g ∈ G such that p = gp for p ∈ S p , and we set ˜ p ). f˜1 ( p ) = h( Since h˜ is G p -invariant, the function f˜1 is well defined on U and is Ginvariant. Moreover, the support of f˜1 is contained in U ∩ ρ −1 (V ). The sets ˜ form a G-invariant open cover of P. We G S p ∩ ρ −1 (V ) and P\G(supp h)

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can extend f˜1 to a smooth G-invariant function f˜ on P, which vanishes on ˜ Since f˜ is G-invariant, there exists a function f ∈ C ∞ (R) P\ρ −1 (ρ(supp h)). such that f˜ = ρ ∗ f . We have ˜ p) = h( p) = 1. f (y) = f (ρ( p)) = f˜( p) = f˜1 ( p) = h( ˜ ⊆ G S p ∩ ρ −1 (V ) ⊆ ρ −1 (V ), it follows that Moreover, since G(supp h) −1 ˜ and f˜|P\ρ −1 (V ) = 0. Hence, f |R\V = 0, which P\ρ (V ) ⊆ P\G(supp h) completes the proof. A left principal fibre bundle with structure group G is a manifold P with an action of G on P such that the orbit space R = P/G is a manifold and the orbit map ρ : P → R is a locally trivial fibration. Moreover, for each x ∈ R, there exist an open neighbourhood V of x in R and a diffeomorphism ψV : π −1 (V ) → G × V , which intertwines the action of G on π −1 (V ) and the action of G on G ×V given by multiplication on the left. In this case, the action of G on P is free; that is, gp = p implies that g is the identity element of G. Theorem 4.3.2 If the action of G on P is free and proper, then P is a left principal fibre bundle with structure group G. Proof We consider the orbit space R = P/G with the differential structure given by equation (4.9). By Proposition 4.3.1, R is a differential space with the quotient topology. Since the action is also free, for each p ∈ P, the orbit Gp through p is diffeomorphic to G. Given p ∈ P, there exists a submanifold S of P containing p and satisfying the condition Ts P = Ts (Gs) ⊕ Ts S

(4.10)

for every s ∈ S ⊂ P. Consider a map of : G × S → P obtained by the restriction of to G × S. For each (g, s) ∈ G × S, we have (g, s) = (g, s) = s (g) = gs. We show first that T(e,s) , where e is the identity in G, is bijective. For each (ξ, v) ∈ Te G × Ts S, we have T(e,s) (ξ, v) = T s (ξ ) + v. If T(e,s) (ξ, v) = 0 for some (ξ, v) ∈ Te G × Ts S, then T s (ξ ) + v = 0, which implies that v = −T s (ξ ) ∈ Ts (Gs). By assumption, v ∈ Ts S, so that v ∈ Ts (Gs) ∩ Ts S. But, by equation (4.10), Ts (Gs) ∩ Ts S = 0, which implies that v = 0 and T s (ξ ) = 0. Hence, T s (tξ ) = 0 for all t ∈ R. Therefore, s is a fixed point of the action restricted to the one-parameter subgroup exp tξ

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of G. The assumption that the action is free implies that ξ = 0, so that T(e,s) . Equation (4.10) ensures that rank T(e,s) = dim G + dim S = dim P. Hence, T(e,s) is surjective. Thus, T(e,s) is bijective. Since (g, s) = g ◦ (e, s), it follows that T(g,s) = Ts g ◦ T(e,s) . Since T(e,s) is bijective and g is a diffeomorphism, we conclude that T(g,s) is bijective for each g ∈ G and s ∈ S. Therefore, : G × S → P is a local diffeomorphism. The next step in the proof is to show that, shrinking S if necessary, we can obtain a diffeomorphism of G × S onto its image in P. We prove this by contradiction. Suppose that there is no neighbourhood S0 of p in S such that | G × S0 is one-to-one. Then, there exists a nested neighbourhood base {Sk } of p in S with the following property. For each k, there are a point sk ∈ Sk and an element gk ∈ G such that gk sk ∈ Uk but gk is bounded away from e. Therefore, gk sk → p as k → ∞. By the properness of , there is a subsequence gkl → g such that gm = m. This contradicts the assumption that the action is free. We have shown that, for each p ∈ P, there exists a submanifold S of P through p satisfying equation (4.10) and the condition that : G × S → P is a diffeomorphism onto its image. It follows from the construction that intertwines the action of G on G × S, given by multiplication on the left, and the action of G on P. In order to show that R is a manifold, observe that each point x ∈ R has a neighbourhood V = ρ(S), where S is a manifold through p ∈ ρ −1 (x) satisfying the conditions in the preceding paragraph. Since the restriction of the orbit map ρ to S is one-to-one, the map ρ|S : S → V is a diffeomorphism of S onto V . Thus, each point x of R has a neighbourhood V diffeomorphic to an open subset of Rdim S . Hence, R is a manifold. Since ρ(G × S) = ρ(S) = V , it follows that maps G × S onto ρ −1 (V ). Hence, ψV : ρ −1 (V ) → G × V = G × ρ(S) : p → (id, ρ) ◦ −1 ( p) is a diffeomorphism intertwining the action of G on ρ −1 (V ) and the action of G on G × V . We now consider a more general case in which the action of G on P need not be free. Let H be a compact subgroup of G, and let p ∈ PH . Consider

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a slice S p = Exp p (B), where Exp p is an H -equivariant diffeomorphism of a neighbourhood of 0 in T p P onto a neighbourhood of p in P, and B is an open ball in hor T p P centred at the origin. Since H is compact, the action of H on S p is proper. Hence, the orbit space S p /H endowed with the differential structure C ∞ (S p /H ) = { f ∈ C 0 (S p /H ) | π ∗ f ∈ C ∞ (S p )}, where π : S p → S p /H is the orbit map, is a differential space with the quotient topology. Lemma 4.3.3 G S p /G is diffeomorphic to S p /H . Proof For every orbit of G in G S p , its intersection with S p is an orbit of the action of H on S p . This gives a natural bijection δ : G S p /G → S p /H. We need to verify that δ is a diffeomorphism of a differential subspace G S p /G of R = P/G onto the quotient of S p by H , where S p is a differential subspace P. Let h be a function in C ∞ (S p /H ). Then, π ∗ h is an H -invariant function in C ∞ (S p ). For q ∈ S p , let W and W be H -invariant neighbourhoods of q in S p such that W ⊆ W ⊆ W ⊆ S p . Let k ∈ C ∞ (S p ) be a function such that k|W = 1 and k|S p \W = 0. The function ∗g (kπ ∗ h) dμ(g) f¯S p = H

on S p , where dμ(g) is the Haar measure on H normalized so that vol H = 1, is smooth and H -invariant. Moreover, the function f¯S p coincides with π ∗ h on W and vanishes on the complement of W in S p . We can extend f¯S p to a G-invariant function f˜ on G S p . Let f be a function on P such that f |G S p = f˜ and f |P\GW = 0. Since f¯S p vanishes on the complement S p \W of W in S p , it follows that f˜ vanishes on G(S p \W ) = G S p \GW . Hence, f is well defined, smooth and G-invariant. Therefore, f pushes forward to a function ρ∗ f in C ∞ (R) such that ρ ∗ (ρ∗ f ) = f. Since W is an H -invariant open neighbourhood of q ∈ S p , W/H is a neighbourhood of π(q) in S p /H and GW/G is a neighbourhood of ρ(q) in R. For each p ∈ W , we have ρ( p ) ∈ GW/G ⊆ R, π( p ) ∈ W/H ⊆ S p /H and δ(ρ( p )) = π( p ). Moreover, ρ∗ f (ρ( p )) = f ( p ) = f˜( p ) = f¯S p ( p ) = k( p ) f S p ( p ) = π ∗ h( p ) = h(π( p )),

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which implies that (δ ∗ h)|GW/G = (ρ∗ f )|GW/G . Since p ∈ S p is arbitrary, Condition 3 of Definition 2.1.1 implies that δ ∗ h ∈ C ∞ (G S p /G). To show that δ −1 is smooth, consider a function h ∈ C ∞ (R). The restriction h |G S p /G of h to G S p /G is in C ∞ (G S p /G). The pull-back ρ ∗ h is a smooth G-invariant function on P. Its restriction to G S p is a smooth H invariant function (ρ ∗ h)|G S p on G S p . Hence, (ρ ∗ h)|G S p pushes forward to a smooth function π∗ ((ρ ∗ h)|G S p /H ) on G S p /H . Moreover, for each p ∈ G S p , β(ρ( p )) = π( p ) implies ρ( p ) = δ −1 (π( p )) and h |G S p /G (δ −1 (π( p ))) = h |G S p /G (ρ( p )) = ρ ∗ h( p ) = (ρ ∗ h)|G S p ( p ) = π∗ ((ρ ∗ h)|G S p )(π( p )). Hence, (δ −1 )∗ h |G S p /G = π∗ ((ρ ∗ h)|G S p ) ∈ C ∞ (S p /H ). Therefore, δ is a diffeomorphism. The next step is to prove that the orbit space R = P/G of a proper action of G on P is subcartesian. The proof uses several classical results, references to which are given in Chapter 1. Theorem 4.3.4 The orbit space R = P/G of a proper action of G on P with the differential structure C ∞ (R) given by Equation (4.9) is subcartesian. Proof (i) Hausdorff property. If R were not Hausdorff, there would be two distinct points x and x in R which cannot be separated by open sets. We choose points p and p in ρ −1 (x) and ρ −1 (x ), respectively. Let (Wk ) be a nested neighbourhood basis of p, and let (Wk ) be a nested neighbourhood basis of p . The assumption that x and x cannot be separated by open sets implies that, for each k ∈ N, there exists pk ∈ GWk ∩ GWk . Hence, there exist gk and gk in G such that gk pk ∈ Wk and gk pk ∈ Wk . By construction, gk pk converges to p and gk pk converges to p . Therefore, (gk gk−1 )(gk pk ) = gk pk converges to p . Since the action of G on P is proper, there exists a subsequence gk n gk−1 in G n that is convergent in G, and −1 )(g p )} = lim (g g ) lim (g p ) p = lim {(gk n gk−1 k k k k n n n n k n kn n n→∞ n→∞ n→∞ −1 = lim (gkn gkn ) p. n→∞

This implies that p and p are in the same G-orbit, which contradicts the assumption that x = x . (ii) Choice of neighbourhood. Next, we need to show that, for each x ∈ R, there exists a neighbourhood of x in R that is diffeomorphic to a subset of Rn , for some n ∈ N. Choose a point p ∈ ρ −1 (x), and let S p be a slice through p constructed as in Proposition 4.2.3. Let H be the isotropy group of p. We have

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shown in Lemma 4.3.3 that G S p /G is diffeomorphic to S p /H . We are going to show that S p /H is diffeomorphic to a subset of Rn . (iii) Differential structure of S p /H . By construction, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a ball in hor T p P centred at the origin. The action of H on T p P is linear, and it leaves hor T p P invariant. Hence, it gives rise to a linear action of H on hor T p P. Moreover, the restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action of H on hor T p P and the action of H on S p . Therefore, S p /H is diffeomorphic to B/H . Since B is an H -invariant open subset of hor T p P and the action of H on hor T p S p is linear, by a theorem of G.W. Schwarz, smooth H -invariant functions on S p are smooth functions of algebraic invariants of the action of H on hor T p P. Let R[hor T p P] H denote the algebra of H -invariant polynomials on hor T p P. Hilbert’s Theorem ensures that R[hor T p P] H is finitely generated. Let σ1 , . . . , σn be a Hilbert basis for R[hor T p P] H consisting of homogeneous polynomials. The corresponding Hilbert map σ : hor T p P → Rn : v → σ (v) = (σ1 (v), . . . , σn (v)) induces a monomorphism σ˜ : (hor T p P)/H → Rn : H v → σ (v), where H v is the orbit of H through v ∈ hor T p P treated as a point in (hor T p P)/H . Let Q be the range of σ . By the Tarski–Seidenberg Theorem, Q is a semi-algebraic set in Rn . Let ϕ : (hor T p P)/H → Q ⊆ Rn be the bijection induced by σ˜ . We want to show that ϕ is a diffeomorphism. Smoothness of σ implies that ϕ is smooth. To show that ϕ −1 : Q → (hor T p P)/H is smooth, consider a function f ∈ C ∞ ((hor T p P)/H ). Let π : hor T p P → (hor T p P)/H denote the orbit map. Then, π ∗ f ∈ C ∞ (hor T p P) H . By the theorem of Schwarz quoted above, there exists F ∈ C ∞ (Rn ) such that, for all v ∈ T p S p , f (π(v)) = π ∗ f (v) = F(σ (v)) = F|Q (σ (v)) = F|Q (ϕ(π(v))). Hence, f = ϕ ∗ F|Q , which implies that F|Q = (ϕ −1 )∗ f . Thus, for every f ∈ C ∞ ((hor T p P)/H ), (ϕ −1 )∗ f ∈ C ∞ (Q). Therefore ϕ : (hor T p P)/H → Q is a diffeomorphism. Since B is an H -invariant open neighbourhood of 0 in hor T p S p , it follows that B/H is open in (hor T p P)/H . Hence, B/H is in the domain of the diffeomorphism ϕ : (hor T p P)/H → Q, which induces a diffeomorphism of B/H onto ϕ(B/H ) ⊆ Q ⊆ Rn . Thus, B/H is diffeomorphic to a subset of Rn . But B/H is diffeomorphic to S p /H , and S p /H is diffeomorphic to G S p /G. Therefore, G S p /G is diffeomorphic to a subset of Rn .

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The arguments given above hold for each point p ∈ P. Therefore, the orbit space R = P/G is subcartesian. Now that we know that the orbit space is subcartesian, we want to show that it is stratified. In the preceding section, we described the orbit type stratification M of P given by connected components of P(H ) = { p ∈ P | ∃g ∈ G such that G p = g H g −1 } as H varies over compact subgroups of G. We are going to show that the orbit map ρ : P → R maps M to a stratification N of R. Theorem 4.3.5 The family N = {ρ(M) ⊆ R | M ∈ M}, where M is the orbit type stratification of P and ρ : P → R is the orbit map, is a stratification of the orbit space R. Proof (i) Strata. Let H be a compact subgroup of G. Consider a point p0 ∈ PH ⊆ P(H ) , and let M be the connected component of P(H ) that contains p0 . We begin with a claim that L = M ∩ PH is the connected component containing p0 , and that M = G L. Clearly, L is an open subset of PH , and p0 ∈ L. Moreover, we can observe that P(H ) = G PH and that the connectedness of G implies that G M = M. Suppose now that L is disconnected. Let (U1 , U2 ) be a disconnection of L. In other words, U1 , U2 are open sets in L such that U1 ∩ U2 = ∅ and U1 ∪ U2 = L. Then, (GU1 , GU2 ) is a disconnection of M, which contradicts the assumption that M is connected. On the other hand, if the connected component L 0 of PH that contains p0 contains L as a proper subset, then G L 0 is a connected open subset of P(H ) , and G L 0 contains M as a proper subset, which contradicts the assumption that M is a connected component of P(H ) . It follows from Proposition 4.2.6 that L is a submanifold of P. Moreover, M = G L is a connected component of P(H ) , and it is a stratum of the orbit type stratification M of P. We want to show that ρ(M) is a submanifold of R. We showed in Proposition 4.2.6 that the action of G L on L is free and proper. It follows that the quotient L/G L is a manifold and that the orbit map π : L → L/G L is a (left) principal fibre bundle projection. Moreover, for every p ∈ L, we have Gp = G L p. Hence, there is a map γ : L/G L → ρ(M), given as follows. Each point in L/G L is of the form π( p), where p ∈ L. Since L ⊆ M ⊆ P, the projection ρ( p) to R is contained in ρ(M). We set γ (π( p)) = ρ( p) ∈ ρ(M). Conversely, consider ρ( p ) ∈ ρ(M) for some p ∈ M. Since M = G L , there exists p ∈ L such that ρ( p) = ρ( p ). Then, γ (π( p)) = ρ( p ). Hence, γ : L/G L → ρ(M) is a bijection. An argument analogous to the proof of Lemma 4.3.3 ensures that γ is a diffeomorphism of the quotient

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differential space L/G L , where L is a differential subspace of M = G L ⊆ P, and the differential subspace ρ(M) of the quotient R = P/G. Since ρ(M) is a differential subspace of R diffeomorphic to the manifold L/G L , it follows that ρ(M) is a submanifold of R. (ii) Frontier condition. Suppose M and M are strata of M such that ρ(M )∩ ρ(M) = ∅. Since M is G-invariant, it follows that ρ(M) = ρ(M). Therefore, M ∩ M = ∅, which implies that either M = M or M ⊂ M\M. Hence, either ρ(M ) = ρ(M) or ρ(M ) ⊂ ρ(M\M) = ρ(M)\ρ(M). Thus, the family N satisfies Frontier Condition 4.1.1. (iii) Local finiteness. For each p ∈ P, there is a neighbourhood V of p in P intersecting a finite number of manifolds M ∈ M. Hence, GV is a Ginvariant neighbourhood of p which intersects a finite number of manifolds M ∈ M. Therefore, ρ(GV ) is a neighbourhood of ρ( p) which intersects a finite number of manifolds ρ(M) ∈ N. This implies that N is locally finite. We have shown that the family N = {N = ρ(M) ⊆ R | M ∈ M} is a locally finite partition of R by submanifolds and satisfies the frontier condition. Hence, N is a stratification of R. The stratification N is called the orbit type stratification of R. Our ultimate goal in this section is to prove that the stratification N of R coincides with the partition O of R by orbits of the family X(R) of all vector fields on R. By Proposition 4.1.5 and Theorem 4.1.6, it suffices to show that N is locally trivial and minimal. First, we show that N of R is locally trivial. According to Definition 4.1.4, N is locally trivial if, for every N ∈ N and each x ∈ N , (i) there exists a neighbourhood U of x in R such that NU is a stratification of U ; (ii) there exists a subcartesian stratified space (S , M ) with a distinguished point y ∈ S such that the singleton {y} ∈ M ; and (iii) there is an isomorphism ϕ : (U, NU ) → ((N ∩ U ) × S , M(N ∩U )×S ) such that ϕ(x) = (x, y). For x ∈ R, there exists p ∈ P such that x = ρ( p). Let H be the isotropy group of p. Consider a slice S p through p constructed as in Proposition 4.2.3. The set U = ρ(G S p ) = G S p /G is an open neighbourhood of x in R. Lemma 4.3.2 ensures that U = G S p /G is diffeomorphic to S p /H . Hence, the restriction to U of the orbit type stratification N of R is a stratification NU isomorphic to the orbit type stratification of S p /H . By construction, S p = Exp p (B), where Exp p is an H -equivariant map from a neighbourhood of 0 in T p P to a neighbourhood of p in P, and B is a ball in

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hor T p P centred at the origin. The action of H on T p P is linear, and it leaves hor T p P invariant. Hence, it gives rise to a linear action of H on hor T p P. Moreover, the restriction of Exp p to B gives a diffeomorphism β : B → S p , which intertwines the linear action of H on hor T p P and the action of H on S p . Therefore, S p /H is diffeomorphic to B/H , and the orbit type stratification of S p /H is isomorphic to the orbit type stratification of B/H . Thus, the stratification NU is isomorphic to the orbit type stratification of B/H , where B is an open ball invariant under a linear action of H in a vector space hor T p P. We begin with a discussion of the orbit type stratification of the space of orbits of a linear action of a compact group on a vector space. Lemma 4.3.6 Consider a linear action of a compact Lie group H on a vector space E endowed with an H -invariant metric k. Let E H be the space of H invariant vectors in E, and let F be the k-orthogonal complement of E H in E. The linear action of H on E induces a linear action of H on F. Let N E/H and N F/H be the orbit type stratifications of E/H and F/H , respectively. Then there is a stratified space isomorphism ψ : (E/H, N E/H ) → (E H × F, M E H ×F/H ) such that ϕ(0) = {0×0}, where M E H ×F/H is the product of the single-stratum stratification {E H } of E H and the orbit type stratification N F/H of F/H . Proof Since F is the k-orthogonal complement of E H in E, we obtain an H -equivariant product structure E = E H × F. For each compact subgroup K of G, E K = E H × FK and E (K ) = H E K = E H × H FK = E H × F(K ) . The product structure E = E H × F induces an isomorphism of the quotient spaces ψ : E/H → (E H /H ) × (F/H ). Since 0 ∈ E is H -invariant, it follows that the quotient map E → E/H maps 0 ∈ E to 0 ∈ E H . Moreover, 0 ∈ F is the only H -invariant vector in F, and the quotient F/H is a cone with vertex 0. Therefore, ψ(0) = (0, 0). For each proper compact subgroup K of H , we have ψ(E (K ) /H ) = E H × (F(K ) )/H. Since E H is connected, it follows that each stratum of N E/H of the orbit type stratification of E/H is mapped by ϕ to the product of E H and a stratum of the orbit type stratification N F/H of F/H . Therefore, ψ is an

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isomorphism of the stratified space (E/H, N E/H ) onto the stratified space (E H × F/H, M E H ×F/H ), where M E H ×F/H is the product of the singlestratum stratification {E H } of E H and the orbit type stratification N F/H of F/H . Next, we can use the local results obtained in Lemma 4.3.6 to show that the stratification N of R is locally trivial. Proposition 4.3.7 The orbit type stratification N of the space R of orbits of a proper action of a connected Lie group G on a manifold P is locally trivial. Proof Let p ∈ PH . Consider a slice S p through p for the action of G on P constructed as in Proposition 4.2.3. The set U = ρ(G S p ) = G S p /G is an open neighbourhood of x in R, and U = G S p /G is diffeomorphic to S p /H . Hence, the restriction to U of the orbit type stratification N of R is a stratification NU isomorphic to the orbit type stratification of S p /H . Moreover, NU is isomorphic to the orbit type stratification of B/H , where B is an open ball invariant under a linear action of H in a vector space hor T p P. In order to use the results of Lemma 4.3.6, we set hor T p P = E so that B is an H -invariant open ball in E. Let B and B be open balls in E H and F, respectively, both centred at the origin and such that B × B ⊆ B. Moreover, we assume that B is invariant under the action of H in F. Note that the H -invariance of B is self-evident because B consists of fixed points. The product B × B is an H -invariant neighbourhood of 0 ∈ E. Since multiplication of vectors in E commutes with the action of H , it follows that (B × B )(K ) = B × (B ∩ F(K ) ) = B × B(K )

for each compact proper subgroup K of H . The isomorphism ψ : E/H → (E H /H ) × (F/H ) restricted to (B × B )/ H gives ψ((B × B )/H ) = (B /H ) × (B /H ) = B × (B /H ). Therefore, for each compact proper subgroup K of H ,

ψ((B × B )(K ) /H ) = B × (B(K ) /H ). Let χ = ψ|B ×B be the restriction of ψ to B × B . This is an isomorphism of the stratified space (B × B , N(B ×B )/H ), where N(B ×B )/H is the orbit type stratification, and the stratified space (B × (B /H ), M B ×(B /H ) ), where M B ×(B /H ) is the product of the single-stratum stratification {B } of B and the orbit type stratification N B /H of B /H .

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Recall that S p = Exp p (B), and U = ρ(S p ) is a neighbourhood of x = ρ( p) ∈ R such that the orbit type stratification N of R restricts to a stratification NU of U , which is isomorphic to the orbit type stratification of B/H . Restricting the neighbourhood of x to U = ρ(B × B ), we obtain an isomorphism ϕ of the stratification NU of U that is induced by the stratification NU of U to the stratification (B × (B /H ), M B ×(B /H ) ) of B × B , which satisfies the conditions of Definition 4.1.4. Since this construction can be performed for every x ∈ R, it follows that the orbit type stratification N of R is locally trivial. Next, we show that the orbit type stratification N of R = P/G is minimal. As before, we begin with the case of a linear action of a compact Lie group. Consider a linear action of a compact Lie group H on a vector space E endowed with an H -invariant metric k. Let E H be the space of H -invariant vectors in E, and let F be the k-orthogonal complement of E H in E. The linear action of H on E induces a linear action of H on F. Moreover, E = E H × F is an H -equivariant product. As in the proof of Theorem 4.3.4, let R[E] H = R[E H × F] H denote the finitely generated algebra of H -invariant polynomials on E. Since E = E H × F, polynomials on E are sums of products of polynomials on E H and on F. We consider a Hilbert basis (σ1 , . . . , σn ) for R[E H × F] H consisting of homogeneous polynomials. For each i = 1, . . . , n, we denote by di the degree of σi . We choose σ1 , . . . , σl , where l = dim E H , to be degree-1 polynomials on E H . These polynomials are H -invariant, since each point of E H is fixed by the action of H on E. Moreover, H -invariant polynomials on E H of degree ≥ 2 are polynomials in σ1 , . . . , σl . Hence, the remaining elements of the basis can be chosen to be polynomials on F. Since F does not contain H -invariant non-zero vectors, there are no H -invariant polynomials on F of degree 1. Therefore, σl+1 , . . . , σn are homogeneous polynomials on F of degree ≥ 2. Moreover, for every v ∈ F, we can set σl+1 (v) = k(v, v), where k is the H -invariant metric on E. Let S = {v ∈ F | k(v, v) = 1} be the unit sphere in F. For each i ≥ l + 2, −1 let Ci be the maximum of |σi (v)| for v ∈ S. For v = 0, |v| v ∈ S, where √ |v| = [k(v, v)] = σl+1 (v) is the norm of v. Moreover, σi (v) = σi (|v| (|v|−1 v)) = |v|di σi (|v|−1 v) ≤ Ci |v|di implies the inequality |σi (v)| ≤ Ci [σl+1 (v)]di /2 , which is also valid for v = 0. Therefore, the image of the Hilbert map σ : E = E H × F → Rn : (u, v) → σ (u, v) = (σ1 (u), . . . , σl (u), σl+1 (v), . . . , σn (v))

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is contained in the set = {(σ1 , . . . , σn ) ∈ Rn | σl+1 ≥ 0 and |σi | ≤ Ci [σl+1 ]di /2 ∀ l + 2 ≤ i ≤ n}. In the description of given here, we have treated σ1 , . . . , σn as coordinates in Rn . Lemma 4.3.8 Let γ : I → σ (E) ⊆ Rn : t → σ (t), where I is an open interval in R containing 0, be a curve in σ (E), differentiable at t = 0 and such that σl+1 (v(0)) = 0. Then σi (0) = 0 for each i = l + 1, . . . , n. Proof The inclusion γ (I ) ⊆ σ (E) and the inequality |σi | ≤ Ci [σl+1 ]di /2 , for all l + 2 ≤ i ≤ n, imply that σl+1 (t) ≥ 0 for each t ∈ I . The assumption that σl+1 (0) = 0 implies that 0 is a local minimum of t → σl+1 (t). Hence, (0) = 0 and σl+1 σl+1 (t) →0 t as t → 0. Moreover, for each i = l + 2, . . . , n, and t ∈ I \{0}, |σi (t)| [σl+1 (t)]d/2 σl+1 (t) . ≤ Ci = Ci [σl+1 (t)](di /2−1) |t| |t| t For i = l +2, . . . , n, the degree di ≥ 2 and the map t → σi (t) is differentiable and continuous at t = 0. This implies that [σl+1 (t)](di /2−1) is bounded as t → 0. Since σl+1 (t) →0 t as t → 0, it follows that |σi (t)| →0 |t| as t → 0. Hence, σi (0) = 0 for i = l + 2, . . . , n. We showed earlier that (0) = 0. Therefore, σ (0) = 0 for i = l + 1, . . . , n. σl+1 i It follows from Lemma 4.3.8 that, if σ (E) is a submanifold of Rn , then the dimension of T0 σ (E) is l, which is equal to the dimension of the space E H of fixed points of H . This implies that E = E H and that the orbit type stratification of E/H has only one stratum. In other words, if the orbit type stratification of E/H has more than one stratum, it cannot be a refinement of a coarser stratification. Thus, the orbit type stratification of E/H is minimal. Theorem 4.3.9 Let R = P/G be the space of G-orbits of the proper action of a connected Lie group G on a manifold P. Then the orbit type stratification of R is minimal.

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Proof Let B be an H -invariant open ball in E centred at the origin. The orbit type stratification N E/H of E/H restricts to the orbit type stratification N B of B. Since B contains the origin in E, the origin in Rn is contained in the range σ (B) of the Hilbert map σ : E → Rn . Hence, the argument above applies to σ (B) and ensures that the orbit type stratification of B/H is minimal. In Proposition 4.2.3, we constructed a slice S p through p for the action of G on P as the image of an open ball B in a vector space E = hor T p P under a local H -equivariant diffeomorphism Exp p of a neighbourhood of 0 in E onto a neighbourhood of p ∈ P, where H = G p . Therefore, the orbit type stratification of S p /H is isomorphic to the orbit type stratification of B/H . Hence, the orbit type stratification of S p /H is minimal. In Lemma 4.3.3, we showed that the orbit type stratification of G S p /G is isomorphic to the orbit type stratification of S p /H . Therefore, the orbit type stratification of S p /H is minimal. However, G S p /G is an open subset of R = P/G and the orbit type stratification of G S p /G is given by the restriction to G S p /P of the orbit type stratification of R. Therefore, the orbit type stratification of R is minimal. Combining the results of this section with Theorem 4.1.6, we obtain the desired result. Theorem 4.3.10 The space R = P/G of G-orbits of the proper action of a connected Lie group G on a manifold P endowed with the differential structure C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}, where ρ : P → R is the orbit map, is a subcartesian space with the quotient space topology. The orbit type stratification of P projects to an orbit type stratification of R. Strata of the orbit type stratification of R are orbits of the family of all vector fields on R. Proof According to Proposition 4.3.1, the differential-space topology of R coincides with the quotient topology. Theorem 4.3.4 ensures that R is subcartesian. In Theorem 4.3.4 we showed that the family N = {ρ(M) ⊆ R | M ∈ M}, where M is the orbit type stratification of P, is a stratification of the orbit space R, called the orbit type stratification of R. According to Proposition 4.3.7, N is locally trivial. Proposition 4.1.5 states that a locally trivial stratification of a subcartesian space R admits local extensions of vector fields. According to Theorem 4.1.6, if N is a stratification of a subcartesian space R which admits local extensions of vector fields, then the partition O of R by orbits of the family X(R) of all vector fields is a stratification of R, and N is a refinement of O. Moreover, if N is minimal, then N = O. Minimality of N has been proven in

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Theorem 4.3.9. Hence, strata of the orbit type stratification of R are orbits of the family of all vector fields on R. Theorem 4.3.10 provides a justification for the technique of singular reduction of symmetries in geometric mechanics. We shall discuss singular reduction of symmetries of symplectic manifolds in Chapter 6.

4.4 Action of a Lie group on a subcartesian space In this section, we show that the space of orbits of a proper action of a Lie group on a locally compact subcartesian space is a Hausdorff, locally compact differential space with the quotient topology. We begin with a review of the original Palais formulation for Hausdorff, locally compact topological spaces. Next, we specialize it to locally compact subcartesian spaces. We consider a continuous action : G × P → P : (g, p) → g ( p) = gp of a Lie group G on a Hausdorff, locally compact topological space P. We assume that P is a proper G-space. This means that each point p ∈ P has a neighbourhood U such that for every q ∈ P, there exists a neighbourhood V of q for which the closure of the set {g ∈ G | gU ∩ V = ∅} is compact. Let H be a closed subgroup of G. A subset S of P is an H -kernel if there exists an equivariant map f : G S → G/H such that f −1 (H ) = S. The following two theorems are quoted, without proof, from Palais (1961). Theorem 4.4.1 Let H be a closed subgroup of G. If S is an H -kernel in P, then: 1. S is closed in G S. 2. S is invariant under H . 3. gS ∩ S = ∅ implies that g ∈ H . If H is compact, then in addition: 4. S has a neighbourhood U in P such that the set {g ∈ G | gU ∩ U = ∅} has compact closure. Conversely, if the above conditions hold, then H is compact and S is an H kernel in P. Proof

See Theorem 2.14 in Palais (1961).

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Let a subset S of P be an H -kernel. If G S is open in P, the set S is called an H -slice in P. If in addition G S = P, then S is a global H -slice in P. Since P is a proper G-space, the isotropy group G p = {g ∈ G | gp = p} of p is compact for each p ∈ P. We say that a subset S of P is a slice at p if S is a G p -slice containing p. In the following, we shall denote a slice at p ∈ P by S p . Theorem 4.4.2 If P is a proper G-space, then for every point p ∈ P there exists a slice at p. Proof

See Proposition 2.3.1 in Palais (1961).

Next, we show that the notions of a ‘proper action’ and a ‘proper G-space’ are equivalent. Proposition 4.4.3 A Hausdorff, locally compact topological space P is a proper G-space if and only if the action of G on P is proper. Proof Given p0 ∈ P, let U be a neighbourhood of p0 in P with compact closure. Take any q0 ∈ P, and let V be a neighbourhood of q0 with compact closure. We want to show that the set W = {g ∈ G | gU ∩ V = ∅} has compact closure. In other words, if gn is a sequence of points in W , then there exists a convergent subsequence. Each gn ∈ W is the limit point of a sequence gn,m ∈ W . That is, for each n, m, there exists pn,m ∈ U such that gn,m pn,m ∈ V . Since V has compact closure, there exists a subsequence gn,m j pn,m j convergent to some qn ∈ V . Similarly, since U has compact closure, there exists a subsequence of pn,m k convergent to pn ∈ U in the limit as k → ∞. Without loss of generality, we may assume that gn,m pn,m → qn and pn,m → pn as m → ∞. By construction, gn,m → gn as m → ∞. The assumption that the action is proper implies that qn = gn pn for every n ∈ N. Let us now consider sequences gn ∈ W , pn ∈ U and qn ∈ V such that qn = gn pn for all n ∈ N. Since U and V are compact, we may assume without loss of generality that the sequences pn and qn are convergent to p ∈ U and q ∈ V , respectively. The properness of the action of G on P implies that there is a subsequence of gn convergent to g ∈ G such that q = gp. However, W is closed, which implies that g ∈ W . Hence, W is compact. Conversely, suppose that P is a proper G-space. Let pn be a sequence of points in P convergent to p, and let gn be a sequence in G such that the sequence gn pn converges to q ∈ P. Let U and V be neighbourhoods of p and q, respectively, such that U , V and W are compact, where W = {g ∈ G | gU ∩ V = ∅}. Since pn → p and gn pn → q, there exists N > 0 such that gn ∈ W ⊆ W for all n > N . The compactness of W ensures that the sequence

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gn has a convergent subsequence gn m with limit g ∈ W . Since the action of G on P is continuous, it follows that lim pn m = gp. q = lim gn pn = lim gn m pn m = lim gn m n→∞

m→∞

m→∞

m→∞

This implies that the action of G on P is proper. From now on, we assume that P is a locally compact subcartesian space and that the action of G on P is smooth. Let H be a compact subgroup of G. We begin with a lemma, which will be needed in what follows. Lemma 4.4.4 Consider an action : H × P → P : (g, p) → g ( p) = gp of a compact Lie group H on a subcartesian differential space P. Let dμ be a Haar measure on H , normalized so that the total volume of H is 1. For each f ∈ C ∞ (P), the H -average ∗g f dμ(g) f˜ = H

is a smooth function on P. Proof The pull-back ∗ f of f ∈ C ∞ (P) by the action is a smooth function on H × P such that ∗ f (g, p) = f (gp) = g f ( p). For each p ∈ P, the function g → g f ( p) on H is smooth. Hence, the integral ∗g f ( p) dμ(g) = ∗ f (g, p) dμ(g) f˜( p) = H

H

exists and f˜ is a function on P. We need to show that f˜ is smooth. Since P is subcartesian, for each p ∈ P there exist a neighbourhood V p of p and a diffeomorphism ϕ p of V p onto a subset of Rn p . Hence, id × ϕ p : H × V p → H × ϕ p (V p ) : (g, q) → (g, ϕ(q)) ∈ H × Rn p is a diffeomorphism. This implies that there exists a function F p ∈ C ∞ (H × Rn p ) such that ((id × ϕ p )−1 )∗ (∗ f |H ×V p ) = F p|H ×ϕ p (V p ) . Therefore, for every (g, q) ∈ H × V p , ∗ f (g, q) = F p (g, ϕ p (q)). Integrating this equation over H , we obtain the following for each q ∈ V p : ∗ f (g, q) dμ(g) = F p (g, ϕ p (q)) dμ(g). f˜|V p (q) = H

H

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Since F p ∈ C ∞ (H × Rn p ) and H is compact, it follows that f˜|V p is smooth. Therefore, there exists a function h p ∈ C ∞ (P) such that f˜|V p = h p|V p . This holds for every p ∈ P, which ensures that f˜ ∈ C ∞ (P). Let be an H -slice at p for the action of G on P. By the definition of a slice, is invariant under the action of H on P. Hence, we have an action of H on H × → : (g, s) → gs = g s.

(4.11)

The differential structure C ∞ () of is generated by restrictions to of smooth functions on P. We denote the space of H -orbits in by /H and the orbit map by ρ : → /H . The orbit space /H is a differential space with a differential structure ∗ C ∞ (/H ) = { f : → R | ρ f ∈ C ∞ ()}.

By the definition of a slice, the space G = {gs ∈ P | g ∈ G and s ∈ } is an open G-invariant neighbourhood of p ∈ P. Its differential structure is generated by the restrictions to G of smooth functions on P. We denote the space of G-orbits in G by G/G and the orbit map by ρG : G → G/G. The differential structure C ∞ (G/G) of G/G consists of functions ∗ f ∈ C ∞ (G). f : G/G → G such that ρG Let ι : → G be the inclusion map. For each s ∈ , the H -orbit H s extends to the unique G-orbits through s. Thus, we have a one-to-one map β : /H → G/G : H s → Gs. Moreover, every G-orbit in G intersects along a unique H -orbit, which implies that β is invertible. We have the following commutative diagram:

ι

ρ

/H

/ G ρG

β

/ G/G.

Theorem 4.4.5 The bijection β : /H → G/G : H s → Gs is a diffeomorphism. Proof For every G-invariant function f ∈ C ∞ (G), the restriction of f to is H -invariant. This implies that β : /H → G/G is smooth.

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In order to demonstrate that β −1 : G/G → /H is smooth, we have to show that every H -invariant function h on extends to a G-invariant function on G. Since each point q ∈ G can be presented as q = gs for some g ∈ G and s ∈ , we can define a function f on G by f (gs) = h(s).

(4.12) g2 s2 , then s2 = g2−1 g1 s1 , of h implies that h(s2 ) =

If (g1 , s1 ) and (g2 , s2 ) ∈ are such that g1 s1 = which implies that g2−1 g1 ∈ H . The H -invariance h(g2−1 g1 s1 ) = h(s1 ). Hence, f is well defined by equation (4.12). Next, we need to show that f is smooth. For each ξ in the Lie algebra g of G, let X ξ be the vector field on P corresponding to the action of exp tξ ξ on P. Since G is G-invariant, the restriction X |G of X ξ to G is a vector field on G. By assumption, P is subcartesian, which implies that G is subcartesian. Hence, for each q ∈ G, there exists an open neighbourhood Uq of q in G and a diffeomorphism ϕq of Uq onto a subset of Rn q . For each ξ ξ ∈ g, ϕq∗ (X |Uq ) is a vector field on ϕ(Uq ). Consider the following system of differential equations on ϕ(Uq ) for functions Fq ∈ C ∞ (Rn q ): ξ

ϕq∗ (X |Uq )(Fq ) = 0 ∀ ξ ∈ g, Fq|ϕq (Uq ∩) = (ϕq−1 )∗ ( f˜|Uq ∩ ). Since every G-orbit in G intersects , there exists a unique solution of this system of equations, and this solution satisfies the condition ϕq∗ (Fq )|Uq ∩Uq = ϕq∗ (Fq )|Uq ∩Uq for every q, q ∈ G. Hence, there exists a unique smooth function on G which coincides with ϕq∗ (Fq )|Uq for every q ∈ G. It is easy to see that this function is the function f defined above. Next, we show that the quotient and differential-space topologies of our orbit spaces coincide. We begin with the action of an isotropy group H of p ∈ P on the slice at p. Proposition 4.4.6 The differential-space topology of C ∞ (/H ) coincides with the quotient topology. Proof Taking Proposition 2.1.11 into account, in order to prove that the topology of the orbit space /H induced by C ∞ (/H ) coincides with the quotient topology, it suffices to show that for each set V in /H which is open in the quotient topology, and each y ∈ V , there exists h ∈ C ∞ (/H ) such that h(y) = 0 and h |(/H )\V = 0. For y ∈ /H , choose q ∈ /H such that ρ (q) = y. Since G is open in P, and P is locally compact and Hausdorff, it follows that there exists an open neighbourhood W of q in G with its closure W contained in ρ −1 (V ),

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where ρ : P → P/G is the orbit map. Moreover, there exists a non-negative function f ∈ C ∞ (G) such that f (q) > 0 and f |P\W = 0, where G\W denotes the complement of W in G. By Lemma 4.4.4, the H -average g f dμ(g) f˜ = H

is in C ∞ (G). The assumption that

f is non-negative and f (q) > of f over H 0 implies that f˜(q) > 0. Since f |G\W = 0, it follows that f˜|G\H W = 0. The compactness of W and H implies that the union H W of all H -orbits through W is compact, and that H W = H W , where H W is the union of all H -orbits through W . Moreover, the assumption that W ⊆ ρ −1 (V ) and the H -invariance of ρ −1 (V ) ensure that H W = H W ⊆ ρ −1 (V ). Thus, f˜ is an H -invariant smooth function on G such that f˜( p) > 0 and f˜ vanishes on G\ρ −1 (V ). Let f˜| be the restriction of f˜ to . Since the differential structure C ∞ () is induced by the restrictions to of smooth functions on P, it follows that f˜| is smooth. Moreover, f˜| (q) = f˜(q) > 0, because q ∈ . On the other hand, f˜ vanishes on G\ρ −1 (V ). Hence, f˜| vanishes on −1 (G\ρ −1 (V )) ∩ = \(ρ −1 (V ) ∩ ) = \ρ (V ).

Furthermore, f˜| is H -invariant because f˜ and are H -invariant. By the definition of the differential structure C ∞ (P/H ) of the orbit space, there exists a function h ∈ C ∞ (P/H ) such that f˜| = ρ ∗ h. Clearly,

h(y) = h(ρ(q)) = ρ h(q) = f˜(q) > 0 ∗

and ∗ 0 = ( f˜| )|\ρ −1 (V ) = (ρ h)|\ρ −1 (V ) = h |ρ ()\V = h |(/H )/V ,

which ensures that the quotient topology and the differential-space topology of /H coincide. Using Theorem 4.4.5, we can extend the result above to the space of orbits of a proper action of G on P. Theorem 4.4.7 For a proper action : G × P → P of a Lie group G on a locally compact, subcartesian differential space P, the differential-space topology of C ∞ (P/G) coincides with the quotient topology. Proof Let V be a neighbourhood of y ∈ P/G that is open in the quotient topology. Choose p ∈ P such that ρ( p) = y. The set ρ −1 (V ) is an open G-invariant neighbourhood of p in P.

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Let be a slice through p for the action of G on P. Then G is an open G-invariant neighbourhood of p in P. We denote the isotropy group of p by H , and the orbit map by ρ : → /H . Since P is locally compact and Hausdorff, there exists an open neighbourhood W of p in P with a compact closure W contained in ρ −1 (V ) ∩ G. Without loss of generality, we may assume that W is H -invariant; see the proof of Proposition 4.4.6. Then, the set ρ −1 (ρ(W )) ∩ ρ −1 (V ) ∩ G is an open G-invariant neighbourhood of p in G. Hence, ρ −1 (ρ(W ) ∩ V ) ∩ = ρ −1 (ρ(W )) ∩ (ρ −1 (V ) ∩ G) ∩ is an H -invariant open neighbourhood of p in . Thus, ρ (ρ(W ) ∩ ρ −1 (V ) ∩ ) is an open neighbourhood of ρ ( p) in the quotient topology of /H . By Proposition 4.4.6, the differential-space topology of C ∞ (/H ) coincides with the quotient topology. Therefore, there exists a smooth function h ∈ C ∞ (/H ) that vanishes on the complement of ρ (ρ −1 (ρ(W ) ∩ V ) ∩ ) in /H and is such that h(ρ ( p)) = 1. Since G-orbits in G intersect along orbits of H in , and W is H -invariant, it follows that ρ −1 (ρ(W )) ∩ = W ∩ . Therefore, our function h vanishes on the complement of ρ (ρ −1 (V )∩W ∩). By Theorem 4.4.5, the map β : /H → G/G : H s → Gs is a diffeomorphism. Therefore, (β −1 )∗ h ∈ C ∞ (G/G), and ρ ∗ (β −1 )∗ h is a G-invariant smooth function on G. By construction, ρ ∗ (β −1 )∗ h( p) = (β −1 )∗ h(ρ( p)) = h(β −1 (ρ( p))) = h(ρ ( p)) = 1, because ρ|G ◦ ι = β ◦ ρ , where ι : → P is the inclusion map. On the other hand, suppose that q ∈ G is in the complement of ρ −1 (ρ(W ) ∩ V ) ∩ G. Hence, ρ(q) is in the complement of ρ(W ) ∩ V ∩ ρ(G) = ρ(W ) ∩ V ∩ (G/G)ρ(W ∩ ρ −1 (V ) ∩ ) in G/G. Since β −1 : G/G → /H is a diffeomorphism, β −1 (ρ(q)) is in the complement of β −1 ◦ ρ(W ∩ ρ −1 (V ) ∩ ) = ρ (W ∩ ρ −1 (V ) ∩ ) in /H . But h vanishes on the complement of ρ (ρ −1 (V ) ∩ W ∩ ) in /H . Therefore, ρ ∗ (β −1 )∗ h = h ◦ β −1 ◦ ρ vanishes on the complement of

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ρ −1 (ρ(W ) ∩ V ) ∩ G in G. Hence, the support of ρ ∗ (β −1 )∗ h is a closed set contained in ρ −1 (ρ(W ) ∩ V ) ∩ G, which is open in P. Now consider a point q in the boundary ∂(G) of G in P. Since G is open in P, it follows that ∂(G) = G\G. We want to show that there exists an open neighbourhood U of q in P such that U ∩ G is contained in the complement of ρ −1 (ρ(W ) ∩ V ) ∩ G. Suppose that there is a sequence Un of neighbourhoods of q such that ∩∞ n=1 Un = {q} and Un ∩ G ∩ ρ −1 (ρ(W ) ∩ V ) = ∅. Then, for each n there is a point qn ∈ Un ∩ G ∩ ρ −1 (ρ(W ) ∩ V ), and the sequence qn convarges to q. Moreover, there exists gn ∈ G such that gn qn ∈ W . Since W is compact, there exists a subsequence gn k qn k convergent to q¯ = limk→∞ gn k qn k ∈ W . By the properness of the action, we may assume without ¯ loss of generality that the sequence gn k is convergent to g¯ ∈ G and that q¯ = gq. This implies that W ∩ ∂(G) = ∅. But, by assumption, W ⊆ G, and G is open in P, so that G ∩ ∂(G) = ∅. Hence, we obtain a contradiction. This implies that there exists a function f ∈ C ∞ (P) such that f |G = ρ ∗ (β −1 )∗ h and f |P\G = 0. Clearly, f is G-invariant, and it pushes forward to a function ρ∗ f ∈ C ∞ (P/G). By construction, ρ∗ f (y) = 1 and (ρ∗ f )|(P/G)\V = 0. The argument above is valid for each point y ∈ P/G and each neighbourhood V of y in P/G that is open in the quotient topology. Hence, the differential-space topology of P/G coincides with its quotient topology. We now show that the orbit space P/G is Hausdorff. First, we observe that the orbit map ρ : P → P/G is open. This can be seen as follows. Let U be an open subset of P. For each g ∈ G, gU = {gp ∈ P | p ∈ P} is open and GU = Ug∈G gU is open. Hence, ρ(U ) = ρ(GU ) is open in P/G. Next, consider the relation ϒ = {( p, q) ∈ P × P | q = gp for some g ∈ G} defined by the partition of P by orbits of G. A convergent sequence of points in ϒ can be written as ( pn , qn ) = ( pn , gn pn ), where the sequences ( pn ) and (gn pn ) converge in P. Since the action of G on P is proper, there exists a convergent subsequence (gn k ) in G, and limn→∞ (gn pn ) = (limk→∞ gn k )(limn→∞ pn ). Therefore, limn→∞ ( pn , gn pn ) ∈ R, which implies that ϒ is closed in P × P. This ensures that P/G is Hausdorff.4 In Theorem 4.4.5, we showed that the bijection β : /H → G/G : H s → Gs is a diffeomorphism. If W is an H -invariant open subset of , then ρ (W ) is an open subset of /H that consists of H -orbits contained in W . 4 See Theorem 11 in Chapter 3 of Kelley (1955).

4.4 Action of a Lie group on a subcartesian space

89

On the other hand, since G is an open subset of P, the set GW , consisting of G-orbits intersecting W, is a G-invariant open subset of G, and ρ(GW ) is an open subset of P/G. Moreover, ρ(W ) = ρ(GW ) = β(ρ (W )). We shall use the above equalities in the arguments below. Proposition 4.4.8 Let G × P → P be a proper action of a Lie group G on a locally compact subcartesian differential space P. The space P/G of G-orbits in P is locally compact. Proof Proposition 4.4.6 ensures that P/G is a differential space with the quotient topology. For p ∈ P, let H be the isotropy group of p and let V be an open neighbourhood of ρ( p) ∈ P/G. We want to show that there exists an open neighbourhood of ρ( p) in P/G that has a compact closure contained in V . Let be the slice at p for the action of G on P. By definition, G is an open G-invariant neighbourhood of p in P. Without loss of generality, we may assume that ρ −1 (V ) ⊆ G. Hence, we may consider V as an open subset of G/G. By Theorem 4.4.5, β : /H → G/G : H s → Gs is a diffeomorphism. Hence, β −1 (V ) is open in /H . Since P is Hausdorff and locally compact, there exists a neighbourhood U of p with a compact closure U contained in ρ −1 (V ). Let g(U ∩ ) = H (U ∩ ). W = {gs ∈ | g ∈ H and s ∈ U ∩ } = g∈H

Since U ∩ is open in and the action of H on is continuous, it follows that g(U ∩ ) is open in for each g ∈ H . Hence, W is an open H -invariant neighbourhood of p in . Therefore, ρ (W ) is an open neighbourhood of ρ ( p) contained in β −1 (V ). This implies that ρ(W ) = β(ρ (W )) is an open neighbourhood of ρ( p) in G/G contained in V . The closure W of W is the set of limit points of sequences in W . Suppose a sequence (gn sn ) in W converges to q ∈ W . Since the sequence (sn ) is contained in U ∩ ⊆ U ∩ and U is compact, there exists a subsequence of sn convergent to q¯ in U ∩ . Compactness of H implies that there is a subsequence of (gn ) convergent to g¯ ∈ H , and q = g¯ q¯ ∈ H (U ∩ ). Conversely, every point of H (U ∩ ) can be presented as a limit of a sequence gn sn for gn ∈ H and sn ∈ U ∩ . Hence, W = H (U ∩ ).

90

Stratified spaces

Since U ⊆ ρ −1 (V ) ⊆ G, it follows that W = H (U ∩ ) ⊆ ρ −1 (V ) ∩ . The action : G × P → P is continuous. Its restriction H : H × P → P to an action of H on P is also continuous. Moreover, the set H (U ∩ ) = H (H × (U ∩ )). Furthermore, U ∩ is compact as a closed subset of a compact set U . Since a product of compact sets is compact and the image of a compact set under a continuous map is compact, it follows that W is compact. Thus, W is an H -invariant neighbourhood of p in such that its closure W is compact and contained in ρ −1 (V ) ∩ . We have shown above that orbit maps of a proper action are open. Hence, ρ (W ) is an open neighbourhood of ρ ( p) in /H . Moreover, ρ (W ) is a compact subset of /H contained in β −1 (V ). Hence, ρ (W ) is closed in /H and contains the closure of ρ (W ). Since every point s ∈ W is the limit of a convergent sequence sn in W , it follows that ρ (s) = limn→∞ ρ (sn ) ∈ ρ (W ). Therefore, ρ (W ) = ρ (W ), which implies that the closure of ρ (W ) is compact. Since β : /H → G/G : H s → Gs is a diffeomorphism, β(ρ (W )) is an open neighbourhood of ρG ( p) in G/G with a compact closure contained in V ⊆ G/G. But G is open in P, so that G/G = ρ(G) ⊆ P/G. Thus, β(ρ (W )) is an open neighbourhood of ρ( p) with a compact closure contained in V ⊆ P. This implies that P/G is locally compact. We have shown that the space of orbits of a proper action of a Lie group G on a locally compact subcartesian space P is a Hausdorff, locally compact differential space P/G with the quotient topology. This result is somewhat disappointing if we compare it with the wealth of information we have about spaces of orbits of proper actions of Lie groups on manifolds. In both cases (manifolds and differential spaces), the starting point is an application of the Slice Theorem, which we have discussed here. In the case of smooth manifolds, the next step is Bochner’s Linearization Lemma. It would be of interest to find a class of subcartesian spaces for which there is an analogue of Bochner’s Linearization Lemma.

5 Differential forms

As in the case of differential forms on a manifold, we can define differential forms on a differential space S either as alternating multilinear maps from the space of derivations of C ∞ (S), called Koszul forms, or pointwise, as maps associating to each point of x ∈ S an alternating multilinear form on Tx S, which we call Zariski forms. These definitions are inequivalent on the singular part of S. Moreover, Koszul forms admit an exterior differential but not a pullback, whereas Zariski forms admit a pull-back but not an exterior differential. There is a third definition, given by Marshall, which leads to forms that admit both pull-backs and exterior differentials. All three definitions agree on the level of 1-forms. It is of interest to see how these forms appear in applications.

5.1 Koszul forms Recall that, for a differential space S, the space of all derivations Der C ∞ (S) is a module over C ∞ (S). Definition 5.1.1 For each k ∈ N, a Koszul k-form is an alternating map ω from (Der C ∞ (S))k to C ∞ (S) that is k-linear over C ∞ (S). We denote the space of Koszul k-forms on S by kK (S). For ω ∈ kK (S), the value of ω on X 1 , X 2 , . . . , X k ∈ Der C ∞ (S) is denoted by ω(X 1 , . . . , X k ). By definition, for every permutation σ of (1, . . . , k), ω(X σ (1) , . . . , X σ (k) ) = (sign σ ) ω(X 1 , . . . , X k ), where sign σ denotes the signature of σ , which is 1 if σ is even and (−1) if σ is odd. Moreover, for every f ∈ C ∞ (S), ω( f X 1 , . . . , X k ) = f ω(X 1 , . . . , X k ).

92

Differential forms

In the remainder of this section, we shall refer to Koszul forms simply as forms. For k = 0, we identify 0-forms with smooth functions; that is, 0K (S) = C ∞ (S). For each f ∈ C ∞ (S), we denote by d f a 1-form given by d f (X ) = X ( f ), for every X ∈ Der C ∞ (S). We shall also use the notation d f (X ) = d f | X , which is convenient if d f or X is given by a lengthy expression. We shall prove in the next section that, for every x in the regular part Sreg of S, there exists an open neighbourhood U of x in Sreg such that for every 1-form θ , there exist functions f 1 , . . . , f n , h 1 , . . . , h n ∈ C ∞ (S), where n is the structural dimension of S at x, such that θ|U =

n

(h i d f i )|U ;

(5.1)

i=1

see the discussion following Proposition 5.2.2. More precisely, for every X ∈ Der C ∞ (S), n h i d fi | X . θ (X )|U = |U

i=1

Since θ (X ) and h i d f i | X are smooth functions on S and the regular component Sreg is dense in S, one can use continuity arguments to find values of θ (X ) at points of the singular component Ssing . For θ1 , . . . , θk ∈ 1K (S), we denote by θ1 ∧ . . . ∧ θk the k-form given by1 (θ1 ∧ . . . ∧ θk )(X 1 , . . . , X k ) = det( θi | X j ),

(5.2)

for every X 1 , . . . , X k ∈ Der C ∞ (S). If f 1 , . . . , f n are local coordinates on U ⊆ Sreg , then the restriction of a k-form ω to U can be given locally by (h i1 ...i k d f i1 ∧ . . . ∧ d fik )|U . (5.3) ω|U = i 1 ,...,i k

We introduce the notation K (S) =

∞

nK (S).

n=0

The sum above is finite if the structural dimension of S is bounded. We have several natural operations on K (S). For each n ≥ 0, n-forms can be added and multiplied by smooth functions. Hence, nK (S) is a module over C ∞ (S). 1 This definition follows the convention adopted by Warner (1971).

5.1 Koszul forms

93

n For ω ∈ m K (S) and ∈ K (S), the wedge product ω ∧ is an (m+n)form in such that for each (X 1 , . . . , X n+m ) ∈ Der C ∞ (S),

(ω ∧ )(X 1 , . . . , X n+m ) (sgn π )ω(X π(1) , . . . , X π(m) ) (X π(m+1) , . . . , X π(m+n) ), =

(5.4)

π

where the summation is taken over permutations π of {1, . . . , m + n} such that π(1) < . . . < π(m) and π(m + 1) < . . . < π(m + n). The next operation on forms is the exterior differential d : K (S) → K (S), which is defined as follows. For X 0 , . . . , X m ∈ Der C ∞ (S), dω(X 0 , . . . , X m ) =

m

(−1)k X k (ω(X 0 , . . . , X k , . . . , X m ))

(5.5)

k=0

+

(−1)k+l ω([X k , X l ], X 0 , . . . , Xk, . . . , X l , . . . , X m ), k

where the terms marked , although written down here, must be omitted. n Proposition 5.1.2 For ω1 ∈ m K (S) and ω2 ∈ K (S),

d(ω1 ∧ ω2 ) = (dω1 ) ∧ ω2 + (−1)m ω1 ∧ dω2 , d 2 ω = ddω = 0. Proof

We leave the proof as an exercise for the reader.

Since d 2 = 0, we can introduce the cohomology for a subcartesian space, which we refer to as the Koszul cohomology of the differential space. However, we shall not pursue this topic here. For a discussion of the Koszul cohomology in the context of algebraic geometry, the reader may consult Aprodu and Nagel (2010) and the references therein. There are two more standard operations on forms that involve derivations of C ∞ (S). If ω is a k-form on S and X is a derivation of C ∞ (S), then the left interior product of X and ω is a (k −1)-form X ω, which is defined as follows. For each x ∈ S and X 1 , . . . , X k−1 ∈ Der C ∞ (S), (X ω)(X 1 , . . . , X k−1 ) = ω(X, X 1 , . . . , X k−1 ). The Lie derivative of a differential form with respect to a derivation X is the anticommutator of the left interior product and the exterior differential £ X ω = X dω + d(X ω). Proposition 5.1.3 For every X ∈ C ∞ (S) and ω1 , ω2 ∈ , £ X (ω1 ∧ ω2 ) = £ X ω1 ∧ ω2 + ω1 ∧ £ X ω2 .

94

Proof

Differential forms

We leave the proof as an exercise for the reader.

Recall that the regular component Sreg of S is open and dense in S and that T Sreg is globally spanned by global derivations; see Theorem 3.3.14 and Proposition 3.3.15. Consider x ∈ Sreg . Given a k-form ω, for every X 1 , . . . , X k ∈ Der C ∞ (S) and each f ∈ C ∞ (S), ω( f X 1 , . . . , X k )(x) = f (x)ω(X 1 , . . . , X k )(x). This implies that there is an alternating k-linear function ω(x) on Tx Sreg such that ω(X 1 , . . . , X k )(x) = ωx (X 1 (x), . . . , X k (x)). In other words, the value of ω(X 1 , . . . , X k ) at a regular point x of Sreg can be expressed as the evaluation of a k-linear function ωx on the derivations X 1 (x), . . . , X k (x) ∈ Tx Sreg . We refer to ωx as the value of ω at x ∈ S. Example 5.1.4 Let S = {(x, y) ∈ R2 | x y = 0}. The singular part of S is the origin (0, 0). We showed in Example 3.1.10 that every derivation X of C ∞ (S) vanishes at the origin (0, 0) ∈ S. Therefore, for any Koszul 1-form ω, the value ω(x,y) is defined for every (x, y) = (0, 0), and ω(0,0) is not defined. Let ϕ : R → S : x → ϕ(x) = (x, 0) be the inclusion map of the x-axis, denoted by R, into S. Clearly, R is a manifold and ϕ is smooth. The derived map T ϕ : T R → T S is an inclusion. If ω is a Koszul 1-form on S, then for each x = 0, we can pull back ω(x,0) to a linear function ω(x,0) ◦ T ϕ on Tx R. In this way, we obtain a 1-form on the complement of 0 in R. Since ω(0,0) is not defined, we do not obtain a 1-form on R. Example 5.1.4 shows that Koszul forms need not have values at singular points. In the next section, we discuss a different type of differential forms, called Zariski forms, which have well-defined values at singular points. However, as we shall see, Zariski forms do not allow exterior differentials.

5.2 Zariski forms Definition 5.2.1 A Zariski k-form on a differential space S is an alternating k-linear smooth map ω from T S to R. We can reformulate this definition as follows. The differential structure of T S, given in Definition 3.3.2, induces a differential structure on the product (T S)k ; see Proposition 2.1.9. The fibre product T k S = {(u 1 , . . . , u k ) ∈ (T S)k | τ (u 1 ) = . . . = τ (u k )}

5.2 Zariski forms

95

is a subset of the differential space (T S)k . Hence, T k S is a differential space with a differential structure generated by restrictions to T k S of smooth functions on (T S)k ; see Proposition 2.1.8. A Zariski k-form on S is a map ω : T k S → R such that ω(u 1 , . . . , u i , . . . , u j , . . . , u k ) = −ω(u 1 , . . . , u j , . . . , u i , . . . , u k ) for every i, j = 1, . . . , k, and ω(u 1 , . . . , au i + bwi , . . . , u k ) = aω(u 1 , . . . , u i , . . . , u k ) + bω(u 1 , . . . , wi , . . . , u k ) for every i = 1, . . . , k, and all a, b ∈ R. Moreover, ω is smooth in the differential structure of T k S described here. We denote by kZ (S) the space of Zariski k-forms on S. As in the case of Koszul forms, the space Z (S) =

∞

kZ (S)

k=0

is a graded module over C ∞ (S), and it admits the following well-defined operations: (i) the wedge product (ω1 , ω2 ) → ω1 ∧ ω2 , such that (ω1 ∧ ω2 )(u 1 , . . . , u n+m ) (sgn π )ω1 (u π(1) , . . . , u π(m) )ω2 (u π(m+1) , . . . , u π(m+n) ); = π

(ii) the left interior product of X ∈ Der C ∞ (S) and ω ∈ kZ (S), i.e. X ω ∈ k−1 Z (S), defined by (X ω)(u 1 , . . . , u k−1 ) = ω(X (x), u 1 , . . . , u k−1 ) for every x ∈ S and u 1 , . . . , u k−1 ∈ Tx S. Let χ : Q → S be a smooth map from a differential space Q to a differential space S, and let ω be a Zariski k-form on S. The pull-back of ω by χ is the Zariski k-form χ ∗ ω on Q such that, for each q ∈ Q and every v1 , . . . , vk ∈ Tq Q, χ ∗ ω(v1 , . . . , vk ) = ω(T χ (v1 ), . . . , T χ (vk )).

(5.6)

It can easily be verified that χ ∗ (ω1 ∧ ω2 ) = χ ∗ ω1 ∧ χ ∗ ω2 for every ω1 , ω2 ∈ kZ (S).

96

Differential forms

A subcartesian space is locally diffeomorphic to an open subset of Rn for some positive integer n that depends on x. We can use this fact to describe Zariski forms on S in terms of differential forms on Rn . Proposition 5.2.2 For a subcartesian space S, an alternating k-linear map ω from T S to R is a Zariski differential form on S if and only if, for each x ∈ S, there exists a smooth map ϕ : U → Rn , where U is an open neighbourhood of x in S, that restricts to a diffeomorphism of U onto ϕ(U ) ⊆ Rn , and a differential form on Rn , such that ω(u 1 , . . . , u k ) = (T ϕ(u 1 ), . . . , T ϕ(u k ))

(5.7)

for every (u 1 , . . . , u k ) ∈ T k U . Proof Since S is subcartesian, for each x ∈ S, there exists a smooth map ϕ : W → Rn , where W is an open neighbourhood x in S, into Rn that restricts to a diffeomorphism of W onto ϕ(W ) ⊆ Rn . The derived map T ϕ : T W → T Rn induces a smooth map T k ϕ : T k W → T k Rn = Rkn × Rn : (u 1 , . . . , u k ) → (T ϕ(u 1 ), . . . , T ϕ(u k )). Since ϕ is a diffeomorphism of W on ϕ(W ) ⊆ Rn , and T ϕ is linear on the fibres of the tangent bundle projection, it follows that the range of T k ϕ is Rnk ×ϕ(W ). Moreover, the restriction of T k ϕ to T k ϕ(T k W ) = Rnk ×ϕ(W ) is invertible. Abusing the notation, we denote by T k ϕ −1 : T k ϕ(T k W ) → T k W the map such that T k ϕ −1 ◦ T k ϕ = identity. Let ω be an alternating k-linear map from T S to R. Equation (5.7) implies that ω is smooth. Hence, ω is a Zariski form on S. Conversely, let ω be a Zariski form on S. We denote by ω|W the pull-back of ω by the inclusion map W → S. We may treat ω|W as a smooth function on T k W . By composing ω|W with T k ϕ −1 , we obtain a smooth function (ω|W ) ◦ T k ϕ −1 on T k ϕ(T k W ) = Rnk ×ϕ(W ). The differential structure of Rnk ×ϕ(W ) is induced by the inclusion map Rnk × ϕ(W ) → Rkn × Rn . Therefore, there exists a neighbourhood V of (0, ϕ(x)) ∈ Rnk × ϕ(W ) and a function F ∈ C ∞ (T k Rn ) = C ∞ (Rkn ×Rn ) such that ((ω|W )◦T k ϕ −1 )|V = F|V . We can pull back this equality to a neighbourhood V˜ = (T k ϕ −1 )(V ) of (0, x) in T k (W ), obtaining (ω|W )|V˜ = (F ◦ T ϕ k )|V˜ = ((T ϕ k )∗ F)|V˜ ,

(5.8)

where ω|W is treated as a smooth function on T k W . Let o : W → T k W be the zero section of T k W , and let U be a neighbourhood of x in W such that o(U ) ⊆ V˜ . Then T k U is the restriction of T k W to

5.2 Zariski forms

97

base points in U . For each z ∈ U , we denote by ωz the restriction of ω|W to the fibre Tzk U . In other words, for each (u 1 , . . . , u k ) ∈ Tzk U , ωz (u 1 , . . . , u k ) = ω|W (u 1 , . . . , u k ). This notation emphasizes that z ∈ U is fixed. For each i = 1, . . . , k, we let πi : T k U → T U : (u 1 , . . . , u i , . . . , u k ) → u i be the projection on the ith factor. The fibres of the tangent bundle projection map τ : T U → U are linear. Hence, for a fixed z ∈ U , ωz (u 1 , . . . , u i , . . . , u k ) is a homogeneous linear function of u i . For every vi ∈ Tz U , we denote by vi ∂iz the derivative of ωz (u 1 , . . . , u i , . . . , u k ) with respect to u i in the direction vi . In other words, d ωz (u 1 , . . . , u i + tvi , . . . , u k )|t=0 dt = ωz (u 1 , . . . , vi , . . . , u k ).

vi ∂iz (ωz (u 1 , . . . , u i , . . . , u k )) =

Therefore, for v1 , . . . , vk ∈ Ty U , vk ∂kz (. . . (v2 ∂2z (v1 ∂1z (ωz (u 1 , . . . , u k )))) . . .) = ωz (v1 , . . . , vk ).

(5.9)

Thus, ωz (v1 , . . . , vk ) is determined by the derivatives of ωz (u 1 , . . . , u i , . . . , u k ) along the image o(U ) of the zero section o of T k U . In the following, we denote variables in T k Rn by boldface symbols. In particular, z = ϕ(z), and ui = Tz ϕ(u i ) for i = 1, . . . , k. Moreover, v i ∂ i = Tz ϕ(vi ∂iz ) denotes the partial derivative given by (v i ∂ i )(F)(u1 , . . . , ui , . . . , uk ) =

d F(u1 , . . . , ui + tv i , . . . , uk )|t=0 . dt

Equations (5.8) and (5.9) give ωz (v1 , . . . , vk ) = vk ∂1z (. . . (v1 ∂1z (ωz (u 1 , . . . , u k ))) . . .)|u i =0 = vk ∂kz (. . . (v1 ∂1z (F ◦ (Tz ϕ(u 1 ), . . . , Tz ϕ(u k )))) . . .)|u i =0 = Tz ϕ(vk ∂kz )(. . . (Tz ϕ(v1 ∂1z )F(Tz ϕ(u 1 ), . . . , Tz ϕ(u k ))) . . .)|Tz ϕ(u i )=0 = (v k ∂ k )(. . . ((v 1 ∂ 1 )F(u1 , . . . , uk )) . . .)|ui =0 .

98

Differential forms

Since ωz is skew-symmetric in the arguments v1 , . . . , vk , the equation will still hold if we antisymmetrize the right-hand side. Therefore, ωz (v1 , . . . , vk ) sgn(π )(v π(k) ∂ π(k) )(. . . (v π(1) ∂ π(1) )F(u1 , . . . , uk ) . . .)|u1 =···=uk =0 , = π

(5.10) where the summation is taken over all permutations π of the ordered set (1, . . . , k). The right-hand side of this equation is an alternating k-linear function of the variables v1 , . . . , v k ∈ Tz Rn that depends smoothly on z ∈ R. In other words, the right-hand side of equation (5.10) is the value on (v 1 , . . . , v k ) of a differential k-form on R, which we denote by . Therefore, equation (5.10) implies that ω|U = ϕ ∗ |U , which is equivalent to equation (5.7). By Proposition 3.3.15, the tangent bundle T Sreg of the regular component of a subcartesian space S is locally spanned by global derivations. Hence, the restriction of a Zariski form on S to Sreg defines a Koszul form on Sreg . Therefore, Proposition 5.2.2 provides a justification for equation (5.1) in the preceding section. The exterior differential of a Zariski form is not defined. For a subcartesian space, we might want to define d : kZ → k+1 Z (S) using Proposition 5.2.2 as follows. If the restriction of ω to Ux is given by equation (5.7), then for y ∈ Ux and u 0 , u 1 , . . . , u k ∈ Ty U x , dω(u 0 , u 1 , . . . , u k ) = d(T ϕx (u 0 ), T ϕx (u 1 ), . . . , T ϕx (u k )),

(5.11)

where d is the exterior differential of the form on Rn . However, this condition does not determine dω uniquely, as can be seen in the following example. Example 5.2.3 Let S = {(x, y) ∈ R2 | x y = 0}, and let ι : S → R2 be the inclusion map. Take 1 = 0 on R2 . Then ω1 = ι∗ = 0 and dω1 = 0. On the other hand, let 2 = y d x on R2 . The pull-back ω2 = ι∗ 2 is identically zero on T S, because y vanishes on the x-axis and d x vanishes on the y-axis. However, the pull-back of d2 = dy ∧ d x by the inclusion map ι : S → R2 does not vanish at the origin (0, 0), because T(0,0) S " R2 . Thus, dω given by equation (5.11) depends on the choice of .

5.3 Marshall forms

99

5.3 Marshall forms We continue with the assumption that S is a subcartesian space. Definition 5.3.1 For each m, we denote by Mm (S) the subset of mZ (S) consisting of Zariski m-forms ω such that, for each x ∈ S, there exists a differential (m − 1)-form " x on Rn , an open neighbourhood Ux of x in S and a smooth map ϕx : U → Rn with the property that ϕx∗ d"x = ω|Ux , ϕx∗ "x

= 0.

(5.12) (5.13)

The conditions (5.12) and (5.13) can be rewritten explicitly in the form ω(u 1 , . . . , u m ) = d"x (T ϕx (u 1 ), . . . , T ϕx (u m )), 0 = "x (T ϕx (u 1 ), . . . , T ϕx (u m−1 )),

(5.14) (5.15)

for every y ∈ U and u 1 , . . . , u m ∈ Ty S. It follows from Example 5.1.4 that Mk (S) need not vanish. Let Mm (S). M(S) = m

Proposition 5.3.2 M(S) is closed under the operations of addition, multiplication by functions in C ∞ (S), the wedge product and the differentiation d given by equation (5.11). Proof Since equations (5.14) and (5.15) are linear homogeneous equations in (ω, "x ), it follows that M is closed under addition and multiplication by numbers. For each ω ∈ Mk (S) and f ∈ C ∞ (S), equation (5.14) yields f (y)ω(u 1 , . . . , u k ) = f (y) d"x (T ϕx (u 1 ), . . . , T ϕx (u k )) = ((ϕx∗ f ) d"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = (d((ϕx∗ f )"x ) − d(ϕx∗ f ) ∧ "x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = d((ϕx∗ f )"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) because equation (5.15) implies that (d(ϕx∗ f )∧"x )(T ϕx (u 1 ), . . . , T ϕx (u k )) = 0. Moreover, equation (5.15) also implies that ((ϕx∗ f )"x )(T ϕx (v1 ), . . . , T ϕx (vk−1 )) = 0 for every v1 , . . . , vk−1 ∈ Ty S. Hence, f ω ∈ Mk (S).

100

Differential forms

Suppose now that ω1 ∈ Mm (S) and ω2 ∈ nZ (S). Equation (5.14) yields (ω1 ∧ ω2 )(u 1 , . . . , u n+m ) = (d"x1 ∧ "x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )) = d("x1 ∧ "x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )) − ("x1 ∧ d"x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m )). On the other hand, ("x1 ∧ d"x2 )(T ϕx (u 1 ), . . . , T ϕx (u n+m−1 )) = 0 because "x1 (T ϕx (v1 ), . . . , T ϕx (vm−1 )) = 0 for every v1 , . . . , vm−1 ∈ Ty S. This implies that ω1 ∧ ω2 ∈ Mm+n (S). Suppose that ω ∈ Mm (S), which implies equations (5.14) and (5.15). Assume also that dω satisfies equation (5.11). Taking equations (5.7) and (5.11) into account, we obtain dω(u 0 , u 1 , . . . , u k ) = d(d"x )(T ϕx (u 0 ), T ϕx (u 1 ), . . . , T ϕx (u k )) = 0. Hence, dω = 0 ∈ M. Definition 5.3.3 Marshall k-forms on S are elements of the quotient kM (S) = kZ (S)/Mk (S). We denote by μ the projection map from the space Z of Zariski forms to the space M (S) = kM (S) k

of Marshall forms. In other words, the Marshall form corresponding to a Zariski form ω is denoted by μ(ω). Below, we show that all of the usual operations on differential forms on manifolds extend to Marshall forms on subcartesian spaces. Proposition 5.3.4 M (S) is a graded module over C ∞ (S). The operation of the wedge product ∧ on Z (S) induces a wedge product and an exterior differential on mM (S) such that μ(ω1 ) ∧ μ(ω2 ) = μ(ω1 ∧ ω2 ). Proof This proposition is a direct consequence of Proposition 5.3.2 and standard properties of differential forms on Rn .

5.3 Marshall forms

101

Proposition 5.3.5 There is an exterior derivation operator d : M (S) → M (S) such that for every Zariski form ω, dμ(ω) = μ(dω),

(5.16)

where dω satisfies equation (5.11). In particular, d2 = 0

(5.17)

and, if ω1 ∈ kZ and ω2 ∈ mZ , then d(μ(ω1 ) ∧ μ(ω2 )) = dμ(ω1 ) ∧ μ(ω2 ) + (−1)k μ(ω1 ) ∧ dμ(ω2 ).

(5.18)

Proof Recall that if ω is a Zariski form, then equation (5.11) defines dω up to an element of M(S). By Proposition 5.3.2, d maps M(S) to zero. Hence, μ(dω) is uniquely determined by μ(ω) for every Zariski form ω. Equations (5.17) and (5.18) are direct consequences of Proposition 5.3.2 and standard properties of differential forms on Rn . Let χ : Q → S be a smooth map from a subcartesian space Q to a subcartesian space S. Lemma 5.3.6 The pull-back χ ∗ : Z (S) → Z (Q) of Zariski forms, defined by equation (5.6), maps M(S) to M(Q). Proof Let ω ∈ Mk (S). For each q ∈ Q and every v1 , . . . , vk ∈ Tq Q, equations (5.6), (5.14) and (5.15) give χ ∗ ω(v1 , . . . , vk ) = ω(T χ (v1 ), . . . , T χ (vk )) = d"x (T ϕx (T χ (v1 )), . . . , T ϕx (T χ (vk ))) = d"x (T (ϕx ◦ χ )(v1 ), . . . , T (ϕx ◦ χ )(vk )) and "x (T ϕx (T χ (v1 )), . . . , T ϕx (T χ (vk−1 ))) = "x (T (ϕx ◦ χ )(v1 ), . . . , T (ϕx ◦ χ )(vk−1 )) = 0, where x = χ (z). Hence, ϕx ◦ χ : χ −1 (Ux ) → Rn is a map of a neighbourhood χ −1 (Ux ) of q in Q that satisfies equations (5.14) and (5.15). Therefore, χ ∗ ω ∈ M(Q). Corollary 5.3.7 The pull-back χ ∗ : Z (S) → Z (Q) of Zariski forms induces a pull-back χ ∗ : M (S) → M (Q) of Marshall forms such that μ(χ ∗ ω) = χ ∗ (μ(ω)) for every ω ∈ Z (S).

102

Differential forms

The notion of Marshall forms has been extended to general differential spaces by Sasin (1986). Some examples of the de Rham cohomology, defined in terms of Marshall forms, were studied by Marshall (1975b). Koszul forms, Zariski forms, Marshall forms and sections of appropriate bundles were compared by Watts (2006). In particular, Marshall and Koszul forms coincide on the set Sreg of regular points of S and are smooth sections of the corresponding bundle. This suggests that, in some sense, Marshall forms are extensions of Koszul forms to the singular part Ssing of S. The nature of this extension requires further investigation.

PART II Reduction of symmetries

6 Symplectic reduction

In Chapter 4, we studied the orbit type stratification N of the orbit space R = P/G of a proper action of a Lie group G on a manifold P. Here, we consider a special case in which P is a symplectic manifold and the action of G on P is Hamiltonian. We show that the symplectic structure of P gives rise to a Poisson structure on the orbit space. Moreover, each stratum of N is a Poisson manifold singularly foliated by symplectic manifolds. We apply our results to Hamiltonian systems with symmetry and show that symplectic reduction leads to a reduced Hamiltonian system in each symplectic leaf of every stratum of N. Since the reduced Hamiltonian systems have a smaller number of degrees of freedom than the original Hamiltonian system, the process of reduction helps us to analyse the equations of motion of the original system. The approach to symplectic reduction of a proper action of the symmetry group presented here is called singular reduction. It was initiated in Cushman (1983). If the action of the symmetry group is free and proper, singular reduction leads to the regular reduction that was introduced by Meyer (1973) and by Marsden and Weinstein (1974). We also discuss algebraic reduction, which is applicable even for an improper action of the symmetry group.

6.1 Symplectic manifolds with symmetry 6.1.1 Co-adjoint orbits The orbits of the co-adjoint action of a connected Lie group are fundamental examples of symplectic manifolds with symmetry. The adjoint action of a Lie group G on its Lie algebra g is Ad : G × g → g : (g, ξ ) → Adg ξ =

d g(exp tξ )g −1 |t=0 , dt

106

Symplectic reduction

where exp tξ is the one-parameter subgroup of G generated by ξ ∈ g. Let g∗ be the dual of g. The co-adjoint action of G on g∗ is Ad ∗ : G × g∗ → g∗ : (g, μ) → Adg∗ μ, where Adg∗ μ | ξ = μ | Adg −1 ξ

for every ξ ∈ g. For each ξ, ζ ∈ g and μ ∈ g∗ , d d ∗ Adexp μ | Adexp(−tζ ) ξ |t=0 tζ μ | ξ |t=0 = dt dt = μ | −ζ ξ + ξ ζ

= μ | [ξ, ζ ] .

(6.1)

The isotropy group of μ ∈ g∗ is G μ = {g ∈ G | Adg∗ μ = μ}. The co-adjoint orbit through μ, given by Oμ = { Adg∗ μ ∈ g∗ | g ∈ G}, is a manifold diffeomorphic to G/G μ , and the inclusion map Oμ → g∗ is an immersion. Hence, Oμ is an immersed submanifold of g∗ . However, Oμ need not be an embedded submanifold; see Pukanszky (1971). We denote by X ξ the vector field on Oμ induced by the co-adjoint action of exp tξ on g∗ . In other words, d ∗ (6.2) f (Adexp tξ λ)|t=0 dt for every f ∈ C ∞ (Oμ ) and λ ∈ Oμ . Since G acts transitively on Oμ , for every vector u ∈ Tλ Oμ there exists ξ ∈ g such that X ξ (λ) = u. Let μ be a 2-form on Oμ such that, for each λ ∈ Oμ and ξ, ζ ∈ g, X ξ ( f )(λ) =

μ (X ξ (λ), X ζ (λ)) = − λ | [ξ, ζ ] .

(6.3)

The form μ is known as the Kirillov–Kostant–Souriau form of the co-adjoint orbit Oμ . Proposition 6.1.1 Equation (6.3) gives a well-defined 2-form μ , which is non-degenerate and closed. Proof Suppose that, for ξ ∈ g and λ ∈ Oμ , X ξ (λ) = 0. That is, d ∗ dt Adexp tξ λ|t=0 = 0. Hence, for each ζ ∈ g, d d ∗ Adexp(−tξ ) ζ Adexp = λ | ζ = 0. λ | [ξ, ζ ] = − λ | tξ dt dt |t=0 |t=0

6.1 Symplectic manifolds with symmetry

107

This ensures that μ is well defined. Conversely, if X ξ (λ) μ = 0, then for each ζ ∈ g, d ∗ λ | ζ = λ | [ξ, ζ ] = −(X ξ (λ), X ζ (λ)) = 0, Adexp tξ dt |t=0 which implies that X ξ (λ) = 0. Therefore, μ is non-degenerate. Next, we show that μ is closed. For each ξ0 , ξ1 , ξ2 ∈ g, dμ (X ξ0 , X ξ1 , X ξ2 ) = X ξ0 (μ (X ξ1 , X ξ2 )) − X ξ1 (μ (X ξ0 , X ξ2 )) + X ξ2 (μ (X ξ0 , X ξ1 )) − μ ([X ξ0 , X ξ1 ], X ξ2 ) + μ ([X ξ0 , X ξ2 ], X ξ1 ) − μ ([X ξ1 , X ξ2 ], X ξ0 ). For each λ ∈ Oμ , − μ ([X ξ0 , X ξ1 ], X ξ2 )(λ) + μ ([X ξ0 , X ξ2 ], X ξ1 )(λ) − μ ([X ξ1 , X ξ2 ], X ξ0 )(λ) = λ | [[ξ0 , ξ1 ], ξ2 ] − λ | [[ξ0 , ξ2 ], ξ1 ] + λ | [[ξ1 , ξ2 ], ξ0 ]

= λ | [[ξ0 , ξ1 ], ξ2 ] + [[ξ1 , ξ2 ], ξ0 ] + [[ξ2 , ξ0 ], ξ1 ] = 0. On the other hand, d ∗ (μ (X ξ j , X ξk ))(Adexp tξi λ)|t=0 dt d ∗ = − Adexp tξi λ | [ξ j , ξk ] |t=0 dt d = − λ | Adexp −tξi ([ξ j , ξk ]) |t=0 dt = λ | [ξi [ξ j , ξk ]] .

X ξi (μ (X ξ j , X ξk ))(λ) =

Hence, dμ (X ξ0 , X ξ1 , X ξ2 )(λ) = λ | [ξ0 , [ξ1 , ξ2 ]]−[ξ1 , [ξ0 , ξ2 ]]+[ξ2 , [ξ0 , ξ1 ]] = 0. Therefore, dμ = 0. Let I : Oμ → g∗ be the inclusion map. Recall that I is an immersion but need not be an embedding. For each ξ ∈ g, the evaluation Iξ = I | ξ

is a smooth function on Oμ . Proposition 6.1.2 For each ξ ∈ g, X ξ μ = −d Iξ .

108

Proof

Symplectic reduction For each ξ ∈ g and λ ∈ Oμ , Iξ (λ) = λ | ξ . Hence, for every ζ ∈ g,

d Iξ | X ζ (λ) = X ζ (Iξ )(λ) =

d d ∗ ∗ I (Adexp Adexp tζ λ) | ξ |t=0 = tζ λ | ξ |t=0 dt dt

d λ | Adexp(−tζ ) ξ |t=0 = − λ | [ζ, ξ ] = μ (X ζ , X ζ )(λ) dt = −μ (X ξ , X ζ )(λ) = − (X ξ μ ) | X ζ (λ). =

Since ζ ∈ g and λ ∈ Oμ are arbitrary, this implies that X ξ μ = −d Iξ . Corollary 6.1.3 If G is connected, then the form μ on Oμ is invariant under the co-adjoint action of G on Oμ . Proof Since G is connected, it suffices to show that £ X ξ μ = 0 for each ξ ∈ g. But £ X ξ μ = X ξ dμ + d(X ξ μ ) = 0, because μ is closed and (X ξ μ ) = −d Iξ is exact.

6.1.2 Symplectic manifolds Symplectic manifolds are generalizations of co-adjoint orbits. Let P be a manifold. A symplectic form on P is a closed, non-degenerate 2-form on P. Non-degeneracy of ω implies that for every f ∈ C ∞ (P), there exists a unique vector field X f such that X f ω = −d f.

(6.4)

The vector field X f is called the Hamiltonian vector field of f .1 For each f ∈ C ∞ (P), the one-parameter local group exp t X f of local diffeomorphisms of P preserves the symplectic form ω, because £ X f ω = X f dω + d(X f ω) = 0. Diffeomorphisms of P that preserve the symplectic form ω are called symplectomorphisms of (P, ω). Proposition 6.1.4 For every f1 and f 2 in C ∞ (P), [X f1 , X f2 ] ω = −d X f1 ( f 2 ) = −dω(X f1 , X f2 ). Proof [X f1 , X f2 ] ω = (£ X f1 X f2 ) ω = £ X f1 (X f2 ω) − X f2 £ X f1 ω = £ X f1 (−d f 2 ) = −d(X f1 ( f 2 )). 1 We follow the notation and sign convention of Sniatycki ´ (1980) here.

6.1 Symplectic manifolds with symmetry

109

Let G be a connected Lie group, and let : G × P → P : (g, p) → g ( p) = gp be a Hamiltonian action of G on P. This means that the action is symplectic. That is, ∗g ω = ω for every g ∈ G. Moreover, there exists an Ad ∗ -equivariant momentum map J : P → g∗ with the following property: for each ξ ∈ g, the action on P of the one-parameter subgroup exp tξ of G is given by translations along the integral curves of X Jξ , where Jξ = J | ξ . The function Jξ is called the momentum corresponding to ξ . For every f ∈ C ∞ (P), d ∗ f |t=0 = X Jξ ( f ). (6.5) dt exp tξ Proposition 6.1.5 The map g → X(P) : ξ → X Jξ is an antihomomorphism of Lie algebras. In other words, [X Jξ , X Jζ ] = −X J[ξ,ζ ] for every ξ, ζ ∈ g. Proof By assumption, the momentum map J : P → g∗ is Ad ∗ -equivariant; that is, ∗ ∗exp tξ J = Adexp tξ ◦ J.

Evaluating this equation on ζ ∈ g, we obtain ∗ ∗exp tξ Jζ = Adexp tξ ◦ J | ζ = J | Adexp −tξ ζ .

Taking equations (6.1) and (6.5) into account, we obtain X Jξ (Jζ ) = J | [ζ, ξ ] = J[ζ,ξ ] .

(6.6)

By Proposition 6.1.4, [X Jξ , X Jζ ] = X X Jξ (Jζ ) = X J[ζ,ξ ] = −X J[ξ,ζ ] , which completes the proof. Comparing the definitions presented above with Section 6.1.1, we see that the Kirillov–Kostant–Souriau form μ on a co-adjoint orbit Oμ is symplectic. Moreover, for each ξ ∈ g, the vector field X ξ given by equation (6.2) is the Hamiltonian vector field of Iξ , so that the inclusion I : O → g∗ is the momentum map of the co-adjoint action of G on O.

110

Symplectic reduction

6.1.3 Poisson algebra The assignment f → X f gives a linear map of the space C ∞ (P) of smooth functions on P into the space X(P) of smooth vector fields on P. If P is connected, the kernel of this map consists of constant functions on P. The symplectic form ω on P induces a bracket on C ∞ (P), called the Poisson bracket, such that for each f 1 , f 2 ∈ C ∞ (P), { f 1 , f2 } = −X f1 f 2 = X f2 f 1 = −ω(X f1 , X f2 ).

(6.7)

The Poisson bracket (6.7) is bilinear and antisymmetric, acts as a derivation { f 1 , f 2 f 3 } = f 2 { f 1 , f 3 } + f 3 { f 1 , f 2 },

(6.8)

and satisfies the Jacobi identity {{ f 1 , f 2 }, f 3 } + {{ f 2 , f 3 }, f 1 } + {{ f 3 , f 1 }, f 2 } = 0.

(6.9)

The associative algebra C ∞ (P) endowed with the Poisson bracket (6.7) is called the Poisson algebra of (P, ω). The map C ∞ (P) → X(P) : f → X f is an antihomomorphism of the Lie algebra structure of C ∞ (P) to the Lie algebra of vector fields on P. In other words, X { f 1 , f2 } = −[X f1 , X f2 ]

(6.10)

for all f 1 , f 2 ∈ C ∞ (P). Proposition 6.1.6 The map g → C ∞ (P) : ξ → Jξ is a homomorphism of Lie algebras. Proof For each ξ, ζ ∈ g, equations (6.6) and (6.7) give J[ζ,ξ ] = X Jξ (Jζ ) = {Jζ , Jξ }. The action of G on P gives rise to the action G × C ∞ (P) → C ∞ (P) : (g, f ) → ∗g f . For each ξ ∈ g, the infinitesimal action of exp tξ on C ∞ (P) is given by d f ◦ exp tξ |t=0 = X Jξ ( f ) = { f, Jξ }. f → dt Since the action of G on P is symplectic, it follows that its action on C ∞ (P) is Poisson. That is, it preserves the Poisson bracket. For each g ∈ G and f 1 , f 2 ∈ C ∞ (P), ∗g { f 1 , f 2 } = {∗g f 1 , ∗g f 2 }.

(6.11)

Therefore, the space C ∞ (P)G of G-invariant smooth functions on P is a Poisson subalgebra of P.

6.2 Poisson reduction

111

6.2 Poisson reduction We assume here that the action of G on P is proper. As before, we denote the space of G-orbits on P by R = P/G and the orbit map by ρ : P → R. The differential structure of R is C ∞ (R) = { f : R → R | ρ ∗ f ∈ C ∞ (P)}. In Section 4.4, we showed that the projection of the orbit type stratification M of P to the orbit space is a stratification N of R, and that the strata of N coincide with orbits of the family X(R) of all vector fields on R. Here, we begin a discussion of the additional structure of R induced by the symplectic structure on P and the existence of the Ad G∗ -equivariant momentum map J : P → g∗ . The pull-back map ρ ∗ : C ∞ (R) → C ∞ (P)G is an isomorphism of associative algebras. Since C ∞ (P)G is a Poisson algebra, we can use ρ ∗ to pull back the Poisson bracket from C ∞ (P)G to C ∞ (R). For each f 1 , f 2 ∈ C ∞ (R), we define { f 1 , f 2 } ∈ C ∞ (R) by ρ ∗ { f 1 , f 2 } = {ρ ∗ f 1 , ρ ∗ f 2 }.

(6.12)

With this definition, C ∞ (R) is a Poisson algebra isomorphic to C ∞ (P)G . Given f ∈ C ∞ (R), let X f ∈ Der C ∞ (R) be defined by X f (h) = {h, f }

(6.13)

for each h ∈ C ∞ (R). We refer to X f as the Poisson derivation or Poisson vector field of f . In Proposition 6.2.2 below, we show that this terminology is consistent with the definition of vector fields on subcartesian spaces in Definition 3.2.2. We denote the family of all Poisson vector fields on R by P(R) = {X f | f ∈ C ∞ (R)}.

(6.14)

Since P(R) ⊆ X(R), it follows that for each stratum N ⊆ R and every x ∈ N , the value at x of the Poisson bracket { f 1 , f 2 } of functions in C ∞ (R) depends only on the restrictions f 1|N and f 2|N to N of f 1 and f 2 , respectively. Hence, the space R(N ) = { f |N | f ∈ C ∞ (R)} of the restrictions to N of smooth functions on R inherits the structure of a Poisson algebra from C ∞ (R). The Poisson bracket on R(N ) is given by { f 1|N , f2|N } = { f 1 , f2 }|N for every f 1 , f 2 ∈ C ∞ (R). By the definition of a stratification, the strata N ∈ N are locally closed connected submanifolds N of R. Proposition 2.1.8 ensures

112

Symplectic reduction

that every f ∈ C ∞ (N ) coincides locally with a function in R(N ). Hence, the Poisson algebra structure of R(N ) extends to C ∞ (N ). Thus, we have proved the following result. Proposition 6.2.1 Each stratum N of the orbit type stratification N of R is a Poisson manifold. In the next proposition, we prove that the Poisson vector fields on R defined by equation (6.13) are vector fields in the sense of Definition 3.2.2, and discuss their relation to Hamiltonian vector fields of G-invariant functions on P. Proposition 6.2.2 For each f ∈ C ∞ (R), the Poisson derivation X f is the push-forward of the Hamiltonian vector field X ρ ∗ f on P by the orbit map ρ : P → R. Moreover, X f is a vector field on R. That is, exp t X f is a local one-parameter local group of diffeomorphisms of R. Proof For f ∈ C ∞ (R), its pull-back ρ ∗ f by ρ is in C ∞ (P)G , and X ρ ∗ f is a G-invariant Hamiltonian vector field on P. Moreover, for each h ∈ C ∞ (R), equation (6.7) implies that X ρ ∗ f ρ ∗ h = {ρ ∗ h, ρ ∗ f } = ρ ∗ {h, f } = ρ ∗ (X f (h)). Thus, X f is the push-forward of X ρ ∗ f by the orbit map ρ. In other words, Tρ ◦ X ρ ∗ f = X f ◦ ρ.

(6.15)

Since X ρ ∗ f generates a local one-parameter local group exp t X ρ ∗ f of diffeomorphisms of P, it follows that translations along integral curves of X f give rise to a local one-parameter local group exp t X f of diffeomorphisms of R such that ρ ◦ (exp t X ρ ∗ f ) = (exp t X f ) ◦ ρ.

(6.16)

Hence, X f is a vector field on R. By Theorem 3.4.5, the orbits of P(R) are smooth manifolds immersed in R. Let Q be the orbit of P(R) through x ∈ R. For each f ∈ C ∞ (R), the restriction X f |Q of the Poisson vector field of f to Q is a vector field on Q, and T Q = {X f (x) | x ∈ Q, f ∈ C ∞ (R)}. This implies that C ∞ (Q) inherits from C ∞ (R) a Poisson algebra structure such that { f 1|Q , f2|Q } = { f 1 , f 2 }|Q

(6.17)

for each f 1 , f 2 ∈ C ∞ (R). Proposition 6.2.3 Each orbit Q of the family P(R) of Poisson vector fields on R is a symplectic manifold with a unique symplectic form ω Q on Q such that ω Q (X f1 |Q , X f2 |Q ) = −{ f 1 , f 2 }|Q

(6.18)

6.2 Poisson reduction

113

for every f 1 , f 2 ∈ C ∞ (R). Moreover, for each p ∈ ρ −1 (Q), ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p).

(6.19)

Proof Let N denote the stratum of R containing Q. In order to see that ω Q is well defined by equation (6.18), take x ∈ Q and consider f 2 , f 3 ∈ C ∞ (R) such that X f 2 (x) = X f3 (x). For each f ∈ C ∞ (R), equation (6.13) gives { f 2 , f }(x) = X f2 ( f )(x) = (X f2 (x))( f ) = (X f3 (x))( f ) = X f3 ( f )(x) = { f 3 , f }. Hence, ω Q is well defined. Clearly, ω Q is bilinear and smooth. Hence, it is a 2-form on Q. Suppose that X f0 (x0 ) ω Q = 0 for f 0 ∈ C ∞ (R) and x0 ∈ Q. Then, X f0 (x0 )( f ) = 0 for each f ∈ C ∞ (R). Since Q is immersed in N and N is a submanifold of R, for each f Q ∈ C ∞ (Q) there exists a neighbourhood U of x in Q, open in the topology of Q, and a function f ∈ C ∞ (R) such that f |U = f Q|U . Therefore, X f0 (x0 )( f Q ) = 0 for every f Q ∈ C ∞ (Q). Therefore, X f0 (x0 ) = 0 ∈ Der C ∞ (Q), and this implies that ω Q is non-degenerate. We have dω Q (X f0 , X f1 , X f2 ) = X f0 (ω Q (X f1 , X f2 )) − X f1 (ω Q (X f0 , X f2 )) + X f2 (ω Q (X f0 , X f1 )) − ω Q ([X f0 , X f1 ], X f2 ) + ω Q ([X f0 , X f2 ], X f1 ) − ω Q ([X f1 , X f2 ], X f0 ). Taking equations (6.7) and (6.10) into account, we obtain dω Q (X f0 , X f1 , X f2 ) = X f0 { f 2 , f 1 } − X f1 { f 2 , f 0 } + X f2 { f 1 , f 0 } + ω Q (X { f0 , f 1 } , X f2 ) − ω Q (X { f0 , f 2 } , X f1 ) + ω Q (X { f 1 , f 2 } , X f0 ) = −{ f 0 , { f 2 , f 1 }} + { f 1 , { f 2 , f 0 }} − { f2 , { f 1 , f 0 }} − {{ f 0 , f 1 }, f 2 } + {{ f 0 , f 2 }, f 1 } − {{ f 1 , f 2 }, f 0 } = 0. Hence, dω Q = 0. Therefore, ω Q is symplectic. Moreover, the definition of ω Q and equations (6.12) and (6.7) imply that for each p ∈ ρ −1 (Q), ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = −{ f 1 , f 2 }|Q (ρ( p)) = −{ f 1 , f2 }(ρ( p)) = −ρ ∗ { f 1 , f 2 }( p) = −{ρ ∗ f 1 , ρ ∗ f 2 }( p) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p), which completes the proof.

114

Symplectic reduction

In Proposition 6.2.3, we showed that Q is a Poisson manifold contained in R. Moreover, the Poisson structure on Q is equivalent to the Poisson structure of C ∞ (Q) defined by the symplectic form ω Q . We shall refer to these facts by saying that Q is a symplectic submanifold of R.

6.3 Level sets of the momentum map We continue with the assumption that the action of G on P is proper. In Hamiltonian mechanics, we often perform the reduction procedure by investigating the structure of the quotient J −1 (μ)/G μ , where G μ = {g ∈ G | Adg∗ μ = μ} is the isotropy group of μ. Since J : P → g∗ is continuous, it follows that J −1 (μ) is a closed subset of P. The local compactness of P implies that J −1 (μ) is locally compact. Moreover, the action G μ on J −1 (μ) is proper because the action of G on P is proper. By the results of Section 4.4, J −1 (μ)/G μ is a locally compact differential space with the quotient space topology and the differential structure C ∞ (J −1 (μ)/G μ ) = { f ∈ C 0 (J −1 (μ)/G μ ) | ρμ∗ f ∈ C ∞ (J −1 (μ))}, where ρμ : J −1 (μ) → J −1 (μ)/G μ is the orbit map. Our aim in this section is to describe the geometric structure of J −1 (μ), the orbit space J −1 (μ)/G μ and the projection ρ(J −1 (μ)) to the orbit space R = P/G. An important tool in this task will be the family E(P) = {X f | f ∈ C ∞ (P)G } of Hamiltonian vector fields of G-invariant smooth functions on P. For each p ∈ P, we denote by E(P) p = span {X f ( p) ∈ T p P | f ∈ C ∞ (P)G }

(6.20)

the subspace of T p P defined by the generalized distribution span E(P) on P spanned by E(P). Proposition 6.3.1 For each p ∈ P, the orbit through p of the family E(P) of Hamiltonian vector fields of G-invariant functions on P is contained in the set PH = {x ∈ P | G x = H }, where H = G p is the isotropy group of p.

6.3 Level sets of the momentum map

115

Proof For f ∈ C ∞ (P)G , let exp t X f denote the local one-parameter group of local diffeomorphisms generated by the Hamiltonian vector field X f of f . The G-invariance of X implies that for each g ∈ G, g ◦ exp t X f = (exp t X f ) ◦ g . Let x = (exp t X f )( p), and let g ∈ G x . Then x = g x implies (exp t X f )( p) = (g ◦ (exp t X f ))( p). Hence, p = ((exp t X f )−1 ◦ g ◦ (exp t X f ))( p) = ((exp t X f )−1 ◦ (exp t X f ) ◦ g )( p) = g p, and g ∈ G p . Thus, G x ⊆ G p = H . In a similar way, we can show that H = G p ⊆ G x . Hence, G x = H , which ensures that the orbit of X f through p is contained in PH . But this is valid for every f ∈ C ∞ (P)G . Therefore, the orbit through p of the family {X f | f ∈ C ∞ (P)G } of all Hamiltonian vector fields of G-invariant functions on P is contained in PH . It is worth noticing that Proposition 6.3.1 does not require properness of the action of G on P. We denote by ker d J the generalized distribution on P given by ker d J = {u ∈ T P | u(J ) = 0}. Hence, for each p ∈ P, ker p d J = {u ∈ T p P | u(J ) = 0}. Theorem 6.3.2 (i) Assume that the action of G on P is proper. Then, for each p ∈ P, E(P) p = ker p d J ∩ T p PG p ,

(6.21)

and the orbit of E(P) through p is the connected component of J −1 (J ( p)) ∩ PG p that contains p.2 (ii) For each compact subgroup H of G, the connected components of PH are symplectic manifolds. (iii) In particular, if p ∈ PH , μ = J ( p) and L is the connected component of PH that contains p, then the connected component of J −1 (μ) ∩ L that contains p is a manifold and its tangent bundle is spanned by Hamiltonian vector fields of G-invariant functions. Proof

(i) The map ω# : T P → T ∗ P : u → ω# (u) = u ω

2 This theorem, due to Ortega and Ratiu, is the foundation of their theory of optimal reduction

(Ortega and Ratiu, 2002).

116

Symplectic reduction

is a vector bundle isomorphism that intertwines the projection maps of τ : T P → P and ϑ : T ∗ P → P. We denote the inverse of ω# by ω$ : T ∗ P → T P. With this notation, we can write, for every f ∈ C ∞ (P), X f = −ω$ ◦ d f. Hamiltonian vector fields X f preserve the symplectic form ω. Therefore, they also preserve the tensor fields ω# and ω$ . Given p ∈ P, for every u ∈ E(P) p there exists h ∈ C ∞ (P)G such that u = X h ( p). Hence, E(P) p = span {X f ( p) ∈ T p P | f ∈ C ∞ (P)G } = {X f ( p) ∈ T p P | f ∈ C ∞ (P)G } = {ω$ ◦ d f ( p) | f ∈ C ∞ (P)G } = ω$ ({d f ( p) | f ∈ C ∞ (P)G }).

(6.22)

Let Gp denote the orbit of G through p. We have T p (Gp) = {X Jξ ( p) | ξ ∈ g}. We denote by T p (Gp)0 the annihilator of T p (Gp) in T p∗ (G P); that is, T p (Gp)0 = {q ∈ T p∗ (G P) | q | X Jξ ( p) = 0 for all ξ ∈ g}. If f ∈ C ∞ (P)G , then d f ( p) | X Jξ ( p) = X Jξ ( f )( p) = 0 for all ξ ∈ g. Therefore, d f ( p) ∈ T p (Gp)0 whenever f ∈ C ∞ (P)G . Let H = G p be the isotropy group of p, and let h be the Lie algebra of H . Since f ∈ C ∞ (P)G and H ⊆ G, for each g ∈ H , we have d f = d∗g f . For every u ∈ T p P, we have T g u ∈ T p P and d f ( p) | u = (d∗g f )( p) | u = u(∗g f ) = T g (u)( f ) = d f ( p) | T g (u) = T Tg (d f ( p)) | u . Hence, d f ( p) is invariant under the linear action of H on T p∗ P induced by the action of H on P. Therefore, d f ( p) ∈ (T p (Gp)0 ) H ∀ f ∈ C ∞ (P)G .

(6.23)

The next step is to show that if q ∈ (T p (Gp)0 ) H , it follows that there exists f ∈ C ∞ (P)G such that d f ( p) = q. Let S p be a slice through p for the action of G on P. There exists a compactly supported function f S on S such that

6.3 Level sets of the momentum map

117

d f S ( p) = q |T p S p . As in the proof of Proposition 4.3.1, we can average f S over the action of H on S p , obtaining an H -invariant function ∗g f S dμ(g). f˜S = H

Since f S has compact support and H is compact, it follows that f˜S has compact support. For each g ∈ H and u ∈ T p S p , we have (d∗g f S )( p) | u = u(∗g f S ) = T g (u)( f S ) = d f S ( p) | T g (u)

= T Tg (d f S ( p)) | u = T Tg (q) | u = q | u

because q is assumed to be H -invariant. This implies that d f˜S ( p)|T p S p = q |T p S p . Following the proof of Proposition 4.3.1, we can extend f˜S to a G-invariant function f on P with support contained in ρ −1 (ρ(S)). By construction, d f ( p)|T p S p = d f˜S ( p)|T p S p = q |T p S p . Since f is G-invariant, it follows that d f ( p)|T p (Gp) = 0. On the other hand, q ∈ T p (Gp)0 is equivalent to q||T p (Gp) = 0. But T p P = T p S p + T p Gp, which implies that d f ( p) = q. Therefore, {d f ( p) | f ∈ C ∞ (P)G } = (T p (Gp)0 ) H .

(6.24)

Combining equations (6.22) and (6.24), we obtain E(P) p = ω$ ({d f ( p) | f ∈ C ∞ (P)G }) = ω$ ((T p (Gp)0 ) H ) = (ω$ (T p (Gp)0 ) H ) = {u ∈ T p P | ω(u, v) = 0 ∀ v ∈ T p (Gp)} H = {u ∈ T p P | ω(u, X Jξ ( p)) = 0 ∀ ξ ∈ g} H = {u ∈ T p P | u(Jξ )( p) = 0 ∀ ξ ∈ g} H = (ker p d J ) H = (ker p d J ) ∩ T p PH , as required. Since this result holds for every p ∈ P, we have proved the first statement of the theorem. (ii) In Proposition 4.2.6, we showed that each connected component of PH is a submanifold of P. It remains to show that, for each p ∈ PH , the form ω p on T p PH is non-degenerate. Note that T p PH coincides with the space (T p P) H of H -invariant vectors in T p P. Since the action of G on P is proper, there exists a G-invariant Riemannian metric k on P. Using k, we can define a G-invariant map j : T P → T P such that, for each p ∈ P and u, v ∈ T p P, k(u, v) = ω(u, j (v)).

118

Symplectic reduction

For a point p ∈ PH , the restriction of k to T p PH is a positive definite bilinear form on T p PH . Moreover, the group H acts on T p P, and the restriction j p to T p P is H -invariant. Hence, j p maps T p PH = (T p P) H to itself. Hence, for every non-zero u ∈ T p PH , we have j p (u) ∈ T p PH and ω(u, j p (u)) = k(u, u) > 0, which implies that the restriction of ω to T p PH is non-degenerate. Hence, the symplectic form ω on P pulls back to a symplectic form on each connected component of PH . (iii) For every f ∈ C ∞ (P)G , equation (6.21) implies that the orbit of E through p is contained in the connected components of the level set J −1 (μ), where μ = J ( p), and of PH , where H = G p . Let L be the connected component of PH that contains p. By Proposition 4.2.6, L is a submanifold of P. For each f ∈ C ∞ (P)G , the Hamiltonian vector field X f is tangent to L. Hence, it defines a vector X f |L on L, which we call the restriction of X f to L. Consider the family E(P)|L = {X f |L | f ∈ C ∞ (P)G }. The orbits of E(P)|L are the orbits of E(P) that are contained in L. On the other hand, (ker d J ) ∩ T L = ker d J|L , where J|L is the restriction of J : P → g∗ to L . Equation (6.21) gives span E(P)|L = ker d J|L , where span E(P)|L is the distribution spanned by E(P)|L . Hence, ker d J|L is a smooth distribution, and integral manifolds of d J|L are connected components of level sets of J|L . If μ = J ( p), then (J|L )−1 (μ) = J −1 (μ) ∩ L. Hence, the connected component of (J|L )−1 (μ) that contains p is the same as the connected component of J −1 (μ) ∩ L that contains p. Since L is the connected component of PH that contains p, it follows that the connected component of J −1 (μ) ∩ L that contains p is the same as the connected component of J −1 (μ) ∩ PH that contains p. This completes the proof. In Proposition 6.2.3, we showed that the orbits of the family P(R) of Poisson vector fields on R are symplectic manifolds. In the proposition below, we show that they are projections to R of intersections of connected components of level sets of J with submanifolds of P with a fixed isotropy group.

6.3 Level sets of the momentum map

119

Proposition 6.3.3 Assume that the action of a connected Lie group G on a symplectic manifold (P, ω) is Hamiltonian and proper. Given p0 ∈ P, let μ = J ( p0 ) and let H = G p0 be the isotropy group of p0 . The connected component K of J −1 (μ) ∩ PH is a submanifold of P, and the projection Q = ρ(K ) is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K , where ρ K : K → Q is the restriction of the orbit map ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Proof Let L be the connected component of PH that contains p0 . According to Theorem 6.3.2, the connected component K of J −1 (μ) ∩ L that contains p0 is the orbit through p0 of the family E(P) of Hamiltonian vector fields of G-invariant functions on P. Hence, the space G C ∞ (P)|K = {h |K | h ∈ C ∞ (P)G }

of restrictions to K of G-invariant functions in C ∞ (P) is a Poisson algebra with a Poisson bracket {h 1|K , h 2|K } = (X h 2 (h 1 ))|K = {h 1 , h 2 }|K . In Proposition 6.2.2, we established that the Poisson vector field X f of f ∈ C ∞ (R) is the push-forward of the Hamiltonian vector field X ρ ∗ f of the pullback ρ ∗ f of f by the orbit map ρ : P → R, and that ρ ◦ (exp t X ρ ∗ f ) = (exp t X f ) ◦ ρ for every f ∈ C ∞ (R); see equation (6.16). Every G-invariant function h ∈ C ∞ (P) pushes forward to a function f = ρ∗ h ∈ C ∞ (R) such that ρ ∗ f = h. Therefore, the projection to R of the orbit K of the family E(P) of Hamiltonian vector fields of G-invariant functions on P is an orbit Q = ρ(K ) of the family P(R) of Poisson vector fields on R. Since Q is an orbit of P(R), it follows that Q is a Poisson manifold with a Poisson bracket such that { f1|Q , f 2|Q } = { f 1 , f 2 }|Q for every f 1 , f 2 ∈ C ∞ (R). Thus, the restriction f → f |Q is a Poisson algebra homomorphism from C ∞ (R) to C ∞ (Q). In Proposition 6.2.3, we established that Q is a symplectic manifold with a symplectic ω Q such that ω Q (X f1 |Q , X f2 |Q ) = −{ f 1 , f 2 }|Q

120

Symplectic reduction

for every f 1 , f 2 ∈ C ∞ (R). Let ρ K : K → Q be the restriction of ρ : P → R to the domain K and codomain Q, and let ω K = ρ K∗ ω Q be the pull-back of ω Q by ρ K : K → Q. For each p ∈ K , we have T p (K ) = {X ρ ∗ f ( p) | f ∈ C ∞ (R)}, and ω K is uniquely determined by ω K (X ρ ∗ f1 , X ρ ∗ f2 )( p) for all f 1 , f 2 ∈ f ∈ C ∞ (R) and p ∈ K . But ω K (X ρ ∗ f1 , X ρ ∗ f2 )( p) = ω K (X ρ ∗ f1 ( p), X ρ ∗ f2 ( p)) = ω Q (Tρ K (X ρ ∗ f1 ( p)), Tρ K (X ρ ∗ f2 ( p))) = ω Q (X f1 (ρ( p)), X f2 (ρ( p))) by equation (6.15) = ω Q (X f1 |Q , X f2 |Q )(ρ( p)) = ω(X ρ ∗ f1 , X ρ ∗ f2 )( p) by equation (6.19) = ω|K (X ρ ∗ f1 , X ρ ∗ f2 )( p). Since the chain of equalities above is valid for every p ∈ K and all f 1 , f 2 ∈ C ∞ (R), it follows that ρ K∗ ω Q = ω K , which completes the proof. In Chapter 4, we described the orbit type stratification M of P given by a proper action of a connected Lie group G on P, and the corresponding orbit type stratification N of the orbit space R = P/G. Our next task is to describe the intersections of the level sets of J with the strata of M and their projections to R. Recall that the strata of M are connected components of P(H ) = { p ∈ P | G p is conjugate to H }. We are now in a position to give a description of the structure of the orbit space induced by the momentum map. For each μ ∈ g∗ , the inverse image J −1 (μ) is a differential subspace of P with a differential structure generated by inclusions into J −1 (μ) of smooth functions on P. Similarly, ρ(J −1 (μ)) is a differential subspace of R with a differential structure generated by the ring R(ρ(J −1 (μ))) = {h |ρ(J −1 (μ)) | h ∈ C ∞ (R)}. Theorem 6.3.4 We assume that the action of G on P is proper, and denote orbit type stratifications of P and R = P/G by M and N, respectively. (i) For each μ ∈ g∗ , the family of sets Mμ = {connected components of J −1 (μ) ∩ M | M ∈ M}

(6.25)

is a stratification of the level set J −1 (μ). The inclusion map J −1 (μ) → P is a morphism of stratified spaces.

6.3 Level sets of the momentum map

121

(ii) The connected components of the sets ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ)) ∩ N , where N = ρ(M), are symplectic orbits of the family P(R) of Poisson vector fields on R. (iii) The family of sets Nμ = {connected components of ρ(J −1 (μ)) ∩ N | N ∈ N} is a stratification of ρ(J −1 (μ)) with symplectic strata. The restriction ρ|J −1 (μ) of ρ to J −1 (μ) is a morphism of stratified spaces. Proof

(i) For each p ∈ P, rank d J p = dim T p P − dim T p Gp = dim P − dim G/G p = dim P − dim G + dim G p .

Let M be a stratum in M. If p1 , p2 ∈ M, then G p1 and G p2 are conjugate. Hence, dim G p is constant on M. Therefore, the restriction J|M : M → g∗ of the momentum map J to M has constant rank. By the Rank Theorem, −1 (μ) are submanifolds for every μ ∈ g∗ , the connected components of J|M

−1 (μ) = J −1 (μ) ∩ M. Therefore, the connected compoof M. Moreover, J|M nents of J −1 (μ) ∩ M are submanifolds of M. Moreover, different connected components of J −1 (μ) ∩ M have empty intersections, and the collection of all connected components is a covering of J −1 (μ) ∩ M. Hence, the family Mμ gives a partition of J −1 (μ) by smooth manifolds. The family Mμ is locally finite, because M is locally finite. Moreover, for each M ∈ M, the connected components of J −1 (μ) ∩ M are locally closed, because J −1 (μ) ∩ M is closed in M and M is locally closed. Finally, the frontier condition for M implies the frontier condition for Mμ . Hence, Mμ is a stratification of J −1 (μ). Moreover, each stratum of Mμ is a connected component of J −1 (μ) ∩ M for some M ∈ M, and J −1 (μ) ∩ M ⊆ M. Therefore, the inclusion map J −1 (μ) → P is a morphism of stratified spaces. (ii) Given p ∈ J −1 (μ) ∩ PH , let L be the connected component of PH that contains p and let M be the connected component of P(H ) that contains p0 . Both M and L are submanifolds of P. Moreover, we have shown in the proof of Theorem 4.3.5 that G L = M, so that ρ(M) = ρ(L). For each g ∈ G, gL is the connected component of Pg H g−1 that contains gp. Then ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ) ∩ G L) = ρ(J −1 (μ) ∩ gL). g∈G

By construction, p ∈ J −1 (μ) ∩ PH . Since J : P → g ∗ is AdG∗ -equivariant, it follows that J (gp) = Adg∗ (J ( p)) = Adg∗ μ. Hence, J −1 (μ) ∩ gL = ∅ only for g ∈ G μ = {g ∈ G | Adg∗ μ = μ}. If g ∈ G μ , then the action g : P → P

122

Symplectic reduction

restricts to a diffeomorphism of J −1 (μ) ∩ L onto J −1 (μ) ∩ gL. Moreover, ρ(J −1 (μ) ∩ L) = ρ(J −1 (μ) ∩ gL) for every g ∈ G μ . Therefore, ρ(J −1 (μ) ∩ M) = ρ(J −1 (μ) ∩ gL) = ρ(J −1 (μ) ∩ L). g∈G μ

In Proposition 6.3.3, we established that connected components of ρ(J −1 (μ)∩ L) are symplectic orbits of the family P(R) of Poisson vector fields on R. (iii) We have Nμ = {connected components of ρ(J −1 (μ)) ∩ N ) | N ∈ N} = {connected components of ρ(J −1 (μ) ∩ M) | M ∈ M} = {connected components of ρ(J −1 (μ) ∩ L) | L = PH M, M ∈ M} = {orbits of P(R) that are contained in ρ(J −1 (μ)) ∩ N ) | N ∈ N}. Since the family Mμ is a stratification of J −1 (μ), it follows that the family Nμ is a partition of ρ(J −1 (μ)). The proof that this partition is a stratification of ρ(J −1 (μ)) is analogous to the proof that Mμ is a stratification of J −1 (μ). We have shown in part (ii) that the strata of Nμ are symplectic manifolds. Since ρ|J −1 (μ) maps connected components of J −1 (μ)∩ M to connected components of ρ(J −1 (μ)) ∩ ρ(M), and N = ρ(M) ∈ N, it follows that ρ|J −1 (μ) : J −1 (μ) → ρ(J −1 (μ)) is a morphism of stratified spaces. We now proceed to investigate the structure of the orbit space J −1 (μ)/G μ with the quotient differential structure C ∞ (J −1 (μ)/G μ ) = { f ∈ C 0 (J −1 (μ)/G μ ) | ρμ∗ f ∈ C ∞ (J −1 (μ))}, where ρμ : J −1 (μ) → J −1 (μ)/G μ is the orbit map. Theorem 6.3.5 There exists a diffeomorphism l : J −1 (μ)/G μ → ρ(J −1 (μ)) such that the following diagram commutes: J −1 (μ) ρu

i

/P ρ

J −1 (μ)/G μ ? R ?? ?? ?? ?? l ??? j ?? −1 ρ(J (μ)),

(6.26)

6.3 Level sets of the momentum map

123

where i : J −1 (μ) −→ P and j : ρ(J −1 (μ)) −→ R denote the inclusion maps. The stratification of ρ(J −1 (μ)) gives rise to a stratification of J −1 (μ)/G μ such that the orbit map ρμ: : J −1 (μ) → J −1 (μ)/G μ is a morphism of stratified spaces. Proof Since ρ(J −1 (μ)) is a subset of R, for each p ∈ J −1 (μ), the orbit G μ p ⊆ J −1 (μ) through p is mapped by the inclusion map i : J −1 (μ) → P into the orbit Gp in P. Hence, ρ(G μ p) in R depends only on ρμ ( p) ∈ J −1 (μ)/G μ . Moreover, ρ(G μ p) is in ρ(J −1 (μ)). Hence, there exists a welldefined map l : J −1 (μ)/G μ → ρ(J −1 (μ)) such that the diagram commutes. We first show that l is invertible. Clearly, l is one-to-one. To show that l maps J −1 (μ)/G μ onto ρ(J −1 (μ)), consider x ∈ ρ(J −1 (μ)). There exists p ∈ J −1 (μ) such that x = ρ( p). Its projection ρμ ( p) is in J −1 (μ)/G μ , and l(ρμ ( p)) = x. Hence, l is a bijection. We want to show that l is a diffeomorphism. Consider first the restriction h |ρ(J −1 (μ)) of h ∈ C ∞ (R) to ρ(J −1 (μ)). For p ∈ J −1 (μ) and x = ρ( p), l ∗ h |ρ(J −1 (μ)) (ρμ ( p)) = h |ρ(J −1 (μ)) (l(ρμ ( p))) = h(x) = h(ρ( p)) = (ρ ∗ h)( p) = (ρ ∗ h)|J −1 (μ) ( p). Since ρ ∗ h ∈ C ∞ (P) is G-invariant, it follows that (ρ ∗ f h)|J −1 (μ) is G μ invariant and that it pushes forward to a smooth function on J −1 (μ)/G μ . Hence, l ∗ h |ρ(J −1 (μ)) ∈ C ∞ (J −1 (μ)/G μ ). This holds for the restriction to ρ(J −1 (μ)) of every function h ∈ C ∞ (R). Proposition 2.1.8 implies that l ∗ maps C ∞ (ρ(J −1 (μ))) to C ∞ (J −1 (μ)/G μ ). Hence, l : J −1 (μ)/G μ → ρ(J −1 (μ)) is smooth. In order to show that l −1 is smooth, it suffices to show that l ∗ maps C ∞ (ρ(J −1 (μ))) onto C ∞ (J −1 (μ)/G μ ). Consider f ∈ C ∞ (J −1 (μ)/G μ ). It follows that ρμ∗ f is a G μ -invariant function in J −1 (μ). The differential structure of J −1 (μ) is generated by its inclusion into P. Hence, for each p ∈ J −1 (μ), there exist a neighbourhood U of p in P and k ∈ C ∞ (P) such that k|U ∩J −1 (μ) = (ρμ∗ f )|U ∩J −1 (μ) . Let S p be a slice for the action of G on P contained in U . The slice S p is invariant under the action on P of the isotropy group H of p. Hence, H ∩G μ acts on S p ∩ J −1 (μ), and the restriction (ρμ∗ f )|S p ∩J −1 (μ) is invariant under this action. Let W1 and W2 be H -invariant neighbourhoods of p in S p such that W 1 ⊆ W2 . There exists a non-negative function h ∈ C ∞ (S p ) such that h |W 1 = 1 and the support of h is contained in W2 . Since H is compact, we may average hk|S p over H , obtaining an H -invariant function

124

Symplectic reduction

k˜ S p =

H

∗g hk|S p dμ(g),

where dμ(g) is the Haar measure on H normalized so that vol H = 1. Since H is compact, W2 is H -invariant and the support of h is contained in W2 , it follows that the support of k˜ is compact and contained in W2 . Now, G S p is a G-invariant neighbourhood of p in P. Since k˜ S p is H -invariant, it extends to a unique smooth G-invariant function kG S p on G S p . Let k P be a function on P such that

k G S p ( p ) if p ∈ G S p . kP(p ) = 0 if p ∈ / GW2 Since k G S p ( p ) = 0 for all p ∈ G S p \GW2 , it follows that k P is well defined. Also, k P is smooth and G-invariant. Moreover, the restriction of k P to J −1 (μ) ∩ W1 coincides with the restriction of ρμ∗ f to W1 ∩ J −1 (μ). Since k P is in C ∞ (P)G , it pushes forward to ρ∗ k P ∈ R. For each p ∈ −1 J (μ) ∩ W1 , we have ρ( p ) ∈ ρ(J −1 (μ)) and f (ρμ ( p )) = ρμ∗ f ( p ) = k P ( p ) = (ρ∗ k P )(ρ( p )) = ρ∗ k P (l(ρμ ( p ))). Hence, f |ρ(J −1 (μ))∩ρ(GW1 ) = l ∗ (ρ∗ k P )|ρ(J −1 (μ))∩ρ(GW1 ) , where (ρ∗ k P )|ρ(J −1 (μ)) is in C ∞ (ρ(J −1 (μ))). A similar result holds for a neighbourhood of every p ∈ J −1 (μ). Using a partition-of-unity argument, we conclude that f is in the range of l ∗ . This implies that l ∗ maps C ∞ (ρ(J −1 (μ))) onto C ∞ (J −1 (μ)/G μ ). Hence, l −1 is smooth. Thus, l is a diffeomorphism. By Theorem 6.3.4, the orbit type stratification N of R induces a stratification {ρ(J −1 (μ)) ∩ N | N ∈ N} of ρ(J −1 (μ)). The diffeomorphism l : J −1 (μ)/G μ → ρ(J −1 (μ)) defines a family {l −1 (ρ(J −1 (μ)) ∩ N ) | N ∈ N} of manifolds contained in J −1 (μ)/G μ , which stratifies J −1 (μ)/G μ . For each M ∈ M, it follows from the commutativity of the diagram (6.26) that ρμ (J −1 (μ)∩ M) = ρ(J −1 (μ)∩ M) = ρ(J −1 (μ))∩ρ(M) = ρ(J −1 (μ))∩ N , where N = ρ(M) ∈ N. Hence, ρμ (J −1 (μ) ∩ M) = l −1 (ρ(J −1 (μ)) ∩ N ), which implies that ρμ is a morphism of stratified spaces. We know that the orbits of the family P(R) of Poisson vector fields on R are symplectic manifolds and that they are strata of the stratification of ρ(J −1 (μ)). Since l : J −1 (μ)/G μ → ρ(J −1 (μ)) is an isomorphism of stratified spaces, for each stratum Q = ρ(J −1 (μ)) ∩ N of ρ(J −1 (μ)), the pull-back l ∗ ω Q of

6.4 Pre-images of co-adjoint orbits

125

the symplectic form ω Q on Q, defined by equation (6.18), is a symplectic form on the stratum ρμ (J −1 (μ) ∩ M) = l −1 (ρ(J −1 (μ)) ∩ N ) of J −1 (μ)/G μ .

6.4 Pre-images of co-adjoint orbits We continue with the assumption that the action of G on P is proper. Let Oμ = { Adg∗ μ | g ∈ G} be the co-adjoint orbit through μ. Since ρ −1 (ρ(J (μ)) is the G-orbit of J −1 (μ), it follows that

ρ −1 (ρ(J −1 (μ))) = G J −1 (μ) = g∈G g J −1 (μ) = g∈G J −1 (Adg∗ μ) = J −1 (Oμ ). Hence, ρ(J −1 (μ)) = J −1 (Oμ )/G. In the following discussion, we identify these two spaces. An analysis of the structure of J −1 (Oμ ) requires caution because, in general, a co-adjoint orbit need not be locally closed in g∗ ; see ?. For example, if the action of G on P is free and proper, but Oμ is not locally closed in g∗ , then J −1 (Oμ ) is only immersed in P. Thus, in discussing the differential structure of J −1 (Oμ ) induced by its inclusion in P, we have to allow for the possibility that the topology of J −1 (Oμ ) may be finer than the subspace topology. Consider the diagram J −1 (Oμ )

i

ρ

ρ Oμ

J −1 (Oμ /G)

/P

j

(6.27)

/ R,

where i : J −1 (Oμ ) → P and j : J −1 (Oμ )/G → R are inclusion maps and ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is the orbit map. The differential structure of J −1 (Oμ ) is defined by its inclusion in P. That is, h : J −1 (Oμ ) → R is in C ∞ (J −1 (Oμ )) if, for each p ∈ J −1 (Oμ ), there exist a neighbourhood V of p in J −1 (Oμ ) and a function h˜ ∈ C ∞ (P) such that h˜ |V = h |V . If the topology of J −1 (Oμ ) is finer than the subspace topology, the neighbourhood

126

Symplectic reduction

V of p in J −1 (Oμ ) need not be equal to the intersection of J −1 (Oμ ) with a neighbourhood of p in P. Nevertheless, the inclusion map i : J −1 (Oμ ) → P is smooth. The orbit space J −1 (Oμ )/G has two differential structures because it is the quotient space of J −1 (Oμ ) and a subset of R. In general, these two differential structures need not coincide. Here, we consider J −1 (Oμ )/G with the differential structure induced by the inclusion map j : J −1 (Oμ )/G = ρ(J −1 (μ)/G μ ) → R. Proposition 6.4.1 The orbit map ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is smooth in the differential structure C ∞ (J −1 (Oμ )/G) generated by the ring R(J −1 (Oμ )/G) = { f |J −1 (Oμ )/G | f ∈ C ∞ (R)}. Proof As before, it suffices to show that for each f ∈ C ∞ (R), the pull∗ f −1 (O )/G is in back ρ O −1 (O )/G of the restriction f |J −1 (O )/G of f to J μ μ μ μ |J C ∞ (J −1 (Oμ )). But

∗ ρO f −1 (Oμ )/G = (ρ ∗ f )|J −1 (Oμ ) ∈ C ∞ (J −1 (Oμ )). μ |J

Hence, ρ Oμ : J −1 (Oμ ) → J −1 (Oμ )/G is smooth. The analogue of Theorem 6.3.5 is trivial, because μ ∈ Oμ implies that J −1 (Oμ )/G = ρ(J −1 (Oμ )) = ρ(J −1 (μ)). For the same reason, the analogues of part (ii) of Theorem 6.3.4 and of the first statement of part (iii) of Theorem 6.3.2 are true. At present, we have no proof that J −1 (Oμ ) is a stratified space, but we have no counterexample either. Hence, we cannot make a statement analogous to part (i) of Theorem 6.3.2.

6.5 Reduction by stages for proper actions In many applications, the symmetry group G of (P, ω) has a normal subgroup H , and it is convenient to perform reduction by stages. First, one passes to the space of H -orbits in P, obtaining a stratified Poisson differential space P/H . The quotient group G/H acts on P/H by isomorphisms of its structure. In reducing the symmetries of P/H , we can no longer use the results obtained for manifolds, because P/H may have more than one stratum. Since P/G = (P/H )/(G/H ), we may expect reduction by stages to give results equivalent to the results of direct reduction when we pass from P to the stratified Poisson differential space P/G.

6.5 Reduction by stages for proper actions

127

We denote the space of G-orbits in P by R = P/G and the space of H orbits in Q by Q = P/H ; these have orbit maps ρ : P → R and η : P → Q, respectively. The action of the quotient group K = G/H on Q is K : K × Q → Q : ([g], η( p)) → [g]η( p) = η(gp),

(6.28)

where [g] = H g is the coset of g in K = G/H . If g = hg is another representative of [g], then η(g p) = η(hgp) = η(gp) because the orbit map η : P → Q = P/H is constant on H -orbits in P. Similarly, if η( p ) = η( p), then p = hp for some h ∈ H . Since H is a normal subgroup of G, for each g ∈ G and h ∈ H there exists h ∈ H such that gh = h g. Hence, Therefore,

K

η(gp ) = η(ghp) = η(h gp) = η(gp). is well defined.

Proposition 6.5.1 If the action of G on P is proper, it follows that the action K of K = G/H on P/H is proper. Proof Let (η( pn )) be a sequence of points in Q convergent to η( p), and let ([gn ]) be a sequence of elements of K such that the sequence ([gn ]η( pn )) = η(gn pn ) is convergent. Since P is a manifold, there exists a diffeomorphism of a neighbourhood U of p that maps p to the origin in Rd and takes U to an open ball Br of radius r > 1, centred at the origin in Rd . Using the Euclidean metric of Rd , we can construct a nested family Vm of neighbourhoods of p in P diffeomorphic to open balls in Rd of radius 1/m centred at the origin. Since η( pn ) → η( p) as n → ∞, for each m ∈ N there exists n m ∈ N such that the orbit H pn m intersects Um . Therefore, there exists h n m ∈ H such that h n m pn m ∈ Vm . This implies that the sequence h n m pn m converges to p. Hence, η(h n m pn m ) = η( pn m ) and h n m pn m → p as m → ∞. Similarly, the assumption that η(gn pn ) is convergent implies that there exists a sequence h n k such that h n k gn k pn k is convergent. Without loss of generality, we may assume that pn → p and gn pn → p as n → ∞. The properness of the action of G on P implies that there exists a convergent subsequence gnl such that lim (gnl pnl ) = lim gnl lim pnl = lim gnl p. l→∞

l→∞

l→∞

l→∞

Since the projection G → G/H is continuous, it follows that lim [gnl ] = lim gnl , l→∞

l→∞

128

Symplectic reduction

so that the sequence [gnl ] is convergent. The continuity of the orbit map η : P → P/H implies that lim η(gnl pnl ) = η lim (gnl pnl ) = η lim gnl p = lim gnl η( p).

l→∞

l→∞

l→∞

l→∞

Hence, the action of K = G/H on Q = P/H is proper. The assumption that the action of G on (P, ω) is Hamiltonian and proper implies that the orbit spaces R = P/G and Q = P/H are stratified Poisson subcartesian spaces. Let S = Q/K be the space of K -orbits in Q, and let σ : Q → S be the corresponding orbit map. Since the action of K on Q is proper and Q is a locally compact subcartesian space, it follows from the discussion in Section 4.4 that S is a Hausdorff, locally compact differential space with the quotient topology and the differential structure C ∞ (S) = { f S : C 0 (S) | σ ∗ f S ∈ C ∞ (Q)}. Proposition 6.5.2 The orbit space S is a Poisson differential space, and there is a unique diffeomorphism ϕ : R → S such that the diagram P

ρ

/R

η

Q

ϕ

σ

/S

commutes and ϕ ∗ : C ∞ (S) → C ∞ (R) is a Poisson algebra isomorphism. Proof The orbit map ρ : P → R is an epimorphism. Given r ∈ R, choose p ∈ ρ −1 (r ) ⊆ P, and set ϕ(r ) = σ (η( p)). If a point p ∈ P is such that ρ( p ) = ρ( p), then there exists g ∈ G satisfying p = gp. Equation (6.28) gives η( p ) = η(gp) = [g]η( p). The fibre of σ : Q → S over η( p) is the K -orbit K η( p) through η( p). Hence, σ (η( p )) = σ ([g]η( p)) = σ (η( p)), which implies that ϕ is well defined. Since the orbit maps η and σ are epimorphisms, it follows that ϕ maps R onto S. Moreover, if ϕ(r ) = ϕ(r ), then there exists p ∈ ρ −1 (r ) such that σ (η( p)) = σ (η( p )). This means that η( p) and η( p ) are in the same fibre of σ . In other words, there exists g ∈ G such that η( p ) = [g]η( p) = η(gp). This implies that there exists h ∈ H such that p = hgp. Therefore, p and p are in the same G-orbit, so that ρ( p ) = ρ( p). This proves that ϕ is one-to-one. To prove that ϕ is smooth, observe that if f S ∈ C ∞ (S), then σ ∗ f S ∈ ∞ C (Q) K ⊆ C ∞ (Q), which implies that η∗ (σ ∗ f S ) ∈ C ∞ (Q) H . Since σ ∗ f S

6.6 Shifting

129

is constant on orbits of K = G/H in Q = P/H , and η∗ (σ ∗ f S ) is constant on orbits on H in P, it follows that η∗ (σ ∗ f S ) is constant on G-orbits in P. Hence, η∗ (σ ∗ f S ) = ρ ∗ f R for a function f R ∈ C ∞ (R). But σ ◦ η = ϕ ◦ ρ implies that ρ ∗ f R = η∗ (σ ∗ f S ) = (σ ◦ η)∗ f S = (ϕ ◦ ρ)∗ f S = ρ ∗ (ϕ ∗ f S ). Since ρ : P → R is an epimorphism, it follows that ρ ∗ : C ∞ (R) → C ∞ (P) is a monomorphism. Therefore, ϕ ∗ f S = f R ∈ C ∞ (R). Hence, ϕ : R → S is smooth. In order to prove that ϕ −1 is smooth, it suffices to show that ϕ ∗ is an epimorphism. Consider f R ∈ C ∞ (R). Then ρ ∗ f R is a G-invariant function on P. Hence, ρ ∗ f R is H -invariant, which implies that ρ ∗ f R = η∗ f Q for f Q ∈ C ∞ (Q). Moreover, the G-invariance of η∗ f Q implies that f Q is K -invariant. Therefore, f Q = σ ∗ f S for a function f S ∈ C ∞ (S). As before, ρ ∗ f R = η∗ (σ ∗ f S ) = ρ ∗ (ϕ ∗ f S ), which implies that f R = ϕ ∗ f S . This proves that ϕ −1 is smooth. Hence, ϕ is a diffeomorphism. It remains to show that ϕ ∗ is a morphism of the Poisson algebra structures of ∞ C (S) and C ∞ (R). For f 1R and f 2R in C ∞ (R), their Poisson bracket satisfies the identity ρ ∗ { f 1R , f 2R } = {ρ ∗ f 1R , ρ ∗ f 2R }. Similarly, the Poisson bracket of f 1S and f 2S in C ∞ (S) satisfies the identities η∗ {σ ∗ f 1S , σ ∗ f 2S } = {η∗ (σ ∗ f 1S ), η∗ (σ ∗ ( f 2S ))} = {ρ ∗ (ϕ ∗ f 1S ), ρ ∗ (ϕ ∗ f 2S )} = ρ ∗ {ϕ ∗ f 1S , ϕ ∗ f 2S }. Hence, if f i R = ϕ ∗ f i S for i = 1, 2, then {ϕ ∗ f 1S , ϕ ∗ f 2S } = { f 1R , f 2R }, which implies that ϕ ∗ is a morphism of Poisson algebras. Corollary 6.5.3 For every p ∈ P, the diffeomorphism ϕ : R → S induces a symplectomorphism of the symplectic leaf of R that contains ρ( p) onto the symplectic leaf of S through σ (η( p)). This result can be paraphrased by saying that for a proper action of a Lie group on a symplectic manifold, the symplectic reduction of the symmetries by stages is equivalent to the symplectic reduction of all symmetries in one stage.

6.6 Shifting The case when μ = 0 is particularly simple because G μ = G and Oμ = {0}, so that the spaces J −1 (μ)/G μ and J −1 (Oμ )/G are identical. Therefore, if μ = 0, it is useful to be able to modify the system so that J −1 (μ) becomes the

130

Symplectic reduction

zero level of the momentum map of the modified system. For a free and proper action, the modification described here is usually called the ‘shifting trick’ of Guillemin and Sternberg. Consider the product P˜ = P × Oμ , with projections π1 : P˜ → P and π2 : P˜ → Oμ and a symplectic form ω˜ = π1∗ ω ⊕ (−π2∗ μ ). The action of G on P˜ is given by ˜ : G × P˜ → P˜ : (g, ( p, λ)) → ˜ g ( p, λ) = (g ( p), Adg∗ λ). This action has an Ad ∗ -equivariant momentum map J˜ = π1∗ J − π2∗ I . ˜ We denote the space of G-orbits in P˜ by R˜ = P/G and the corresponding ˜ ˜ orbit map by ρ˜ : P → R. The restriction of the domain of π1 : P˜ → P : ( p, λ) → p to J˜−1 (0) gives a map from J˜−1 (0) to P with range J −1 (Oμ ), because J˜( p, λ) = 0 implies J ( p) = λ ∈ Oμ . Thus, we obtain a map : J˜−1 (0) → J −1 (Oμ ) : ( p, J ( p)) → p.

(6.29)

Now consider a map P → P × g∗ : p → ( p, J ( p)). Restricting the domain to J −1 (Oμ ), we obtain a map from J −1 (Oμ ) to P × g∗ with range J˜−1 (0). Hence, we have a map % : J −1 (Oμ ) → J˜−1 (0) : p → ( p, J ( p)).

(6.30)

For each p ∈ J −1 (Oμ ), (%( p)) = ( p, J ( p)) = p. Similarly, for each ( p, J ( p)) ∈ J˜−1 (0), %(( p, J ( p)) = %( p) = ( p, J ( p)). Therefore, % = −1 . For each g ∈ G, (gp, Adg∗ J ( p)) = (gp, J (gp)) = gp and %(gp) = ˜ g %( p). Thus, and % are G-equivariant, (gp, J (gp)) = (gp, Adg∗ J ( p)) = and they induce maps of the corresponding orbit spaces

and

γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) : ρ( ˜ p, J ( p)) → ρ( p)

(6.31)

˜ J˜−1 (0)) : ρ( p) → ρ( ˜ p, J ( p)), δ : ρ(J −1 (Oμ )) → ρ(

(6.32)

which are called shifting maps. Proposition 6.6.1 If Oμ is an embedded submanifold of g∗ , then the shifting ˜ J˜−1 (0)) are maps γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) and δ : ρ(J −1 (Oμ )) → ρ( diffeomorphisms.

6.6 Shifting

131

Proof The map : J˜−1 (0) → J −1 (Oμ ) is obtained from π1 : P˜ = P × Oμ → P by restricting its domain to J˜−1 (0) and its codomain to J −1 (Oμ ). Since Oμ is a submanifold of g∗ , we take the subspace topology on J −1 (Oμ ). By Proposition 2.1.8, in order to prove the smoothness of , it suffices to show that for each f ∈ C ∞ (P), the map ∗ f |J −1 (Oμ ) = f |J −1 (Oμ ) ◦ : J˜−1 (0) → R is smooth. Since ( p, λ) ∈ J˜−1 (0) implies that J ( p) = λ ∈ Oμ , we have f |J −1 (Oμ ) ◦( p, λ) = f |J −1 (Oμ ) ◦π1 ( p, λ) = π1∗ f ( p, λ) = (π1∗ f )| J˜−1 (0) ( p, λ). Therefore, ∗ f |J −1 (Oμ ) = (π1∗ f )| J˜−1 (0) ∈ C ∞ ( J˜−1 (0)), which implies the smoothness of . We show smoothness of % : J −1 (Oμ ) → J˜−1 (0) in a similar manner. ˜ We need to show that %∗ f ˜−1 is smooth. For each Consider f˜ ∈ C ∞ ( P). | J (0) p ∈ J −1 (Oμ ), %∗ f˜| J˜−1 (0) ( p) = f˜| J˜−1 (0) ◦ %( p) = f˜| J˜−1 (0) ( p, J ( p)) = f˜( p, J ( p)) = f˜ ◦ (id P × J )( p) = (id P × J )∗ f˜( p) = ((id P × J )∗ f˜)|J −1 (O ) ( p). μ

Hence, %∗ f˜| J˜−1 (0) = ((id P × J )∗ f˜)|J −1 (Oμ ) ∈ C ∞ (J −1 (Oμ )). Since Oμ is an embedded submanifold of g∗ , it is locally compact. Hence, −1 J˜ (0) and J −1 (Oμ ) are locally compact differential spaces. Moreover, the actions of G on J˜−1 (0) and J −1 (Oμ ) are proper. By Proposition 4.4.6 and Theorem 4.4.7, the orbit spaces ρ( ˜ J˜−1 (0)) and ρ(J −1 (Oμ )) are locally compact differential spaces endowed with the quotient topologies. This implies that the shifting maps γ and δ are smooth. Since they are inverses of each other, it follows that they are diffeomorphisms. It follows from Theorem 6.3.4 that for each stratum N of the orbit type stratification of R, the intersection N ∩ ρ(J −1 (Oμ )) coincides with a symplectic leaf Q of N . Hence, for each x ∈ ρ(J −1 (Oμ )) ⊆ R and every pair f 1 , f 2 of functions in C ∞ (R), the Poisson bracket { f 1 , f 2 }(x) depends only on the restrictions of f 1 and f 2 to Q = N ∩ ρ(J −1 (Oμ )), where N is the stratum of R through x. Hence, { f 1 , f 2 }(x) depends on the restrictions of f 1 and f 2 to ρ(J −1 (Oμ )). This implies that ρ(J −1 (Oμ )) is a Poisson differential space; that is, C ∞ (ρ(J −1 (Oμ ))) has the structure of a Poisson algebra. Similarly, ρ( ˜ J˜−1 (0)) is a Poisson differential space. ˜ J˜−1 (0))) and δ ∗ : We want to show that γ ∗ : C ∞ (ρ(J −1 (Oμ ))) → C ∞ (ρ( ∞ −1 ∞ −1 ˜ C (ρ( ˜ J (0))) → C (ρ(J (Oμ ))) are isomorphisms of Poisson algebras. Since the symplectic form ω˜ on P˜ = P×Oμ is given by ω˜ = π1∗ ω⊕(−π2∗ μ ), → P : ( p, λ) → p is Poisson. Moreover, π1 is it follows that π1 : P

132

Symplectic reduction

G-equivariant. Hence, it induces a Poisson map π1 : R˜ → R such that the following diagram commutes: P

π1

/P ρ

ρ˜

˜ R

π1

/ R.

˜ is a Poisson algebra homomorphism. The The map π1∗ : C ∞ (R) → C ∞ ( R) −1 ˜ is also a Poisson map. Therefore, the cominclusion map ι : ρ( ˜ J (0)) → R −1 ˜ J˜ (0)) → R is a Poisson map. Note that π1 ◦ ι has values position π1 ◦ ι : ρ( in ρ(J −1 (Oμ )), and γ is obtained by the restriction of the codomain of π1 ◦ ι −1 to ρ(J (Oμ )). The differential structure of ρ(J −1 (Oμ )) is generated by the ring R(ρ(J −1 (Oμ ))) = { f |ρ(J −1 (Oμ )) | f ∈ C ∞ (R)}, which we identify with C ∞ (R)/N (ρ(J −1 (Oμ ))), where N (ρ(J −1 (Oμ ))) = { f ∈ C ∞ (R) | f |ρ(J −1 (Oμ )) = 0} is the null ideal of ρ(J −1 (Oμ )) in C ∞ (R). Lemma 6.6.2 The associative ideal N (ρ(J −1 (Oμ ))) in C ∞ (R) is a Poisson ideal. In other words, if f ∈ N (ρ(J −1 (Oμ ))) and h ∈ C ∞ (R), then { f, h} ∈ N (ρ(J −1 (Oμ ))). Proof

If h ∈ C ∞ (R), then ρ ∗ h ∈ C ∞ (P)G and, for every ξ ∈ g, X ρ ∗ h Jξ = −X Jξ ρ ∗ h = 0.

Hence, X ρ ∗ h preserves the level sets of the momentum map J. In particular, X ρ ∗ h preserves J −1 (Oμ ). If f ∈ N (ρ(J −1 (Oμ ))), then ρ ∗ f vanishes on J −1 (Oμ ) and X ρ ∗ h (ρ ∗ f ) also vanishes on J −1 (Oμ ). Since ρ{ f, h} = {ρ ∗ f, ρ ∗ h} = X ρ ∗ h (ρ ∗ f ), it follows that π ∗ { f, h} vanishes on J −1 (Oμ ). Thus, the bracket { f, h} vanishes on ρ(J −1 (Oμ )). This implies that N (ρ(J −1 (Oμ ))) is a Poisson ideal in C ∞ (R). Since N (ρ(J −1 (Oμ ))) is a Poisson ideal in C ∞ (R), it follows that the quotient R(ρ(J −1 (Oμ ))) = C ∞ (R)/N (ρ(J −1 (Oμ )))

6.6 Shifting

133

inherits the structure of a Poisson algebra. For every f1 , f 2 ∈ C ∞ (R), the Poisson bracket of the restrictions f 1|ρ(J −1 (Oμ )) and f 2|ρ(J −1 (Oμ )) of f 1 and f 2 , respectively, to ρ(J −1 (Oμ )), evaluated at x ∈ ρ(J −1 (Oμ )) ⊆ R, is given by { f 1|ρ(J −1 (Oμ )) , f2|ρ(J −1 (Oμ )) }(x) = { f 1 , f 2 }(x).

(6.33)

Every function in C ∞ (ρ(J −1 (Oμ ))) coincides locally with a function in R(ρ(J −1 (Oμ ))). Hence, it follows that equation (6.33) specifies uniquely the Poisson bracket on C ∞ (ρ(J −1 (Oμ ))). ˜ J˜−1 (0))) vanishes on Next, we show that ( π1 ◦ ι)∗ : C ∞ (R) → C ∞ (ρ( the null ideal N (ρ(J −1 (Oμ ))). Let f ∈ C ∞ (R) be a function vanishing on ρ(J −1 (Oμ )). For each ( p, J ( p)) ∈ J˜−1 (0), ( π1 ◦ ι)∗ f (ρ( ˜ p, J ( p))) = f ( π1 (ι(ρ( ˜ p, J ( p))))) = f ( π1 (ρ( ˜ p, J ( p)))) = f (ρ(π1 ( p, J ( p)))) = f (ρ( p)) = 0, because ( p, J ( p)) ∈ J˜−1 (0) implies that p ∈ J −1 (Oμ ) and f vanishes on ρ(J −1 (Oμ )). The vanishing of ( π1 ◦ ι)∗ : C ∞ (R) → C ∞ (ρ( ˜ J˜−1 (0))) on the null −1 ∗ π1 ◦ ι) induces a Poisson algebra homoideal N (ρ(J (Oμ ))) implies that ( morphism C ∞ (R)/N (ρ(J −1 (Oμ ))) → C ∞ (ρ( ˜ J˜−1 (0))). Recall that the ∞ −1 quotient C (R)/N (ρ(J (Oμ ))) is the subspace C ∞ (ρ(J −1 (O))) consisting of smooth functions on ρ(J −1 (O)), which extend to smooth functions on R. Hence, the Poisson homomorphism induced by ( π1 ◦ ι)∗ is the restriction of ∗ ∞ −1 ∞ −1 ∞ ˜ γ : C (ρ(J (Oμ ))) → C (ρ( ˜ J (0))) to C (R)/N (ρ(J −1 (Oμ ))). ∗ In order to show that γ is a Poisson algebra homomorphism, note that for every f 1 , f 2 ∈ C ∞ (ρ(J −1 (Oμ ))) and x ∈ ρ( ˜ J˜−1 (0)), there exist a neighbourhood U of γ (x) in R and h 1 , h 2 ∈ C ∞ (R) such that for i = 1, 2, h i|U ∩ρ(J −1 (Oμ )) = f i|U ∩ρ(J −1 (Oμ )) and {γ ∗ h 1|ρ(J −1 (Oμ )) , γ ∗ h 2|ρ(J −1 (Oμ )) }(x) = γ ∗ {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(x), because h i|ρ(J −1 (O)) ∈ C ∞ (R)/N (ρ(J −1 (Oμ ))). Therefore, {ρ ∗ f 1 , ρ ∗ f 2 }(x) = {ρ ∗ h 1|ρ(J −1 (Oμ )) , ρ ∗ h 2|ρ(J −1 (Oμ )) }(x) = γ ∗ {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(x)

= {h 1|ρ(J −1 (Oμ )) , h 2|ρ(J −1 (Oμ )) }(γ (x)) = { f 1 , f2 }(γ (x)) = γ ∗ { f 1 , f 2 }(x).

Hence, γ ∗ is a Poisson algebra homomorphism. Since γ is a diffeomorphism and δ = γ −1 , it follows that γ ∗ and δ ∗ are Poisson algebra isomorphisms. We have proved the following result.

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Symplectic reduction

Theorem 6.6.3 If Oμ is an embedded submanifold of g∗ , then the shifting ˜ J˜−1 (0)) are maps γ : ρ( ˜ J˜−1 (0)) → ρ(J −1 (Oμ )) and δ : ρ(J −1 (Oμ )) → ρ( diffeomorphisms of Poisson differential spaces.

6.7 When the action is free In the case of a free and proper action of G on P, the reduction described here is called regular reduction or Marsden–Weinstein reduction (Marsden and Weinstein, 1974). In this case, the stratification N of R consists of a single stratum P, the stratification M of P also consists of a single stratum R and we can rephrase the results of the preceding sections as follows. Corollary 6.7.1 Let J : P → g∗ be an equivariant momentum map for a free and proper action of a connected Lie group G on a symplectic manifold (P, ω). (i) The orbit space R = P/G is a Poisson manifold, and the orbit map ρ : P → R is a submersion. The pull-back map ρ ∗ : C ∞ (R) → C ∞ (P) gives rise to a Poisson algebra isomorphism from C ∞ (R) to C ∞ (P)G . For each f ∈ C ∞ (R), the Poisson vector field X f on R is ρ-related to the Hamiltonian vector field X ρ ∗ f on P; that is, ρ∗ X ρ ∗ f = X f . (ii) The kernel of d J : T P → g∗ is spanned by Hamiltonian vector fields of G-invariant functions. The connected components of level sets of J are orbits of the family {X h | h ∈ C ∞ (P)G } of Hamiltonian vector fields of G-invariant functions on P. (iii) Each orbit Q of the family {X f | f ∈ C ∞ (R)} of Poisson vector fields on R is endowed with a symplectic form ω Q on Q induced by the Poisson structure of C ∞ (Q). The pre-image ρ −1 (Q) is a connected component of J −1 (μ), where μ is the constant value of J on ρ −1 (Q). In other words, Q is a connected component of ρ(J −1 (μ)). The pull-back of ω Q to ρ −1 (Q) coincides with the restriction of ω to ρ −1 (Q). (iv) For each μ ∈ g∗ , each connected component of the orbit space J −1 (μ)/G μ , where G μ = {g ∈ G | Adg∗ μ = μ}, is a symplectic manifold symplectomorphic to the corresponding connected component of ρ(J −1 (μ)). (v) If the co-adjoint orbit Oμ through μ is locally closed,3 consider the ˜ ω), symplectic manifold ( P, ˜ where P˜ = P × Oμ

and

ω˜ = π1∗ ω ⊕ (−π2∗ μ ),

3 Regular reduction of the pre-images of co-adjoint orbits that are not locally closed was

discussed by Ortega and Ratiu (2004).

6.8 When the action is improper

135

and μ is the Kirillov–Kostant–Souriau symplectic form on Oμ . The ˜ given by action of G on P, ˜ : G × P˜ → P˜ : (g, ( p, ν)) → ˜ g ( p, ν) = (g ( p), Adg∗ ν), is Hamiltonian with the momentum map J˜ : (P × Oμ ) → g∗ : ( p, ν) → (J ( p), −ν). The manifolds J˜−1 (0), J −1 (Oμ )/G, ρ(J −1 (Oμ )), ρ(J −1 (μ)) and J −1 (μ)/G μ are symplectic manifolds, and they are symplectomorphic to each other.

6.8 When the action is improper If the action of G on P is not proper, then invariant functions need not separate G-orbits. This implies that the differential-space topology of the orbit space R = P/G may not be Hausdorff. Also, the differential-space topology may differ from the quotient topology. We illustrate this fact with a simple example. Example 6.8.1 We consider here an example of an action of G = R on P = R2 # (x, y), given by : R × P → P : (t, (x, y)) → (x, y + xt). This action is Hamiltonian with respect to the canonical symplectic form ω = d x ∧ dy on R2 , and the corresponding momentum map is J (x, y) = 12 x 2 . The action describes a free motion of a particle with one degree of freedom and unit mass, but this fact is not relevant here. Each point of the y-axis is fixed by the action of R. Hence, the fixed-point set is PR = {(0, y) | y ∈ R} = J −1 (0). Since R is not compact, it follows that the action is not proper. For μ = 0, the level set is given by J −1 (μ) = {(x, y) | x 2 = 2μ, y ∈ R}. Thus, J −1 (μ) consists of two lines in R2 parallel to the y-axis. These lines tend to the y-axis as μ → 0. It is easy to see that the action is free and proper on the complement of the fixed-point set. Hence, the orbit space is {(a, y) | y ∈ R} {(0, y)} . R= a=0 y

136

Symplectic reduction

The orbit map ρ : R2 → R is given by

{(x, y) | y ∈ R} ρ(x, y) = {(0, y)}

if x = 0 if x = 0.

Let f ∈ C ∞ (P) be an R-invariant function. The restriction of f to R) depends only on x. Hence, by continuity, the value of f is constant on the fixed-point set J −1 (0). Thus, the differential-space topology of R is not Hausdorff. In fact, the differential-space topology does not even satisfy the separation axiom T1 , because every open set containing a fixed point contains all fixed points. The quotient topology of R is also not Hausdorff. However, the quotient topology is T1 because, for every pair of distinct points (x 1 , y1 ) and (x2 , y2 ), there exists an open set V ∈ R containing ρ(x 1 , y1 ) but not ρ(x 2 , y2 ). This is obvious if x 1 = 0 and x2 = 0. If x 1 = x2 = 0, the complement U of the point (0, y2 ) in R2 is open and R-invariant, and contains the point (0, y1 ). Hence, V = ρ(U ) is open in R, contains ρ(0, y1 ) and does not contain ρ(0, y2 ). This shows that the quotient topology is different from the differential-space topology. R2 \({0} ×

In most applications to dynamics, the action of the symmetry group is proper except for some exceptional values of the momentum map. Hence, in the case of an improper action, it suffices to define the reduction for fixed values of the momentum map. Thus, we have arrived at the following problem. Let G be a connected Lie group with a Hamiltonian action on a symplectic manifold (P, ω) corresponding to an Ad ∗ -equivariant momentum map J : P → g∗ . Suppose that for μ ∈ g∗ , the action of the isotropy group G μ of μ on the μ-level J −1 (μ) is improper. In this case, we can use the algebraic reduction described below.

6.9 Algebraic reduction For μ ∈ g∗ , let Jμ be the ideal in C ∞ (P) generated by the components of J − μ : P → g∗ . Thus,

k ∞ Jμ = J − μ | ξi fi | ξ1 , . . . , ξk ∈ g and f 1 , . . . , f k ∈ C (P) , i=1

where (ξ1 , . . . , ξk ) is a basis in g. If our functions were polynomials, we would refer to the μ-level of J as the variety generated by the ideal Jμ . We also know that some problems, especially problems related to multiplicities of roots, have

6.9 Algebraic reduction

137

no solutions in terms of varieties, but can be solved in terms of schemes. Extrapolating this insight from algebraic geometry to differential geometry, we proceed as follows. We denote by C ∞ (P)/Jμ the quotient of C ∞ (P) by the ideal Jμ . For each f ∈ C ∞ (P), we denote by [ f ]μ the class of f in C ∞ (P)/Jμ . Since the momentum map J is Ad ∗ -equivariant, it follows that for every g ∈ G μ , k k ∗g J − μ | ξi fi = ∗g ( J | ξi − μ | ξi ) ∗g f i i=1

i=1

=

k

( ∗g J | ξi ∗g f i − μ | ξi )∗g f i

i=1

=

k

( J | Adg ξi ∗g f i − μ | Adg ξi )∗g f i

i=1

=

k

J − μ | Adg ξi ∗g f i .

i=1

Hence, Jμ is G μ -invariant. This implies that the action of G on P induces an action ∗ μ : G μ ×(C ∞ (P)/Jμ ) → C ∞ (P)/Jμ : (g, [ f ]μ ) → μ g [ f ]μ = [g −1 f ]μ

of G μ on C ∞ (P)/Jμ . We denote by (C ∞ (P)/Jμ )G μ the set of G μ -invariant elements of C ∞ (P)/Jμ ; that is, (C ∞ (P)/Jμ )G μ = {[ f ]μ ∈ C ∞ (P)/Jμ ) | [∗g−1 f ]μ = [ f ]μ ∀ g ∈ G μ }. It follows from the definition that [ f ]μ ∈ (C ∞ (P)/Jμ )G μ ⇐⇒ ∗g−1 f − f ∈ Jμ ∀ g ∈ G μ .

(6.34)

In particular, [ f ]μ ∈ (C ∞ (P)/Jμ )G μ =⇒ X Jξ ( f ) ∈ Jμ ∀ ξ ∈ gμ , where gμ is the Lie algebra of G μ . If G μ is connected, then the reverse implication holds. Proposition 6.9.1 The Poisson algebra structure in C ∞ (P) induces a Poisson algebra structure in (C ∞ (P)/Jμ )G μ , with a Poisson bracket {[ f 1 ]μ , [ f 2 ]μ } such that {[ f 1 ]μ , [ f 2 ]μ } = [{ f 1 , f 2 }]μ .

(6.35)

138

Proof

Symplectic reduction For f ∈ C ∞ (P) and h =

{ f, h} =

k

=

i=1

i=1 J

{ f, J − μ | ξi h i } =

i=1 k

k

− μ | ξi h i ∈ Jμ ,

k k { f, Jξi }h i + J − μ | ξi { f, h i } i=1

X Jξi ( f )h i +

k

i=1

J − μ | ξi { f, h i }.

i=1

Hence, { f, h} ∈ Jμ , provided [ f ]μ ∈ (C ∞ (P)/Jμ )G μ . In particular, if f ∈ Jμ , then [ f ]μ = 0 ∈ (C ∞ (P)/Jμ )G μ , so that { f, h} ∈ Jμ for every h ∈ Jμ . Let f 1 , f 2 be functions in C ∞ (P) such that [ f 1 ]μ , [ f 2 ]μ ∈ (C ∞ (P)/Jμ )G μ . Equation (6.34) implies that for each g ∈ G μ , there exist h 1 , h 2 ∈ Jμ such that ∗g { f 1 , f 2 } = {∗g f 1 , ∗g f 2 } = { f 1 + h 1 , f 2 + h 2 } = { f 1 , f 2 } + { f 1 , h 2 } − { f 2 , h 1 } + {h 1 , h 2 }. Therefore, ∗g { f 1 , f 2 } − { f 1 , f2 } ∈ Jμ , which implies that { f 1 , f 2 } is in (C ∞ (P)/Jμ )G μ . The same calculation shows that for every h 1 , h 2 ∈ Jμ , { f 1 + h 1 , f 2 + h 2 } − { f 1 , f 2 } ∈ Jμ . Hence, the class [{ f 1 , f 2 }]μ depends only on [ f 1 ]μ and [ f 2 ]μ , which implies that the bracket {[ f 1 ]μ , [ f 2 ]μ } on (C ∞ (P)/Jμ )G μ is well defined by equation (6.35). Since the Poisson bracket on C ∞ (P) is bilinear and antisymmetric, acts as a derivation, and satisfies the Jacobi identity, it follows that the bracket {[ f 1 ]μ , [ f 2 ]μ } has the same properties. Therefore, {[ f 1 ]μ , [ f 2 ]μ } is a Poisson bracket on (C ∞ (P)/Jμ )G μ . Definition 6.9.2 Let G × P → P be a Hamiltonian action of a connected Lie group on a symplectic manifold corresponding to an Ad ∗ -equivariant momentum map J : P → g∗ . Algebraic reduction assigns the Poisson algebra (C ∞ (P)/Jμ )G μ to each level J −1 (μ) of J . For μ ∈ g∗ , we refer to (C ∞ (P)/Jμ )G μ as the Poisson algebra of algebraic reduction at μ. It should be noted that algebraic reduction at μ encodes not only information about the level set J −1 (μ) but also some information about its inclusion in (P, ω). It is of interest to compare algebraic reduction with other reduction techniques. Proposition 6.9.3 Let G × P → P be a free and proper Hamiltonian action of a connected Lie group on a symplectic manifold corresponding to an Ad ∗ equivariant momentum map J : P → g∗ . For each μ ∈ g∗ , the Poisson algebra (C ∞ (P)/Jμ )G μ is naturally isomorphic to the Poisson algebra of the symplectic manifold (Q, ω Q ), where Q = ρ(J −1 (μ)) and ρ ∗ ω Q is the restriction of ω to J −1 (μ); see Proposition 6.3.3.

6.9 Algebraic reduction

139

Proof The assumption that the action of G on P is free and proper implies that for each p ∈ P, the differential d J ( p) maps T p P onto g∗ . This can be seen as follows. If d J ( p) : T p P → g∗ is not a submersion, there exists ξ ∈ g such that d J (v) | ξ = 0 for all v ∈ T p P. This is equivalent to X Jξ ( p) ω = 0. Since ω is non-degenerate, this implies that X Jξ ( p) = 0. Hence, p is a fixed point of the action of the one-parameter subgroup exp tξ of G. This contradicts the assumption that the action of G on p is free. For each μ in the range of the momentum map J : P → g∗ , the level set J −1 (μ) is a submanifold of P. Moreover, every function that vanishes on J −1 (μ) is in the ideal Jμ . That is, N (J −1 (μ)) = Jμ and R(J −1 (μ)) = C ∞ (P)/N (J −1 (μ)) = C ∞ (P)/Jμ . Furthermore, J −1 (μ) is a closed subset of P, which implies that C ∞ (J −1 (μ)) = R(J −1 (μ)) = C ∞ (P)/Jμ . Since Q is a closed subset of R, each function f ∈ C ∞ (Q) extends to a function h ∈ C ∞ (R); that is, f = h |Q . The pull-back ρ ∗ h is in C ∞ (P)G , which implies that its class [ρ ∗ h]μ in C ∞ (P)/Jμ is G μ -invariant. Thus, [ρ ∗ h]μ ∈ (C ∞ (P)/Jμ )G μ . On the other hand, (ρ|J −1 (μ) )∗ h |Q = (ρ ∗ h)|J −1 (μ) , where ρ|J −1 (μ) : J −1 (μ) → Q is the restriction of the orbit map ρ : P → R to J −1 (μ). Therefore, [ρ ∗ h]μ = (ρ|J −1 (μ) )∗ (ρ ∗ h)|J −1 (μ) = h |Q = f . For h 1 , h 2 ∈ C ∞ (R), f 1 = h 1|Q and f 2 = h 2|Q are in C ∞ (Q). The Poisson bracket in (C ∞ (P)/Jμ )G μ is given by {[ρ ∗ h 1 ]μ , [ρ ∗ h 2 ]μ } = [{ρ ∗ h 1 , ρ ∗ h 2 }]μ = (ρ|J −1 (μ) )∗ {ρ ∗ h 1 , ρ ∗ h 2 }|J −1 (μ) = (ρ|J −1 (μ) )∗ (ρ ∗ {h 1 , h 2 })|J −1 (μ) = {h 1 , h 2 }|Q = {h 1|Q , h 2|Q } = { f 1 , f 2 }. It follows from equation (6.18) that the Poisson bracket in C ∞ (Q) is given by the symplectic form ω Q on Q. It follows from Proposition 6.9.3 that for a free and proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω), algebraic reduction is equivalent to regular reduction. However, when the action of G on (P, ω) is proper but not free, then algebraic reduction and singular reduction need not be equivalent. This can be seen in the following example. Example 6.9.4 Let P be the space of 2 × 2 complex matrices z = (z i j ) with the symplectic form ω = i(d z¯ 11 ∧ dz 11 + d z¯ 21 ∧ dz 21 + d z¯ 12 ∧ dz 12 + d z¯ 22 ∧ dz 22 ).

140

Symplectic reduction

The group G = SU (2) of unitary matrices g = (gi j ) with determinant 1 acts on P by multiplication on the left. Consider a basis i 1 0 0 0 2i 2 2 , , (ξ1 , ξ2 , ξ3 ) = i 0 − 2i 0 − 12 0 2 of the Lie algebra su(2) of SU (2). The vector fields X ξi corresponding to the action on P of the one-parameter subgroup exp tξi of SU (2) are X ξ1 =

1 (z 21 ∂11 + z¯ 21 ∂¯11 − z 11 ∂21 − z¯ 11 ∂¯21 ) 2 1 + (z 22 ∂12 + z¯ 22 ∂¯12 − z¯ 12 ∂¯22 − z 12 ∂22 ), 2

X ξ2 =

i (z 21 ∂11 + z 22 ∂12 + z 11 ∂21 + z 12 ∂22 ) 2 i ¯ z¯ 21 ∂11 + z¯ 22 ∂¯12 + z¯ 11 ∂¯21 + z¯ 12 ∂¯22 − 2

and X ξ3 =

i (z 11 ∂11 + z 12 ∂12 − z 21 ∂21 − z 22 ∂22 ) 2 i ¯ z¯ 11 ∂11 + z¯ 12 ∂¯12 − z¯ 21 ∂¯21 − z¯ 22 ∂¯22 . − 2

Each X ξi is the Hamiltonian vector field of a function Ji , where i J1 = − [¯z 11 z 21 − z¯ 21 z 11 + z¯ 12 z 22 − z¯ 22 z 12 ], 2 1 J2 = − [z 21 z¯ 11 + z 22 z¯ 12 + z 11 z¯ 21 + z 12 z¯ 22 ] 2 and 1 [−¯z 11 z 11 − z¯ 12 z 12 + z¯ 21 z 21 + z¯ 22 z 22 ]. 2 The functions J1 , J2 , J3 are components of a momentum map J : P → su(2)∗ with respect to the basis in su(2)∗ dual to (ξ1 , ξ2, ξ3 ). The ideal J0 is generated by J1 , J2 , and J3 . Consider the following functions: J3 =

i K 1 = − [¯z 11 z 12 − z¯ 12 z 11 + z¯ 21 z 22 − z¯ 22 z 21 ], 2 1 K 2 = − [z 12 z¯ 11 + z¯ 12 z 11 + z 22 z¯ 21 + z¯ 22 z 21 ] 2 and K3 =

1 [−z 11 z¯ 11 + z¯ 12 z 12 − z 21 z¯ 21 + z¯ 22 z 22 ]. 2

6.9 Algebraic reduction

141

It is easy to check that these functions are SU (2)-invariant and that they satisfy the identity J12 + J22 + J32 = K 12 + K 22 + K 32 . Hence, they vanish on the zero level set of J . However, none of the functions K 1 , K 2 and K 3 is contained in the ideal J0 in C ∞ (P) generated by J1 , J2 , J3 . Hence, the Poisson algebra (C ∞ (P)/J0 )G of algebraic reduction at 0 ∈ su(2)∗ contains non-zero elements [K 1 ]0 , [K 2 ]0 and [K 3 ]0 , which correspond to functions that vanish identically on J −1 (0). The fact that in the case of a proper non-free action the algebraic reduction need not be equivalent to the singular reduction is somewhat disturbing. We shall see in the following chapter, using examples, that quantization of the two reductions leads to equivalent results. We need further examination of this phenomenon. We now return to the case in which the action of the group G on P is not proper. In this case, since singular reduction does not apply, we have no alternative to algebraic reduction at present. We give a simple example of algebraic reduction for an improper action; this example is a continuation of Example 6.8.1. Example 6.9.5 Consider the symplectic manifold with symmetry discussed in Example 6.8.1. We describe here the algebraic reduction of the action of G = R on P = R2 at 0 ∈ g∗ = R. The ideal J0 is given by J0 = {x 2 f (x, y) | f ∈ C ∞ (R2 )}. Hence, C ∞ (P)/J0 can be parametrized as C ∞ (P)/J0 = {h(y) + xk(y)0 | h, k ∈ C ∞ (R)}, where the functions on the right-hand side have been chosen as representatives of their J0 -equivalence classes. The action of R on C ∞ (P)/J0 associates to t ∈ R and h(y) + xk(y) ∈ C ∞ (P) the class [h(y − xt) + xk(y − xt)]0 = [h(y) − xth (y) + xk(y)]0 in C ∞ (P)/J0 . Hence, [h(y) + xk(y)]0 is R-invariant if h (y) = 0. This means that (C ∞ (P)/J0 )R = {c + xk(y) | c ∈ R and k ∈ C ∞ (R)}. Recall that the zero level of the momentum map is the fixed-point set J −1 (0) = {(0, y) | y ∈ R}.

142

Symplectic reduction

Given y1 = y2 ∈ R, there exists a function k ∈ C ∞ (R) separating y1 from y2 . Hence, the elements of (C ∞ (P)/J0 )R separate points in J −1 (0). For each f 1 , f 2 ∈ C ∞ (R2 ), we have { f1 , f 2 } = −X f1 f 2 =

∂ f1 ∂ f2 ∂ f1 ∂ f2 − . ∂y ∂x ∂x ∂y

In particular, {c1 + xk1 (y), c2 + xk2 (y)} = xk1 (y)k2 (y) − k1 (y)xk2 (y). Hence, the Poisson bracket in (C ∞ (P)/J0 )R is {c1 + xk1 (y), c2 + xk2 (y)} = x(k1 (y)k2 (y) − k1 (y)k2 (y)). Thus, algebraic reduction at 0 gives rise to a non-trivial Poisson algebra.

As in the case of singular reduction, we have a shifting theorem for algebraic reduction, proved by Arms (1996), which we state without proof. Theorem 6.9.6 If G and G μ are connected and Oμ is an embedded submanifold of g∗ , then the Poisson algebra (C ∞ (P)/Jμ )G μ is isomorphic to the G ˜ Poisson algebra (C ∞ ( P)/J 0) . Example 6.9.7 Let P = R4 # (x, y), where x = (x1 , x 2 ) and y = (y1 , y2 ) are in R2 , and let ω = d x1 ∧ dy1 + d x 2 ∧ dy2 . The corresponding Poisson bracket is 2 ∂ f1 ∂ f2 ∂ f1 ∂ f2 { f1 , f2} = (6.36) − ∂ yi ∂ xi ∂ xi ∂ yi i=1

for all f 1 , f 2 ∈ C ∞ (R4 ). We consider an action of R on P given by : R × P → P : (t, (x, y)) → t (x, y) = (x, y + t x). This action is Hamiltonian, with the momentum map given by the kinetic energy J (x, y) = 12 x 2 = 12 ((x1 )2 + (x 2 )2 ). As in the case of one degree of freedom, the action is improper. The zero level of the momentum map P0 ≡ J −1 (0) = {(0, 0, y1 , y2 ) | (y1 , y2 ) ∈ R2 } ∼ = R2 consists of fixed points of the action. Its complement P\P0 is G-invariant, and the induced action of R on P\P0 is free and proper. Hence, every R-invariant function f ∈ C ∞ (P) restricted to P\P0 can be expressed as a smooth function of the algebraic invariants x1 , x2 and j = x 1 y2 − x 2 y1 .

6.9 Algebraic reduction

143

In the remainder of this section, we split the discussion of Example 6.9.7 into a sequence of lemmas, propositions etc., which are numbered from 6.9.7/1 to 6.9.7/7. Lemma 6.9.7/1 For each f ∈ C ∞ (P)R , − j x2 j x1 , f (x1 , x2 , y1 , y2 ) = f x1 , x2 , (x1 )2 + (x 2 )2 (x1 )2 + (x2 )2 whenever J = 12 (x1 )2 + (x 2 )2 = 0.

(6.37)

Proof For (x1 , x2 ) = (0, 0), the action is free and proper. Moreover, f is constant along the line x1 y2 − x2 y1 = j. The line x 1 y2 − x2 y1 = j has a unique point − j x2 j x1 , (y1 , y2 ) = (x1 )2 + (x2 )2 (x1 )2 + (x 2 )2 that is closest to the origin. Substituting this expression for (y1 , y2 ) into f (x 1 , x2 , y1 , y2 ), we obtain equation (6.37). Proposition 6.9.7/2 For each f ∈ C ∞ (P)R , the restriction of f to P0 is a constant function. Proof If f ∈ C ∞ (P)R , then f ((x1 , x2 , y1 +t x 1 , y2 +t x2 ) = f (x 1 , x2 , y1 , y2 ) for all t ∈ R. Differentiating with respect to t and setting x 2 = 0, we obtain x1

∂f (x1 , 0, y1 , y2 ) = 0 ∂ y1

for all x1 = 0 and (y1 , y2 ) ∈ R2 . By continuity, ∂f (0, 0, y1 , y2 ) = 0 ∂ y1 for all (y1 , y2 ) ∈ R2 . Similarly, ∂f (0, 0, y1 , y2 ) = 0 ∂ y2 for all (y1 , y2 ) ∈ R2 . Hence, the restriction of f to P0 is constant. It follows from Proposition 6.9.7/2 that the R-invariant functions on P do not separate points in P0 . Hence, singular reduction does not apply. The ideal J0 is given by 1 2 2 ˜ ∞ ˜ J0 = (x 1 ) + (x2 ) f (x, y) | f ∈ C (P) . 2

144

Symplectic reduction

In order to describe the equivalence class [ f ]0 ∈ C ∞ (P)/J0 of a function f ∈ C ∞ (P), we introduce the variables z = x1 + i x2 and z¯ = x1 − i x 2 , and the corresponding derivations ∂ ∂ 1 ∂ −i = ∂z 2 ∂ x1 ∂ x2

and

∂ 1 = ∂ z¯ 2

∂ ∂ +i ∂ x1 ∂ x2

.

We have z z¯ = (x 1 )2 + (x2 )2

and

z

∂ ∂ ∂ ∂ + x2 . + z¯ = x1 ∂z ∂ z¯ ∂ x1 ∂ x2

Therefore, the Taylor series for f with respect to x = (x1 , x 2 ) can be written as follows: ∞ 1 d n f (t x, y) |t=0 n! dt n n=0 ∞ 1 ∂ ∂ n = + x2 f (x, y) |x=0 x1 n! ∂ x1 ∂ x2 n=0 ∞ 1 ∂ n ∂ = f (x, y) |x=0 + z¯ z n! ∂z ∂ z¯ n=0 ∞ n 1 n! ∂ n−k ∂ k z¯ = f (x, y) |x=0 z n! k!(n − k)! ∂z ∂ z¯

f (x, y) = f (t x, y) |t=1 ∼

n=0

k=0

∞ n z k z¯ n−k = f k(n−k) (y), k!(n − k)!

(6.38)

n=0 k=0

where f mn (y) =

∂m ∂n f (x, y) |x=0 . ∂z m ∂ z¯ n

(6.39)

Proposition 6.9.7/3 A function f ∈ C ∞ (R4 ) is divisible by z z¯ = (x1 )2 + (x2 )2 if and only if the coefficients f n0 (y) and f 0n (y) vanish for every n = 0, 1, 2, . . . , and y = (y1 , y2 ) ∈ R2 . Proof If f ∈ C ∞ (R4 ) is divisible by (x 1 )2 + (x2 )2 , then f = ((x1 )2 + (x2 )2 )h for some h ∈ C ∞ (R4 ), and the Taylor series for f is the product of (x 1 )2 + (x2 )2 = z z¯ and the Taylor series for h. Hence, f n0 (y) and f 0n (y) vanish identically. Conversely, suppose that f n0 (y) and f 0n (y) vanish for every n = 0, 1, 2, . . . , and that y = (y1 , y2 ) ∈ R2 . Hence, the Taylor series for f is

6.9 Algebraic reduction

145

divisible by z z¯ = (x 1 )2 + (x2 )2 . By the Borel Extension Lemma, there exists a function h ∈ C ∞ (R4 ) with its Taylor series in x given by ∞ n−1 k−1 n−1−k z z¯ f k,n−k (y). k!(n − k)! n=1 k=1

Hence, f and ((x1 )2 + (x2 )2 )h have the same Taylor series, which implies that f − ((x1 )2 + (x2 )2 )h vanishes at x = 0 together with all partial derivatives with respect to x1 and x2 . This implies that f − ((x1 )2 + (x 2 )2 )h is divisible by (x1 )2 + (x2 )2 . Hence, f is divisible by (x 1 )2 + (x2 )2 , as required. It follows from Proposition 6.9.7/3 that for every f ∈ C ∞ (R4 ), the representative of f in C ∞ (P)/J0 is the formal power series [ f ]0 = f 0,0 (y) +

∞ 1 f n,0 (y)z n + f 0,n (y)¯z n , n!

(6.40)

n=1

where the coefficients fmn are given by equation (6.39). Since f is real-valued, it follows that f 00 (y) = f (0, y) is real and f n0 (y) = f 0n (y). By the Borel Extension Lemma, for every formal power series a(y) +

∞

(a¯ n (y)z n + an (y)¯z n ),

(6.41)

n=1

where a(y) and an (y) are smooth functions of y, and a0 (y) is real, there exists a function f ∈ C ∞ (R)4 such that a = f 00 and a¯ n = f n0 /n!. The behaviour of Taylor series under addition and multiplication of functions implies that the map

f → [ f ]0 = f 00 (y) +

∞ 1 f n0 (y)z n + f 0n (y)¯z n n! n=1

is a homomorphism of associative algebras. Proposition 6.9.7/4 [ f ]0 ∈ (C ∞ (P)/J0 )R if and only if ∂ ∂ z + z¯ f (x, y) ∂w ∂ w¯ is in J0 , where ω = y1 + i y2

and

ω¯ = y1 − i y2 .

146

Symplectic reduction

Proof Since R is connected, it follows that the class [ f ]0 is G-invariant if there exists a function λ such that X J f = λJ . But, ∂f ∂f d (x, y) + x2 (x, y) f (x, y + t x)|t=0 = x1 dt ∂ y1 ∂ y2 i 1 ∂f ∂f (x, y) − (z − z¯ ) (x, y) = (z + z¯ ) 2 ∂ y1 2 ∂y 2 ∂ ∂ ∂ ∂ z z¯ −i +i = f (x, y) + f (x, y) 2 ∂ y1 ∂ y2 2 ∂ y1 ∂ y2 ∂ ∂ + z¯ f (x, y). = z ∂w ∂ w¯

X J f (x, y) =

Hence, [ f ]0 ∈ (C ∞ (P)/J0 )R if and only if ∂ ∂ z + z¯ f (x, y) ∈ J0 . ∂w ∂ w¯ Proposition 6.9.7/5 An element [ f ]0 of C ∞ (P)/J0 is R-invariant if and only if the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 . Proof

For f ∈ C ∞ (P), let

∂ ∂ h(x, y) = z + z¯ ∂w ∂ w¯ Equation (6.39) implies that

∂ ∂ + z¯ h 00 (y) = h(0, y) = z ∂w ∂ w¯ and ∂n ∂n h 0n (y) = n h(x, y) |x=0 = n ∂ z¯ ∂ z¯ =

f (x, y). f (x, y)|x=0 = 0,

∂ ∂ + z¯ z ∂w ∂ w¯

f (x, y) |x=0

∂ ∂ n−1 ∂ f 0,n−1 (y) f (x, y)|x=0 = n−1 ∂ w¯ ∂ z¯ ∂ w¯

for n ≥ 1. By Proposition 6.9.7/3, h ∈ J0 if and only if the coefficients h n0 (y) and h 0n (y) vanish identically in y. If [ f ]0 is R-invariant, then h ∈ J0 by Proposition 6.9.7/4, and h 0n (y) =

∂ f 0,n−1 (y) = 0 ∂ w¯

for all y ∈ R2 and n = 1, 2, . . . . Hence, all the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 .

6.9 Algebraic reduction

147

Suppose that the coefficients f 0n (y), n = 0, 1, 2, . . . , are entire analytic functions of w = y1 + i y2 . Then h 0n (y) = 0 for n = 0, 1, . . . and all y. Since h is a real-valued function, it follows from equation (6.39) that h n0 (y) = h 0n (y) = 0. Hence, h ∈ J0 , which implies that f mod J0 is G-invariant. Corollary 6.9.7/6 (C ∞ (P)/J0 )R consists of formal power series a+

∞

(a¯ n (w)z n + an (w)¯z n ),

n=1

where a ∈ R and, for every n ∈ N , an (w) is an entire analytic function of w. Next we want to describe the Poisson bracket in (C ∞ (P)/J0 )G . A straightforward but tedious computation from the definition yields 2 ∂ f ∂h ∂ f ∂h { f, h} ≡ (6.42) − ∂ xi ∂ yi ∂ yi ∂ xi i=1 ∂ f ∂h ∂ f ∂h ∂ f ∂h ∂ f ∂h + − − . =2 ∂z ∂ w¯ ∂ z¯ ∂w ∂ w¯ ∂z ∂w ∂ z¯ A calculation using equation (6.39) gives m m! ∂ ∂ { f, h}m,0 (y) = 2 h m−k,0 + f k1 h m−k,0 f k+1,0 k!(m − k)! ∂ w¯ ∂w k=0 m m! ∂ ∂ −2 f m−k,0 + h k1 f m−k,0 . h k+1,0 k!(m − k)! ∂ w¯ ∂w k=0

(6.43) Similarly, { f, h}0,m (y) = 2

m

m! k!(m − k)!

k=0 m

−2

k=0

f 0,k+1

∂ ∂ h 0,m−k + f 1,k h 0,m−k ∂w ∂ w¯

m! ∂ ∂ f 0,m−k + h 1,k f 0,m−k . h 0,k+1 k!(m − k)! ∂w ∂ w¯ (6.44)

Therefore, [{ f, h}]0 = { f, h}0,0 +

∞ 1 { f, h}n,0 z n + { f, h}0,n z¯ n , n! n=1

where { f, h}0,0 = { f, h}(0, y) and the coefficients of z n and z¯ n are given by equations (6.43) and (6.44), respectively.

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For G-invariant [ f ]0 and [h]0 , Proposition 6.9.7/5 implies ∂ fl,0 = 0, ∂w

∂ f 0,l = 0, ∂ w¯

∂ h 0,l = 0 ∂ w¯

and

∂ h l,0 = 0, ∂w

(6.45)

for l = 0, 1, . . . , and f 0,0 and h 0,0 are constant. Setting m = 0 in equation (6.43), we obtain the result that { f, h}0,0 = 0. Therefore, { f, h} mod J0 n ∞

(6.46)

∂ ∂ h n−k,0 − h k+1,0 f n−k,0 z n ∂ w¯ ∂w n=1 k=0 n ∞ 1 ∂ ∂ +2 h 0,n−k − h 0,k+1 f 0,n−k z¯ n . f 0,k+1 k!(n − k)! ∂w ∂ w¯

=2

1 k!(n − k)!

f k+1,0

n=1 k=0

It follows from equation (6.45) that ∂ ∂ [ f ]0 [h]0 ∂z ∂ w¯ ∞ 1 ∂ n n = f 0,0 + f n,0 (y)z + f 0,n (y)¯z ∂z n! n=1 ∞ ∂ 1 n n h 0,0 + h n,0 (y)z + h 0,n (y)¯z × ∂ w¯ n! n=1 ∞ ∞ 1 ∂ 1 n−1 n f n,0 (y)z h n,0 (y)z = (n − 1)! n! ∂ w¯ n=1 n=1 n ∞ 1 ∂ n z = f k+1,0 (y) h n−k,0 (y) . k!(n − k)! ∂ w¯ n=1

k=0

Similarly, n ∞ ∂[ f ]0 ∂[h]0 1 ∂h n−k,0 (y) n = f 0,k+1 (y) , z¯ ∂ z¯ ∂w k!(n − k)! ∂w n=1 k=0 n ∞ ∂[h]0 ∂[ f ]0 1 (y) ∂ f n−k,0 zn = h k+1,0 (y) , ∂z ∂ w¯ k!(n − k)! ∂ w¯ n=1 k=0 n ∞ 1 ∂ f n−k,0 (y) ∂[h]0 ∂[ f ]0 n = h 0,k+1 (y) . z¯ ∂ z¯ ∂w k!(n − k)! ∂w n=1

k=0

Taking equation (6.46) into account, we arrive at the following result.

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149

Proposition 6.9.7/7 The Poisson bracket of f mod J0 and h mod J0 in (C ∞ (P)/J0 )G is given by {[ f ]0 , [h]0 }J0 ∂[h]0 ∂[ f ]0 ∂[ f ]0 ∂[h]0 ∂[h]0 ∂[ f ]0 ∂[ f ]0 ∂[h]0 + + −2 . =2 ∂z ∂ w¯ ∂ z¯ ∂w ∂z ∂ w¯ ∂ z¯ ∂w This completes our discussion of Example 6.9.7.

7 Commutation of quantization and reduction

The theory of geometric quantization forms a bridge between quantum mechanics and the representation theory of Lie groups. In representation theory, geometric quantization is a geometric technique for obtaining a unitary representation of a connected Lie group from its action on a symplectic manifold. In quantum mechanics, geometric quantization provides a geometric method of quantization of a classical system. This dual role of geometric quantization enables us to use representation theory to test hypotheses in quantum mechanics and vice versa. Geometric quantization of a Hamiltonian action of a connected Lie group G on a connected symplectic manifold (P, ω) gives rise to a representation of the Lie algebra g of G by skew-adjoint operators on a Hilbert space H. We assume that this representation of g can be integrated to a unitary representation U of G on H F . This representation can be decomposed into a direct integral of irreducible unitary representations of U λ of G. Hence, Hλ dμ(λ), (7.1) HF =

where is the space of equivalence classes of irreducible unitary representations of G, Hλ is the representation space of U λ and dμ(λ) is a spectral measure on ; see Kirillov (1962). If G is compact, then every irreducible unitary representation U λ of G is finite-dimensional and can be obtained by geometric quantization of the corresponding co-adjoint orbit Oλ (Kostant, 1966). The term ‘commutation of quantization and reduction’ was coined by Guillemin and Sternberg (1982), who investigated a quantization representation corresponding to an action of a compact connected Lie group G on a compact Kähler manifold P. In this case, the quantization representation of G is finite-dimensional, the set is discrete and the spectral decomposition (7.1) reads

7.1 Review of geometric quantization HF =

m λ Hλ ,

151

(7.2)

λ∈

where the multiplicities m λ are positive integers. Assuming that the action of G on J −1 (Oλ ) is free, Guillemin and Sternberg showed that the multiplicity m λ of Uλ in equation (7.2) is equal to the dimension of the representation space obtained by quantization of the Marsden–Winstein reduction of J −1 (Oλ ). Sjamaar (1995) extended the results of Guillemin and Sternberg by removing the condition of freeness of the action of G on J −1 (Oλ ). Taken together, the results of Guillemin and Sternberg and those of Sjamaar show that for an action of a compact connected Lie group G on a compact Kähler manifold, all the multiplicities m λ occurring in equation (7.2) are completely determined by the quantization of the reduced spaces J −1 (Oλ )/G, where Oλ is the co-adjoint orbit corresponding to Uλ . We use the term ‘commutation of quantization and reduction’ to denote a programme of determining the spectral measure dμ(λ) in terms of the quantization of the symplectic reduction of the inverse images of co-adjoint orbits under the momentum map J : P → g∗ . We begin with a brief review of geometric quantization. Next, we discuss some partial results on the commutation of quantization and singular reduction of the zero level of the momentum map, which generalize the results of Guillemin, Sternberg and Sjamaar to noncompact symplectic manifolds that need not be Kähler, and to non-compact Lie groups. We continue with a discussion of the commutation of quantization and reduction of non-zero co-adjoint orbits. Next, we proceed to a discussion of the commutation of quantization and algebraic reduction. We conclude with an example of commutation of quantization and algebraic reduction at J = 0 for an improper action of a connected Lie group.

7.1 Review of geometric quantization Geometric quantization of a symplectic manifold consists of three stages: prequantization, polarization and unitarization. In the prequantization stage, we construct a faithful unitary representation of the Poisson algebra of the symplectic manifold, called a prequantization representation. The prequantization representation is reducible and fails to satisfy Heisenberg’s Uncertainty Principle. In the next stage, polarization, we restrict the prequantization representation to the kernel of a subalgebra of the Poisson algebra. In this way, we obtain a smaller representation, but may lose the scalar product. The last stage, unitarization, deals with the recovery of a scalar product in the representation space.

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7.1.1 Prequantization Let λ : L → P be a complex line bundle with a connection and a connectioninvariant Hermitian form · | · . A connection on L is given by a covariant derivative operator ∇, which associates to each section σ of L and each vector field X on P a section ∇ X σ of L such that for each f ∈ C ∞ (P), ∇ X ( f σ ) = X ( f )σ + f ∇ X σ and ∇ f X σ = f ∇ X σ . For every section σ of L, f ∈

C ∞ (P),

(7.3)

and X 1 , X 2 ∈ X(P),

(∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )( f σ ) = f (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ. Hence, there is a 2-form α on P such that (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = 2πiα(X 1 , X 2 )σ .

(7.4)

The form α is the pull-back by the section σ of the curvature form of the connection ∇. A Hermitian form · | · on L is connection-invariant if, for every pair of sections σ1 , σ2 of L and every vector field X on P, X ( σ1 | σ2 ) = ∇ X σ1 | σ2 + σ1 | ∇ X σ2 . The quantization of a mechanical system is defined in terms of an additional free parameter . In quantum mechanics, is the value of Planck’s constant divided by 2π . However, in the quasi-classical approximation, we consider limits of various expressions as → 0. The line bundle L over P with a connection ∇ and a connection-invariant Hermitian form on L is a prequantization line bundle of (P, ω) if the following prequantization condition is satisfied: i (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = − ω(X 1 , X 2 )σ, (7.5) for every X 1 , X 2 ∈ X(P) and each section σ of L. In other words, we require −1 that 2π ω is the curvature of ∇. The prequantization condition (7.5) requires that the de Rham cohomology class [(2π )−1 ω] on P is in H 2 (Z). If σ is a non-zero section of L, the covariant derivative ∇ X σ is proportional to σ , and there is a complex-valued 1-form θ on P such that ∇ X σ = −i−1 θ |X σ for every vector field X on P. Hence, ∇σ = −i−1 θ ⊗ σ. The 1-form θ is called the pull-back by σ of the connection form of ∇. Equation (7.5) implies that dθ = ω. C ∞ (P)

(7.6)

generates a local one-parameter group exp t X f of A function f ∈ local symplectomorphisms of (P, ω). The Hamiltonian vector field X f on f

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153

can be lifted to a vector field X f on L such that exp t X f is a lift of exp t X f that preserves the connection ∇. For each p ∈ P and non-zero z ∈ L p , the horizontal component of X f (z) is the horizontal lift of X f at z, and the vertical X f acts component of X f (z) is proportional to f ( p). If X f ( p) = 0, then exp t on the fibre L p by multiplication by e2πi f ( p) . We denote by S ∞ (L) the space of smooth sections of L. For each σ ∈ S ∞ (L), we set d (exp t X f ◦ σ ◦ exp(−t X f ))|t=0 . dt

(7.7)

P f σ = (−i∇ X f + f )σ ;

(7.8)

P f σ = i Direct computation yields

´ see Sniatycki (1980). We refer to P f as the prequantization operator corresponding to f. The map P : C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → P f σ is called the prequantization map. Proposition 7.1.1 For each f 1 , f2 ∈ C ∞ (P) and σ ∈ S ∞ (L) [ P f1 , P f2 ] = i P { f1 , f 2 } .

(7.9)

Proof [ P f 1 , P f2 ]σ = ((−i∇ X f1 + f 1 )(−i∇ X f2 + f 2 ) + −(−i∇ X f2 + f 2 )(−i∇ X f1 + f 1 ))σ = (−2 (∇ X f1 ∇ X f 2 − ∇ X f2 ∇ X f1 ) − i(X f1 ( f 2 ) − X f2 ( f 1 )))σ i = −2 ∇[X f1 ,X f2 ] − ω(X f1 , X f2 ) −i(X f1 ( f 2 ) − X f2 ( f 1 )) σ = (2 ∇ X { f1 , f2 } + iω(X f1 , X f2 ) − i(X f1 ( f 2 ) − X f2 ( f 1 )))σ = (i(−i∇ X { f1 , f2 } + { f 1 , f 2 } − { f 1 , f 2 }) + +iω(X f1 , X f2 ) − iX f1 ( f 2 ) + iX f2 ( f 1 ))σ = i P { f1 , f 2 } σ, because −{ f 1 , f 2 } + ω(X f1 , X f2 ) −X f1 ( f 2 ) +X f2 ( f 1 ) = 0 by equation (6.7).

Corollary 7.1.2 The map C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → − i P f σ is a representation of the Lie algebra structure of C ∞ (P) on S ∞ (L).

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Proof For f 1 , f 2 ∈ C ∞ (P) and σ ∈ S ∞ (L), equation (7.9) yields i2 i i2 i i − P f1 , − P f2 σ = 2 [ P f1 , P f2 ]σ = 2 (i) P { f1 , f 2 } σ = − P { f1 , f 2 } σ, as required. The space S0∞ (L) of compactly supported smooth sections of L has a Hermitian scalar product (7.10) (σ1 | σ2 ) = σ1 | σ2 ωn , P

C ∞ (P),

where n = dim P. For each f ∈ the prequantization operator P f is symmetric with respect to the scalar product (7.10). If the Hamiltonian vector field X f of f is complete, then P f is self-adjoint on the Hilbert space H obtained by completion of S0∞ (L) with respect to the norm given by equation (7.10). Equation (7.9) gives the usual commutation relations imposed in quantum mechanics. However, prequantization does not correspond to a quantum theory, because an interpretation of (σ | σ )( p) as the probability density of localizing the state σ at a point p ∈ P fails to satisfy Heisenberg’s Uncertainty Principle.1 Now consider a Hamiltonian action of a connected Lie group G on (P ω) with an equivariant momentum map J :P → g∗ . Proposition 6.1.6 in Chapter 4 states that the map ξ → Jξ is a homomorphism of g of G to the Poisson algebra C ∞ (P). Hence, the map ξ → (−i/) P Jξ is a linear representation of the Lie algebra g on the space S ∞ (L), which we call the prequantization representation of g. Since the Hamiltonian vector fields X Jξ are complete, each operator (−i/)PJξ is skew-adjoint on the Hilbert space H obtained by the completion of S0∞ (L) with respect to the norm given by equation (7.10). Recall that the action of g on L is given by vector fields X Jξ on L; see equation (7.7). We assume that this action integrates to an action of G on L that covers the action of G on P. We refer to this action as the prequantization action of G on P. This assumption implies that the prequantization representation of g described above integrates to a representation of G. That is, we have a linear representation 1 2

R : G × S ∞ (L) → S ∞ (L) : (g, σ ) → R g σ

(7.11)

d (Rexp tξ σ )|t=0 = (−i/) P Jξ σ dt

(7.12)

such that

1 Heisenberg’s Uncertainty Principle states that for every quantum state of a particle, the accu-

racies %x and %p with which one can determine the position x and momentum p satisfy the inequality %x %p ≥ .

7.1 Review of geometric quantization

155

for each ξ ∈ g. The linear representation R induces a unitary representation U : G × H → S ∞ (L) : (g, σ ) → U g σ

(7.13)

such that U g σ = R g σ for each σ ∈ S ∞ (L) ∩ H. We refer to R and U as prequantization representations of G. In general, the unitary prequantization representation U fails to be irreducible.

7.1.2 Polarization A complex distribution F ⊂ T C P = C⊗T P on a symplectic manifold (P, ω) is Lagrangian if for each p ∈ P, the restriction of the symplectic form ω to the subspace F p ⊂ T pC P vanishes identically and rankC F = 12 dim P. If F is a complex distribution on P, we denote its complex conjugate by F. Let D = F ∩ F ∩ T P and E = (F + F) ∩ T P. A polarization of (P, ω) is an involutive complex Lagrangian distribution F such that D and E are involutive distributions on P. Let C ∞ (P)0F be the space of smooth complex-valued functions on P that are constant along F; that is, C ∞ (P)0F = { f ∈ C ∞ (P) ⊗ C | u f = 0 for all u ∈ F}.

(7.14)

The polarization F is said to be strongly admissible if the spaces P/D and P/E of integral manifolds of D and P, respectively, are quotient manifolds of P and the natural projection P/D → P/E is a sumbersion. A strongly admissible polarization F is locally spanned by Hamiltonian vector fields of ¯ ≥ 0 for every functions on C ∞ (P)0F . A polarization F is positive if iω(w, w) w ∈ F. A positive polarization F is semi-definite if ω(w, w) ¯ = 0 for w ∈ F implies that w ∈ D C . Let C F∞ (P) denote the space of functions on P whose Hamiltonian vector fields preserve F. In other words, f ∈ C F∞ (P) if, for every h ∈ C ∞ (P)0F , the Poisson bracket { f, h} ∈ C ∞ (P)0F . If f 1 , f 2 ∈ C F∞ (P) and h ∈ C ∞ (P)0F , then the Jacobi identity implies that {{ f 1 , f 2 }, h} = −{ f 2 , { f 1 , h}} + { f 1 , { f 2 , h}} ∈ C ∞ (P)0F . Hence, the ring C F∞ (P) is a Poisson subalgebra of C ∞ (P). Let S F∞ (L) denote the space of smooth sections of L that are covariantly constant along F, namely, S F∞ (L) = {σ ∈ S ∞ (L) | ∇u σ = 0 for all u ∈ F}.

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Commutation of quantization and reduction

We shall refer to S F∞ (L) as the space of polarized sections. For each h ∈ C ∞ (P)0F , f ∈ C F∞ (P) and σ ∈ S F∞ (L), we have ∇ X h ( Q f σ ) = 0. Thus, for every f ∈ C F∞ (P), the prequantization operator P f maps S F∞ (L) to itself. Definition 7.1.3 The quantization map Q relative to a polarization F is the restriction of the prequantization map P : C ∞ (P) × S ∞ (L) → S ∞ (L) : ( f, σ ) → P f σ = (−i∇ X f + f )σ to the domain C F∞ (P) × S F∞ (L) ⊂ C ∞ (P) × S ∞ (L) and codomain S F∞ (L) ⊂ S ∞ (L). In other words, Q : C F∞ (P) × S F∞ (L) → S F∞ (L) : ( f, σ ) → Q f σ = (−i∇ X f + f )σ . (7.15) We assume that the action : G × P → P preserves the polarization F. Hence, for each ξ ∈ g, the momentum Jξ is in C F∞ (P). Restricting the prequantization representation to the Poisson algebra spanned by Jξ , for ξ ∈ g, we obtain a representation ξ → (i)−1 Q Jξ of g on S F∞ (L). If the action of G on P lifts to an action of G on L by connection-preserving automorphisms, then this representation of g integrates to a linear representation R : G × S F∞ (L) → S F∞ (L) : (g, σ ) → R g σ

(7.16)

of G on S F∞ (L). For each g ∈ G, f ∈ C ∞ (P)0F and σ ∈ S F∞ (L), R g ( f σ ) = (∗g−1 f )R g σ. We refer to R : G × S F∞ (L) → S F∞ (L) as the quantization representation of G. Note that the quantization representation of G introduced in equation (7.16) is the restriction of the prequantization representation R introduced in equation (7.11) to the domain G × S F∞ (L) and codomain S F∞ (L). Therefore, using the same symbol R for both representations should not lead to any contradiction.

7.1.3 Examples of unitarization S ∞ (L)

The space of smooth sections of L is endowed with a scalar product given by equation (7.10). In general, polarized sections in S F∞ (L) need not have a finite norm corresponding to this scalar product. Unitarization deals with the choice of the scalar product and a modification of the quantization representation R leading to a unitary representation. Here, we consider two special cases for which unitarization is straightforward.

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157

Kähler polarization A Kähler polarization of (P, ω) is a strongly admissible polarization F such that F ⊕ F¯ = T C P and iω(w, w) ¯ > 0 for all non-zero w ∈ F. These assumptions imply that there is a complex structure J on P such that F is the space of antiholomorphic directions. Moreover, P is a Kähler manifold such that −ω is the Kähler form on P. For a Kähler polarization F on (P, ω), the prequantization line bundle L over P is holomorphic and the space S F∞ (L) of polarized sections coincides with the space of holomorphic sections. Moreover, the holomorphic sections of L which are normalizable with respect to the scalar product (7.10) form a Hilbert space H F . In other words, H F = H ∩ S F∞ (L). Hence, the linear representation R of G on S F∞ (L) gives rise to a unitary representation U of G on H. Thus, a Kähler polarization does not require additional unitarization. Proposition 7.1.4 A co-adjoint orbit (O, ) of a compact connected Lie group G admits a Kähler polarization. Proof Since G is compact, its Lie algebra g admits a positive definite AdG invariant metric k, which allows an identification of g with g∗ . Under this identification, co-adjoint orbits go to adjoint orbits. Hence, we can treat O as an adjoint orbit. For each ξ ∈ O, the tangent space Tξ O is the quotient of g by the Lie algebra hξ of the isotropy group Hξ = {g ∈ G | Adg ξ = ξ }. The map adξ : g → g : ζ → [ξ, ζ ] preserves hξ and induces a map Aξ of Tξ O onto itself. The map Aξ is skew-symmetric with respect to k. Hence, the eigenvalues of Aξ are purely imaginary, and half of them lie on the positive imaginary axis. Let Fξ ⊂ Tξ O ⊗ C be the space spanned by these positive eigenvalues. It can be shown that the set F = ∪ξ ∈O Fξ ⊂ T C O is a Kähler polarization of the symplectic manifold (O, ). Theorem 7.1.5 Let O be a quantizable co-adjoint orbit. The unitary representation U of G on the Hilbert space H F obtained by the quantization of (O, ) with respect to the Kähler polarization F described in Proposition 7.1.4 is irreducible. Moreover, the map O → U O is a bijection of the space of quantizable co-adjoint orbits of G onto the space of irreducible representations of G. Theorem 7.1.5 is the Borel–Weil Theorem in the formulation due to Kostant (1966).

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Commutation of quantization and reduction

Cotangent polarization We assume here that P = T ∗ Q is the cotangent bundle of a manifold Q, ω is the canonical symplectic form of T ∗ Q, and the polarization F is the complexification of ker T ϑ, where ϑ : T ∗ Q → Q is the cotangent bundle projection. The canonical 1-form θ of the cotangent bundle T ∗ Q is defined as follows. For each p ∈ T ∗ Q and every u ∈ T p T ∗ Q, θ (u) = p | T ϑ(u) . The canonical symplectic form of T ∗ Q is ω = dθ . Since ω is exact, the prequantization line bundle is trivial; that is, L = C × P. We denote by σ0 : P → L : p → (1, p) the trivializing section of L. We choose the covariant derivative operator ∇ such that ∇σ0 = −i−1 θ ⊗ σ0 . Moreover, we normalize the Hermitian form σ1 , σ2 appearing in equation (7.10) so that σ0 , σ0 = 1. The space C ∞ (P)0F consists of complex-valued functions on P = T ∗ Q that are constant along the fibres of the cotangent bundle projection. In other words, C ∞ (P)0F = {ϑ ∗ f | f ∈ (C ⊗ C ∞ (Q))}. The space S F∞ (L) of polarized sections of L is given by S F∞ (L) = {ϑ ∗ (ψ)σ0 | ψ ∈ C ⊗ C ∞ (Q)}. For each σ = ϑ ∗ (ψ)σ0 , we have ¯ ¯ ϑ ∗ (ψ)σ0 , ϑ ∗ (ψ)σ0 = ϑ ∗ (ψψ) = (ψψ) ◦ ϑ. Since the fibres of the cotangent bundle projection ϑ are not compact, it follows that ¯ σ, σ ωn = (ψψ) ◦ ϑ ωn = ∞ T∗Q

T∗Q

unless σ = 0. Let D = {ϑ ∗ (ψ)σ0 ∈ S F∞ (L) | ψ ∈ C ⊗ C0∞ (Q)}, where C ⊗ C0∞ (Q) is the space of compactly supported complex-valued smooth functions on Q. We endow D with a topology of uniform convergence with all derivatives on compact sets. We may introduce an alternative scalar product on D by setting ∗ ∗ ψ¯ 1 (q)ψ2 (q) dμ(q), (7.17) (σ1 | σ2 ) Q = (ϑ (ψ1 )σ0 | ϑ (ψ2 )σ0 ) Q = Q

7.1 Review of geometric quantization

159

where dμ(q) is a Lebesgue measure on Q. We denote the completion of D with respect to the scalar product (7.17) by H F , and the topological dual of D by D . Then, D ⊂ H F ⊂ D . We consider a Hamiltonian action : G ×T ∗ Q → T ∗ Q of a connected Lie group such that the momentum map J : T ∗ Q → g∗ is constant along the fibres of the cotangent bundle projection. Hence, there exists a map j : Q → g∗ such that J = ϑ ∗ j = j ◦ ϑ. For each ξ ∈ g, the differential d Jξ annihilates ker T ϑ, which implies that X Jξ has values in ker T ϑ. Therefore, the action of G on T ∗ Q preserves each fibre of the cotangent bundle projection. Since the fibres of the cotangent bundle projection are Lagrangian submanifolds of T ∗ Q, it follows that the action of G on T ∗ Q is abelian. We may assume that G is an abelian group. Examples of such an action may be found in Examples 6.9.5 and 6.9.7. In both of these examples, the action of G on P was improper on the zero level set of the momentum map. Half-densities and half-forms Linear frames of the tangent bundle T M of a manifold M form a principal G L(n, R)-fibre bundle over M, where n = dim M; see Kobayashi and Nomizu (1963). Similarly, the complexified tangent bundle T C M = T M ⊗ C is a principal G L(n, C)-fibre bundle F T C M over M. If (v1 , . . . , vn ) ∈ F T C M and A = (ai j ) ∈ G L(n, C), then n n vi ai1 , . . . , vi ain (v1 , . . . , vn )A = i=1

i=1

is also a linear frame on T C M. A density of weight s on a manifold M, of dimension n, is a function d on the bundle F T C M of linear frames of T C M such that, for every linear frame (v1 , . . . , vn ) ∈ F T C M and A ∈ G L(n, C), d((v1 , . . . , vn )A) = |det A|s d(v1 , . . . , vn ).

(7.18)

We can generalize the notion of densities to frame bundles of arbitrary vector bundles on a manifold. In particular, a half-density on a polarization F of a symplectic manifold (P, ω) is a density of weight 12 on the bundle of linear frames F F of F. In other words, a half-density ν on F associates to each frame (v1 , . . . , vn ) of F a number ν(v1 , . . . , vn ) such that, for every A ∈ G L(n, C), 1

ν(v1 , . . . , vn ) = |det A| 2 ν(v1 , . . . , vn ).

(7.19)

The half-densities on a polarization F of (P, ω) form a complex line bundle √ over P, which we denote by |∧n F|. This bundle has a flat partial connection

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Commutation of quantization and reduction

√ over F. Thus, covariant derivatives of sections of |∧n F| are well defined. There is a canonical pairing of sections of the bundle of half-densities that √ associates to sections ν1 and ν2 of |∧n F| a density ν1 | ν2 on F, defined by ν1 | ν2 (v1 , . . . , vn ) = (v1 (v1 , . . . , vn ))ν2 (v1 , . . . , vn ). Unitarization of the quantization representation in terms of half-densities is obtained by tensor multiplication of the prequantization line bundle L on √ (P, ω) by the bundle |∧n F|. The representation space of the quantization √ representation is modified from S F∞ (L) to S F∞ (L ⊗ |∧n F|). If F = D ⊗ C is a strongly admissible real polarization, and σ1 , σ2 ∈ S F∞ (L) and ν1 , ν2 are √ sections of |∧n F| that are covariantly constant along F, then a pairing of σ1 ⊗ ν1 and σ2 ⊗ ν2 gives a density σ1 ⊗ ν1 | σ2 ⊗ ν2 of weight 1 on P/D defined as follows. Recall that the assumption that F is strongly admissible implies that P/D is a quotient manifold of P. Let ϑ : P → P/D denote the canonical projection. Consider a linear frame (u 1 , . . . , u n ) in Tq (P/D). Let p ∈ ϑ −1 ( p), and let u˜ i be the lift of u i to T p P. Choose a linear frame (v1 , . . . , vn ) in D p so that (v1 , . . . , vn ; u˜ 1 , . . . , u˜ n ) is a symplectic frame in T p P. Therefore, ω(vi , u˜ j ) = δi j and ω(vi , v j ) = ω(u i , u j ) = 0, where δi j is the Kronecker symbol. We set σ1 ⊗ ν1 | σ2 ⊗ ν2 (u 1 , . . . , u n ) = σ1 | σ2 (q)ν1 (v1 , . . . , vn )ν2 (v1 , . . . , vn ). We can show that σ1 ⊗ ν1 | σ2 ⊗ ν2 is a well-defined density of weight √ 1 on P/D. Hence, we can define a scalar product on S F∞ (L ⊗ |∧n F|) by integration of σ1 ⊗ ν1 | σ2 ⊗ ν2 over P/D. If P = T ∗ Q and F is the cotangent polarization defined in the preceding section, then the scalar product (7.17) is equivalent to the scalar product defined here. A half-form μ on F associates to each frame (v1 , . . . , vn ) of F a number μ(v1 , . . . , vn ) such that, for every A ∈ G L(n, C), 1

ν(v1 , . . . , vn ) = (det A) 2 ν(v1 , . . . , vn ).

(7.20)

Note that the difference between half-densities and half-forms is the replacement of the square root of the absolute value of det A in equation (7.19) by the square root of the determinant of A in equation (7.20). In order to 1 make (det A) 2 well defined, we need to introduce a double covering of the ´ symplectic frame bundle of (P, ω). For details, see Sniatycki (1980). The use of half-forms is essential for obtaining the metaplectic representation corresponding to the action of the symplectic group Sp(n) on R2n .

7.2 Commutation of quantization and singular reduction at J = 0 161

7.2 Commutation of quantization and singular reduction at J = 0 In this section, we consider a quantization representation of a connected Lie group G corresponding to a proper Hamiltonian action of G on a symplectic manifold (P, ω). Let J : P → g∗ be the equivariant momentum map corresponding to this action. The quantization structure on (P, ω) considered here consists of a polarization F, and a prequantization line bundle L over P with a connection ∇ that satisfies the prequantization condition (7.5). We assume that the action of G on P preserves the polarization F and lifts to a connection-preserving action on L. Geometric quantization gives rise to a linear representation R of G on the space S F∞ (L) of smooth sections of L that are covariantly constant along F. A G-invariant scalar product on S F∞ (L) leads from the linear representation R to a unitary representation U of G on a Hilbert space H. The unitary representation U can be decomposed into a direct sum or integral of irreducible unitary representations of G. If an irreducible unitary representation U λ of G can be obtained by the geometric quantization of a co-adjoint orbit Oλ , we would like to describe the contribution of U λ to U in terms of a quantization of J −1 (Oλ )/G. If the action of G on P is not free and proper, then the orbit space J −1 (Oλ )/G is not a symplectic manifold, and we have first to define what we mean by quantization of J −1 (Oλ )/G. First, we shall concentrate on the definition of the quantization of J −1 (0)/G, and discuss the commutation of quantization and reduction at the zero level of the momentum map. For non-zero co-adjoint orbits, we shall use the shifting trick discussed in Section 6.6. Consider the diagram below:

F

→

L ↓ P

=⇒ F|J −1 (0)

L|J −1 (0) ↓ −1 → J (0) ⇓

F|J −1 (0) /G

→

(7.21) L|J −1 (0) /G , ↓ J −1 (0)/G

where L is the prequantization line bundle and F is a G-invariant polarization of (P, ω). The horizontal double arrow symbolizes the restriction to

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Commutation of quantization and reduction

J −1 (0), and the vertical double arrow symbolizes the projection to the space of G-orbits in J −1 (0). Our aim in this chapter is to describe the geometric structure of the spaces and maps occurring in the diagram (7.21) in the case when the action of G on P is proper, without assuming that G and P are compact. The case when P and G are compact, the action of G on J −1 (0) is free, and the polarization F is Kähler was the object of the work of Guillemin and Sternberg (1982) referred to at the beginning of this chapter. These authors showed that all of the spaces in the diagram (7.21) are manifolds and all of the maps are complex line bundle projections. In particular, J −1 (0)/G is symplectic. Moreover, the restriction F|J −1 (0) of F to J −1 (0) projects to a Kähler polarization (F|J −1 (0) )/G of J −1 (0)/G. Thus, geometric quantization of J −1 (0)/G is a quantization of a symplectic manifold with respect to a Kähler polarization. Furthermore, the restriction to J −1 (0) gives rise to a linear isomorphism from the space S F∞ (L)G of G-invariant polarized sections of L → P onto the space S F∞ (L|J −1 (0) )G of G-invariant sections of L|J −1 (0) → J −1 (0) that are covariantly constant along F|J −1 (0) . Similarly, we might expect that the orbit map projection J −1 (0) → J −1 (0)/G will give rise to a linear isomorphism from the space S F∞ (L|J −1 (0) )G to the space S F∞ (L|J −1 (0) /G) of polarized sections of L|J −1 (0) /G → J −1 (0)/G. In the general case considered here, when the action of G on P is proper but not free and neither G nor P need be compact, we cannot hope to obtain results as strong as those of Guillemin and Sternberg. Nevertheless, we can use the tools developed in Part I to investigate the singularities of the various spaces and mappings occurring in the diagram (7.21). Moreover, we can try to determine relationships between the spaces S F∞ (L)G , S F∞ (L|J −1 (0) )G and S F∞ (L|J −1 (0) /G). We are now going to discuss the structure on J −1 (0) induced by the quantization data on (P, ω). First, we need a few additional results, which were not given in Chapter 6 because they are specific to the zero level of the momentum map. Recall that, for each compact subgroup H of G, we have introduced the sets PH = { p ∈ P | G p = H }, P(H ) = { p ∈ P | G p = g H g −1 for some g ∈ G}, where G p = {g ∈ G | gp = p} is the isotropy group of p ∈ P. Connected components of ρ(J −1 (0) ∩ P(H ) ), where ρ : P → R = P/G is the orbit map, give rise to a stratification of ρ(J −1 (0)) = J −1 (0)/G. Each stratum Q of the stratification of J −1 (0)/G is the orbit of a family of Poisson derivations of

7.2 Commutation of quantization and singular reduction at J = 0 163 C ∞ (R), and it is the projection to R of a connected component K of J −1 (0) ∩ PH for some subgroup H of G. In other words, Q = ρ(K ), where K is a submanifold of P. Moreover, Q is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K ,

(7.22)

where ρ K : K → Q is the restriction of the orbit map ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Let L be a connected component of PH . Then the pull-back ω L of ω to L by the inclusion map L → P is symplectic. Moreover, the group N L = {g ∈ G | gp ∈ L ∀ p ∈ L} is a closed subgroup of G containing H as a normal subgroup. Hence, G L = N L /H is a Lie group. For each g ∈ N L , we denote the class of g in G L by [g]. Since H acts trivially on L, the action of N L on L induces an action L of G L on L, given by L : G L × L → L : ([g], p) −→ [g] p = gp. The action L is Hamiltonian provided L ∩ J −1 (0) = ∅. Proposition 7.2.1 (i) For each connected component L of PH , the action of the group G L on L preserves the symplectic form ω L on L given by the pullback to L of the symplectic form ω on P. If L ∩ J −1 (0) = ∅, then the action of G L on L is Hamiltonian with an Ad ∗ -equivariant momentum map JL : L → g∗L . (ii) J −1 (0) ∩ L = JL−1 (0). (iii) The action of G L on L induces on K the structure of a left principal fibre bundle with structure group G L , base space Q and principal bundle projection ρ K : K → Q. Proof (i) Let L be a connected component of PH . Recall that G L = N L /H , where N L = {g ∈ G | p ∈ L =⇒ gp ∈ L}.

(7.23)

We have shown that the action of G L on L, given by G L × L → L : ([g], p) → gp, is free and proper. Since the action of N L on P preserves ω and L, it follows that it preserves the pull-back ω L of ω to L. Let h and n L denote the Lie algebras of H and N L , respectively. The Lie algebra of G L is the quotient n L /h. For ξ ∈ n L , we denote the equivalence class of ξ in G L by [ξ ]. The action of the one-parameter group exp t[ξ ] of G L is given by the action of exp tξ on L. If we denote the restriction of X Jξ to L by X Jξ |L , we have

164

Commutation of quantization and reduction X Jξ |L ω L = −d Jξ |L .

Each point p ∈ L ⊆ PH is a fixed point of H . Hence, for every η ∈ h, the Hamiltonian vector field X Jη ( p) vanishes. This implies that d Jη ( p) = 0. Since L is a connected component of PH , it follows that Jη is constant on L. Moreover, L ∩ J −1 (0) = ∅ implies that Jη|L = 0. Hence, for every η ∈ h, X Jξ +η |L ω L = −d Jξ +η|L = −d Jξ |L − d Jη|L = −d Jξ |L . Therefore, we can define a momentum map JL : L → g∗L such that for each [ξ ] ∈ g L , JL[ξ ] = Jξ |L .

(7.24)

Since Jη|L = 0 for all η ∈ h, the map JL : L → g∗L is AdG∗ L -equivariant. (ii) Clearly, J −1 (0) ∩ L ⊆ JL−1 (0). Suppose that there is a ξ ∈ g such that Jξ |J −1 (0) is not identically zero. Then d Jξ does not vanish on JL−1 (0), and X Jξ L

does not vanish on JL−1 (0). We can average X Jξ over H , obtaining a vector field (X Jξ ) H ( p) = T g ◦ X Jξ (g −1 ( p)) dμ(g). H

The momentum map J : P →

g∗

is AdG∗ -equivariant. That is,

J (gp) = Adg∗ J ( p) for all g ∈ G. Therefore, for every ξ ∈ g, ∗g Jξ ( p) = Jξ (gp) = J (gp) | ξ = Adg∗ J ( p) | ξ = J ( p) | Adg −1 ξ . If η = H Adg −1 ξ is the average of ξ over G, then (X Jξ ) H is the Hamiltonian vector field of Jη . In other words, (X Jξ ) H = X Jη . But η is Ad H -invariant, which implies that η is in the Lie algebra of n L . Therefore, we have a decomposition X Jξ = X Jη +X J(ξ −η) . By part (iii) of Proposition 4.2.6, X Jη ( p) ∈ T p L for each p ∈ L. (iii) Every connected component of PH ∩ P0 has to be contained in a connected component of PH . Let K be a connected component of PH ∩ P0 contained in L. Then, K = L ∩ P0 . The action of N L on P preserves P0 , because N L ⊆ G and P0 = J −1 (0) is G-invariant. Hence, K = L ∩ P0 is N L -invariant, which implies that the action of G L = N L /H on L preserves K . Therefore, G L acts on K . Since the action of G L on L is free and proper, it follows that the action of G L on K is free and proper. Therefore, K is a

7.2 Commutation of quantization and singular reduction at J = 0 165

left principal fibre bundle with structure group G L . Its base space is the space K /G L of G L -orbits in K . By Proposition 4.2.6, for each p ∈ K , Gp ∩ L = G L p. Hence, K /G L = ρ(K ) = Q, and the principal bundle projection is ρ K :K → Q, obtained by restricting ρ : P → R to the domain K and codomain Q. We begin with a discussion of the structure induced on J −1 (0)/G by the G-invariant polarization F of (P, ω). The intersection F ∩ T C (J −1 (0)) is a linear subset of the complexified tangent bundle space of J −1 (0). Hence, it is a complex distribution on J −1 (0) in the sense of Definition 3.4.6. Moreover, F ∩ T C (J −1 (0)) is invariant under the action of G on J −1 (0). Recall that, in general, only the regular component of a differential space has its tangent bundle spanned by global derivations; see Proposition 3.3.15. Therefore, F ∩ T C (J −1 (0)) need not be spanned by global derivations in neighbourhoods of singular points of J −1 (0). In other words, F ∩T C (J −1 (0)) need not be smooth. Nevertheless, we shall refer to F0 = F ∩ T C (J −1 (0))

(7.25)

as the polarization of J −1 (0). The orbit type stratification M of P enables us to discuss F stratum by stratum. For each M ∈ M, consider the restriction F|M of F to points in M. Recall that M is a connected component of M(H ) for a compact subgroup H of G. Let L be a connected component of PH contained in M. Then M = G L is the union of G-orbits through L. The G-invariance of F ensures that F|M is uniquely determined by the restriction F|L of F to L. Recall that, for every p ∈ L, we have a decomposition of the tangent space to P at p, given by T p P = T p L ⊕ T p⊥ L , where T p L = T p PH , because L is an open subset of PH , and T p⊥ L consists of vectors v ∈ T p P such that the H -average of v vanishes; see Proposition 4.2.6. Proposition 7.2.2 The space T p⊥ L is the symplectic complement of T p L. In other words, T p⊥ L = {v ∈ T p P | ω(u, v) = 0 for all u ∈ T p L}. Proof

Since ω is G-invariant, for every u, v ∈ T p P and g ∈ H , we have ω(T g (u), T g (v)) = ω(u, v).

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Commutation of quantization and reduction

If u ∈ T p L, then u is H -invariant, and T g (u), T g (v) dμ(g) = ω u, T g (v) dμ(g) ω(u, v) = ω H

H

= ω(u, v H ), where v H = H T g (v) dμ(g) is the H -average of v. Since v H ∈ T p L, we have ω(u, v H ) = ω L (u, v H ). Therefore, ω(u, v) = 0 for all u ∈ T p L if and only if ω L (u, v H ) = 0 for all u ∈ T p L. Since ω L is symplectic, it follows that ω(u, v) = 0 for all u ∈ T p L if and only if v H = 0. Thus, ω(u, v) = 0 for all u ∈ T p L if and only if v ∈ T p⊥ L. The restriction ω|L of ω to points in L decomposes as the sum ω|L = ω L + ω⊥ L, where ω L is the pull-back of ω by the inclusion map L → P, and ω⊥ L = ω|L − ω L is the restriction of ω to vectors in T ⊥ L. Moreover, (T ⊥ L , ω⊥ ) L is a symplectic vector bundle over L. Let F|L be the restriction of F to points in L. We introduce the notation FL = F|L ∩ T C L and FL⊥ = F ∩ T ⊥C L . Proposition 7.2.3 If F is a G-invariant polarization of (P, ω), then: (i) F|L = FL ⊕ FL⊥ ; (ii) FL is a G L -invariant polarization of (L , ω L ); (iii) FL⊥ is a G L -invariant complex Lagrangian subbundle of the complexification of (T ⊥ L , ω⊥ L ). Proof (i) Since F is G-invariant, it follows that F|L is H -invariant and G L -invariant. For p ∈ L, let (w1 , . . . , wn ) be a basis in F|L . For each i = 1, . . . , n, we denote by wi H the H -average of wi . The H -invariance of ⊥ = FL implies that the vectors w1H , . . . , wn H are in FL and the vectors w1H ⊥ ⊥ w1 − w1H , . . . , wn H = wn − wn H are in FL . Since the basis (w1 , . . . , wn ) in FL is arbitrary, it follows that FL and FL⊥ span F|L . But FL ∩ FL⊥ = 0, and therefore F|L = FL ⊕ FL⊥ . (ii) For every w1 , w2 ∈ FL ⊆ F, ω L (w1 , w2 ) = ω(w1 , w2 ) = 0, which implies that FL is an isotropic complex distribution on (L , ω L ). Moreover, if w ∈ T pC L is such that ω L (w, u) = 0 for all u ∈ FL attached at p, then ω(w, u) = 0 for all u ∈ F p . Since F is a maximal isotropic complex distribution on P, it follows that w ∈ F. Hence, w ∈ FL , which ensures that FL is a maximal isotropic complex distribution on (PL , ω L ). In other words, FL is Lagrangian.

7.2 Commutation of quantization and singular reduction at J = 0 167

By definition, F is an involutive distribution. Since L is a manifold, it follows that FL = F ∩ T C L is involutive. Thus, FL is an involutive complex Lagrangian distribution on (L , ω L ), which means that FL is a polarization of (L , ω L ). Moreover, FL is G L -invariant because L and F|L are G L -invariant. (iii) We can show, as above, that FL⊥ is a maximal isotropic subbundle of ⊥ ⊥ (T ⊥ L , ω⊥ L ). Moreover, FL is G L -invariant because T L is G L -invariant. Let K be a connected component of J −1 (0) ∩ L = JL−1 (0), where JL : L → g L is the momentum map for a free and proper action of G L on L; see Proposition 7.2.1. Since the action of G L on L is free and proper, K is a submanifold of L, and the orbit space K /G L is a quotient manifold of K . We can identify K /G L with the projection ρ(K ) of K by the orbit map ρ : P → R. We know that Q = ρ(K ) = K /G L is a symplectic manifold with a symplectic form ω Q such that ρ K∗ ω Q = ω K ,

(7.26)

where ρ K : K → Q is the restriction of ρ : P → R to the domain K and codomain Q, and ω K is the pull-back of ω by the inclusion map K → P. Since K ⊆ L ⊆ P, we can say that ω K is the pull-back of ω L by the inclusion map K → L. Definition 7.2.4 A G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0) if, for every compact subgroup H of G and each connected component K of J −1 (0) ∩ PH , the intersection FK = F ∩ T C K has constant rank. Let T C ρ K : T C K → T C Q be the complexification of the derived map Tρ K : T K → T Q of ρ K : K → Q. Proposition 7.2.5 Suppose a G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0). Then, for every compact subgroup H of G and each connected component K of J −1 (0) ∩ PH , Fˆ Q = T C ρ K (FK )

(7.27)

is a polarization of (Q, ω Q ), where Q = ρ K (K ) and ρ K∗ ω Q = ω K . Proof Since FK has constant rank, it follows that FK is an involutive complex distribution on K . If the projection T C ρ K (FK ) has constant rank on Q, then

168

Commutation of quantization and reduction

T C ρ K (FK ) is a complex distribution on Q. However, T C ρ K (FK ) has constant rank if the rank of FK ∩ ker T C ρ K is constant. Let FL|K denote the restriction of FL to points of K . Since FL|K ⊇ FK ⊇ (FK ∩ ker T C ρ K ), we obtain rankC FL|K = rankC (FL|K /FK ) + rankC FK = rankC (FL|K /FK ) + rankC FK /(FK ∩ ker T C ρ K ) + rankC (FK ∩ ker T C ρ K ) = rankC FL|K − rankC FK + rankC FK /(FK ∩ ker T C ρ K ) + rankC (FK ∩ ker T C ρ K ). Therefore, rankC (FK ∩ ker T C ρ K ) = rankC FK − rankC FK /(FK ∩ ker T C ρ K ). Moreover, FL is a maximal isotropic distribution on L. On the other hand, equation (7.26) implies that T C K / ker T C ρ K is symplectic. Since FL is a maximal isotropic distribution on L, it follows that the projection FK /(FK ∩ ker T C ρ K ) is a maximal isotropic subbundle in T C K / ker T C ρ K . In other words, rank FK /(FK ∩ ker T C ρ K ) = dim Q. Therefore, rankC (FK ∩ ker T C ρ K ) = rankC FK − dim Q. This implies that rankC (FK ∩ ker T C ρ K ) is constant provided rankC FK is constant. Thus, F has a clean intersection with J −1 (0), and T C ρ K (FK ) is a maximal isotropic complex distribution on Q. The involutivity of FK implies that T C ρ K (FK ) is involutive. Hence, C T ρ K (FK ) is a polarization of (Q, ω Q ). We shall refer to Fˆ = T C ρ(F ∩ T C (J −1 (0)))

(7.28)

as the polarization of J −1 (0)/G. Let Fˆ|Q be the restriction of Fˆ to points in Q. Since FK = F ∩ T K and ρ K : K → Q is the restriction of ρ to the domain K and codomain Q, we have T C ρ K (FK ) = T C ρ(F ∩ T C (J −1 (0)))|Q ∩ T C Q = Fˆ|Q ∩ T C Q.

7.2 Commutation of quantization and singular reduction at J = 0 169 Proposition 7.2.5 states that the part of Fˆ|Q that is tangential to Q is a polarization Fˆ Q of (Q, ω Q ). It is of interest to interpret the quotient Fˆ|Q /FQ , which describes directions in Fˆ|Q that are not tangent to Q. Since Q is a stratum of the stratification of J −1 (0)/G, directions in Fˆ|Q that are not tangent to Q describe how the polarization Fˆ varies from stratum to stratum. Directions in Fˆ|Q that are not tangent to Q can be characterized by quotients FL|K /FK ⊥ / ker T C ρ, where F ⊥ is the restriction of F ⊥ to points of K , and FL|K |K L|K L and ker|K T C ρ is the restriction of ker T C ρ to points of K . The quotient T C ρ(FL|K )/ Fˆ Q corresponds to the directions in TQ R which are tangent to the stratum N of the orbit type stratification N of R = P/G that contains ⊥ ) corresponds to directions in T R that are not Q. The projection T C ρ(FL|K Q tangent to the stratum N . We now proceed to a discussion of the reduction of the prequantization line bundle λ : L → P of (P, ω) under the following simplifying assumption. Assumption 7.2.6 For every connected component L of PH such that L ∩ J −1 (0) = ∅, the action of G on the prequantization line bundle L induces a trivial action of H on the restriction L L of L to L. Consider next the restriction of the prequantization line bundle λ : L → P to P0 = J −1 (0). In order to simplify the notation, we shall write L0 for the restriction of L to J −1 (0). In other words, L0 = λ−1 (J −1 (0)).

(7.29)

The continuity of λ implies that L0 is a closed subset of L. Hence, smooth functions on L0 are restrictions to L0 of smooth functions on L. Let λ0 : L0 → J −1 (0) be the restriction of λ to the domain L0 and codomain J −1 (0). If f |J −1 (0) is the restriction of f ∈ C ∞ (P) to J −1 (0), then λ∗0 f |J −1 (0) = f |J −1 (0) ◦ λ0 = ( f ◦ λ)|L0 ∈ C ∞ (L0 ). Therefore, λ0 is a smooth map. We shall refer to L0 as a complex line bundle with base space J −1 (0) and projection map λ0 . A map σ0 : J −1 (0) → L0 is a section of L0 if λ0 ◦ σ0 is the identity on −1 J (0). By definition, a section σ0 of L0 is smooth if, for every f ∈ C ∞ (L), the pull-back σ0∗ f |L0 = f ◦ σ0 is in C ∞ (J −1 (0)). We denote the space of smooth sections of L0 by S(L0 ). In other words, S(L0 ) = {σ0 : P0 → L0 | λ0 ◦ σ0 = id P0 and σ0∗ f |L0 ∈ C ∞ (P0 ) ∀ f ∈ C ∞ (L).

170

Commutation of quantization and reduction

Proposition 7.2.7 A section of L0 is smooth if and only if it is a restriction to P0 of a smooth section of L. Proof Let σ0 be a section of L0 . Suppose σ0 = σ|J −1 (0) for some section of L. Then, for every f ∈ C ∞ (L) in C ∞ (J −1 (0)), σ0∗ f |L0 = f ◦ σ0 = f ◦ σ|J −1 (0) = ( f ◦ σ )|J −1 (0) . Since f ◦ σ ∈ C ∞ (P), it follows that ( f ◦ σ )|P0 ∈ C ∞ (J −1 (0)). Therefore, σ0 is smooth. Conversely, suppose that σ0 is smooth. For each p ∈ J −1 (0), a local trivialization of L in an open neighbourhood U of p in P is given by a smooth map τ : λ−1 (U ) → U × C such that λ|λ−1 (U ) = π1 ◦ τ , where π1 : U × C → U is the projection on the first factor. The restriction of σ0 to J −1 (0) ∩ U gives a smooth function π2 ◦ τ ◦ σ0|U ∩P0 on J −1 (0) ∩ U with values in C. Therefore, there is a neighbourhood V of p0 in U and a complex-valued function ϕ : V → C such that π2 ◦ τ ◦ σ0|U ∩J −1 (0) = ϕ|V ∩J −1 (0) . The map σV : V → L : q → σV (q) = τ −1 (q, ϕ(q)) is a smooth section of the restriction of L to V . Moreover, if q ∈ J −1 (0) ∩ V , then σV (q) = τ −1 (q, π2 ◦ τ ◦ σ0|U ∩J −1 (0) ) = σ0|U ∩J −1 (0) (q). Hence, σV |V ∩J −1 (0) = σ0|V ∩J −1 (0) . This implies that, for every p ∈ J −1 (0), there is an extension σV of σ0 to a local section σV of L defined in a neighbourhood V of p in P. Using a partition-of-identity argument, we can extend σ0 to a smooth section σ of L. The prequantization action of G on L leaves L0 invariant. Hence, it induces an action of G on L0 such that every G-invariant section of L restricts to a G-invariant section of L0 . Proposition 7.2.8 Every smooth G-invariant section of L0 extends to a smooth G-invariant section of L. Proof Let σ0 be a smooth G-invariant section of L0 . By Proposition 7.2.7, there exists a smooth section σ of L that extends σ0 . For each p ∈ P, let S p be a slice through p for the action of G on P. We denote by σ p the restriction of σ to S p . Averaging σ p over G p , we obtain a G p -invariant section σ˜ p of L|S p such that for each q ∈ S p , σ˜ p (q) = gσ (g −1 q) dμ(g). Gp

7.2 Commutation of quantization and singular reduction at J = 0 171 Let h p ∈ C ∞ (S p ) be a compactly supported G p -invariant non-negative function such that h p = 1 in a neighbourhood U p of p in S p and f = 0 in the complement of an open neighbourhood V p of p that contains the closure of U p . The product gives a new section h p σ˜ p of L|S p , which coincides with σ p in U p and vanishes in the complement of V p . The set G S p is a G-invariant neighbourhood of p in P. We extend h p σ˜ p to a G-invariant section σˇ p of L|G S p as follows. For each q ∈ S p and g ∈ G, we set σˇ p (gq) = h p (q)g σ˜ p (q). If g1 , g2 ∈ G and q1 , q2 ∈ S p are such that g1 q1 −1 g2 g1 q1 , which implies that g2−1 g1 ∈ G p . Since h p and

(7.30) = g2 q2 , then q2 = σ˜ p are G p -invariant,

it follows that h p (q2 )g2 σ˜ p (q2 ) = h p (g2−1 g1 q1 )g2 σ˜ p (g2−1 g1 q1 ) = h p (q1 )g2 (g2−1 g1 σ˜ p (q1 )) = h p (q1 )g1 σ˜ p (q1 ). Hence, σˇ p is well defined by equation (7.30). The collection {GU p | p ∈ P} is a covering of P by G-invariant open sets. Using a G-invariant partition of unity subordinate to this covering, we obtain a G-invariant section σˇ of L. We want to show that σˇ |J −1 (0) = σ0 . Suppose that p ∈ J −1 (0). Then, σ p|S p ∩J −1 (0) = σ |S p ∩J −1 (0) = σ0|S p ∩J −1 (0) . The G-invariance of σ0 implies that σ˜ p|S p ∩J −1 (0) = σ0|S p ∩J −1 (0) . Since h p = 1 on U p ⊆ S p , we have h p σ˜ p|U p ∩J −1 (0) = σ0|U p ∩J −1 (0) , and σˇ p|GU p ∩J −1 (0) = σ0|GU p ∩J −1 (0) . Therefore, σˇ |GU p ∩J −1 (0) = σ0|GU p ∩J −1 (0) . The connection ∇ on L induces a connection ∇ 0 on L0 . Since J −1 (0) has singularities, we have to define what we mean by a connection on a singular space. We interpret ∇ 0 as the covariant differential of sections of L0 . In other words, for each σ0 ∈ S(L0 ), we have a linear map ∇ 0 σ0 : T (J −1 (0)) → L0 : u → ∇u0 σ0 such that ∇ 0 ( f 1 σ1 + f 2 σ2 ) = f 1 ∇ 0 σ1 + d f 1 ⊗ σ1 + f 2 ∇ 0 σ2 + d f 2 ⊗ σ2 for all f 1 , f 2 ∈ C ∞ (J −1 (0)) and σ1 , σ2 ∈ S(L0 ). With this interpretation, the connection ∇ 0 on L0 induced by the connection ∇ on L is given by ∇u0 σ0 = ∇u σ, where σ is an extension of σ0 to a section in S(L).

(7.31)

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Commutation of quantization and reduction

With this definition, the connection ∇ 0 can be described in terms of a Zariski form on J −1 (0); see Chapter 5. Hence, the curvature of ∇ 0 is not defined globally on J −1 (0), but only on the regular part J −1 (0)reg of J −1 (0), because T (J −1 (0)reg ) is locally spanned by global derivations; see Proposition 3.3.15. By assumption, the connection ∇ satisfies the prequantization condition (7.5). Therefore, for every pair of global derivations X 0 , Y0 of C ∞ (J −1 (0)) and each section σ0 of L0 , we have i 0 )σ0 = − ω0K (X 0 , Y0 )σ0 , (∇ X0 0 ∇Y00 − ∇Y00 ∇ X0 0 − ∇[X 0 ,Y0 ]

(7.32)

where ω0K is the Koszul form on Der C ∞ (P0 ), defined by evaluation of ω on vector fields X and Y on P that extend X 0 and Y0 , respectively. In other words, ω0K (X 0 , X 0 ) = ω(X, Y )|J −1 (0) . The ∇-invariant Hermitian form · | · on L restricts to a ∇ 0 -invariant Hermitian form · | · 0 on L0 . We may consider the line bundle λ0 : L0 → P0 with the connection ∇ 0 and the ∇ 0 -invariant Hermitian form · | · 0 as the prequantization structure on (P0 , ω0K ) induced by the prequantization of (P, ω). Since J −1 (0) is G-invariant, the prequantization action of G on L induces an action of G on L0 = L|J −1 (0) , which preserves ∇ 0 and · | · 0 . Let σ0 be a section of L0 . We say that σ0 is covariantly constant along F0 = F ∩ T C (J −1 (0)) if ∇w0 σ0 = 0 for every w ∈ F0 . We denote by S F0 (L0 ) the space of sections of L0 that are covariantly constant along F0 . In other words, S F0 (L0 ) = {σ0 ∈ S(L0 ) | ∇w0 σ0 = 0 for all w ∈ F0 = F ∩ T C (J −1 (0))}. For simplicity, we shall refer to sections in S F0 (L0 ) as polarized sections of L0 . Similarly, we denote by S F (L0 )G the space of G-invariant polarized sections of L0 . Let σ be a polarized section of L; that is, σ is a section of L that is covariantly constant along F. In other words, ∇ X σ = 0 for every vector field X on P with values in F. By the definition of the connection ∇ 0 on L0 , it follows that the restriction σ|J −1 (0) of σ to J −1 (0) is a section of L0 which is covariantly constant along F0 . Therefore, the restriction of sections of L to J −1 (0) gives rise to a linear map F : S F (L) → S F0 (L0 ) : σ → σ|J −1 (0) .

(7.33)

Moreover, if σ is G-invariant, then σ|J −1 (0) is G-invariant. Hence, the restriction of F to G-invariant sections gives a linear map FG : S F (L)G → S F0 (L0 )G : σ → σ|J −1 (0) .

(7.34)

7.2 Commutation of quantization and singular reduction at J = 0 173

Remark 7.2.9 The horizontal double arrow in the diagram (7.21) corresponds to a study of the following questions: (i) Under what conditions is FG : S F (L)G → S F0 (L0 )G one-to-one? (ii) Under what conditions is FG : S F (L)G → S F0 (L0 )G onto? Our next task is to describe the vertical double arrow in the diagram (7.21). We have already described the polarization Fˆ = T C ρ(F ∩ T C (J −1 (0))) of J −1 (0)/G. Now, we need to consider the projection to J −1 (0)/G of the complex line bundle L0 on J −1 (0). We begin with the space Lˆ = L0 /G

(7.35)

ˆ Since L0 is the of G-orbits in L0 , and denote the orbit map by κ : L0 → L. restriction of L to J −1 (0), which is a locally compact subcartesian space, it follows that L0 is a locally compact subcartesian space. Theorem 4.4.7 and Proposition 4.4.8 ensure that Lˆ is a locally compact differential space with the quotient topology. On the other hand, L0 is a subset of L, which is a manifold. Therefore, Lˆ is a subset of a stratified space L/G, which implies that Lˆ is a subcartesian space. Since the projection map λ0 : L0 → J −1 (0) is G-equivariant, there exists a map λˆ : Lˆ → J −1 (0)/G such that the following diagram commutes: L0

κ

λ0

J −1 (0)

/ Lˆ λˆ

ρ0

(7.36)

/ J −1 (0)/G.

Proposition 7.2.10 The map λˆ : Lˆ → J −1 (0)/G is smooth. Proof We need to show that for each f ∈ C ∞ (J −1 (0)/G), the pull-back λˆ ∗ f ˆ But ρ ∗ f ∈ C ∞ (J −1 (0))G , and λ∗ ρ ∗ f ∈ C ∞ (L0 )G . Moreover, is in C ∞ (L). 0 0 0 the commutativity of the diagram (7.36) yields κ ∗ λˆ ∗ f = λ∗0 ρ0∗ f ∈ C ∞ (L0 )G , ˆ which implies that λˆ ∗ f ∈ C ∞ (L). Given p ∈ J −1 (0), we denote the fibre of L0 over p by L0| p ; that is, L0| p =

λ−1 0 ( p).

Proposition 7.2.11 For each p ∈ P, the isotropy group G p of p acts on L0| p as a finite group of C-linear transformations.

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Proof Recall that the action of G on L is given by translations along the integral curves of lifts to L of Hamiltonian vector fields X Jξ , where ξ ∈ g. The infinitesimal action on L p of the Lie algebra g p of Gp is given by lifts Xˆ Jξ of Hamiltonian vector fields X Jξ such that X Jξ ( p) = 0. Hence, the horizontal component of Xˆ Jξ vanishes. The vertical component of X Jξ is proportional to Jξ ( p), which vanishes because p ∈ J −1 (0). Hence, the action of g p on L0| p is trivial. Therefore, G p acts on L0| p as a discrete group. To be more precise, let K p be the maximal subgroup of G p that acts on L0| p by the identity transformation. Then, K p is a normal subgroup of G p and we have an action G p /K p × L0| p → L0| p : ([g], z) → gz of the quotient group G p /K p on L0| p . Since the action of g p on L0| p is trivial, the Lie algebra k p of K p coincides with the Lie algebra g p of G p . Hence, G p /K p is a discrete group, and it is finite because G p is compact. Finally, the action of G p /K p on L0| p is C-linear because G acts on L by complex line bundle isomorphisms. Quotients of Cn by finite groups of C-linear transformations are called Vmanifolds, in the sense of Satake (1957). It follows from Proposition 7.2.11 that the fibres of λˆ : Lˆ → J −1 (0)/G have the structure of V-manifolds. However, Assumption 7.2.6 implies that for each p ∈ J −1 (0), the isotropy group G p of p acts trivially on the fibre L0| p of L0 . Hence, for each orbit Gp in J −1 (0), the fibre of Lˆ over Gp is a complex line C. Corollary 7.2.12 Assumption 7.2.6 ensures that λˆ : Lˆ → J −1 (0)/G is a complex line bundle with a singular base. Consider the restriction L L of the prequantization line bundle L to a symplectic manifold (L , ω L ), where L is a connected component of PH contained in J −1 (0). We denote by λ L : L L → L the restriction to L of the projection map λ : L → P, and by ∇ L the restriction to L L of the connection ∇ on L. It is easy to see that the complex line bundle L L with the connection ∇ L is a prequantization line bundle for (L , ω L ). By construction, prequantization gives an action of the Lie algebra g of G on L. We have assumed that this action integrates to an action of G on L that covers the action of G on P. Let H be a compact subgroup of G, and let L be a connected component of PH . The action on P of the subgroup N L of G given by equation (7.23) preserves L, by definition. Since the action of G on L is a lift of the action of G on P, it follows that the action of N L on L preserves L L . Therefore, we have an action of N L on L L that covers the action of N L on L.

7.2 Commutation of quantization and singular reduction at J = 0 175 Proposition 7.2.13 If L is a connected component of PH contained in J −1 (0), then the action of N L on L L induces a free and proper action of G L = N L /H on L L . Proof

The action of G L = N L /H on L L is given by G L × L L → L L : ([g], z) → gz

(7.37)

for every g ∈ N L . For every g ∈ N L such that g −1 g ∈ H , Assumption 7.2.6 implies that g −1 g z = z for all z ∈ L L . Hence, g z = gz, and [g]z is well defined by equation (7.37). For g1 , g2 ∈ N L and z ∈ L L , we have [g1 ]([g2 ]z) = g1 (g2 z) = (g1 g2 )z = ([g1 ][g2 ])z. Together with Assumption 7.2.6, this implies that equation (7.37) defines an action of G L on L L . Let λ L : L L → L be the restriction of the complex line bundle projection λ : L → P to the domain L and codomain L L . Since λ intertwines the action of G on L and the action of G on P, it follows that λ L intertwines the action of N L on L L and the action of N L of L. Therefore, for every g ∈ N L and z ∈ L L , we have λ L ([g]z) = λ L (gz) = gλ L (z) = [g]λ L (z). This implies that λ L intertwines the action of G L on L L and the action of G L on L. In other words, the action of G L on L L is a lift of the action of G L on L. Since the action of G L on L is proper, it follows that the action of G L on L L is proper. Moreover, suppose that [g]z = z for some [g] ∈ G L , and that z ∈ L L . Then, [g]λ L (z) = λ L ([g]z) = λ L (z). But the action of G L on L is free. Hence, [g] is the identity in G L , which implies that the action of G L on L L is free. Let K be a connected component of L ∩ J −1 (0) ⊆ PH , and let Q = ρ0 (K ) be the corresponding stratum of J −1 (0)/G. We denote the restriction of the prequantization line bundle L to K by L K , and the restriction of λ0 to the domain L K and codomain K by λ K : L K → K . In other words, −1 L K = L0|K = λ−1 0 (K ) = λ K (K ).

We denote the restriction of Lˆ to Q by Lˆ Q ; that is, Lˆ Q = λˆ −1 (Q). Similarly, we denote the restriction of λˆ : Lˆ → J −1 (0)/G by λˆ Q : Lˆ Q → Q. Since λˆ ◦ κ = ρ0 ◦ λ0 , it follows that λˆ ◦ κ(L K ) = ρ0 ◦ λ0 (L0|K ) = ρ0 (K ) = Q.

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Therefore, κ(L K ) ⊆ λˆ −1 (Q) = Lˆ Q , and the restriction of κ to L K has its range in Lˆ Q . We denote the restriction of κ : L0 → Lˆ to the domain L K and codomain Lˆ Q by κ K : L K → Lˆ Q . Proposition 7.2.14 If the action of G on the prequantization line bundle L induces a trivial action of H on the restriction L L of L to L, then: (i) Lˆ Q is a complex line bundle over Q with projection map λˆ Q : Lˆ Q → Q. (ii) Every smooth section σˆ of Lˆ Q has a unique lift to a G L -invariant section σ : K → L K such that σˆ ◦ ρ K = κ K ◦ σ . (iii) The G-invariant connection ∇ on L induces a connection ∇ Q on Lˆ Q . (iv) The Hermitian form · | · on L induces a ∇ Q -invariant Hermitian form · | · Q on L|Q . Proof (i) Consider a point q0 ∈ Q. By Proposition 7.2.1, the action of G L on L induces on K the structure of a left principal fibre bundle with structure group G L , base space Q and principal bundle projection ρ K : K → Q. Hence, there is a neighbourhood U of q in Q and a section τ : U → K of ρ K : K → Q. We denote by G L τ (U ) the collection of points in K that lie on orbits of G L intersecting τ (U ). In other words, G L τ (U ) = {gτ (q) ∈ K | g ∈ G L and q ∈ U }. G L U is an open G-invariant neighbourhood of τ (q0 ) in K . Moreover, the map G L τ (U ) → G L × U : gτ (q) → (g, q)

(7.38)

is a trivialization of G L U = ρ K−1 (U ). The restriction L K of L to K is a complex line bundle over K . Hence, there exist a neighbourhood V of τ (q0 ) in K and a trivializing section σ0 : V → L K of λ K : L K → K . Without loss of generality, we may assume that τ (U ) ⊆ V . We can construct a G L -invariant section σU : G L τ (U ) → L K that extends the restriction of σ0 to τ (U ). For each p = gτ (q) ∈ G L τ (U ), σU ( p) = σU (gτ (q)) = gσ0 (τ (q)). The smoothness of the trivialization map (7.38) implies that σU is smooth. By construction, Lˆ Q = L K /G L is the space of G L -orbits in L K . Hence, the G L -invariant section σU : G L τ (U ) → L K gives rise to a section σˆ U : U → Lˆ Q defined as follows. Points in Lˆ Q are G L -orbits in L K . Similarly, each q ∈ U is the G L -orbit in K through τ (q). In other words, q = G L τ (q). We set σˆ U : U → Lˆ Q : q = G L τ (q) → σˆ U (q) = G L σU (τ (q)).

(7.39)

7.2 Commutation of quantization and singular reduction at J = 0 177 Clearly, λˆ Q ◦ σˆ U is the identity on U . Moreover, σˆ U ◦ ρ K = κ K ◦ σU . Furthermore, if fˆ ∈ C ∞ (Lˆ K |U ) and κU is the restriction of κ K to L K |τ (U ) , then f = fˆ ◦ κU is in C ∞ (L K |τ (U ) ). Hence, for each q ∈ U , fˆ ◦ σˆ U (q) = fˆ ◦ σˆ (ρ K (τ (q))) = fˆ ◦ σˆ U ◦ ρ K (τ (q)) = fˆ(κ K (σU (τ (q))) = fˆ ◦ κU ◦ σU ◦ τ (q). Therefore, σˆ U∗ fˆ = fˆ ◦ σˆ U ∈ C ∞ (U ) because κU , σU and τ are smooth. This implies that the section σˆ U : U → Lˆ Q is smooth. Since G L acts on L K by complex line bundle automorphisms, it follows that the fibres of Lˆ Q are complex lines. By assumption, σ0 : V → L K is nowhere zero. Hence, σU : G L τ (U ) → L K is nowhere zero, which implies that σˆ U : U → Lˆ Q is nowhere zero. Thus, the bundle Lˆ Q is a locally trivial complex line bundle. (ii) Consider the following commutative diagram: LK

κK

/ Lˆ Q Lˆ Q

λK

K

ρK

/ Q,

where λ K : L K → K is the restriction of λ : L → P to the domain L K and codomain K . Here, the vertical arrows denote complex line bundle projections and the horizontal arrows denote principal G L -bundle projections. Let σˆ : Q → Lˆ Q be a section of Lˆ Q . We define its G-invariant lift σ : K → L K to a section of L K as follows. For each p ∈ K , ρ K ( p) = G L p is the orbit through p of the action of G L on L K . Moreover, σˆ (ρ K ( p)) is the orbit of the action of G L on L |K that covers G L ( p). Since the action of G L on L K is the lift of a free and proper action of G L on K , it follows that the fibre λ−1 K ( p) and the orbit σˆ (ρ K ( p)) in L K intersect at a single point. We denote by σ ( p) the unique point of intersection of the fibre λ−1 ˆ (ρ K ( p)) K ( p) and the orbit σ in L K . Clearly, λ K (σ ( p)) = p and κ K (σ ( p)) = σˆ (ρ K ( p)). Repeating this construction at every p ∈ K , we obtain a section σ : K → L K such that κ K ◦ σ = σˆ ◦ ρ K . It remains to show that σ is smooth. We have shown in part (i) of this proof that for each q0 ∈ Q, there exist a neighbourhood U of q0 in Q and trivializing sections σˆ U : U → Lˆ Q and σU : G L τ (U ) → L K such that σˆ U ◦ρ K = κ K ◦σU .

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Since σˆ is smooth, there exists a smooth complex-valued function fˇ on U such that the restriction σˆ |U of σˆ to U can be expressed in the form σˆ |U = fˇσˆ U . We can extend the function fˇ on U to a G L -invariant function on G L τ (U ) by setting f (gq) = fˇ(q) for every gq ∈ G L τ (U ). By construction, for every p ∈ K , σ ( p) is the unique point of intersection of the fibre λ−1 ˆ (ρ K ( p)) in L K . On the other hand, equation K ( p) and the orbit σ (7.39) implies that σU ( p) is the unique intersection of the orbit σˆ U (q) and the fibre λ−1 K ( p) for every p ∈ τ (U ). Since the action of G L on L K is linear, it follows that for every p ∈ τ (U ), σ ( p) = σ|U ( p) = f (ρ K ( p))σU (q). Moreover, the sections σ : K → L K and σU : G L τ (U ) → L K are G L invariant. Hence, σ restricted to G L τ (U ) coincides with a smooth section (ρ K∗ f )σU . This holds for a neighbourhood U of every q ∈ Q. Therefore, σ is smooth. (iii) Since K is a submanifold of P, the G-invariant connection ∇ on L restricts to a G L -invariant connection ∇ K on L K . We define a connection ∇ Q on L|Q as follows. Let σ be a G L -invariant section of L K . We denote by ρ K ∗ σ the section of L|Q such that ρ K ∗ σ ◦ ρ K = κ K ◦ σ . Conversely, if σˆ is a section of L|Q , then we denote by lift σˆ the G L -invariant section σ constructed above. This satisfies the condition σˆ ◦ ρ K = κ K ◦ lift σˆ . A G L -invariant vector field X on K pushes forward to a unique vector field Xˆ = ρ K ∗ X on Q. Conversely, every vector field Xˆ on Q lifts to a G L -invariant vector field X K on K , but this lift is not unique. Given a section σˆ of L |Q and a vector field Xˆ on Q, we set ∇ Qˆ σˆ = ρ K ∗ (∇ XK (lift σˆ )). X

(7.40)

Since ∇ K , X and lift σˆ are G L -invariant, it follows that ∇ XK (lift σˆ ) pushes forward to a section of L|Q . We need to show that the right-hand side of equation (7.40) does not depend on the choice of the G L -invariant vector field X such that Xˆ = ρ K ∗ X . Suppose that X is another G L -invariant vector field on K satisfying Xˆ = ρ K ∗ X . Then Y = X − X is a vertical G L -invariant vector field on K . Since the vertical directions are spanned by the restrictions to K of Hamiltonian vector fields of J L , equation (7.24) gives Y =

k i=1

f i X JL[ξi ] |K =

k i=1

f i X Jξi |L |K ,

7.2 Commutation of quantization and singular reduction at J = 0 179 where ξ1 , . . . , ξk ∈ n L . The G-invariance of lift σˆ implies that (P Jξ )|K lift σˆ = 0 for every ξ ∈ n L , where (P Jξ )|K is the restriction of the partial differential operator P Jξ to points of K . Taking equation (7.8) into account, we obtain (−i∇ XKJ + Jξ |K )lift σˆ = 0 ξ

for all ξ ∈ n L . But K ⊆ J −1 (0), which implies that Jξ |K = 0. Therefore, ∇ XKJ (lift σˆ ) = 0 for every ξ ∈ n L , which ensures that ∇YK (lift σˆ ) = 0. Hence, ξ

∇ XK (lift σˆ ) is independent of the choice of the G L -invariant vector field X such Q that Xˆ = ρ K ∗ X . Hence, ∇ ˆ σˆ is well defined by equation (7.40). X For f ∈ C ∞ (Q), ρ K∗ f is a G L -invariant function on K , and lift f σˆ = ∗ (ρ K f )lift σˆ , so that K ∇ Q ˆ σˆ = ρ K ∗ (∇(ρ ∗ fX

K

ˆ )) f )X (lift σ

= ρ K ∗ ((ρ K∗ f )∇ XK (lift σˆ ))

= f ∇ Qˆ σˆ , X

and ∇ Qˆ ( f σˆ ) = ρ K ∗ (∇ XK ((ρ K∗ f )lift σˆ )) X

= ρ K ∗ (X (ρ K∗ f )) lift σˆ + (ρ K∗ f )∇ XK (lift σˆ ) Q = Xˆ ( f )σˆ + f ∇ ˆ σˆ . X

Hence, ∇ Q gives a connection on L Q . (iv) The Hermitian form · | · on L is invariant under the parallel transport defined by the connection ∇. Since the action of G L on L K is given by parallel transport along integral curves of the Hamiltonian vector fields of components of JL , it follows that · | · induces a Hermitian form · | · Q on L|Q such that ρ K∗ σˆ 1 | σˆ 2 Q = lift σˆ 1 | lift σˆ 2 . We have shown that for every symplectic stratum (Q, ω Q ) of J −1 (0)/G, the restriction Lˆ Q of Lˆ to Q is a prequantization line bundle of (Q, ω Q ). In Proposition 7.2.5, we have shown that if our G-invariant polarization F of (P, ω) has a clean intersection with J −1 (0), then F induces a polarization Fˆ Q of (Q, ω). Thus, J −1 (0)/G is stratified by prequantized, polarized symplectic manifolds.2 It is tempting to define quantum states of the quantization of J −1 (0)/G to be sections σˆ of Lˆ such that for each symplectic stratum (Q, ω Q ), the restriction of σˆ to Q gives a section of Lˆ Q that is covariantly constant along FQ . 2 This result was obtained for a Kähler polarization by Huebschmann (2006).

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However, such a definition is not sufficiently restrictive, because it ignores the components of Fˆ = T C ρ(F ∩ T C (J −1 (0))) that are not tangential to strata of J −1 (0)/G. In order to define a quantization of J −1 (0)/G that might satisfy the principle of commutation of quantization and reduction, we need to extend some of the results of Proposition 7.2.14 to the complex line bundle Lˆ over the singular space J −1 (0). Proposition 7.2.15 Each G-invariant section σ0 of L0 pushes forward to a smooth section σˆ of Lˆ such that κ ◦ σ0 = σˆ ◦ ρ0 . Proof The projection of a point p ∈ J −1 (0) to J −1 (0)/G is the G-orbit through p. In other words, ρ0 ( p) = Gp. If z ∈ L0 , then κ(z) = Gz. Therefore, κ(gz) = Ggz = Gz for every g ∈ G. Define σˆ : J −1 (0)/G → Lˆ by σˆ (ρ0 ( p)) = κ(σ0 ( p)). For every g ∈ G, we have σˆ (ρ0 (gp)) = κ(σ0 (gp)) = κ(gσ0 ( p)) = κ(σ0 ( p)) = σˆ (ρ0 ( p)). Hence, σˆ : Pˆ → Lˆ is well defined, and κ ◦ σ0 = σˆ ◦ ρ0 . In order to show that σˆ is smooth, consider the pull-back σˆ ∗ f of a function ˆ We need to verify that σˆ ∗ f ∈ C ∞ (J −1 (0)/G). By the definif ∈ C ∞ (L). tion of the quotient differential structure, it suffices to show that ρ0∗ σˆ ∗ f ∈ C ∞ (J −1 (0)). However, κ ◦ σ0 = σˆ ◦ ρ0 implies that ρ0∗ σˆ ∗ f = f ◦ σˆ ◦ ρ0 = f ◦ κ ◦ σ0 = σ0∗ κ ∗ f . But κ ∗ f ∈ C ∞ (L0 ) and σ0∗ κ ∗ f ∈ C ∞ (J −1 (0)), because κ and σ0 are smooth. Proposition 7.2.16 Every smooth section σˆ of Lˆ can be lifted to a unique G-invariant section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . Proof

Recall that we have the following commutative diagram: L0

κ

λ0

J −1 (0)

/ Lˆ λˆ

ρ0

/ J −1 (0)/G.

For each p ∈ J −1 (0), the projection of p to J −1 (0)/G is the G-orbit through p; that is, ρ0 ( p) = Gp. Similarly, for each z ∈ L0 , we have κ(z 0 ) = Gz and λˆ (Gz) = G λˆ (z).

7.2 Commutation of quantization and singular reduction at J = 0 181 ˆ For each p ∈ J −1 (0), σˆ (Gp) Let σˆ : J −1 (0)/G → Lˆ be a section of L. −1 is the orbit in L0 covering the orbit Gp in J (0). Hence, there exists a point z ∈ σˆ (Gp) such that p = λ0 (z). Therefore, σˆ (Gp) = Gz. We are going to show that z is uniquely defined by the conditions z ∈ σˆ (Gp) and p = λ0 (z). Suppose that there exists z ∈ σˆ (Gp) such that p = λ0 (z ). Since z and z are in Gz, there exists g ∈ G such that z = gz. Moreover, λ0 (z) = λ0 (z ) = p implies that g is in the isotropy group H = {g ∈ G | gp = p} of p. Let L be the connected component of PH containing p. Assumption 7.2.6 ensures that the action of g on the restriction L L of L to L is trivial. Hence, z = z. Since z is uniquely defined by the conditions z ∈ σˆ (Gp) and p = λ0 (z), there is a well-defined map σ0 : J −1 (0) → L0 such that σ0 ( p) = z. The condition p = λ0 (z) implies that σ0 is a section of L0 , which need not be smooth. For every g ∈ G, σ0 (gp) is the unique element of σˆ (Ggp) = σˆ (Gp) = Gz such that λ0 (σ0 (gp)) = gp. Since gz ∈ Gz and λ0 (gz) = gλ0 (z) = gp, it follows that σ0 (gp) = gσ0 ( p). Hence, σ0 is G-invariant. In Proposition 7.2.16, we have shown that every smooth section σˆ of Lˆ can be lifted to a unique G-invariant section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . However, we have not proved that the section σ0 is smooth. Throughout the remainder of this section, we make the following assumption. Assumption 7.2.17 Each section σˆ of Lˆ lifts to a unique G-invariant smooth section σ0 of L0 such that κ ◦ σ0 = σˆ ◦ ρ0 . Our next step is to discuss the notion of a connection ∇ˆ on the complex line bundle Lˆ over a singular space J −1 (0)/G induced by the connection ∇ 0 on L0 ; see equation (7.31). The main difficulty is that the map Tρ0 : T (J −1 (0)) → T (J −1 (0)/G) need not be onto. However, in geometric quantization, we are interested mainly in a partial connection on the complex prequantization line ´ bundle covering the polarization (Sniatycki, 1980). Definition 7.2.18 Let L be a complex line bundle over a singular space S, and let F ⊆ T C S be a complex distribution on S. A partial connection on L over F is given by a covariant derivative operator ∇ which associates to each section σ of L a C-linear map ∇σ : F → L : w → ∇w σ that commutes with the projection to S. Moreover, we assume that ∇(σ1 + σ2 ) = ∇σ1 + ∇σ2 , and for every section σ of L and every f ∈ C ∞ (S), ∇w ( f σ ) = w( f )σ + f (x)∇w σ, for all w ∈ Fx .

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Clearly, the restriction of the connection ∇ on the prequantization line bundle L of (P, ω) to L0 = L |J −1 (0) and F0 = F ∩ T C J −1 (0) gives rise to a connection ∇ 0 on L0 over F0 in the sense of Definition 7.2.18. This connection induces a connection ∇ˆ on Lˆ over Fˆ such that for every w ∈ F0 , ∇ˆ T C ρ0 (w) σˆ = κ(∇w0 σ0 ),

(7.41)

where σ0 is the unique G-invariant section of L 0 such that κ ◦ σ0 = σˆ ◦ ρ0 , which exists by Assumption 7.2.17. Proposition 7.2.19 The connection ∇ˆ is well defined by equation (7.41). Proof Let w1 and w2 be two vectors in F0 attached at the same point p and such that T C ρ0 (w1 ) = T C ρ0 (w2 ). Then, T C ρ0 (w1 − w2 ) = 0 and w1 − w2 ∈ T pC (Gp). But T p (Gp) is spanned by Hamiltonian vector fields X Jξ of momenta Jξ corresponding to elements ξ ∈ g. By assumption, σ0 is a G-invariant section of L0 = L|J −1 (0) . This implies that (i∇ X Jξ + Jξ )σ0 = 0 for every ξ ∈ g; see

equation (7.8). But Jξ ( p) = 0 for every p ∈ J −1 (0). Hence, ∇ X Jξ σ0 = 0 for all ξ ∈ g. Therefore, ∇w1 −w2 σ0 = 0, which implies that ∇w1 σ0 = ∇w2 σ0 . Thus, the right-hand side of equation (7.41) is independent of the choice of the vector w ∈ F0 attached at p that projects to T C ρ0 (w). Suppose now that w1 and w2 are vectors in F0 , attached at p1 and p2 , respectively, such that T C ρ0 (w1 ) = T C ρ0 (w2 ). Then, there exists g ∈ G such that gp1 = p2 . Moreover, the G-invariance of F0 implies that T C g (w1 ) is also in F0 , that it is attached at p2 and that T C g (w1 ) − w2 ∈ ker T C ρ0 . The G-invariance of σ0 and ∇ 0 implies that g(∇w0 1 σ0 ) = ∇T0 C Since ∇T0 C

g (w1 )−w2

g (w1 )

σ0 = ∇T0 C

g (w1 )−w2

σ0 + ∇w0 2 σ0 .

σ0 = 0, it follows that κ(∇w0 1 σ0 ) = κg((∇w0 2 σ0 )).

Hence, ∇ˆ is well defined by equation (7.41). Proposition 7.2.20 A smooth section σˆ of Lˆ is covariantly constant along Fˆ if and only if the lift of σˆ to a unique G-invariant section σ0 of L 0 is covariantly constant along F0 . Proof The proof is a straightforward consequence of the definition of the ˆ partial connection ∇ˆ over F.

7.2 Commutation of quantization and singular reduction at J = 0 183 Let S(L0 )G denote the space of smooth G-invariant sections of L0 , and let ˆ be the space of smooth sections of L. ˆ Proposition 7.2.15 ensures that S(L) there is a linear map ˆ : σ0 → (σ0 ) such that κ ◦ σ0 = (σ0 ) ◦ ρ0 . (7.42) : S(L0 )G → S(L) Clearly, the map is one-to-one. In Assumption 7.2.17, we assumed that ˆ is onto. This assumption enables us to define a partial : S(L0 )G → S(L) ˆ connection ∇ˆ on Lˆ that covers F.

Let S F0 (L0 )G denote the space of smooth G-invariant sections of L0 that are ˆ be the space of smooth sections covariantly constant along F0 , and let S Fˆ (L) ˆ Proposition 7.2.20 ensures that the of Lˆ that are covariantly constant along F. G restriction of to S(L0 ) gives rise to a linear isomorphism

ˆ : σ0 → F (σ0 ) such that κ ◦ σ0 = (σ0 ) ◦ ρ0 . F0 : S F0 (L0 )G → S Fˆ (L) 0 (7.43) Composing F0 with the restriction map FG : S F (L)G → S F0 (L0 )G : σ → σ|J −1 (0) given in equation (7.34), we obtain a linear map ˆ : σ → F0 (σ|J −1 (0) ). F0 ◦ FG : S F (L)G → S Fˆ (L) If F0 ◦ FG is an isomorphism, then the space S F (L)G that corresponds to ˆ of a trivial representation of G on S F (L) is isomorphic to the space S Fˆ (L) −1 quantization data on J (0)/G. Recall that we have defined the term ‘commutation of quantization and reduction’ as a programme of determining the spectral measure dμ(λ) in terms of the quantization of the symplectic reduction of the inverse images of co-adjoint orbits under the momentum map J : P → g∗ . Here, we have considered the trivial representation corresponding to λ = 0. If F0 ◦ FG is an isomorphism, then the space S F (L)G that corresponds to a trivial representaˆ of quantization data on tion of G on S F (L) is isomorphic to the space S Fˆ (L) −1 J (0)/G. We would like to determine the contribution of the trivial representation to the spectral measure dμ(λ) at λ = 0. If F0 ◦ FG is a monomorphism but not an isomorphism, then we might still be able to determine the space ˆ of quantization data on J −1 (0)/G. However, S F (L)G from the space S Fˆ (L) in this case we would have to identify the range of F0 ◦ FG . If F0 ◦ FG has a non-zero kernel, there would be a loss of information in passing from S F (L)G ˆ to S Fˆ (L). The discussion above was given under Assumption 7.2.17, which requires ˆ be a vector space epimorphism. By Proposithat : S(L0 )G → S(L) ˆ is one-to-one. If is not an tion 7.2.15, the map : S(L0 )G → S(L)

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epimorphism, we cannot use equation (7.41) to define a connection ∇ˆ on Lˆ ˆ of sections of Lˆ that ˆ and we have no notion of the space S ˆ (L) over F, F ˆ are covariantly constant along F. In this case, we can use the restriction map FG : S F (L)G → S F0 (L0 )G to relate the space S F (L)G of G-invariant sections of L that are covariantly constant along F to the space S F0 (L0 )G of their restrictions to J −1 (0). If FG is a monomorphism, then we might still be able to determine the space S F (L)G from the a knowledge of the range of FG in S F0 (L0 )G .

7.3 Special cases 7.3.1 The results of Guillemin and Sternberg We discuss first the case investigated by Guillemin and Sternberg (1982). These authors considered a Kähler polarization F on a compact symplectic manifold (P, ω), and a Hamiltonian action of a compact connected Lie group G that is free on the zero level of the momentum map J . Hence, P is a Kähler manifold, the prequantization line bundle L of (P, ω) is holomorphic and the space S F (L) consists of holomorphic sections of L. Moreover, J −1 (0) is a submanifold of P, and J −1 (0)/G is a quotient manifold of J −1 (0). This means that the stratification of J −1 (0)/G has only one stratum, which we denote by Q. In other words, Q = J −1 (0)/G. With this notation, Lˆ is a line bundle over Q. By Proposition 7.2.5, the polarization F of (P, ω) gives rise to a polarization FQ of (Q, ω Q ). It is easy to show that the assumption that F is Kähler implies that FQ is Kähler. Hence, Q has the structure of a Kähler manifold. By Proposition 7.2.14, the prequantization structure of L, consisting of a connection ∇ and a connection-invariant Hermitian form · | · , induces ˆ With this structure, Lˆ is a a prequantization structure ∇ Q and · | · Q on L. ˆ of polarized sections holomorphic line bundle over Q, and the space S FQ (L) ˆ of L consists of holomorphic sections. ˆ Proposition 7.2.14 ensures that the monomorphism : S(L0 )G → S(L) defined by equation (7.42) is an isomorphism of vector spaces. Moreover, the results of Guillemin and Sternberg show that the restriction of to polarized ˆ sections gives a vector space isomorphism F : S F (L0 )G → S FQ (L). G G G Now consider the restriction map F : S F (L) → S F (L0 ) : σ → σ|J −1 (0) ; see equation (7.33). Guillemin and Sternberg showed that FG is a vector space isomorphism. Their approach makes use of the action on P of the complex Lie group G C such that G is the real form of G C . It should be noted that the compactness of P and G implies that the spaces of polarized sections considered here are finite-dimensional. Hence, the results of

7.3 Special cases

185

Guillemin and Sternberg are formulated in terms of equality of the dimensions of the spaces under consideration. Sjamaar (1995) extended the results of Guillemin and Sternberg to the case when the action of the compact Lie group G on J −1 (0) need not be free. He retained the assumption that P is a compact Kähler manifold. For a sufficiently ‘positive’ polarization, the multiplicity m α can be obtained from the Riemann–Roch formula; see Guillemin and Sternberg (1982). This result has been studied by several authors. We shall not pursue this line of research here, however, because we are interested in cases in which the multiplicities may be infinite.

7.3.2 Kähler polarization without compactness assumptions We consider here the case when F is a Kähler polarization of a symplectic manifold (P, ω) and G is a connected Lie group with a proper Hamiltonian action on (P, ω) that preserves F. We do not assume that P or G is compact. In this case, we have no information about whether the projection : S(L0 )G → ˆ is an epimorphism. Similarly, we do not know whether G : S F (L)G → S(L) F S F (L0 )G : σ → σ|J −1 (0) is an epimorphism. However, we have the following result. Proposition 7.3.1 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a Kähler polarization F, and that J −1 (0) contains a Lagrangian submanifold of (P, ω). Then the restriction map F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism. Similarly, FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. Proof We show first that the complexified tangent space TC P of P along the Lagrangian manifold is the direct sum of T C and F| . Consider a point p ∈ . Suppose that w ∈ F p ∩ T pC . Since F is Kähler, iω(w, w) ¯ ≥ 0, and iω(w, w) ¯ = 0 implies that w = 0. On the other hand, from the assumption that is a Lagrangian submanifold, it follows that if w ∈ T C , then w¯ ∈ T C and ω(w, w) ¯ = 0. Hence, F p ∩ T pC = 0. Since dimC F = dimC T C = 1 2 dim P, it follows that TC P = F| ⊕ T C . Since F is a Kähler polarization of (P, ω), it follows that P has the structure of a Kähler manifold, that the prequantization line bundle L is holomorphic and that polarized sections of L are holomorphic. Let σ ∈ S F∞ (L) be in the kernel of . This means that σ is a holomorphic section of L that vanishes on J −1 (0). In particular, σ is identically zero on . Hence, all derivatives of σ in

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the directions in T C are zero. On the other hand, all derivatives of σ in the directions in F are zero because σ is holomorphic. Hence, all derivatives of σ vanish on ⊂ P. Since σ is holomorphic, it follows that σ = 0. It follows that ker F = 0 and that F is a monomorphism. Since F (σ ) is G-invariant if σ is G-invariant, it follows that FG : ∞ S F (L)G → S F∞0 (L0 )G is a monomorphism. If the Hamiltonian action of G on (P, ω) is free and proper, then it is easy to show that the zero level of the momentum map contains a Lagrangian submanifold of (P, ω). Hence, Proposition 7.3.1 implies that FG is a monomorphism whenever the action of G is free and proper. On the other hand, for a proper non-free Hamiltonian action of a compact group G on a symplectic manifold (P, ω), the zero level of the momentum map need not contain a Lagrangian submanifold of (P, ω). Hence, we cannot use Proposition 7.3.1 to obtain Sjamaar’s results (Sjamaar, 1995). If P and G are not compact, then the quantization representation R of G on S F∞ (L) may be infinite-dimensional, and the space S F∞ (L)G of G-invariant sections in S F∞ (L) may also have infinite dimension. Unitarization leads from R to a unitary representation U on a Hilbert space H F . With some abuse of terminology, we can say that a section σ in S F∞ (L) is in the Hilbert space H F if unitarization leads to a vector in H F . In other words, we ignore the details of the unitarization process and continue the discussion as if the Hilbert space H F were the completion of a dense open subset of S F∞ (L) with respect to some scalar product (· | ·). The aim of the programme of commutation of quantization and reduction is to discuss the decomposition of the unitary representation U into irreducible unitary representations U α of G; see equation (7.1). Here, we concentrate on the trivial representation of G occurring in U. Let HG F denote the closed subspace of H F consisting of G-invariant vectors in H F . If we denote the trivial representation of G by U 0 , the space HG F corresponds to the atomic part of the spectral measure dμ(λ) at λ = 0. Suppose that quantization commutes with reduction on the zero level of the momentum map. In other words, we assume that the restriction map FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. This assumption means that all sections in S F∞ (L)G can be uniquely determined in terms of sections in range FG ⊆ S F∞0 (L0 )G . The scalar product (· | ·) on S F∞ (L), restricted to S F∞ (L)G , can be pushed forward by to a scalar product (· | ·)0 on range FG . For each normalizable σ1 and σ2 in S F∞ (L)G , we have (σ1 | σ2 ) = ( FG (σ1 ) | FG (σ2 ))0 .

7.3 Special cases

187

It would be of great help if we could describe ( FG (σ1 ) | FG (σ2 ))0 directly in terms of sections FG (σ1 ) and FG (σ2 ) in range FG . This would allow us to G G determine HG F in terms of the reduced data in range F . Sections in range F that are not normalizable in terms of (· | ·)0 correspond to the continuous part of the spectral measure dμ(λ) at λ = 0. Using unitarization in terms of half-forms, Hall and Kirwin obtained an explicit expression for ( FG (σ1 ) | FG (σ2 ))0 in terms of FG (σ1 ) and FG (σ2 ) in the case when F is a Kähler polarization of a compact symplectic manifold (P, ω) and G is a compact group with a free action on J −1 (0) (Hall and Kirwin, 2007). The case when the action of G on J −1 (0) is not free was investigated by Li (2008), who found estimates for (· | ·)0 . For compact G and P, all representations considered here are finite-dimensional and the spectral measure dμ(λ) is atomic. This implies that equation (7.1) can be rewritten as a direct sum. Nevertheless, the results of Hall, Kirwin and Li may serve as a starting point for investigations of the case when neither G nor P is compact. It is not necessary to know the scalar product (· | ·)0 on range FG in order G to determine HG F in terms of the reduced data in range F . It would suffice if we knew any other scalar product (· | ·)1 such that m(σ0 | σ0 )1 ≤ (σ0 | σ0 )0 ≤ M(σ0 | σ0 )1

(7.44)

for some positive numbers m and M, and all σ0 ∈ range . The inequalities (7.44) imply that σ0 is normalizable with respect to (· | ·)1 if and only if it is normalizable in terms of (· | ·)0 .

7.3.3 Real polarization We say that a polarization F of (P, ω) is real if F = D ⊗ C, where D is an involutive Lagrangian distribution on (P, ω). A real polarization F is strongly admissible if the space P/D of integral manifolds of D is a quotient manifold of P. In this case, D is spanned by Hamiltonian vector fields. Moreover, for each integral manifold of D, the restriction ∇| of ∇ to is a flat connection on L | . Proposition 7.3.2 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a strongly admissible real polarization F = D ⊗ C. If every integral manifold of D intersects J −1 (0), then the restriction maps F : S F∞ (L) → S F∞0 (L0 ) and FG : S F∞ (L)G → S F∞0 (L0 )G are monomorphisms. If the integral manifolds of D are simply connected, then F and FG are epimorphisms.

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Proof Each section σ ∈ S F∞ (L)G is covariantly constant along F = D ⊗ C. Hence, σ is covariantly constant along D. This means that, for every integral manifold of D, the restriction σ| is covariantly constant on . Suppose that σ ∈ ker ; that is, σ|J −1 (0) = 0. Let be an integral manifold of D. By assumption, the intersection ∩ J −1 (0) is not empty, and there exists p ∈ ∩ J −1 (0). Then, σ ( p) = 0. Since σ is covariantly constant on , it follows that σ| = 0. This holds for every integral manifold of D. Therefore, σ = 0. Hence, F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism, which implies that FG : S F∞ (L)G → S F∞0 (L0 )G is a monomorphism. For each integral manifold of D, the connection ∇| on L| is flat. The assumption that is simply connected implies that the holonomy group of this connection is trivial. This means that for every point z ∈ L, we can use parallel transport along the integral manifold through the point p = λ(z) to obtain a covariantly constant section σ p of L| such that σ p ( p) = z. Now consider a section σ0 ∈ S F∞0 (L0 )G . For each p ∈ J −1 (0), we denote by p the integral manifold p of D through p, and by σ p the unique section of L| p such that σ p ( p) = σ0 ( p). Since σ0 is covariantly constant along D|J −1 (0) , it follows that σ p is independent of the choice of the point p in p ∩ J −1 (0). By assumption, all integral manifolds of D intersect J −1 (0). Hence, we have a map σ : P → L such that for every integral manifold of D, the restriction σ| of σ to coincides with σ p for any p ∈ ∩ J −1 (0). By construction, σ is invariant under parallel transport along D. The smoothness of the parallel transport and the assumed smoothness of σ0 imply that σ is smooth. Hence, F : S F∞ (L) → S F∞0 (L0 ) is an epimorphism. The G-invariance of D and ∇ implies that the action of G commutes with parallel transport along integral manifolds of D. Hence, the assumed G-invariance of σ0 implies that σ is G-invariant. Therefore, FG : S F∞ (L)G → S F∞0 (L0 )G is an epimorphism. Example 7.3.3 Consider the case when P = T ∗ R3 is the cotangent bundle R3 , ω is the canonical symplectic form of T ∗ R3 and the polarization F is the complexification of ker T ϑ, where ϑ : T ∗ R3 → R3 is the cotangent bundle projection. If x, y, z are the canonical coordinates on R3 and px , p y . pz are the corresponding conjugate momenta, then the canonical 1-form θ of T ∗ R3 is θ = px d x + p y dy + pz dz, and ω = dθ = dpx ∧ d x + dp y ∧ dy + dpz ∧ dz.

7.3 Special cases

189

The polarization F = D C , where D = span

∂ ∂ ∂ , , ∂ p x ∂ p y ∂ pz

,

and integral manifolds of D are fibres of the cotangent bundle projection ϑ : T ∗ R5 → R3 : ( p, x) = ( px , p y . pz , x, y, z) → x = (x, y, z). Consider the action of G = SU (2) on P = T ∗ R3 given by (eiϕ , ( px , p y . pz , x, y, z)) → (cos ϕpx + sin ϕp y , − sin ϕpx + cos ϕp y , pz , cos ϕx + sin ϕy, − sin ϕx + cos ϕy, z). This action is Hamiltonian, and its momentum map is the z-component of the angular momentum for the action of the rotation group on T R3 ; that is, J = px y − p y x. Moreover, this action preserves the cotangent bundle polarization. For x ∈ R3 , the cotangent space Tx∗ R3 is the integral manifold of D projecting to x. The intersection of Tx∗ R3 with J −1 (0) depends on x as follows. If x = (0, 0, z), then J −1 (0) ∩ Tx∗ R3 = {( px , p y , pz , 0, 0, z) ∈ Tx∗ R3 | ( px , p y , pz ) ∈ R3 }. If x = (0, y, z), where y = 0, then J −1 (0) ∩ Tx∗ R3 = {(0, p y , pz , 0, y, z) ∈ Tx∗ R3 | ( p y , pz ) ∈ R2 }. Similarly, if x = (x, 0, z), where x = 0, then J −1 (0) ∩ Tx∗ R3 = {( px , 0, pz , 0, y, z) ∈ Tx∗ R3 | ( px , pz ) ∈ R2 }. Finally, if x = (x, y, z), where x = 0 and y = 0, then J −1 (0) ∩ Tx∗ R3 = {(xt, yt, pz , x, y, z) ∈ Tx∗ R3 | (t, pz ) ∈ R2 }. Thus, every integral manifold of D is simply connected and has a nonempty intersection with J −1 (0). By Proposition 7.3.2, the restriction map FG : S F∞ (L)G → S F∞0 (L0 )G is an isomorphism.

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Commutation of quantization and reduction

7.4 Non-zero co-adjoint orbits In this section, we use the shifting trick discussed in Section 6.6 to extend the results for J −1 (0)/G obtained in the previous section to quantization of J −1 (O)/G, where O is a quantizable co-adjoint orbit. Let O ⊆ g∗ be a co-adjoint orbit of G, and let be the Kirillov–Kostant– Souriau form on O; see Section 6.1.1. We assume that O is an embedded submanifold of g∗ . Moreover, we assume that is quantizable, which means that it admits a prequantization line bundle λ O : L O → O with a connec−1 tion ∇ O and a connection-invariant Hermitian form · | · O such that 2π ˜ ω), ˜ where is the curvature of L O . We construct a symplectic manifold ( P, P˜ = P × O, with projections π1 : P˜ → P and π2 : P˜ → Oμ , and ω˜ = π1∗ ω ⊕ (−π2∗ ). ˜ given by The action of G on P, ˜ : G × P˜ → P˜ : (g, ( p, λ)) → ˜ g ( p, λ) = (g ( p), Adg∗ λ), is Hamiltonian with an AdG∗ -equivariant momentum map J˜ = π1∗ J − π2∗ I , where I : O → g ∗ is the inclusion map. We denote the space of G-orbits in ˜ ˜ P˜ by R˜ = P/G and the corresponding orbit map by ρ˜ : P˜ → R. According to Theorem 6.6.3, J −1 (O)/G and J˜−1 (0) are isomorphic Poisson differential spaces. We use this result to define the geometric quantization of J −1 (O)/G as being given by the geometric quantization of J˜−1 (0)/G discussed in the preceding sections. Our aim in this section is to express the quantization data for J˜−1 (0)/G in terms of data obtained by the geometric quantization of (P, ω) and (O, ). Let us consider geometric quantization of (O, ) in terms of the prequantization line bundle λ O : L O → O and a positive polarization FO . We denote by C F∞ (O) the space of functions in C ∞ (O) such that their Hamiltonian vector fields preserve the polarization FO . Similarly, we denote by S F∞ (L O ) the space of smooth sections of L O that are covariantly constant along FO . Quantization associates to each f ∈ C F∞ (O) a linear operator Q Of on the space S F∞ (L O ) given by Q Of σ O = (−i(∇ O ) X f + f )σ O ,

(7.45)

for every σ O ∈ S F∞ (L O ). Here, X f denotes the Hamiltonian vector field on O defined in terms of the symplectic form , and ∇ O is the covariant derivative operator on sections of L O . We denote by R O the representation of G on

7.4 Non-zero co-adjoint orbits

191

S F∞ (L O ) obtained by integration of the representation ξ → (i)−1 Q O Iξ of g. Note that Iξ = I | ξ is a smooth function on O. Complex conjugation z → z¯ in L O is an automorphism of L O considered as a real vector bundle over O, but it conjugates the complex structure. We denote by L O the complex line bundle over O with the conjugated complex structure. Note that for every section σ O of L O , the complex conjugate σ¯ O is O a section of L O . We denote by P the prequantization map given by the line O bundle L O with connection ∇ . For every ξ ∈ g∗ , the co-adjoint action of exp tξ on O is given by translations along integral curves of the Hamiltonian vector field of X Iξ relative to the symplectic form , where I : O → g∗ is the inclusion map. We denote by X − Iξ the Hamiltonian vector field of Iξ relative to the symplec-

−ξ . This means tic form − on O. We have X − Iξ = −X Jξ = X J−ξ = X

that X − Iξ corresponds to the action of exp(−tξ ) on O. On the other hand, − − ξ X− I−ξ = X −Iξ = −X Iξ = X Iξ = X , which means that the momentum map for the action of exp tξ on (O, −) is −Iξ .

Proposition 7.4.1 The connection ∇ O on L O induces a connection ∇ O with curvature 2π1 such that, for all σ O ∈ S ∞ (L O ), O

∇ σ¯ O = ∇ O σ O .

(7.46)

For each f ∈ C ∞ (O) and σ O ∈ S ∞ (L O ), O

P f σ¯ O = P Of σ O .

(7.47)

Moreover, the ∇ O -invariant Hermitian form · | · O on L O induces a ∇ O invariant Hermitian form · | · − O on L O such that O O σ¯ 1O | σ¯ 2O − O = σ2 | σ1 O

(7.48)

for all σ1O , σ2O ∈ S ∞ (L O ). Proof For every real f ∈ C ∞ (O) and X ∈ X(O), and each σ O ∈ S ∞ (L O ), equation (7.46) gives O

O

∇ f X σ¯ O = ∇ OfX σ O = f ∇ XO σ O = f ∇ XO σ O = f ∇ X σ¯ O , and O

∇ X ( f σ¯ O ) = ∇ XO ( f σ O ) = (X f )σ O + f ∇ XO σ O = (X f )σ¯ O + f ∇ XO σ O O

= (X f )σ¯ O + f ∇ X σ¯ O . Hence, equation (7.46) defines a connection ∇

O

on L O .

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Commutation of quantization and reduction

−1 O Since 2π is the curvature form of ∇ , equation (7.5) reads (∇ X 1 ∇ X 2 − −i O ∇ X 2 ∇ X 1 − ∇[X 1 ,X 2 ] )σ = (X 1 , X 2 )σ O . Taking into account equation (7.46), we get (∇ X 1 ∇ X 2 − ∇ X 2 ∇ X 1 − ∇ [X 1 ,X 2 ] )σ¯ O = i (X 1 , X 2 )σ¯ O , which implies that 2π1 is the curvature of ∇. For each f ∈ C ∞ (O), let X −f be the Hamiltonian vector field of f with respect to the symplectic form −. Equation (7.8) implies that O

P f σ¯ O = (−i∇ X − + f )σ¯ O = (i∇ X f + f )σ¯ O f

= i∇ X f σ¯ O + f σ¯ O = i∇ X f σ O + f σ¯ O

because X −f = −X f by equation (7.46)

= −i∇ X f σ O + f σ O = (−i∇ X f + f )σ O = P Of σ O . Here, we follow Dirac’s convention Dirac (1950), and our Hermitian forms are antilinear in the first argument and linear in the second argument. Therefore, σ2O | σ1O O is antilinear in σ2O and linear in σ1O , which implies that σ2O | σ1O O is linear in σ¯ 2O and antilinear in σ¯ 1O . Hence, equation (7.48) defines a Hermitian form on L O . Denoting by d O the differential on O, we have O O O O O O O O d O σ¯ 1O | σ¯ 2O − O = d O σ2 | σ1 O = ∇ σ2 | σ1 O + σ2 | ∇ σ1 O O O ¯ O − = σ = σ¯ 1O | ∇ O σ2O − ¯ 1O | ∇ σ¯ 2O − ¯ 1O | σ¯ 2O − 2 O O + ∇ σ1 | σ O + ∇ σ O. O

O

This implies that the Hermitian form · | · − O on L O is ∇ -invariant. O

Proposition 7.4.2 (i) If FO is a positive polarization of (O, ), then F O is a positive polarization of (O, −). (ii) If a function f ∈ C ∞ (O) is quantizable with respect to a polarization FO , then it is quantizable with respect to the polarization F O . In other words, ∞ O C F∞ (O) = C F∞ ¯ O ∈ S F∞ ¯ (P). Similarly, if σ ∈ S F (L O ), then σ ¯ (L O ). ∞ ∞ O (iii) For each f ∈ C F (O) and σ ∈ S F (L O ), O

Q Of σ O = Q f σ¯ O .

(7.49)

Proof (i) FO is a positive polarization of (O, ) if i(w, w) ¯ ≥ 0 for all w ∈ FO . However, for each w ∈ FO , its complex conjugate w¯ is in F O , and i(−)(w, ¯ w) ≥ 0, which implies that F O is positive. ∞ (ii) C F (O) denotes functions in C ∞ (O) whose Hamiltonian vector fields preserve F. If X f preserves F O , then X −f = −X f preserves F O . Therefore, (O). C F∞ (O) = C ∞ F Recall that if σ O ∈ S F∞ (L O ), then ∇w σ O = 0 for all w ∈ FO . Hence, ∇ w¯ σ¯ O = ∇w σ O = 0 for all w¯ ∈ F O .

7.4 Non-zero co-adjoint orbits

193

O

(iii) By Definition 7.1.3, the quantization map Q is the restriction of the O ∞ prequantization map P to the domain C F∞ ¯ (O) × S F¯ (L O ) and codomain S F∞ ¯ O (L O ). Equation (7.49) is a consequence of the results in part (ii) above and equation (7.47). ˜ ω) The next step is to relate the quantization structure on ( P, ˜ to the quantization structures on (P, ω) and (O, ). We take a prequantization line bundle to be the tensor product of bundles λ : L → P and λ¯ O : L O → O. More precisely, we first have to pull back the bundles λ : L → P and λ¯ O : L O → O by the projection maps π1 : P × O → P and π2 : P × O → O, respectively. We obtain bundles π1∗ L and π2∗ L O such that the following diagrams commute: π1∗ L

λ∗ π1

π1∗ λ

P×O

/L λ

π1

/P

π2∗ L O and

∗

λ0 π2

π2∗ λ0

P×O

/ L0 λ0

π2

/ O.

Here, π1∗ L is the complex line bundle over P × O such that its fibre over ( p, μ) ∈ P × O is the fibre of L over p = π1 ( p, μ). If z ∈ π1∗ L is a point in the fibre over ( p, μ) ∈ P × O, then the projection map π1∗ λ(z) is equal to ( p, μ). By definition, a point z in the fibre of π1∗ L over ( p, μ) ∈ P × O is a point in the fibre of L over p, which we denote by λ∗ π1 (z). Similarly, π2∗ L O is the complex line bundle over P × O such that its fibre over ( p, μ) ∈ P × O is the fibre of L O over μ = π2 ( p, μ). The maps π1∗ λ¯ O : π2∗ L O → P × O and λ¯ O∗ π1 : π2∗ L O → L O are defined in the same way as for π1∗ L. We define the prequantization line bundle L˜ over P˜ = P × O to be L˜ = π1∗ L ⊗ π2∗ L O . ˜ This is a complex line bundle over P˜ = (P × O) with projection map λ˜ : L˜ → P, ˜ such that for each ( p, μ) ∈ P, λ˜ −1 ( p, μ) = λ−1 ( p) ⊗ (λ¯ O )−1 (μ), where the tensor product of the complex lines λ−1 ( p) and (λ¯ O )−1 (μ) is taken over the field C of complex numbers. If z 1 ∈ λ−1 ( p) and z 2 ∈ (λ¯ O )−1 (μ), we can identify z 1 ⊗ z 2 ∈ λ˜ −1 ( p, μ) with the product z 1 z 2 of complex numbers. Let σ : P → L be a section. The pull-back of σ by π1 is the π1∗ σ : P ×O → ∗ π1 L that associates to ( p, μ) ∈ P × O the element σ ( p) ∈ L, considered as a point in the fibre of π1∗ L over ( p, μ). Similarly, we define the pull-back of a section σ¯ O : O → L O by π2 : P × O → O. If σ is a section of L and σ¯ O is

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Commutation of quantization and reduction

a section of L O , we may define a section π1∗ σ ⊗ π2∗ σ¯ O : P × O → L˜

(7.50)

= π1∗ L ⊗ π2∗ L O : ( p, μ) → σ ( p) ⊗ σ¯ O (μ) = σ ( p)σ¯ O (μ) ˜ Moreover, a local section σ˜ of L˜ can be expressed in the form of L. σ˜ = f˜ π1∗ σ ⊗ π2∗ σ¯ O ,

(7.51)

where f˜ ∈ C ∞ (P × O) ⊗ C, and σ and σ¯ iO are local non-zero sections of L ˜ the space of smooth sections of L. ˜ and L O , respectively. We denote by S ∞ (L) O

Proposition 7.4.3 (i) The connections ∇ on L and ∇ on L O give rise to a ˜ a section σ˜ = connection ∇˜ on L˜ such that for every vector field X on P, ∗ ∗ O ˜ we have f˜π σ ⊗ π σ¯ of L˜ and a point ( p, μ) ∈ P, 1

2

∇˜ X ( p,μ) σ˜ = ∇˜ X ( p,μ) ( f˜π1∗ σ ⊗ π2∗ σ¯ O ) = X ( f )( p, μ)σ ( p)σ¯ O (μ) + f ( p, μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O

+ f ( p, μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O .

(7.52)

−1 ∗ ∗ The curvature of ∇˜ is 2π {π1 ω ⊕ (−π2 )}. (ii) The Hermitian forms · | · and · | · − O on L and L O , respectively, give rise to a Hermitian form · | · P˜ on L˜ such that, if σ˜ 1 = f˜1 π1∗ σ ⊗ π2∗ σ¯ O and σ˜ 2 = f˜2 π1∗ σ ⊗ π2∗ σ¯ O , then

σ˜ 1 | σ˜ 2 P˜ = f˜1 f˜2 π1∗ σ1 | σ2 π2∗ σ¯ 1O | σ¯ 2O − O ,

(7.53)

where f˜2 denotes the complex conjugate of f˜2 . The Hermitian form · | · P˜ on ˜ L˜ is ∇-invariant. (iii) If F and F¯ O are positive polarizations of (P, ω) and (O, −), respectively, then F˜ = π1∗ F ⊕ π2∗ F¯ O ˜ ω). is a positive polarization of ( P, ˜ For every section σ of L and section σ¯ O of ∗ L O , the tensor product σ˜ = π1 σ ⊗ π2∗ σ¯ O is covariantly constant along F˜ if σ is covariantly constant along F and if σ¯ O is covariantly constant along F¯ O . Proof that

(i) First, we show that ∇˜ σ˜ is well defined by equation (7.52). Suppose σ˜ = f˜π1∗ σ ⊗ π2∗ σ¯ O = f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O .

7.4 Non-zero co-adjoint orbits

195

Then, σ1 = hσ and σ1O = h O σ O for some h ∈ C ∞ (P) and h O = C ∞ (O), and f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O = f˜1 π1∗ (hσ ) ⊗ π2∗ (h O σ¯ O ) = f˜1 (π1∗ h)(π2∗ h O )π1∗ σ ⊗ π2∗ σ¯ O , which implies that f˜ = f˜1 (π1∗ h)(π2∗ h O ). Moreover, (X ( f˜))( p, μ) = X ( f˜1 (π1∗ h)(π2∗ h O ))( p, μ) = (X ( f˜1 )( p, μ))h( p)h O (μ) + f˜1 ( p, μ)(X (π1∗ h)( p, μ))h O (μ) + f˜1 ( p, μ)h( p)(X (π2∗ h O )( p, μ)) = (X ( f˜1 )( p, μ))h( p)h O (μ) + f˜1 ( p, μ)(T π1 (X ( p, μ))(h))h O (μ) + f˜1 ( p, μ)h( p)(T π2 (X ( p, μ))(h O )). Therefore, ∇˜ X ( p,μ) σ˜ = ∇ X ( p μ) ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = (X ( f˜)( p, μ))σ ( p)σ¯ O (μ) + f˜( p, μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O + f˜( p, μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O (μ)

= (X ( f˜1 )( p, μ))h( p)h O (μ)σ ( p)σ O (μ) + f˜1 ( p, μ)(T π1 (X ( p, μ))(h))h O (μ)σ ( p)σ¯ O (μ) + f˜1 ( p, μ)h( p)(T π2 (X ( p, μ))(h O ))σ ( p)σ¯ O (μ) + f˜1 ( p, μ)h( p)h O (μ)(∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) O + f˜1 ( p, μ)h( p)h O (μ)σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O (μ).

Since ∇ and ∇

O

satisfy equation (7.3), we have

(T π1 (X ( p, μ))(h))σ ( p) + h( p)(∇T π1 (X ( p,μ)) σ ) = ∇T π1 (X ( p,μ)) (hσ ) and (T π2 (X ( p, μ))(h O ))σ O (μ) + h O (μ)(∇T π2 (X ( p,μ)) σ ) = ∇T π2 (X ( p,μ)) (h O σ O ).

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Commutation of quantization and reduction

Hence, ∇˜ X ( p,μ) σ˜ = (X ( f˜1 )( p, μ))(hσ )( p)(h O σ O )(μ) + f˜1 ( p, μ)(∇T π1 (X ( p,μ)) (hσ ))(h O σ O )(μ) + f˜1 ( p, μ)(∇T π2 (X ( p,μ)) (h O σ O ))(hσ )( p) = (X ( f˜1 )( p, μ))(σ1 ( p))(σ1O (μ)) + f˜1 ( p, μ)(∇T π1 (X ( p,μ)) (σ1 ))(σ1O )(μ) + f˜1 ( p, μ)(∇T π2 (X ( p,μ)) (σ1O ))σ1 ( p) = ∇˜ X ( p,μ) ( f˜1 (π1∗ σ1 ) ⊗ (π2∗ σ O )), which implies that ∇˜ X ( p,μ) σ˜ is well defined by equation (7.52). ˜ For every h˜ ∈ C ∞ ( P), ˜ p, μ)∇˜ X ( p,μ) σ˜ , ∇˜ (h˜ X )( p,μ) σ˜ = h( ˜ σ˜ + h( ˜ p, μ)∇˜ X ( p,μ) σ˜ . ∇˜ X ( p,μ) h˜ σ˜ = (X ( p, μ)(h)) ˜ Therefore, ∇˜ is a connection on L. −1 −1 We know that 2π ω is the curvature of ∇ and 2π is the curvaO ture of ∇ . This leads by a straightforward computation to the result that −1 ∗ ∗ ˜ 2π {π1 ω ⊕ (−π2 )} is the curvature of ∇. (ii) As before, we have to show that the form σ˜ 1 | σ˜ 2 P˜ is well defined by equation (7.53). Suppose that σ˜ 1 = f˜1 π1∗ σ ⊗ π2∗ σ¯ O = f˜11 π1∗ σ1 ⊗ π2∗ σ¯ 1O , σ˜ 2 = f˜2 π1∗ σ ⊗ π2∗ σ¯ O = f˜21 π1∗ σ1 ⊗ π2∗ σ¯ 1O , where σ1 = hσ and σ1O = h O σ O , for some h ∈ C ∞ (P) and h O = C ∞ (O). Then f˜1 = (π1∗ h)(π2∗ h O ) f˜11 and f˜2 = (π1∗ h)(π2∗ h O ) f˜21 . Hence, σ˜ 1 | σ˜ 2 P˜ = f˜1 f˜2 π1∗ σ | σ π2∗ σ¯ O | σ¯ O − O = (π1∗ h)(π2∗ h O ) f˜11 (π1∗ h)(π2∗ h O ) f˜21 π1∗ σ | σ π2∗ σ¯ O | σ¯ O − O = f˜11 f˜21 π1∗ hσ | hσ π2∗ h O σ¯ O | h O σ¯ O − O = f˜11 f˜21 π1∗ σ1 | σ1 π2∗ σ¯ 1O | σ¯ 1O − O , which implies that σ˜ 1 | σ˜ 2 P˜ is well defined. Since the forms · | · and · | · − O ˜ are Hermitian, it follows that · | · P˜ is Hermitian. The ∇-invariance of · | · P˜ follows by straightforward but tedious computations from the ∇-invariance of O · | · and the ∇ -invariance of · | · − O.

7.4 Non-zero co-adjoint orbits

197

(iii) Let 2n = dim P and 2k = dim O. Then dimC F˜ = n + k. Consider ˜ Without loss of generality, we may assume that a basis (w1 , . . . , wn+k ) in F. (w1 , . . . , wn ) is a basis in π1∗ F, and (wn+1 , . . . , wn+k ) is a basis in π2∗ F¯ O . For i, j = 1, . . . , n and r, s = 1, . . . , k, ω(w ˜ i , wn+r ) = (π1∗ ω ⊕ (−π2∗ ))(wi , wn+r ) = (π1∗ ω)(wi , wn+r ) − (π2∗ )(wi , wn+r ) = ω(T π1 (wi ), T π1 (wn+r )) − (T π2 (wi ), T π2 (wn+r )) = 0 because T π1 (wn+r ) = 0 and T π2 (wi ) = 0. Moreover, ω(w ˜ i , w j ) = ω(T π1 (wi ), T π1 (w j )) = 0, because T π1 (wi ) and T π1 (w j ) are in F. Similarly, T π2 (wr ) and T π2 (ws ) are in F¯ O , which implies that ω(w ˜ r , ws ) = −(T π2 (wr ), T π2 (ws )) = 0. ˜ The positivity of the polarization Hence, F˜ is a Lagrangian distribution on P. ¯ F of (P, ω) and the polarization FO of (O, −) ensures that F˜ is a positive ˜ ω). polarization of ( P, ˜ It remains to show that F˜ is involutive. Let X 1 , . . . , X n be vector fields on P locally spanning F. Similarly, we consider vector fields X 1O , . . . , X kO on O which locally span F¯ O . We can lift these vector fields to vector fields on P˜ as follows. For each i = 1, . . . , n, we define π1∗ X i to be the unique vector field on P˜ such that π1∗ X i (π1∗ f ) = π1∗ (X i ( f )) for all f ∈ C ∞ (P), π1∗ X i (π2∗ f O ) = 0 for all f O ∈ C ∞ (O). Similarly, for every r = 1, . . . , k, we define π2∗ X rO to be the unique vector field on P˜ such that π2∗ X rO (π1∗ f ) = 0 for all f ∈ C ∞ (P), π2∗ X rO (π2∗ f O ) = π2∗ (X rO ( f O )) for all f O ∈ C ∞ (O). The vector fields π1∗ X 1 , . . . , π1∗ X n locally span π1∗ F, and the vector fields π2∗ X 1O , . . . , π2∗ X kO locally span π2∗ F¯ O . For every i = 1, . . . , n and r = 1, . . . , k, the vector fields π1∗ X i and π2∗ X rO commute because second partial derivatives of smooth functions commute. On the other hand,

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Commutation of quantization and reduction

[π1∗ X i , π1∗ X j ](π1∗ f ) = π1∗ X i ((π1∗ X j )(π1∗ f )) − π1∗ X j ((π1∗ X i )(π1∗ f )) = π1∗ X i (π1∗ (X j ( f ))) − π1∗ X j (π1∗ (X i ( f ))) = π1∗ (X i (X j ( f )) − X j (X i ( f ))) = π1∗ ([X i , X j ]( f )) = (π1∗ [X i , X j ])(π1∗ f ), and [π1∗ X i , π1∗ X j ](π1∗ f O ) = 0. Since F is involutive, the commutator [X i , X j ] is a linear combination of the vector fields X 1 , . . . , X n , which implies that [π1∗ X i , π1∗ X j ] is a linear combination of the vector fields π1∗ X 1 , . . . , π1∗ X n . Hence, π1∗ F is involutive. In a similar way, we can show that π2∗ F¯O is involutive. Therefore, F˜ = π1∗ F ⊕ π2∗ F¯ O is involutive. Thus, we ˜ ω). have shown that F˜ is a positive polarization of ( P, ˜ Consider σ˜ = π1∗ σ ⊗ π2∗ σ¯ O . By equation (7.52), ∇˜ X ( p,μ) σ˜ = ∇˜ X ( p,μ) (π1∗ σ ⊗ π2∗ σ¯ O ) O

= (∇T π1 (X ( p,μ)) σ ) ⊗ σ¯ O (μ) + σ ( p) ⊗ ∇ T π2 (X ( p,μ)) σ¯ O . ˜ then T π1 (X ( p, μ)) ∈ F and T π2 (X ( p, μ)) ∈ F¯ O . ThereIf X ( p, μ) ∈ F, fore, σ˜ π1∗ σ ⊗π2∗ σ¯ O is covariantly constant along F˜ if σ is covariantly constant along f and if σ O is covariantly constant along F¯ O . ˜ of smooth sections of It follows from equation (7.51) that the space S ∞ (L) ˜ L can be presented locally as follows: ˜ = {σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O | f˜ ∈ C ∞ ( P), ˜ σ¯ O ∈ S ∞ (L˜ O )}. ˜ σ ∈ S ∞ (L), S ∞ (L) 1 2 Similarly, ˜ ˜ σ¯ O ∈ S ∞ (L˜ O )}. ˜ ˜ 0 , σ ∈ S F∞ (L), S F∞ ¯ O | f˜ ∈ C ∞ ( P) ˜ (L) = { f σ ⊗ σ F¯ F˜ Proposition 7.4.4 ˜ is given by For each ξ ∈ g∗ , the prequantization operator P˜ J˜ξ on S ∞ (L) P˜ J˜ξ ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( P O Iξ σ )).

(7.54) ˜ Similarly, the quantization operator on S ∞ ˜ (L) is given by F

Q˜ J˜ξ ( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( Q Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( Q O Iξ σ )).

(7.55)

7.4 Non-zero co-adjoint orbits

199

Proof Since P˜ = P × O, ω˜ = π1∗ ω ⊕ (−π2∗ ), and J˜ = π1∗ J − π2∗ I , for each ξ ∈ g∗ , the Hamiltonian vector field X J˜ξ of P˜ξ = π1∗ Jξ −π2∗ Iξ is given by X J˜ξ = X π1∗ Jξ − X π2∗ Iξ = π1∗ X Jξ + π2∗ X Iξ = π1∗ X Jξ − π2∗ X − Iξ , where X − Iξ is the Hamiltonian field of Iξ on (O, −). Hence, the prequantization operator P˜ ˜ is given by Jξ

P˜ J˜ξ σ˜ = (−i∇˜ X J˜ + J˜ξ )σ˜ = (−i∇˜ X J˜ + J˜ξ )( f˜ π1∗ σ ⊗ π2∗ σ¯ O ) ξ

ξ

= −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + f˜ (−i∇˜ X J˜ + J˜ξ )(π1∗ σ ⊗ π2∗ σ¯ O ) ξ

= −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + − f˜ (i∇˜ X π ∗ J − i∇˜ X −∗ )(π1∗ σ ⊗ π2∗ σ¯ O ) 1 ξ

π2 I ξ

+ f˜ (π1∗ Jξ − π2∗ Iξ )(π1∗ σ ⊗ π2∗ σ¯ O ) = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + +(π1∗ (−i∇ X Jξ + Jξ )σ ) ⊗ π2∗ (σ¯ O ) + +π1∗ σ ⊗ π2∗ ((i∇ X − − Iξ )σ¯ O ) Iξ

=

−i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O O − f˜π1∗ σ ⊗ π2∗ ( P Iξ σ¯ O ).

+ f˜π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O

Taking into account Proposition 7.4.1, we get P˜ J˜ξ σ˜ = −i(X J˜ξ ( f ))π1∗ σ ⊗ π2∗ σ¯ O + O + f˜ (π1∗ ( P Jξ σ ) ⊗ π2∗ σ¯ O − π1∗ σ ⊗ π2∗ ( P O Iξ σ )).

Equation (7.55) follows from the fact that Q˜ is the restriction of P˜ to the ∞ ˜ ∞ ˜ ∞ ˜ ∞ ˜ ∞ ˜ ˜ domain C ∞ ˜ ( P)× S ˜ (L) ⊂ C ( P)× S (L) and codomain S ˜ (L) ⊂ S (L). F

F

F

By assumption, the prequantization actions of g on L and L O lift to actions of G on L and L O , respectively. Hence, the prequantization action of G on ˜ For each z ∈ λ−1 ( p) ⊆ L and L˜ = π1∗ L ⊗ π2∗ L O lifts to an action of G on L. O −1 z O ∈ (λ¯ ) (μ) ⊆ L O , the product z ⊗ z O satisfies z ⊗ z O ∈ λ˜ −1 ( p, μ) ⊆ π1∗ L ⊗ π2∗ L O . For every g ∈ G, we have g(z ⊗ z O ) = gz ⊗ gz O ∈ λ˜ −1 (gp, gμ). In order to continue, we need a scalar product on the space S F∞ (L O ). For simplicity, we assume that the polarization FO of (O, ) is Kähler. This

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Commutation of quantization and reduction

implies that L O is a holomorphic line bundle and that sections in S F∞O (L O ) are holomorphic. The scalar product in S F∞O (L O ) is given by (σ1O | σ2O ) O = σ1O | σ2O O k , (7.56) O

where k = 12 dim O; see equation (7.10). We denote by H FO the Hilbert space of sections in S F∞ (L O ) for which the scalar product (7.56) exists. ˜ where σ ∈ S ∞ (L), Theorem 7.4.5 Consider σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O ∈ S ∞ (L), 1

F˜

2

F

σ O ∈ H FO and f˜|π −1 ( p) σ O ∈ H FO for every p ∈ P.

(i) For each σ0O ∈ H FO , the map 'σ˜ (σ0O ) : P → L : p → σ O | f˜|π −1 ( p) σ0O O k σ ( p) = (σ

(7.57)

1

O O

| f˜|π −1 ( p) σ0O ) O σ ( p), 1

is in S F∞ (L). ˜ as the product (L) (ii) 'σ˜ (σ0 ) is independent of the presentation of σ˜ ∈ S ∞ F˜ f˜ π ∗ σ ⊗ π ∗ σ¯ O , where σ ∈ S ∞ (L) and f˜|π −1 ( p) σ O ∈ H FO for every 1

F

2

p ∈ P. ˜ (iii) 'σ˜ : σ0O → 'σ˜ (σ0 ) is a linear homomorphism of H FO to S ∞ ˜ (L). F

(iv) If σ˜ is G-invariant, then 'σ˜ : σ0O → 'σ˜ (σ0 ) intertwines the actions of ˜ G on H FO and S ∞ ˜ (L). F

Proof (i) Since σ0O ∈ H FO , and f˜|π −1 ( p) σ O ∈ H FO for every p ∈ P, it follows that the function p → (σ O | f˜|π −1 ( p) σ0O ) O is in C ∞ (P). The 1 ˜ implies that ∇˜ u˜ σ˜ = 0 for assumption that σ˜ = f˜ π ∗ σ ⊗ π ∗ σ¯ O ∈ S ∞ (L) 1

2

F˜

˜ f˜) = 0 for all u˜ ∈ π1∗ F. Therefore, the funceach u˜ ∈ π1∗ F. Hence, u( O ˜ tion p → (σ | f |π −1 ( p) σ0O ) O is constant along F. The assumption that 1

σ ∈ S F∞ (L) ensures that 'σ˜ (σ0O ) ∈ S F∞ (L). (ii) Suppose that

f˜π1∗ σ ⊗ π2∗ σ¯ O = f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O , where σ and σ1 are in S F∞ (L), and σ O and σ1O are in S F∞ (L O ). Then there exist h ∈ C F∞ (P) and h O ∈ C F∞O (O) such that σ1 = f σ and σ1O = h O σ O . Therefore, f˜1 π1∗ σ1 ⊗ π2∗ σ¯ 1O = ( f˜1 π1∗ hπ1∗ h¯ O )π1∗ σ ⊗ π2∗ σ¯ O = f˜π1∗ σ ⊗ π2∗ σ¯ O ,

7.4 Non-zero co-adjoint orbits

201

and f˜ = ( f˜1 π1∗ h π1∗ h¯ O ). Hence, (σ1O | f˜1|π −1 ( p) σ0O ) O σ1 ( p) = (h O σ O | f˜1|π −1 ( p) σ0O ) O h( p)σ ( p) 1

1

= (σ O | h¯ O f˜1|π −1 ( p) h( p)σ0O ) O σ ( p) 1

= (σ O | ( f˜1 π1∗ h π1∗ h¯ O )|π −1 ( p) σ0O ) O σ ( p) 1

= (σ O | f˜|π −1 ( p) σ0O ) O σ ( p), 1

which shows that 'σ˜ (σ0O ) is well defined by equation (7.57). (iii) Since 'σ˜ (σ0O ) is linear in σ0O ∈ H FO , it follows that 'σ˜ : σ0O → ˜ 'σ˜ (σ0 ) is a linear homomorphism of H FO to S ∞ (L). F˜ ∗ ∗ O ˜ (iv) Suppose that σ˜ = f π σ ⊗π σ¯ is G-invariant. Then, for every g ∈ G, 1

2

p ∈ P and μ ∈ O, σ˜ (gp, Adg∗ μ) = g(σ˜ ( p, μ)). Hence,

f˜(gp, Adg∗ μ)σ (gp)(σ¯ O (Adg∗ μ)) = g( f˜( p, μ)σ ( p)σ¯ O (μ))

(7.58)

= f˜( p, μ)g(σ ( p))g(σ¯ (μ)). O

Recall that R O denotes the quantization representation of G on S F∞O (L O ). For σ0O ∈ S F∞O (L O ), and each g ∈ G and μ ∈ O, (R O σ0O )(μ) = g −1 (σ0O (Adg∗ μ)). Moreover, R gO ( f O σ0O ) = ((gO−1 )∗ f O )R gO σ0O . If σ0O ∈ H O , then U O σ0O = R O σ0O defines a unitary transformation on H FO , so that (σ O | U gO σ0O ) O = (U gO−1 σ O | σ0O ) O . Therefore, (σ O | f˜| π −1 ( p) U gO σ0O ) O = ( f˜ | π −1 ( p) σ O | U gO σ0O ) O 1

1

=

(U gO−1 ( f˜ | π −1 ( p) σ O ) | σ0O ) O 1

= (((gO−1 )∗ f˜ | π −1 ( p) )U gO−1 σ O | σ0O ) O 1 f˜ −1 (Adg −1 (μ)) gσ O (Ad ∗−1 μ) | σ O O k = O

= O

| π1 ( p)

g

0

f˜| π −1 ( p) (Adg∗−1 μ) gσ O(Adg∗−1 μ) | σ0O (μ) Ok . 1

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Commutation of quantization and reduction

Equation (7.58) implies that f˜(g −1 p, Adg∗−1 μ)σ (g −1 p)(gσ O (Adg∗−1 μ)) = f˜( p, μ)g −1 (σ ( p))g −1 (gσ O (μ)) = f˜( p, μ)g −1 (σ ( p)σ¯ O (μ)).

Replacing p by gp, we obtain f˜( p, Adg∗−1 μ)σ ( p)(gσ O (Adg∗−1 μ)) = f˜(gp, μ)g −1 (σ (gp)σ¯ O (μ)). Hence, 'σ˜ (U gO σ0O )( p) = (σ O | f˜|π −1 ( p) U g σ0O ) O σ ( p) 1 ˜ = f |π −1 ( p) (Adg∗−1 μ) gσ O (Adg∗−1 μ) | σ0O (μ) O k σ ( p) 1 O f˜|π −1 (gp) (μ) σ O (μ) | σ0O (μ) O k g −1 σ (gp) = =g

O −1

1

'σ˜ (σ0O )(gp) = (R g 'σ˜ (σ0O ))( p).

Therefore, 'σ˜ ◦ U gO = R g ◦ 'σ˜ , as required. Suppose that we have a scalar product on S F∞ (L) leading to a Hilbert space H F of square-integrable polarized sections of L. Then, the quantization representation G can be decomposed into a direct integral of irreducible unitary representations of G. In particular, we have the following decomposition of H F : Hλ dμ(λ), (7.59) HF =

where is the space of equivalence classes of irreducible unitary representations of G, Hλ is the representation space of a representation U λ corresponding to the class λ, and dμ(λ) is a spectral measure on . Suppose that U O is the irreducible unitary representation corresponding to a quantizable coadjoint orbit O, and λ O is the equivalence class of U O . If 'σ˜ (σ O ) = 0, then the spectral measure dμ(λ) does not vanish at λ = λ O . If, in addition, 'σ˜ (σ O ) ∈ Hλ O , then the spectral measure dμ(λ) contains an atomic part of the form m λ δ(λ − λ O ), where m λ O is the multiplicity of λ O , possibly infinite, in the decomposition of U F . If m λ O is finite, there exists a closed subspace m λ O Hλ O of H F on which U F is equivalent to the direct sum of m α copies of U λ O . For infinite m λ O , m λ O Hλ O is still a closed invariant subspace of H F , and U F restricted to m λ O Hλ O is equivalent to an appropriately defined direct sum of copies of U λ O . Thus, sections 'σ˜ (σ O ) in S F∞ (L) that are not normalizable correspond to the continuous part of the spectrum of dμ(λ) in the decomposition (7.59).

7.5 Commutation of quantization and algebraic reduction

203

7.5 Commutation of quantization and algebraic reduction In order to discuss commutation of quantization and algebraic reduction, we first have to define what we mean by quantization of algebraic reduction. As before, we consider only the zero level set of the momentum map.

7.5.1 Quantization of algebraic reduction Recall that algebraic reduction at the zero level of the momentum map is defined in terms of the ideal

k ∞ J = Jξi f i | f 1 , . . . , f k ∈ C (P) , i=1

where (ξ1 , . . . , ξk ) is a basis in g. The reduced Poisson algebra (C ∞ (P)/J )G consists of G-invariant equivalence classes [ f ] of functions f ∈ C ∞ (P). Since G is connected, [ f ] ∈ C ∞ (P)/J ⇐⇒ X Jξ ( f ) ∈ J ∀ ξ ∈ g. The prequantization of (P, ω) gives rise to a representation f → i1 P f of the Poisson algebra C ∞ (P) on the space S ∞ (L) of smooth sections of L. Let J S ∞ (L) = span { f σ | f ∈ J and σ ∈ S ∞ (L)}. S ∞ (L),

For each σ ∈ S ∞ (L)/J S ∞ L) .

(7.60)

we denote the equivalence class of σ by [σ ] ∈

Proposition 7.5.1 For each ξ ∈ g and σ ∈ S ∞ (L), the class [ P Jξ σ ] ∈ S ∞ (L)/J S ∞ (L) is independent of the choice of the representative σ of the class [σ ] ∈ S ∞ (L)/J S ∞ (L). Proof If [σ ] = [σ ] then σ = σ + i f i Jξi σi . Taking into account equations (7.8) and (6.6), we get P Jξ = −i∇ X Jξ + Jξ , which implies that P Jξ Jζ = Jζ P Jξ − iX Jξ (Jζ ) = Jζ P Jξ − iJ[ζ,ξ ] . Hence, P Jξ σ = P Jξ σ + P Jξ ( = P Jξ σ +

f i Jξi σi ) = P Jξ σ +

i

Jξi P Jξ ( f i σi ) − i

i

= P Jξ σ +

P Jξ (Jξi f i σi )

i

X Jξ (Jξi ) f i σi

i

Jξi P Jξ ( f i σi ) − i

i

Therefore, [ P Jξ σ ] = [ P Jξ σ ].

i

J[ξi ,ξ ] f i σi .

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Commutation of quantization and reduction

We have assumed that the action of g on S ∞ (L), given by (ξ, σ ) → i1 P Jξ σ integrates to a representation of G on S ∞ (L). Hence, the action of g on S ∞ (L)/J S ∞ (L), given by (ξ, [σ ]) → i1 [ P Jξ σ ], integrates to a representation of G on S ∞ (L)/J S ∞ (L). We denote (S ∞ (L)/J S ∞ (L))G the space of G-invariant elements in S ∞ (L)/J S ∞ (L). Since G is connected, it follows that [σ ] ∈ (S ∞ (L)/J S ∞ (L))G ⇐⇒ P Jξ σ ∈ J S ∞ (L) for all ξ ∈ g. Proposition 7.5.2 The map P : (C ∞ (P)/J )G × (S ∞ (L)/J S ∞ (L))G → (S ∞ (L)/J S ∞ (L))G : ([ f ], [σ ]) → P [ f ] [σ ] = [ P f σ ] is well defined. Proof For [ f ] ∈ (C ∞ (P)/J )G , we have X Jξ ( f ) ∈ J . Hence, for each Jξ σ ∈ J S ∞ (L), P f (Jξ σ ) = (−i∇ X f + f )(Jξ σ ) = Jξ (−i∇ X f + f )σ − i(X f (Jξ ))σ = Jξ (−i∇ X f + f )σ − i(X Jξ ( f ))σ ∈ J S ∞ (L). This implies that, for [ f ] ∈ (C ∞ (P)/J )G , the operator P f maps J S ∞ (L) to itself. Hence, [ P f σ ] does not depend on the representative σ of [σ ]. For k Jξ ∈ J and [σ ] ∈ (S ∞ (L)/J S ∞ (L))G , we have X k Jξ = k X Jξ + Jξ X k . Therefore, P k Jξ σ = (−i∇ X k Jξ + k Jξ )σ

= (−ik∇ X k Jξ − iJξ ∇ X k + k Jξ )σ

= −iJξ ∇ X k σ + ik P Jξ σ ∈ J S ∞ (L).

Hence, [ P f σ ] does not depend on the representative f of [ f ]. Combining these results, we obtain that an equivalence class [ P f σ ] ∈ S ∞ (L)/J S ∞ (L) depends only on [σ ] ∈ (S ∞ (L)/J S ∞ (L))G and [ f ] ∈ (C ∞ (P)/J )G . It remains to show that [ P f σ ] is G-invariant. For [ f ] ∈ (C ∞ (P)/J )G , [σ ] ∈ (S ∞ (L)/J S ∞ (L))G and ξ ∈ g, P Jξ P f σ = P f P Jξ σ + [ P Jξ , P f ]σ. But [σ ] ∈ (S ∞ (L)/J S ∞ (L))G implies P Jξ σ ∈ J S ∞ (L) so that P f P Jξ σ ∈ J S ∞ (L) by the first part of the proof. On the other hand [ P Jξ , P f ]σ = i P {Jξ , f } σ = i(−i∇ X {Jξ , f } + {Jξ , f })σ.

7.5 Commutation of quantization and algebraic reduction

205

But, {Jξ , f } = −X Jξ f ∈ J because [ f ] is G-invariant. Hence, {Jξ , f } = ∞ j f j Jζ j for some f j ∈ C (P) and ζ j ∈ g. Moreover, X f 1 f 2 = f 1 X f 2 + f 2 X f1 implies that ∇ X f j Jζ = ( f j ∇ X Jζ + Jζ j ∇ X f j ). j

j

Therefore, (−i∇ X {Jξ , f } + {Jξ , f })σ =

−i( f j ∇ X Jζ + Jζ j ∇ X f j ) + f j Jζ j σ j

=

j

j

j

f j P Jζ j − iJζ j ∇ X f j σ ∈ J S ∞ (L)

j

because [σ ] is G-invariant. Therefore, [ P f σ ] ∈ (S ∞ (L)/J S ∞ (L))G . Definition 7.5.3 The map that associates to each [ f ] ∈ (C ∞ (P)/J )G an operator P [ f ] on the space (S ∞ (L)/J S ∞ (L))G is a prequantization of the reduced Poisson algebra. Let F be a strongly admissible polarization of (P, ω), and let C F∞ (P) be a subalgebra of C ∞ (P) consisting of functions f such that X f preserves F. Quantization in terms of the polarization F assigns to each f ∈ C F∞ (P) an operator Q f on the space S F∞ (L) = {σ ∈ S ∞ (L) | ∇u σ = 0 for all u ∈ F}. Moreover, Q f σ = P f σ for each f ∈ C F∞ (P) and σ ∈ S F∞ (L). By analogy, we expect quantization after reduction to give a representation of the restriction of (C ∞ (P)/J )G to the set (C F∞ (P)/J )G consisting of G-invariant J -equivalence classes of functions in C F∞ (P). The representation space will be the space (S F∞ (L)/J S ∞ (L))G consisting of G-invariant J S ∞ (L)-equivalence classes of sections in S F∞ (L). We shall use the following identifications: (C F∞ (P)/J )G = (C F∞ (P)/(C F∞ (P) ∩ J ))G , (S F∞ (L)/J

∞

S (L)) = G

(S F∞ (L)/(S F∞ (L) ∩ J

∞

(7.61)

S (L)) . G

(7.62)

Proposition 7.5.4 The map Q : (C F∞ (P)/J )G × (S F∞ (L)/J S ∞ (L))G → (S F∞ (L)/J S ∞ (L))G : ([ f ], [σ ]) → Q [ f ] [σ ] = [ P f σ ] is well defined. Proof We know that if f ∈ C F∞ (P) and σ ∈ S F∞ (L), then P f σ ∈ S F∞ (L). In Proposition 7.5.2, we proved that [ P f σ ] ∈ (S ∞ (L)/J S ∞ (L))G is independent of the representatives f of [ f ] ∈ (C ∞ (P)/J )G and σ of

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Commutation of quantization and reduction

[σ ] ∈ (S ∞ (L)/J S ∞ (L))G . Therefore, [ P f σ ] ∈ (S F∞ (L)/J S ∞ (L))G . Moreover, [ P f σ ] is independent of the representative f ∈ C F∞ (P) of [ f ] ∈ (C F∞ (P)/J )G . Similarly, [ P f σ ] is independent of the representative σ ∈ S F∞ (L) of [σ ] ∈ (S F∞ (L)/J S ∞ (L))G . Hence, the map Q : ([ f ], [σ ]) → Q [ f ] [σ ] = [ P f σ ] is well defined. Definition 7.5.5 A map that associates to each [ f ] ∈ (C F∞ (P)/J )G an operator Q [ f ] on the space (S F∞ (L)/J S ∞ (L))G is a quantization of the subalgebra (C F∞ (P)/J )G of the reduced Poisson algebra.

7.5.2 Kähler polarization Now consider the case when F is a Kähler polarization of (P, ω). We denote by F : S F∞ (L) → S F∞ (L)/J S ∞ (L) the projection map that associates to each σ in S F∞ (L) its class [σ ] in S F∞ (L) → S F∞ (L)/J S ∞ (L). If σ is G-invariant, then its class [σ ] is G-invariant. Hence, the restriction of F to S F∞ (L)G gives a projection map ∞ G ∞ ∞ G G F : S F (L) → (S F (L)/J S (L)) : σ → [σ ].

(7.63)

Proposition 7.5.6 Assume that the Hamiltonian action of G on (P, ω) preserves a Kähler polarization F of (P, ω). If J −1 (0) contains a Lagrangian submanifold of (P, ω), then the projection G F is a vector space isomorphism such that G G F ◦ Q f = Q[ f ] ◦ F

for every f ∈ C F∞ (P)G . Proof Consider σ ∈ S F∞ (L) ∩ J S ∞ (L). Since σ ∈ J S ∞ (L), it follows that σ|J −1 (0) = 0. By Proposition 7.3.1, σ = 0, because J −1 (0) contains a Lagrangian submanifold of (P, ω). Hence, S F∞ (L) ∩ J S ∞ (L) = 0. Using the identification (7.62), we obtain S F∞ (L)/J S ∞ (L) = S F∞ (L)/(S F∞ (L) ∩ J S ∞ (L)) = S F∞ (L)/(0) = S F∞ (L). Hence, (S F∞ (L)/J S ∞ (L))G = S F∞ (L)G , ∞ ∞ G ∞ G which implies that the projection map G F : S F (L) → (S F (L)/J S (L)) is an isomorphism. For every G-invariant function f ∈ C F∞ (P), the class [ f ] is in f ∈ ∞ C F (P)G . Moreover, for each σ ∈ S F∞ (L)G , we have Q f σ ∈ S F∞ (L)G and ∞ ∞ G [σ ] = G F σ ∈ (S F (L)/J S (L)) . Hence,

7.5 Commutation of quantization and algebraic reduction

207

G Q [ f ] ◦ G F σ = Q [ f ] [σ ] = [ Q f σ ] = F ◦ Q f σ,

which completes the proof.

7.5.3 Real polarization We consider here a Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) that preserves a real strongly admissible polarization F = D ⊗ C, as in Section 7.3.3. Proposition 7.5.7 Assume that a proper Hamiltonian action of a connected Lie group G on a symplectic manifold (P, ω) preserves a strongly admissible real polarization F = D⊗C. If every integral manifold of D intersects J −1 (0), then the projection map ∞ G ∞ ∞ G G F : S F (L) → (S F (L)/J S (L)) : σ → [σ ]

is an isomorphism such that G G F ◦ Q f = Q[ f ] ◦ F

for every f ∈ C F∞ (P)G . Proof Consider σ ∈ S F∞ (L) ∩ J S ∞ (L). Since σ ∈ J S ∞ (L), it follows that σ|J −1 (0) = 0. By Proposition 7.3.2, the assumptions made here ensure that the restriction map F : S F∞ (L) → S F∞0 (L0 ) is a monomorphism. Hence, σ|J −1 (0) = 0 implies that σ = 0. Therefore, S F∞ (L) ∩ J S ∞ (L) = 0. We continue as in the proof of Proposition 7.5.4. The identification (7.62) yields S F∞ (L)/J S ∞ (L) = S F∞ (L)/(S F∞ (L) ∩ J S ∞ (L)) = S F∞ (L)/(0) = S F∞ (L). Hence, (SF∞ (L)/J S ∞ (L))G = S F∞ (L)G , ∞ ∞ G ∞ G which implies that the projection map G F : S F (L) → (S F (L)/J S (L)) is an isomorphism. For every G-invariant function f ∈ C F∞ (P), the class [ f ] is in f ∈ ∞ C F (P)G . Moreover, for each σ ∈ S F∞ (L)G , we have Q f σ ∈ S F∞ (L)G and ∞ ∞ G [σ ] = G F σ ∈ (S F (L)/J S (L)) . Hence, G Q [ f ] ◦ G F σ = Q [ f ] [σ ] = [ Q f σ ] = F ◦ Q f σ,

which completes the proof.

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Commutation of quantization and reduction

We see in the cases considered here that the same assumptions lead to commutation of quantization and algebraic reduction and to commutation of quantization and singular reduction. In general, for a proper group action, algebraic reduction may give a larger Poisson algebra than singular reduction; see Example 6.9.4. However, in that example also, quantization of singular reduction and quantization of algebraic reduction give equivalent results (Bates et al., 2009).

7.5.4 Improper action Unlike singular reduction, algebraic reduction does not require properness of the action of G on (P, ω). The action of G on the cotangent bundle space endowed with the canonical symplectic form, which we have described in Section 7.1.3, is improper. In this section, we discuss a slight generalization of the special case described in Section 7.1.3. We assume that the polarization F is real and that the momentum map J : P → g∗ is constant along D. In this case, for each ξ ∈ g and σ ∈ S F∞ (L), we have Q Jξ σ = Jξ σ .

(7.64)

Hence, a section σ ∈ S F∞ (L) is G-invariant only if J σ = 0. Thus, the support of σ is contained in J −1 (0). We assume that J −1 (0) is nowhere dense in P. This implies that the equation J σ = 0 has only weak solutions. In order to discuss generalized sections of S F∞ (L), we need to describe the topology on S F∞ (L). We assume that the space Q of integral manifolds of the distribution D is a quotient manifold of P and that the leaves of D are complete affine spaces. Let ϑ : P → Q be the map that associates to each p ∈ P the maximal integral manifold of D through p. We assume that ϑ is a submersion. Moreover, we assume that the prequantization bundle L is trivial, that is, L = P × C, and that there exists a section σ0 ∈ S F∞ (L) such that σ0 ( p) = ( p, 1). All of these assumptions are satisfied by the system described in Section 7.1.3. Under the assumptions made here, S F∞ (L) = {ϑ ∗ (ψ)σ0 | ψ ∈ C ⊗ C ∞ (Q)}.

(7.65)

Let D = {ϑ ∗ (ψ)σ0 ∈ S F∞ (L) | ψ ∈ C ⊗ C0∞ (Q)}, where C⊗C0∞ (Q) is the space of compactly supported complex-valued smooth functions on Q. We endow D with a topology of uniform convergence on

7.5 Commutation of quantization and algebraic reduction

209

compact sets of all derivatives of functions ψ ∈ C ⊗ C0∞ (Q). We define a scalar product on D as follows: ∗ ∗ (σ1 | σ2 ) Q = (ϑ (ψ1 )σ0 | ϑ (ψ2 )σ0 ) Q = ψ¯ 1 (q)ψ2 (q) dμ(q). (7.66) Q

Let H F be the completion of D with respect to the norm given by the scalar product on D, and let D be the topological dual of D. Clearly, D ⊂ H F ⊂ D .3 For each ξ ∈ g, the quantum operator Q Jξ on S F∞ (L) preserves D. Hence, it extends to a self-adjoint operator on H and gives rise to a dual operator Q Jξ on D such that, for every ξ ∈ g, ϕ ∈ D and σ ∈ D, we have Q Jξ ϕ | σ = ϕ | Q Jξ σ , where · | · denotes the evaluation map. The space of generalized invariant vectors is ker Q J = {ϕ ∈ D | Q Jξ ϕ = 0 for all ξ ∈ g}. On the other hand, the range of Q J in D is range Q J = { Q Jξ σ1 + Q Jξ σ2 + . . . + Q Jξ σk | σ1 , . . . , σk ∈ D}, 1

2

k

where (ξ1 , . . . , ξk ) form a basis of g. There is a duality between ker Q J and D/range Q J such that for every ϕ ∈ ker Q J and every class [σ ] ∈ D/range Q J , we have ϕ | [σ ] = ϕ | σ . Since each Q Jξ is a multiplication operator, it follows that range Q J = {Jξ1 σ1 + Jξ2 σ2 + . . . + Jξk σk | σ1 , . . . , σk ∈ D} = J D. In the following discussion, we look for conditions under which D/range Q J = D/J D and (S F∞ (L)/J S ∞ (L))G are isomorphic. We begin with a simple lemma. Lemma 7.5.8 The class [σ ] ∈ S ∞ (L)/J S ∞ (L) of σ ∈ S ∞ (L) is uniquely determined by the restriction of σ to any open set containing J −1 (0). Proof

For σ ∈ S ∞ (L), if (support σ ) ∩ J −1 (0) = ∅, then [σ ] = 0.

Theorem 7.5.9 If Q is locally compact and ϑ(J −1 (0)) is compact, then D/J D and (S F∞ (L)/J S ∞ (L))G are isomorphic. 3 The triplet of spaces D ⊂ H ⊂ D obtained here is usually called a Gelfand triplet.

210

Commutation of quantization and reduction

Proof For every σ ∈ S F∞ (L), there exists an open set Vσ ⊂ Q = P/D such that Vσ ⊇ ϑ(J −1 (0)), [σ ] is uniquely determined by ϑ −1 (Vσ ), and V¯σ is compact. Hence, there exists σ ∈ D = S F∞ (L) ∩ H F such that σ|ϑ −1 (Vσ ) = σ|ϑ −1 (V ) and [σ ] = [σ ]. Therefore, σ

S F∞ (L)/J

S ∞ (L) = S F∞ (L)/(J S ∞ (L) ∩ S F∞ (L)) = D/(J S ∞ (L) ∩ D).

For σ ∈ D, the class [σ ] ∈ D/(J S ∞ (L) ∩ D) is given by

[σ ] = σ + f i Jξi σi | f i Jξi σi ∈ D . i

But f i Jξi σi ∈ D implies that fi Jξi σi = ϑ ∗ (ψi )σ0 , where ψi has compact support in Q. There exists a function χi ∈ C ⊗ C0∞ (Q) such that χi (q) = 1 for every q in the support of ψi . Then, ψi = ψi χi and f i Jξi σi = ϑ ∗ (ψi )σ0 = ϑ ∗ (ψi χi )σ0 = ϑ ∗ (ψi )ϑ ∗ (χi )σ0 = f i Jξi ϑ ∗ (χi )σi ∈ J D. This implies [σ ] = [σ ] ∈ D/J D, so that S F∞ (L)/J S ∞ (L) = D/J D.

(7.67)

By definition, (S F∞ (L)/J S ∞ (L))G = {[σ ] ∈ S F∞ (L)/J S ∞ (L) | [ Q Jξ σ ] = 0 for all ξ ∈ g}. But Q Jξ σ = Jξ σ . Hence, [ Q Jξ σ ] = 0 for all σ ∈ S F∞ (L) and ξ ∈ g. Therefore, (S F∞ (L)/J S ∞ (L))G = S F∞ (L)/J S ∞ (L).

(7.68)

Equations (7.67) and (7.68) yield (S F∞ (L)/J S ∞ (L))G = D/J D, which completes the proof. We have shown that in the case under consideration, the representation space (S F∞ (L)/J S ∞ (L))G of the quantization of the singular reduction at J = 0 is naturally isomorphic to the space D/J D of generalized invariant vectors of the geometric quantization of the original phase space (P, ω).

8 Further examples of reduction

8.1 Non-holonomic reduction ‘Non-holonomic reduction’ is the term used for the reduction of symmetries of non-holonomically constrained Hamiltonian systems. The dynamics of such a system can be described in terms of a symplectic distribution (D, ) on a manifold P. Here, D is a distribution on P and is a symplectic form on D. In other words, associates to each p ∈ P a linear symplectic form p on D p ⊆ T p P. We assume that is smooth in the following sense. For each pair of smooth vector fields X 1 and X 2 with values in D, (X 1 , X 2 ) ∈ C ∞ (P). The symplectic form on D associates to each f ∈ C ∞ (P) a unique vector field Y f on P, with values in D, such that (Y f , X ) = X ( f )

(8.1)

for each vector field X on P with values in D. We call Y f the distributional Hamiltonian vector field of f . There is a bracket operation [·, ·] on the space C ∞ (P) of smooth functions on P, called the almost-Poisson bracket, such that [ f 1 , f 2 ] = Y f1 ( f2 )

(8.2)

for every f 1 , f 2 ∈ C ∞ (P). The almost-Poisson bracket need not satisfy Jacobi’s identity. The ring C ∞ (P) endowed with the almost-Poisson bracket [·, ·] is called an almost-Poisson algebra. Consider a proper action of a Lie group G on P which preserves the distribution D, the symplectic form on D and the Hamiltonian h. Non-holonomic reduction is concerned with the description of the reduced dynamics on the space P/G of orbits of G in P. Since the action of G on P preserves , it follows that for each g ∈ G, the pull-back ∗g : C ∞ (P) → C ∞ (P) preserves the almost-Poisson bracket [·, ·]. In other words, for each f 1 , f 2 ∈ C ∞ (D), ∗g [ f 1 , f 2 ] = [∗g f 1 , ∗g f 2 ].

(8.3)

212

Further examples of reduction

The orbit space P/G, endowed with a differential structure C ∞ (P/G) = { f¯ ∈ C 0 (P/G) | ρ ∗ f ∈ C ∞ (P)},

(8.4)

where ρ : P → P/G is the orbit map, is a subcartesian differential space with the quotient topology. Since the action of G on P is proper, the orbit space P/G is stratified by orbit type. As before, we denote this stratification by N. Each stratum N ∈ N is an orbit of the family X(P/G) of all vector fields on P/G. Let C ∞ (P)G denote the ring of G-invariant functions in C ∞ (P). Equation (8.3) implies that C ∞ (P)G is closed under the almost-Poisson bracket. Since ρ ∗ gives an isomorphism of C ∞ (P/G) onto C ∞ (P)G , it follows that C ∞ (P/G) inherits an almost-Poisson bracket [·, ·] P/G such that [ρ ∗ f¯1 , ρ ∗ f¯2 ] = ρ ∗ [ f¯1 , f¯2 ] P/G .

(8.5)

Thus, C ∞ (P/G) is an almost-Poisson algebra. We can proceed as in Section 6.2, even though the almost-Poisson algebra C ∞ (P/G) does not satisfy the Jacobi identity and the action of G on P fails to have a momentum map. In a sense, the non-holonomic reduction discussed here consists of picking out the results of symplectic reduction which do not depend on the Jacobi identity and the existence of the momentum map. This is the reason why this section is so short. A comprehensive presentation of the current state of the geometry of non-holonomically constrained Hamiltonian systems can be found in a recent book Cushman et al. (2010), which contains several examples. Following the convention adopted in Cushman et al. (2010), we use a bar above a symbol in this chapter to denote that the symbol refers to an object on the orbit space. This should not lead to confusion, because we have no operation of complex conjugation in this chapter. For each f¯ ∈ C ∞ (P/G), we denote by Y f¯ the derivation of C ∞ (P/G) such that Y ¯ ( f¯ ) = [ f¯, f¯ ] P/G (8.6) f

for each

f¯

∈

C ∞ (P/G).

Proposition 8.1.1 The derivation Y f¯ is the push-forward of the distributional Hamiltonian vector field Yρ ∗ f¯ on P by the orbit map ρ : P → P/G. Moreover, Y f¯ is a vector field on P/G; that is, exp tY f¯ is a local one-parameter local group of diffeomorphisms of P/G. Proof For f¯ ∈ C ∞ (P/G), its pull-back ρ ∗ f¯ by ρ is in C ∞ (P)G and Yρ ∗ f0 is a G-invariant distributional Hamiltonian vector field on P. Moreover, for each f¯ ∈ C ∞ (P/G), equation (8.2) implies that

8.1 Non-holonomic reduction

213

Yρ ∗ f¯ (ρ ∗ f¯ ) = [ρ ∗ f¯, ρ ∗ f¯ ] = ρ ∗ [ f¯, f¯ ] P/G = ρ ∗ (Y f¯ ( f¯ )). Thus, Y f¯ is the push-forward of Yρ ∗ f¯ by the orbit map ρ. Since Yρ ∗ f¯ generates a local one-parameter local group exp tYρ ∗ f¯ of diffeomorphisms of P, it follows that translations along integral curves of Y f¯ give rise to a local one-parameter local group exp tY f¯ of diffeomorphisms of P/G such that ρ ◦ (exp t Tρ ∗ f¯ ) = (exp tY f¯ ) ◦ ρ.

(8.7)

Hence, Y f¯ is a vector field on P/G. For each f¯ ∈ C ∞ (P/G), we refer to Y f¯ as the almost-Poisson vector field of f¯ and denote by P(P/G) = {Y f¯ | f¯ ∈ C ∞ (P/G)}

(8.8)

the family of all almost-Poisson vector fields on P/G. Since P(P/G) ⊆ X(P/G), it follows that for each stratum N ⊆ P/G and every x ∈ N , the value at x of the almost-Poisson bracket [ f¯1 , f¯2 ] P/G of functions in C ∞ (P/G) depends only on the restrictions f¯1|N and f¯2|N of f¯1 and f¯2 , respectively, to N . Hence, the space R(N ) = { f¯|N | f¯ ∈ C ∞ (P/G)} of the restrictions to N of smooth functions on P/G inherits the structure of an almost-Poisson algebra from C ∞ (P/G). The almost-Poisson bracket on R(N ) is given by [ f¯1|N , f¯2|N ] N = ([ f¯1 , f¯2 ] P/G )|N for every f¯1 , f¯2 ∈ C ∞ (P/G). By the definition of a stratification, strata N ∈ N are locally closed connected submanifolds N of P/G. Proposition 2.1.8 ensures that every f¯ ∈ C ∞ (N ) coincides locally with a function in R(N ). Hence, the almost-Poisson algebra structure of R(N ) extends to C ∞ (N ). Thus, we have proved the following result. Proposition 8.1.2 Each stratum N of the orbit type stratification N of D/G is an almost-Poisson manifold. Now consider the family P(P/G) of almost-Poisson vector fields on P/G, given by equation (8.8). By Theorem 3.4.5, orbits of P(P/G) are smooth manifolds. Since P(P/G) ⊆ X(P/G), for each x ∈ P/G, the orbit Q of P(P/G) through x is a manifold immersed in the stratum N of P/G that contains x. Moreover, for each f¯ ∈ C ∞ (D/G), the restriction Y f¯|Q of the almost-Poisson vector field of f¯ to Q is a vector field on Q. The set D Q = {Y f¯ (x) | x ∈ Q, f¯ ∈ C ∞ (D/G)}

(8.9)

214

Further examples of reduction

is a generalized distribution on Q locally spanned by vector fields. For each x ∈ Q, consider a form Q on D Q defined by Q (Y f¯ (x), Y f¯ (x)) = (Y f¯ ( f¯))(x).

(8.10)

Proposition 8.1.3 Equation (8.10) gives a well-defined symplectic form Q on the generalized distribution D Q . Proof Suppose that f¯ and f¯ are functions in C ∞ (P/G) such that Y f¯ (x) = Y f¯ (x). Then (Y f¯ ( f¯))(x) = (Y f¯ (x))( f¯) = (Y f¯ (x))( f¯) = (Y f¯ ( f¯))(x), which implies that Q (Y f¯ (x), Y f¯ (x)) does not depend on the choice of f¯ in equation (8.10). Equation (8.6) implies that Q is skew-symmetric. Hence, Q (Y f¯ (x), Y f¯ (x)) depends on Y f¯ (x) but not on the choice of f¯ defining Y f¯ . Therefore, Q is well defined. It remains to show that Q is non-singular. Suppose Y f¯ (x) is a vector in D Q such that Q (Y f¯ (x), Y f¯ (x)) = 0 for all vectors Y f¯ (x) ∈ D Q ∩ Tx Q. Then, (Y f¯ (x))( f¯) = 0 for all f¯ ∈ C ∞ (P/G). But Q is an orbit of the family P(P/G) of almost-Poisson vector fields. Hence, for each function f˜ ∈ C ∞ (Q), there exist a neighbourhood U of x in Q and a function f¯ ∈ C ∞ (P/G) such that f˜|U = f¯|U . Thus, (Y f¯ (x))( f˜) = 0 for all f˜ ∈ C ∞ (Q). Therefore, Y f¯ (x) = 0 ∈ Tx Q, and Q is non-singular. A distributional Hamiltonian system on a manifold P is a symplectic distribution (D, ) on P and a Hamiltonian h ∈ C ∞ (P). The evolution of the system is given by integral curves of the distributional Hamiltonian vector field Yh of the Hamiltonian h. The action of G on P is a symmetry of the distributional Hamiltonian system (D, , h) on P if preserves H , and h. ¯ where h¯ ∈ C ∞ (P/G) is called In this case h ∈ C ∞ (P)G , so that h = ρ ∗ h, the reduced Hamiltonian. The vector field Yh pushes forward to a vector field Yh¯ on P/G. Consider the maximal integral curve c : I → P of Yh through p = c(0) ∈ P. Let Q be the orbit of P(P/G) through ρ( p), and let h¯ |Q be the restriction of h¯ to Q. The reduced motion c¯ = ρ ◦ c : I → Q is the maximal integral curve of the distributional Hamiltonian vector field Yh¯ |Q on Q defined in terms of the symplectic distribution (D Q , Q ).

8.2 Dirac structures The Pontryagin bundle of a manifold Q is the direct sum P of the tangent and cotangent bundles of Q. A Dirac structure is an isotropic subbundle D; it describes a class of dynamical systems. For example, Hamiltonian systems on

8.2 Dirac structures

215

Q corresponding to a given symplectic form ω on Q are elements of the Dirac structure {(u, p) ∈ T Q ⊕ T ∗ Q | p = u ω}. Similarly, distributional Hamiltonian systems corresponding to a distribution H and a symplectic form on H are elements of the Dirac structure {(u, p) ∈ H ⊕ T ∗ Q | p|H = u }.

8.2.1 Symmetries of the Pontryagin bundle The Pontryagin bundle of a manifold Q is the direct sum T Q ⊕ T ∗ Q of the tangent and cotangent bundles of Q. It is naturally isomorphic to the fibre product P = T Q × Q T ∗ Q. Let τ : T Q → Q and ϑ : T ∗ Q → Q be the tangent and the cotangent bundle projections, respectively, and let π : P = T Q × Q T ∗ Q → Q : (u, p) → π(u, p) = (τ (u), ϑ( p)). The Pontryagin bundle carries a symmetric form ·, ·

defined as follows. For each (u 1 , p1 ) and (u 2 , p2 ) in the same fibre of π , (u 1 , p1 ), (u 2 , p2 )

= p1 | u 2 + p2 | u 1 .

(8.11)

The form ·, ·

is indefinite, with signature (dim Q, dim Q). Moreover, the space (P) of smooth sections of the Pontryagin bundle carries a bilinear skew-symmetric bracket [(X, α), (Y, β)] = ([X, Y ], £ X β − £Y α + 12 d(α(Y ) − β(X ))),

(8.12)

called the Courant bracket. A Dirac structure on Q is a maximal isotropic subbundle D of (P, ·, ·

); that is, D is a subbundle of P = T Q × Q T ∗ Q, the form ·, ·

vanishes on all pairs of elements of D, and rank D = dim Q. We denote the inclusion map by ι : D → P and the projection of D onto Q by δ = π ◦ ι : D → Q. The space (D) of smooth sections of D consists of sections (X, α) of P = T Q × Q T ∗ Q with values in D. We assume that D is locally spanned by smooth sections (X 1 , α1 ), . . . , (X m , αm ). A Dirac structure is said to be closed (or integrable) if, for every pair (X, α) and (Y, β) of sections of D, the bracket [(X, α), (Y, β)] is a section of D.1 Let : G × Q → Q : (g, x) → g (x) = gx be an action of a connected Lie group G on the manifold Q. This induces an action T : G × T Q → T Q : (g, u) → T g (u) 1 There is an alternative terminology also used in the literature, in which the ‘Dirac structure’

defined here is called an ‘almost-Dirac structure’, whereas a ‘closed Dirac structure’ is called simply a ‘Dirac structure’.

216

Further examples of reduction

of G on the tangent bundle T Q of Q. The push-forward of a vector field X on Q by g is given by (g )∗ X = T g ◦ X ◦ g −1 , where we treat X as a section of the tangent bundle projection τ : T Q → Q. A vector field X is G-invariant if (g )∗ X = X for each g ∈ G. Similarly, we have an induced action T ∗ : G × T ∗ Q → T ∗ Q : (g, p) → T ∗ g ( p), where T ∗ g ( p) | u = p | T g−1 (u)

for every pair (u, p) ∈ P = T Q × Q T ∗ Q. This definition implies that the action of G on P preserves the evaluation. In other words, T ∗ g ( p) | T g (u) = p | u

(8.13)

for all g ∈ G. If α is a 1-form on Q, considered as a section of the cotangent bundle ϑ : T ∗ Q → Q, then (g )∗ α = T ∗ g ◦ α ◦ g −1 is a section of ϑ, which we shall also call the push-forward of α by g . A form α is G-invariant if (g )∗ α = α for every g ∈ G. For every vector field X and a 1-form α on Q, (g )∗ α | (g )∗ X (x) = ((g )∗ α)(x) | ((g )∗ X )(x)

= T ∗ g (α(g −1 x)) | T g (X (g −1 x))

= α(g −1 x) | X (g −1 x) = α | X (g −1 (x)) = ∗g −1 α | X (x). Therefore, (g )∗ α = ∗g−1 α. The product of T and T ∗ gives rise to the action : G × P → P : (g, (u, p)) → g (u, p) = (T g (u), T ∗ g ( p)). (8.14) For a section σ = (X, α) of π : P → Q, we denote by (g )∗ σ the section of π given by (g )∗ σ = g ◦ σ ◦ g −1 = ((g )∗ X, (g )∗ α) = ((g )∗ X, ∗g−1 α). (8.15) A section σ of π is G-invariant if (g )∗ σ = σ for each g ∈ G. We consider here a Dirac structure D ⊂ P that is invariant under the action of G on P.

8.2 Dirac structures

217

8.2.2 Free and proper action If the action of G on Q is free and proper, the action of G on the Pontryagin bundle is also free and proper. Therefore, Q is a left principal fibre bundle with structure group G, base manifold Q/G and projection map ρ Q : Q → Q/G.2 Similarly, P is a left principal G-bundle with base manifold P/G and projection map ρ P : P → P/G. Since the Pontryagin bundle projection π : P → Q intertwines the actions of G on P and on Q, that is, for each g ∈ G, π ◦ g = g ◦ π , it follows that there exists a map π : P/G → Q/G such that the following diagram commutes: P π

Q

ρP

/ P/G π

ρQ

/ Q/G.

Moreover, the action on P is linear on fibres of the projection π . This implies, as in Proposition 7.2.14, that π : P/G → Q/G is a vector bundle (in the category of manifolds). In general, we have the following result. Remark 8.2.1 Let ε : E → M be a vector bundle, and let G be a Lie group acting freely and properly on E by vector bundle automorphisms. This means that the action of G on E covers a free and proper action of G on M, and the action of G on fibres of ε is linear. Then, the space E/G of G-orbits in E has the structure of a vector bundle over the space M/G of the space of G-orbits in M. Proof The proof of this remark is analogous to the proof of part (i) of Proposition 7.2.14. If σ = (X, α) : Q → P is a G-invariant section of π , then there exists a section σ = (X , α) : Q/G → P/G of π such that the following diagram commutes: PO

ρP

σ

/ P/G O σ

Q

ρQ

/ Q/G.

Here, X : Q/G → T (Q/G) is the section induced by the G-invariant vector field X , and α : Q/G → T ∗ (Q/G) is the section induced by the G-invariant 2 In order to keep the notation simple, since there are several spaces involved here, we do not

assign separate symbols to the quotients.

218

Further examples of reduction

1-form α. In order to give an interpretation of the sections X and α, we introduce a connection on Q. A connection on the principal bundle Q is a G-invariant distribution hor T Q that is complementary to the vertical distribution ver T Q = ker T π . This implies that we have a direct-sum decomposition T Q = ver T Q ⊕ hor T Q.

(8.16)

Every vector u ∈ Tq Q can be decomposed into a vertical part ver u and a horizontal part hor u; that is, u = ver u + hor u. Similarly, every covector p ∈ Tq∗ Q can be decomposed into a vertical part ver p and a horizontal part hor p such that p | u = ver p + hor p | ver u + hor u = ver p | ver u + hor p | hor u . (8.17) This gives rise to a decomposition of the cotangent bundle T ∗ Q = ver T ∗ Q ⊕ hor T ∗ Q.

(8.18)

The decompositions (8.16) and (8.18) lead to a decomposition of the Pontryagin bundle P = ver P ⊕ hor P, where the vertical Pontryagin bundle ver P and the horizontal Pontryagin bundle hor P are given by ver P = ver T Q ⊕ ver T ∗ Q

and

hor P = hor T Q ⊕ hor T ∗ Q.

Equations (8.11) and (8.17) imply that the bilinear form on P decomposes into its vertical and horizontal components (u 1 , p1 ), (u 2 , p2 )

= (ver u 1 , ver p1 ), (ver u 2 , ver p2 )

+ (hor u 1 , hor p1 ), (hor u 2 , hor p2 )

.

(8.19) (8.20)

We denote the restrictions of ·, ·

to ver P and hor P by ver ·, ·

and hor ·, ·

, respectively. Equation (8.19) implies that ·, ·

= ver ·, ·

+ hor ·, ·

.

(8.21)

The Courant bracket (8.12) on the space of sections of P does not decompose into horizontal and vertical parts, because the bracket of a horizontal section of P with a vertical section of P need not vanish. The orbit spaces (ver P)/G and (hor P)/G are vector bundles over P/G. We call (ver P)/G the reduced vertical Pontryagin bundle and (hor P)/G the reduced horizontal Pontryagin bundle. The adjoint bundle of a principal fibre bundle Q is Q[g] = (Q × g)/G. Similarly, the co-adjoint bundle of Q is Q[g∗ ] = (Q × g∗ )/G.3 3 A comprehensive discussion of the associated bundles of a principal bundle can be found in

Kobayashi and Nomizu (1963).

8.2 Dirac structures

219

Proposition 8.2.2 (i) The reduced vertical Pontryagin bundle (ver P)/G is isomorphic to the direct sum Q[g] ⊕ Q[g∗ ] of the adjoint and co-adjoint bundles of Q. (ii) The reduced horizontal Pontryagin bundle is isomorphic to the Pontryagin bundle of the orbit space Q/G. Proof (i) For each ξ ∈ g, we denote by X ξ the vector field on Q generating the action on Q of the one-parameter subgroup exp tξ of G. We shall refer to X ξ as the fundamental vector field corresponding to ξ . For each g ∈ G, we have g(exp tξ )g −1 = exp t Adg ξ . Hence, d ξ ξ −1 −1 exp tξ (g q) ((g )∗ X )(q) = T g (X (g (q))) = T g dt |t=0 d d = g (exp tξ (g −1 q))|t=0 = exp t Adg ξ (q)|t=0 dt dt = X Adg ξ (q). The vertical distribution ver T Q consists of values of vertical vector fields; that is, ver T Q = {X ξ (q) | ξ ∈ g, q ∈ g}. The connection 1-form on Q is a g-valued 1-form θ such that θ (X ξ ) = ξ for every ξ ∈ g. For each g ∈ G, θ (T g ◦ X ξ ◦ g −1 ) = θ ((g )∗ X ξ ) = θ (X Adg ξ ) = Adg ξ . Hence, θ ◦ T g = Adg ◦ θ ; that is, θ intertwines the action of G on ver T Q and the adjoint action of G on g. Consider the map (τ, θ ) : ver T Q → Q × g : u → (τ (u), θ (u)). This is an isomorphism of the vector bundles ver T Q and Q × g over Q. For each g ∈ G and u ∈ ver T Q, we have (τ, θ )(T g (u)) = (τ (T g (u)), θ (T g (u))) = (gτ (u), Adg θ (u)). Hence, (τ, θ ) : ver T Q → Q × g intertwines the action of G on ver T Q with the action G × (Q × g) → Q × g : (g, (q, ξ )) → (gq, Adg ξ ). Therefore, (τ, θ ) induces an isomorphism (τ, θ ) : (ver T Q)/G → (Q × g)/G of the corresponding orbit spaces. In other words, (ver T Q)/G is isomorphic to the adjoint bundle Q[g∗ ]. Now consider the vertical subbundle of T ∗ Q given by ver T ∗ Q = {ver p | p ∈ T ∗ Q}. Let ϕ : ver T ∗ Q → g∗ be a map such that for each p ∈ ver Tq∗ Q and ξ ∈ g∗ ,

220

Further examples of reduction ϕ( p) | ξ = p | X ξ (q) .

For each g ∈ G, ϕ(T ∗ g ( p)) | ξ = T ∗ g ( p) | X ξ (gq) = p | T g −1 (X ξ (gq))

= p | X

Adg−1 ξ

(q) = ϕ( p) | Adg−1 ξ = Adg∗ ϕ( p) | ξ .

Hence, ϕ intertwines the action of G on ver T ∗ Q and the co-adjoint action of G on g∗ . Hence, the map (τ, ϕ) : ver T ∗ Q → Q × g∗ : p → (τ ( p), ϕ( p)) is a G-equivariant isomorphism of vector bundles. Therefore, (τ, ϕ) induces an isomorphism of the orbit spaces (ver T ∗ Q)/G and Q × g∗ )/G, which we denote by (τ, ϕ) : (ver T ∗ Q)/G → (Q × g∗ )/G. The orbit space (Q × g∗ )/G is a vector bundle over Q/G, called the co-adjoint bundle of Q and denoted by Q[g∗ ]. This is the dual bundle of the adjoint bundle Q[g]. The preceding discussion shows that (τ, θ, ϕ) : ver P → Q × g ⊕ g∗ : p → (τ ( p), θ ( p), ϕ( p)) is a isomorphism of vector bundles which preserves the direct-sum structure of the fibres and intertwines the actions of G on ver P and on Q × g ⊕ g∗ . Hence, (τ, θ, ϕ) induces an isomorphism (τ, θ, ϕ) : (ver P)/G → (Q × g ⊕ g∗ )/G = Q[g] ⊕ Q[g∗ ] of bundles over Q/G. (ii) Consider next the horizontal distribution hor T Q on Q. The tangent map Tρ Q maps hor T Q onto T (Q/G). For each g ∈ G, Tρ Q ◦ T g = Tρ Q . Moreover, Tρ Q (u) = 0 if and only if u ∈ ver T Q. Hence, Tρ Q induces a vector bundle isomorphism η : (hor T Q)/G → T (Q/G). −1 (x), we denote by liftq w the For each w ∈ Tx (Q/G) and q ∈ ρ Q unique vector in hor Tq Q that projects on w; that is, Tρ Q (liftq w) = w. The vector liftq w is called the horizontal lift of w to q. The dual map η ∗ : (hor T ∗ Q)/G → T ∗ (Q/G) of η is defined as follows. For each p ∈ Tq∗ Q and w ∈ Tρ Q (q) Q, we have η∗ ( p) | w = p | liftq w . The map (η⊕η∗ ) : (hor P)/G = (hor T Q)/G⊕(hor T ∗ Q)/G → T (Q/G)⊕T ∗ (Q/G) is an isomorphism of bundles over Q/G. The bundle T (Q/G) ⊕ T ∗ (Q/G) is the Pontryagin bundle of the orbit space Q/G. Corollary 8.2.3 The reduced Pontryagin bundle P/G is isomorphic to the direct sum of Q[g] ⊕ Q[g∗ ] and T (Q/G) ⊕ T ∗ (Q/G).

8.2 Dirac structures

221

The bilinear form ·, ·

on P is G-invariant and gives rise to a bilinear form ·, ·

P/G on the reduced Pontryagin bundle such that p1 , p2

= ρ P ( p1 ), ρ P ( p2 )

P/G

(8.22)

for every p1 , p2 in the same fibre of P. Similarly, ver ·, ·

defines a bilinear form ver ·, ·

P/G in (ver P)/G, and hor ·, ·

defines a bilinear form hor ·, ·

P/G in (hor P)/G. Equations (8.21) and (8.22) imply that ·, ·

P/G = ver ·, ·

P/G ⊕ hor ·, ·

P/G .

(8.23)

The Courant bracket (8.12) evaluated on G-invariant sections of P → Q gives a G-invariant section of P → G. Hence, there is a bracket [·, ·] P/G on the space (P/G) of sections of π : P/G → Q/G such that if σ1 and σ2 are G-invariant sections of P → G, then [σ 1 , σ 2 ] P/G = [σ1 , σ2 ].

(8.24)

The bracket [σ1 , σ2 ] of two sections of P, σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ), can be decomposed into a vertical component ver [σ1 , σ2 ] and a horizontal component hor [σ1 , σ2 ]. If σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ) are G-invariant sections of ver P, then ver [σ1 , σ2 ] is a G-invariant section of ver P. Therefore, ver [σ1 , σ2 ] projects to a section of (ver P)/G → Q/G, which we denote by ver [σ 1 , σ 2 ] P/G ; this depends only on the sections σ 1 and σ 2 . Hence, the bracket ver [·, ·] gives rise to a bracket ver [·, ·] P/G on the space of sections (ver P)/G. Similarly, if σ1 = (X 1 , α1 ) and σ2 = (X 2 , α2 ) are G-invariant sections of hor P, then hor [σ1 , σ2 ] is a G-invariant section of hor P. Therefore, hor [σ1 , σ2 ] projects to a section of (hor P)/G → Q/G, which we denote by hor [σ 1 , σ 2 ] P/G ; this depends only on the sections σ 1 and σ 2 . Hence, the bracket hor [·, ·] gives rise to a bracket hor [·, ·] P/G on the space of sections (hor P)/G. Lemma 8.2.4 We assume that the action of G on Q is free and proper. A G-invariant Dirac structure D ⊂ P = T Q ⊕ T ∗ Q is locally spanned by G-invariant sections. Proof Consider q0 ∈ Q, and let m = dim Dq0 = dim Q. Since D is smooth, there exists an m-tuple ((X 1 , α 1 ), . . . , (X m , α m )) of smooth sections of D that span D in a neighbourhood W of q0 . Since Q is a principal bundle over Q/G, there exists a local section σ : U → Q of ρ Q : Q → Q/G passing through q0 . Let S = σ (U ). Without loss of generality, we may assume that S is contained in W. Let m 1 1 m m ((X 1S , α S1 ), . . . , (X m S , α S )) be the restrictions of ((X , α ), . . . , (X , α )) to

222

Further examples of reduction

points of S. These are sections of TS Q ⊕ TS∗ Q over S. Moreover, they span the restriction D S of D to S. Let f : S → R be a bump function with compact support equal to 1 in a neighbourhood S0 of q0 in S. Then, m (( f X 1S , f α S1 ), . . . , ( f X m S , f α S )) span D S in S0 . Since Q is a principal G-bundle over Q/M and σ : U → M is a section of −1 (U ) → G × S intertwining ρ Q : Q → Q/G, there is a diffeomorphism δ : ρ Q

−1 the action of G on ρ Q (U ) and multiplication on the left in G. Without loss of generality, we may assume that δ maps each point s ∈ S to (e, s), where e is −1 (U ) the identity in G. For each i = 1, . . . , m, let Y˜ i be a vector field on ρ Q −1 i i defined as follows. For each q ∈ ρ (U ), Y˜ (q) = T g ( f (s)X (s)), where Q

−1 q ∈ δ −1 (g, s) ∈ (G, S). Clearly, Y˜ i is a G-invariant vector field on ρ Q (U ) −1 i i ˜ extending X S . Similarly, we define a G-invariant 1-form β on ρ Q (U ) by setting β˜ i (q) | u = f (s)α iS (s) | T g −1 u ∀ u ∈ Tq Q.

Since f has compact support in S, the vector fields Y˜ i and the forms β˜ i vanish −1 (U ) in Q. Hence, they can be extended by zero to near the boundary of ρ Q G-invariant globally defined vector fields Y i and 1-forms β i on Q. In other words,

−1 (U ) Y˜ i (q) for q ∈ ρ Q i Y (q) = −1 0 for q ∈ / ρ Q (U ),

−1 β˜ i (q) for q ∈ ρ Q (U ) β i (q) = −1 0 for q ∈ / ρ Q (U ). m Since (( f X 1S , f α S1 ), . . . , ( f X m S , f α S )) span D S0 and D is G-invariant, it −1 1 1 m m follows that ((Y , β ), . . . , (Y , β )) span D in ρ Q (ρ Q (S0 )).

Proposition 8.2.5 Let G be a Lie group acting freely and properly on a manifold Q, and let D be a G-invariant Dirac structure on Q. (i) The space D/G of G-orbits in D is a maximal isotropic subbundle of (P/G, ·, ·

P/G ). (ii) If the Dirac structure D is closed in the sense that, for each pair σ1 and σ2 of G-invariant sections of D → Q, the bracket [σ1 , σ2 ] has values in D, then [σ 1 , σ 2 ] P/G has values in D/G for every pair of sections σ 1 , σ 2 of (D/G) → Q/G. Proof By Corollary 8.2.3, Q/G is a manifold and P/G is a vector bundle over Q/G isomorphic to the direct sum of Q[g] ⊕ Q[g∗ ] and the Pontryagin bundle of Q/G. By assumption, the Dirac structure D is a G-invariant

8.2 Dirac structures

223

subbundle of the Pontryagin bundle P. Moreover, G acts freely and properly on Q, and it acts freely and properly on P by vector bundle automorphisms. By Lemma 8.2.4, D is locally spanned by G-invariant sections. Following an argument analogous to that in the proof of part (i) of Proposition 7.2.14, we conclude that the orbit space D/G is a locally trivial vector bundle over Q/G; see Remark 8.2.1. Equation (8.22) ensures that D/G is isotropic with respect to the bilinear form ·, ·

P/G . Consider a G-invariant section σ of P such that for some q ∈ Q, σ (ρ Q (q)), σ 1 (ρ Q (q))

P/G = 0 for all G-invariant sections σ1 of D. Equation (8.22) implies that σ (q), σ1 (q)

= 0 for all G-invariant sections σ1 of D. Since D is maximal, it follows that σ (q) ∈ D. Hence, there exists a G-invariant section σ of D such that σ (q) = σ (q). Therefore, σ (ρ Q (q)) = σ (ρ Q (q)). This ensures that D/G is a maximal isotropic subbundle of (P/G, ·, ·

P/G ). This completes the proof of part (i). Part (ii) follows from equation (8.24).

8.2.3 Proper non-free action In this subsection, we drop the assumption that the action of G on Q is free. We begin by showing that the freeness of the action of G on Q is not a necessary condition for the conclusion of Lemma 8.2.4. Proposition 8.2.6 A Dirac structure D ⊂ P = T Q ⊕ T ∗ Q, invariant under a proper action of a connected Lie group G on Q, is locally spanned by G-invariant sections. Proof Consider q0 ∈ Q, and let m = dim Dq0 = dim Q. Since D is smooth, there exists an m-tuple (σ 1 , . . . , σ m ) of smooth sections of D that span D in a neighbourhood W of q0 . Let H be the isotropy group of q0 , and let S be the slice at q0 for the action of G on Q. Without loss of generality, we may assume that S is contained in W . The restrictions σ S1 , . . . , σ Sm of σ 1 , . . . , σ m to S span D in S. Let f : S → R be a bump function with compact support equal to 1 in a neighbourhood V0 of q0 in S. The sections f σ1 , . . . , f σn have compact support in S and span D in V0 . Since D is G-invariant, the restriction of D to S is H -invariant. Averaging the sections f σ S1 , . . . , f σ Sm over H , we obtain H -invariant sections σ˜ S1 , . . . , σ˜ Sm of D, given by

224

Further examples of reduction σ˜ Si =

H

(g )∗ ( f σ Si ) dμ(g)

for each i = 1, . . . , m. The sections σ˜ S1 , . . . , σ˜ Sm have compact support in S and span D in a neighbourhood of q0 in S. The sections σ˜ S1 , . . . , σ˜ Sm can be extended to G-invariant sections σ˜ U1 , . . . , σ˜ Um of D in a G-invariant neighbourhood U = G S of q0 in Q, which are defined as follows. Each q ∈ U is of the form q = gs, where g ∈ G and s ∈ S. For each i = 1, . . . , m, and q ∈ U , we set σ˜ Ui (q) = σ˜ Ui (gs) = g (σ Si (s)). If g1 s1 = g2 s2 for g1 , g2 ∈ G and s1 , s2 ∈ S, then s1 = g1−1 g2 s2 , which implies that h = g1−1 g2 ∈ H . Hence, g1 (σ Si (s1 )) = g1 (σ Si (hs2 )) = g1 (h (σ Si (s2 ))) because σ Si is H -invariant. Moreover, g1 (h (σ Si (s2 ))) = g1 h (σ Si (s2 )) = g1 g−1 g2 (σ Si (s2 )) = g2 (σ Si (s2 )), 1

which implies that σ˜ Ui is well defined. Also, for each q = gs ∈ Q and g¯ ∈ G, (( g¯ )∗ σ˜ Ui )(gs) = g¯ ◦ σ˜ Ui ◦ g¯ −1 (gs) = g¯ (σ˜ Ui (g¯ −1 gs)) = g¯ (g¯ −1 g (σ Si (s))) = g¯ g¯ −1 g (σ Si (s)) = g (σ Si (s)) = σ˜ Ui (gs). Therefore, σ˜ Ui is G-invariant. We can extend each section σ˜ Ui to a global section σ˜ i of π : P → Q by setting σ˜ i (q) = 0 for every q in the complement of U = G S in P. Since U is G-invariant and σ˜ Ui is G-invariant, it follows that σ˜ i is G-invariant. Moreover, the support of σ˜ Ui is the G-orbit of the support of σ˜ Si , which is compact in S. Hence, σ˜ i is a smooth global section. Finally, each section σ˜ i has values in D because D is G-invariant. We have obtained smooth, G-invariant, global sections σ˜ 1 , . . . , σ˜ m of π : P → Q, with values in D, which span D in a neighbourhood of q0 . This completes the proof. Lemma 8.2.7 Let σ = (X, α) be a G-invariant section of the Pontryagin bundle P of Q. Given q ∈ Q, let H be the isotropy group of q and let Q H = {q ∈ Q | G q = H }. Let k be a G-invariant Riemannian metric on Q,, and let Tq Q ⊥ H be the k-orthogonal complement of Tq Q H in Tq Q. Then, X (q) ∈ Tq Q H and α(q) annihilates Tq Q ⊥ H.

8.2 Dirac structures

225

Proof For q ∈ Q, let H be the isotropy group of q, and let Q H ⊆ Q be the set of fixed points of the action of H in Q. According to Proposition 4.2.5, Q H is a local submanifold of Q, and Tq Q H = (Tq Q) H is the space of H -invariant vectors in Tq Q. It follows from the proof of Proposition 4.2.6 that Q H is an open subset of Q H . Hence, Tq Q H = Tq Q H = (Tq Q) H . Therefore, if X is a G-invariant vector field on Q, then X (q) is invariant under the action of H on Tq Q, which implies that X (q) ∈ Tq Q H . A G-invariant Riemannian metric k on Q gives rise to a unique vector bundle isomorphism k$ : T ∗ Q → T Q such that if p ∈ Tq∗ Q, then k$ ( p) ∈ Tq Q and k(k$ ( p), u) = p | u . Since α and k are G-invariant, Y = k$ ◦ α is a G-invariant vector field on Q. Hence, Y (q) ∈ Tq Q H . Therefore, for each u ∈ Tq Q ⊥ H, α(q) | u = k(k$ (α( p)), u) = k(Y ( p), u) = 0, which completes the proof. Theorem 8.2.8 Let D ⊂ P = T Q ⊕ T ∗ Q be a Dirac structure. We assume that D is invariant under a proper action of a connected Lie group G on Q. Let H be a compact subgroup of G, and let L be a connected component L of Q H . (i) If (X, α) and (Y, β) are G-invariant sections of P, then their restrictions to points of L give rise to sections (X L , α L ) and (Y L , β L ) of the Pontryagin bundle PL of L that are invariant under the action of G L on L, where G L is defined by equation (4.7). (ii) The restriction to L of the Courant bracket of the G-invariant sections (X, α) and (Y, β) gives the Courant bracket of the sections (X L , α L ) and (Y L , β L ) of PL ; that is, [(X, α), (Y, β)] L = [(X L , α L ), (Y L , β L )]. (iii) The Dirac structure D on Q restricts to a Dirac structure D L on L. (iv) The Dirac structure D is uniquely determined by the collection of all Dirac structures D L as L varies over connected components of Q H and H varies over compact subgroups of G. Proof (i) Let L be a connected component of Q H , where H is a compact subgroup of G. By Proposition 4.2.6, L is a submanifold of Q. By Proposition 8.2.6, every p ∈ D is in the image of a smooth G-invariant section σ = (X, α) of D. In other words, p = (X (q), α(q)), where q = π( p). Proposition 4.2.6 ensures that X (q) ∈ Tq L. Let α L be the pull-back of α to L, and let X L be the

226

Further examples of reduction

restriction of X to points in L. Then, (X L , α L ) is a section of the Pontryagin bundle PL = T L ⊕ T ∗ L. (ii) If (X, α) and (Y, β) are G-invariant, then [X, Y ] is G-invariant, and [X, Y ] restricted to points in L is a vector field [X, Y ] L tangent to L. Moreover, [X, Y ] L = [X L , Y L ]. Similarly, all forms appearing in the expression £ X β − £Y α + 12 d(α(Y ) − β(X )) are G-invariant. Hence, 1 £ X β − £Y α + d(α(Y ) − β(X )) 2 L 1 = (£ X β) L − (£Y α) L + (d(α(Y ) − β(X )) L ) 2 1 = £ X L β L − £YL α L + d(α L (Y L ) − β L (X L )). 2 Therefore, [(X, α), (Y, β)] L = [(X L , α L ), (Y L , β L )]. (iii) We denote by D L the subset of PL spanned by the sections (X L , α L ), which are the restrictions to L of G-invariant sections of D. Let ·, ·

L be the bilinear form of the Pontryagin bundle PL of L. If (X, α) and (X , α ) are G-invariant sections of D, then for each q ∈ L, (X L , α L ), (X L , α L )

L (q) = α L (q) | X L (q) + α L (q) | X L (q)

= α(q) | X (q) + α (q) | X (q)

= (X, α), (X , α )

(q) = 0. This implies that D L is isotropic. Suppose now that (u L , a L ) ∈ Tq L ⊕Tq∗ L satisfies (u L , a L ), (X L (q), α L (q))

L = 0 for every G-invariant section (X, α) of D. For each q ∈ Q, let Tq L ⊥ be the k-orthogonal complement of Tq L in Tq Q, where k is a G-invariant Riemannian metric on Q. We can extend a L ∈ Tq∗ L to a covector a ∈ Tq∗ Q by setting a | v = 0 for each v ∈ Tq L ⊥ . Since Tq L ⊆ Tq Q, the vector u L ∈ Tq L is in Tq Q, and (u L , a) ∈ P. For each G-invariant section (X, α) of D, (X (q), α(q)), (u L , a)

= α(q) | u L + a | X (q)

= α L (q) | u L + a L | X L (q)

= (X L (q), α L (q)), (u L , a L )

L = 0. Hence, (u L , a) ∈ D because D is a maximal isotropic subbundle of (P, ·, ·

). Therefore, (u L , a L ) ∈ D L , which implies that D L is a maximal isotropic subbundle of (PL , ·, ·

L ). Thus, D L is a Dirac structure on L. (iv) We still need to prove that D is uniquely determined by the collection of all Dirac structures D L as L varies over connected components of

8.2 Dirac structures

227

Q H and H varies over compact subgroups of G. It suffices to show that for each q ∈ L ⊆ Q H and a G-invariant section (X, α) of D, the value of α at q is uniquely determined by α L (q), where α L is the pull-back of α to L. Lemma 8.2.7 implies that α(q) | v = 0 for each v ∈ Tq L ⊥ . Hence, α(q) is determined by its restriction α L (q) to Tq L. In Proposition 4.2.6, we showed that the action of G L = N L /H on L is free and proper. Here, N L is the closed subgroup of G consisting of elements g ∈ G that preserve the manifold L. Since the Dirac structure D is G-invariant, it follows that the Dirac structure D L is G L -invariant. Hence, we can apply the results obtained in the preceding subsection to the Dirac structure D L on L. Corollary 8.2.9 For each compact subgroup H of G and each connected component L of Q H , we have the following result. (i) The reduced Pontryagin bundle PL /G L decomposes into a direct sum of the reduced vertical Pontryagin bundle (ver PL )/G L and the reduced horizontal Pontryagin bundle (hor PL )/G L . (ii) (ver PL )/G L is isomorphic to L[g L ]⊕ L[g∗L ], where g L is the Lie algebra of G L , and (hor PL )/G L is isomorphic to T (L/G L ) ⊕ T ∗ (L/G L ). (iii) The reduced Dirac structure D L /G L is a maximal isotropic subbundle (PL /G L , ·, ·

PL /G L ). (iv) If the Dirac structure D L is closed, then the Dirac structure D L /G L is closed. Recall that the connectedness of G implies that M = G L is the stratum of the orbit type stratification M of Q. Hence, the stratum N = ρ(M) of the orbit type stratification of Q/G can be identified with L/G L . Therefore, the restriction D M of D to M can be identified with the G-orbit G D L of D L in P, which implies that D M /G = (G D L )/G = D L /G L . Thus, for each stratum N of the orbit type stratification of Q/G, we have a complete description of D/G restricted to N .

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Index

adjoint action, 105 algebraic reduction, 138 almost-Poisson algebra, 211 almost-Poisson bracket, 211 Arms, J., 6, 8, 142 Aronszajn, N., 2 Bates, L., 4, 6–10 Bierstone, E., 5 Blankenstein, G., 11 Bleuler, K., 9 Bochner, S., 5 Borel Extension Lemma, 145 Buchner, K., 2 Bursztyn, H., 11 Cavalcanti, G.R., 11 Cegiełka, K., 3 co-adjoint action, 106 co-adjoint orbit, 106 commutation of quantization and reduction, 151 connection, 152 on principal bundle, 218 cotangent polarization, 158 Courant, T., 11 covariant derivative, 152 Cushman, R., 4–7, 9, 10, 105 density, 159 derivation, 25 at a point, 26 derived map, 39 diffeomorphism, 16 differential space, 15 differential structure, 15 generated by family of functions, 17 differential subspace, 18 Dirac, P.A.M., 9 Dirac structure, 214

distribution, 51 Duistermaat, J.J., 4, 5, 10 Duval, C., 9 Elhadad, M.J., 9 exponential map, 57, 59 frontier condition, 52 global derivation, 26 Gotay, M., 6, 8, 9 Gualtieri, M., 11 Guillemin, V., 7, 9, 10, 130, 150, 162, 184 Gupta, S.N., 9 H -slice, 82 half-density, 159 half-form, 160 Hall, B., 187 Hamilton, M., 7, 9 Hamiltonian action, 109 Hamiltonian vector field, 108 Heller, M., 2 Hilbert, D., 5 horizontal Pontryagin bundle, 218 Huebschmann, J., 9 immersed manifold, 18 integral curve, 31 integral manifold, 51 isotropy group, 56 isotropy type, 56 Jacobi identity, 110 Jennings, G., 8 Jotz, M., 11 Kähler polarization, 157 Kimura, T., 8 Kirillov, A.A., 6, 9 Kirillov–Kostant–Souriau form, 106 Kirwin, W., 187

234

Koch curve, 41 Kolk, J.A.C., 4 Koon, W.S., 10 Kostant, B., 6, 9, 157 Koszul, J.L., 5 Koszul form, 91 Lagrangian distribution, 155 Leibniz’s rule, 25 Li, H., 187 Libermann, P., 7 local extension, 54 local one-parameter group, 34 local submanifold, 60 locally complete, 46 locally trivial stratification, 53 Lusala, T., 4 Marle, C.-M., 7 Marsden, J.E., 6, 7, 10, 11, 105 Marsden–Weinstein reduction, 134 Marshall, C.D., 3, 5, 102 Marshall form, 100 Maschke, B.M., 10 Meyer, K.R., 6, 105 minimal stratification, 53 Misiołek, G., 7 momentum, 109 momentum map, 109 Moncrief, V., 6 Multarzynski, P., 2 Noether, E., 8 non-holonomic reduction, 211 normalizer, 61 orbit of group action, 56 of vector fields, 45 orbit type, 56 orbit type stratification, 67, 75 Ortega, J.-P., 6, 7 Palais, R., 5, 81 partition of unity, 23 Pasternak-Winiarski, Z., 3 Perlmutter, M., 7 Pflaum, M.J., 4 Poisson algebra, 110 bracket, 110 derivation, 111 differential space, 131

Index

ideal, 132 reduction, 111 vector field, 111 polarization, 155 Pontryagin bundle, 214 positive polarization, 155 prequantization action, 154 condition, 152 map, 153 operator, 153 representation, 154, 155 principal fibre bundle, 69 proper action, 56 push-forward, 27 quantization map, 156 representation, 156 Ratiu, T., 6, 7, 11 refinement, 53 regular component, 41 regular point, 41 regular reduction, 134 Sasin, W., 2, 102 Satake, I., 174 Schwarz, G., 5 second countable, 22 section, 38 Seidenberg, A., 5 semi-definite polarization, 155 shifting maps, 130 shifting trick, 130 Sikorski, R., 1, 2, 15, 16 singular point, 41 singular reduction, 105 singular set, 41 Sjamaar, R., 9, 151, 185 slice, 57 smooth functions, 15 smooth map, 16 ´ Sniatycki, J., 3–12 Souriau, J.-M., 6, 9 space of derivations, 25 Stefan, P., 4 Sternberg, S., 7, 9, 10, 130, 150, 162, 184 strata, 52 stratification, 52 strongly admissible polarization, 155 structural dimension, 41

Index

subcartesian differential space, 21 submanifold, 18 Sussmann, H., 4 symplectic complement, 165 symplectic form, 108 symplectomorphism, 108 tangent bundle, 37 projection, 37 tangent map, 40 tangent space, 26 tangent vector, 26 Tarski, A., 5 Thom, R., 4 Tuynman, G.M., 9 unitarization, 156

van der Schaft, A.J., 10, 11 vector field, 34 vertical Pontryagin bundle, 218 Walczak, P., 2 Watts, J., 3, 5, 102 Wazewski, T., 20 Weinstein, A., 6–8, 10, 11, 105 Weyl, H., 5 Whitney, H., 4 Wilbour, D.C., 8 Woodhouse, N., 9 Yoshimura, H., 11 Zariski, O., 5 Zariski form, 94

235

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