Lectures on the geometry of manifolds

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Author: Liviu I Nicolaescu

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Lectures on the

Geometry of Manifolds

This page is intentionally left blank

Lectures on the

Geometry of Manifolds

Liviu I Nicolaescu University of Michigan, USA

V | f e World Scientific «l^

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Nicolaescu, Liviu I. Lectures on the geometry of manifolds / Liviu I. Nicolaescu. p. cm. Includes bibliographical references and index. ISBN 9810228368 1. Geometry, Differential. 2. Manifolds (Mathematics) I. Title. QA649.N53 1996 516.3'6-dc20 96-28020 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

First published 1996 Reprinted 1999

Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. This book is printed on acid-free paper.

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Thinking of Ia§i, my hometown

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Introduction

Shape is a fascinating and intriguing subject which has stimulated the imagination of many people. It suffices to look around to become curious. Euclid did just that and came up with the first pure creation. Relying on the common experience, he created an abstract world that had a life of its own. As the human knowledge progressed so did the ability of formulating and answering penetrating questions. In particular, mathematicians started wondering whether Euclid's "obvious" absolute postulates were indeed obvious and/or absolute. Scientists realized that Shape and Space are two closely related concepts and asked whether they really look the way our senses tell us. As Felix Klein pointed out in his Erlangen Program, there are many ways of looking at Shape and Space so that various points of view may produce different images. In particular, the most basic issue of "measuring the Shape" cannot have a clear cut answer. This is a book about Shape, Space and some particular ways of studying them. Since its inception, the differential and integral calculus proved to be a very versatile tool in dealing with previously untouchable problems. It did not take long until it found uses in geometry in the hands of the Great Masters. This is the path we want to follow in the present book. In the early days of geometry nobody worried about the natural context in which the methods of calculus "feel at home". There was no need to address this aspect since for the particular problems studied this was a non-issue. As mathematics progressed as a whole the "natural context" mentioned above crystallized in the minds of mathematicians and it was a notion so important that it had to be given a name. The geometric objects which can be studied using the methods of calculus were called smooth manifolds. Special cases of manifolds are the curves and the surfaces and these were quite well understood. B. Riemann was the first to note that the low dimensional ideas of his time were particular aspects of a higher dimensional world. The first chapter of this book introduces the reader to the concept of smooth manifold through abstract definitions and, more importantly, through many we believe relevant examples. In particular, we introduce at this early stage the notion of Lie group. The main geometric and algebraic properties of these objects will be gradually described as we progress with our study of the geometry of manifolds. Besides their obvious usefulness in geometry, the Lie groups are academically very friendly. They provide a marvelous testing ground for abstract results. We have consistently taken advantage of this feature throughout this book. As a bonus, by the end of these lectures the reader will feel comfortable manipulating basic Lie theoretic concepts. To apply the techniques of calculus we need "things to derivate and integrate". These "things" are introduced in Chapter 2. The reason why smooth manifolds have many differentiable objects attached to them is that they can be locally very well approximated by linear spaces called tangent spaces . Locally, everything looks like vn

Vlll

traditional calculus. Each point has a tangent space attached to it so that we obtain a "bunch of tangent spaces" called the tangent bundle. We found it appropriate to introduce at this early point the notion of vector bundle. It helps in structuring both the language and the thinking. Once we have "things to derivate and integrate" we need to know how to explicitly perform these operations. We devote the Chapter 3 to this purpose. This is perhaps one of the most unattractive aspects of differential geometry but is crucial for all further developments. To spice up the presentation, we have included many examples which will found applications in later chapters. In particular, we have included a whole section devoted to the representation theory of compact Lie groups essentially describing the equivalence between representations and their characters. The study of Shape begins in earnest in Chapter 4 which deals with Riemann manifolds. We approach these objects gradually. The first section introduces the reader to the notion of geodesies which are defined using the Levi-Civita connection. Locally, the geodesies play the same role as the straight lines in an Euclidian space but globally new phenomena arise. We illustrate these aspects with many concrete examples. In the final part of this section we show how the Euclidian vector calculus generalizes to Riemann manifolds. The second section of this chapter initiates the local study of Riemann manifolds. Up to first order these manifolds look like Euclidian spaces. The novelty arises when we study "second order approximations " of these spaces. The Riemann tensor provides the complete measure of how far is a Riemann manifold from being flat. This is a very involved object and, to enhance its understanding, we compute it in several instances: on surfaces (which can be easily visualized) and on Lie groups (which can be easily formalized). We have also included Cartan's moving frame technique which is extremely useful in concrete computations. As an application of this technique we prove the celebrated Theorema Egregium of Gauss. This section concludes with the first global result of the book, namely the Gauss-Bonnet theorem. We present a proof inspired from [21] relying on the fact that all Riemann surfaces are Einstein manifolds. The Gauss-Bonnet theorem will be a recurring theme in this book and we will provide several other proofs and generalizations. One of the most fascinating aspects of Riemann geometry is the intimate correlation "local-global". The Riemann tensor is a local object with global effects. There are currently many techniques of capturing this correlation. We have already described one in the proof of Gauss-Bonnet theorem. In Chapter 5 we describe another such technique which relies on the study of the global behavior of geodesies. We felt we had the moral obligation to present the natural setting of this technique and we briefly introduce the reader to the wonderful world of the calculus of variations. The ideas of the calculus of variations produce remarkable results when applied to Riemann manifolds. For example, we explain in rigorous terms why "very curved manifolds" cannot be "too long" . In Chapter 6 we leave for a while the "differentiable realm" and we briefly discuss

DC

the fundamental group and covering spaces. These notions shed a new light on the results of Chapter 5. As a simple application we prove Weyl's theorem that the semisimple Lie groups with definite Killing form are compact and have finite fundamental group. Chapter 7 is the topological core of the book. We discuss in detail the cohomology of smooth manifolds relying entirely on the methods of calculus. In writing this chapter we could not, and would not escape the influence of the beautiful monograph [14], and this explains the frequent overlaps. In the first section we introduce the DeRham cohomology and the Mayer-Vietoris technique. Section 2 is devoted to the Poincare duality, a feature which sets the manifolds apart from many other types of topological spaces. The third section offers a glimpse at homology theory. We introduce the notion of (smooth) cycle and then present some applications: intersection theory, degree theory, Thorn isomorphism and we prove a higher dimensional version of the Gauss-Bonnet theorem at the cohomological level. The fourth section analyzes the role of symmetry in restricting the topological type of a manifold. We prove Elie Cartan's old result that the cohomology of a symmetric space is given by the linear space of its bi-invariant forms. We use this technique to compute the lower degree cohomology of compact semisimple Lie groups. We conclude this section by computing the cohomology of complex grassmannians relying on Weyl's integration formula and Schur polynomials. The chapter ends with a fifth section containing a concentrated description of Cech cohomology. Chapter 8 is a natural extension of the previous one. We describe the Chern-Weil construction for arbitrary principal bundles and then we concretely describe the most important examples: Chern classes, Pontryagin classes and the Euler class. In the process, we compute the ring of invariant polynomials of many classical groups. Usually, the connections in principal bundles are defined in a global manner, as horizontal distributions. This approach is geometrically very intuitive but, at a first contact, it may look a bit unfriendly in concrete computations. We chose a local approach build on the reader's experience with connections on vector bundles which we hope will attenuate the formalism shock. In proving the various identities involving characteristic classes we adopt an invariant theoretic point of view. The chapter concludes with the general Gauss-Bonnet-Chern theorem. Our proof is a variation of Chern's proof. Chapter 9 is the analytical core of the book. Many objects in differential geometry are defined by differential equations and, among these, the elliptic ones play an important role. This chapter represents a minimal introduction to this subject. After presenting some basic notions concerning arbitrary partial differential operators we introduce the Sobolev spaces and describe their main functional analytic features. We then go straight to the core of elliptic theory. We provide an almost complete proof of the elliptic a priori estimates (we left out only the proof of the Calderon-Zygmund inequality). The regularity results are then deduced from the a priori estimates via a simple approximation technique. As a first application of these results we consider a Kazhdan-Warner type equation which recently found applications in solving the

X

Seiberg-Witten equations on a Kahler manifold. We adopt a variational approach. The uniformization theorem for compact Riemann surfaces is then a nice bonus. This may not be the most direct proof but it has an academic advantage. It builds a circle of ideas with a wide range of applications. The last section of this chapter is devoted to Fredholm theory. We prove that the elliptic operators on compact manifolds are Fredholm and establish the homotopy invariance of the index. These are very general Hodge type theorems. The classical one follows immediately from these results. We conclude with a few facts about the spectral properties of elliptic operators. The last chapter is entirely devoted to a very important class of elliptic operators namely the Dirac operators. The important role played by these operators was singled out in the works of Atiyah and Singer and, since then, they continue to be involved in the most dramatic advances of modern geometry. We begin by first describing a general notion of Dirac operators and their natural geometric environment, much like in [10]. We then isolate a special subclass we called geometric Dirac operators. Associated to each such operator is a very concrete Weitzenbock formula which can be viewed as a bridge between geometry and analysis, and which is often the source of many interesting applications. The abstract considerations are backed by a full section describing many important concrete examples. In writing this book we had in mind the beginning graduate student who wants to specialize in global geometric analysis in general and gauge theory in particular. The second half of the book is an extended version of a graduate course in differential geometry we taught at the University of Michigan during the winter semester of 1996. The minimal background needed to successfully go through this book is a good knowledge of vector calculus and real analysis, some basic elements of point set topology and linear algebra. A familiarity with some basic facts about the differential geometry of curves of surfaces would ease the understanding of the general theory, but this is not a must. Some parts of Chapter 9 may require a more advanced background in functional analysis. The theory is complemented by a large list of exercises. Quite a few of them contain technical results we did not prove so we would not obscure the main arguments. There are however many non-technical results which contain additional information about the subjects discussed in a particular section. We left hints whenever we believed the solution is not straightforward. P e r s o n a l n o t e It has been a great personal experience writing this book and I sincerely hope I could convey some of the magic of the subject. Having access to the remarkable science library of the University of Michigan and its computer facilities certainly made my job a lot easier and improved the quality of the final product. I learned differential equations from Professor Viorel Barbu, very generous and enthusiastic person who guided my first steps in this field of research. He stimulated my curiosity by his remarkable ability of unveiling the hidden beauty of this highly technical subject. My thesis advisor, Professor Tom Parker, introduced me to more

XI

than the fundamentals of modern geometry. He played a key role in shaping the manner in which I regard mathematics. In particular, he convinced me that behind each formalism there must be a picture and uncovering it is a very important part of the creation process. Although I did not directly acknowledge it, their influence is present throughout this book. I only hope the filter of my mind captured the full richness of the ideas they so generously shared with me. My friends Louis Funar and Gheorghe Ionesei read parts of the manuscript. I am grateful to them for their effort, their suggestions and for their friendship. I want to thank Arthur Greenspoon for his advice, enthusiasm and relentless curiosity which boosted my spirits when I most needed it. Also, I appreciate very much the input I received from the graduate students of my "Special topics in differential geometry" course at the University of Michigan which had a beneficial impact on the style and content of this book. At last, but not the least, I want to thank my family who supported me from the beginning to the completion of this project.

Ann Arbor, 1996.

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Contents Introduction

i

1

Manifolds 1.1 Preliminaries §1.1.1 Space and Coordinatization §1.1.2 The implicit function theorem 1.2 Smooth manifolds §1.2.1 Basic definitions §1.2.2 Partitions of unity §1.2.3 Examples §1.2.4 How many manifolds are there?

1 1 1 3 6 6 8 9 15

2

Natural Constructions on Manifolds 2.1 The tangent bundle §2.1.1 Tangent spaces §2.1.2 The tangent bundle §2.1.3 Vector bundles §2.1.4 Some examples of vector bundles 2.2 A linear algebra interlude §2.2.1 Tensor products §2.2.2 Symmetric and antisymmmetric tensors §2.2.3 The "super" slang §2.2.4 Duality §2.2.5 Some complex linear algebra 2.3 Tensor fields §2.3.1 Operations with vector bundles §2.3.2 Tensor fields §2.3.3 Fiber bundles

18 18 18 21 23 26 30 30 33 39 42 49 52 52 53 56

3

Calculus on Manifolds 3.1 The Lie derivative §3.1.1 Flows on manifolds §3.1.2 The Lie derivative

61 61 61 63

xiii

XIV

3.2

3.3

3.4

§3.1.3 Examples Derivations of Q*(M) §3.2.1 The exterior derivative §3.2.2 Examples Connections on vector bundles §3.3.1 Covariant derivatives §3.3.2 Parallel transport §3.3.3 The curvature of a connection §3.3.4 Holonomy §3.3.5 Bianchi identities §3.3.6 Connections on tangent bundles Integration on manifolds §3.4.1 Integration of 1-densities §3.4.2 Orientability and integration of differential forms §3.4.3 Stokes formula §3.4.4 Representations and characters of compact Lie groups §3.4.5 Fibered calculus

67 69 69 74 75 75 80 82 85 88 89 91 91 95 101 105 112

4

Riemannian Geometry 4.1 Metric properties §4.1.1 Definitions and examples §4.1.2 The Levi-Civita connection §4.1.3 The exponential map and normal coordinates §4.1.4 The minimizing property of geodesies §4.1.5 Calculus on Riemann manifolds 4.2 The Riemann curvature §4.2.1 Definitions and properties §4.2.2 Examples §4.2.3 Cartan's moving frame method §4.2.4 The geometry of submanifolds §4.2.5 The Gauss-Bonnet theorem

116 116 116 120 125 128 134 144 144 147 150 154 160

5

Elements of the calculus of variations 5.1 The least action principle §5.1.1 1-dimensional Euler-Lagrange equations §5.1.2 Noether's conservation principle 5.2 The variational theory of geodesies §5.2.1 Variational formula? §5.2.2 Jacobi fields

169 169 169 175 179 179 183

XV

6

The fundamental group and covering spaces 6.1 The fundamental group §6.1.1 Basic notions §6.1.2 Of categories and functors 6.2 Covering spaces §6.2.1 Definitions and examples §6.2.2 Unique lifting property §6.2.3 Homotopy lifting property §6.2.4 On the existence of lifts §6.2.5 The universal cover and the fundamental group

191 192 192 196 198 198 200 201 202 204

7

Cohomology 7.1 DeRham cohomology §7.1.1 Speculations around the Poincare lemma §7.1.2 Cech vs. DeRham §7.1.3 Very little homological algebra §7.1.4 Functorial properties of DeRham cohomology §7.1.5 Some simple examples §7.1.6 The Mayer-Vietoris principle §7.1.7 The Kunneth formula 7.2 The Poincare duality §7.2.1 Cohomology with compact supports §7.2.2 The Poincare duality 7.3 Intersection theory §7.3.1 Cycles and their duals §7.3.2 Intersection theory §7.3.3 The topological degree §7.3.4 Thom isomorphism theorem §7.3.5 Gauss-Bonnet revisited 7.4 Symmetry and topology §7.4.1 Symmetric spaces §7.4.2 Symmetry and cohomology §7.4.3 The cohomology of compact Lie groups §7.4.4 Invariant forms on grassmannians and Weyl's integral formula §7.4.5 The Poincare polynomial of a complex grassmannian 7.5 Cech cohomology §7.5.1 Sheaves and presheaves §7.5.2 Cech cohomology

206 206 206 210 213 218 222 224 227 230 230 234 239 239 243 249 251 253 258 258 261 265 267 273 279 279 284

8

Characteristic classes 8.1 Chern-Weil theory §8.1.1 Connections in principal G-bundles

293 293 293

XVI

8.2

8.3

9

§8.1.2 G-vector bundles §8.1.3 Invariant polynomials §8.1.4 The Chern-Weil theory Important examples §8.2.1 The invariants of the torus Tn §8.2.2 Chern classes §8.2.3 Pontryagin classes §8.2.4 The Euler class §8.2.5 Universal classes Computing characteristic classes §8.3.1 Reductions §8.3.2 The Gauss-Bonnet-Chern theorem

Elliptic equations on manifolds 9.1 Partial differential operators: algebraic aspects §9.1.1 Basic notions §9.1.2 Examples §9.1.3 Formal adjoints 9.2 Functional framework §9.2.1 Sobolev spaces in MN §9.2.2 Embedding theorems: integrability properties §9.2.3 Embedding theorems: differentiability properties §9.2.4 Functional spaces on manifolds 9.3 Elliptic partial differential operators: analytic aspects §9.3.1 Elliptic estimates in MN §9.3.2 Elliptic regularity §9.3.3 An application: prescribing the curvature of surfaces 9.4 Elliptic operators on compact manifolds §9.4.1 The Fredholm theory §9.4.2 Spectral theory §9.4.3 Hodge theory

10 Dirac operators 10.1 The structure of Dirac operators §10.1.1 Basic definitions and examples §10.1.2 Clifford algebras §10.1.3 Clifford modules: the even case §10.1.4 Clifford modules: the odd case §10.1.5 A look ahead §10.1.6 Spin §10.1.7 Spin0 §10.1.8 Low dimensional examples

297 298 300 304 304 305 308 310 313 320 320 325 335 335 335 341 344 350 350 358 362 367 370 371 376 381 392 392 402 407 411 411 411 414 417 422 423 425 433 436

xvii §10.1.9 Dirac bundles 10.2 Fundamental examples §10.2.1 The Hodge-DeRham operator §10.2.2 The Dolbeault operator §10.2.3 The spin Dirac operator §10.2.4 The spin0 Dirac operator

441 445 445 451 457 462

Bibliography

468

Index

473

Chapter 1 Manifolds 1.1 §1.1.1

Preliminaries Space and Coordinatization

Mathematics is a natural science with a special modus operandi. It replaces concrete natural objects with mental abstractions which serve as intermediaries. One studies the properties of these abstractions in the hope they reflect facts of life. So far, this approach proved to be very productive. The most visible natural object is the Space, the place where all things happen. The first and most important abstraction of math is the notion of number. Loosely speaking, the aim of this book is to illustrate how these two concepts, Space and Number, fit together. It is safe to say that geometry as a rigorous science is a creation of ancient Greeks. Euclid proposed a method of research which was later adopted by the entire mathematics. We refer of course to the axiomatic method. He viewed the space as a collection of points and distinguished some basic objects in the space such as lines, planes etc. He then postulated certain (natural) relations between them. All other properties were derived from these simple axioms. Euclid's work is a masterpiece of mathematics and has produced many interesting results but it has its own limitations. For example, the most complicated figures one could reasonably study using this method are the conies and/or quadrics and the Greeks certainly did this. A major breakthrough in geometry was the discovery of coordinates by Rene Descartes in the 17th century. Numbers were put to work in the study of space. Their idea of producing what is now commonly referred to as Cartesian coordinates is familiar to any undergraduate. These coordinates are obtained using a very special method (in this case using three concurrent, pairwise perpendicular lines each one endowed with an identification with M). What is important here is that they produced an one-to-one mapping Euclidian Space -> K3

P ^ (x{P), y{P), 1

z(P)).

2

Manifolds

Figure 1.1: Polar coordinates We call such a process coordinatization. The corresponding map is called (in this case) Cartesian system of coordinates. A line or a plane becomes via coordinatization an algebraic object (more precisely an equation). In general, any coordinatization replaces geometry by algebra and we get a twoway correspondence Study of Space <—> Study of Equations. The shift from geometry to numbers is beneficial to geometry as long as one has efficient tools do deal with numbers and equations. Fortunately, about the same time with the introduction of coordinates Newton created the differential and integral calculus which opened new horizons in the study of equations. The Cartesian system of coordinates is by no means the unique or the most useful coordinatization. Concrete problems dictate other choices. For example, the polar coordinates represent another coordinatization of (a piece of the plane) (see Figure 1.1). PH*(r(P),0(P))e(O,oo)x(-7r,7r). This choice is related to the Cartesian choice by the well known formulae :r = rcos0

y = rsin9.

(1.1.1)

A remarkable feature of (1.1.1) is that x(P) and y(P) depend smoothly upon r(P) and 0(P). As science progressed so did the notion of Space. One can think of Space as a configuration set, i.e. the collection of all possible states of a certain phenomenon. For example, we know from the principles of dynamics that the motion of a particle in the

Preliminaries

3

ambient space can be completely described if we know the position and the velocity of the particle at a given moment. The space associated with this problem consists of all pairs (position, velocity) a particle can possibly have. We can coordinatize this space using 6 functions: three of them will describe the position and the other three of them will describe the velocity. We say the configuration space is 6-dimensional. We cannot visualize this space but it helps to think of it as an Euclidian space, only "roomier". There are many ways to coordinatize the configuration space of a motion of a particle and for each choice of coordinates we get a different description of the motion. Clearly, all these descriptions must "agree" in some sense sense since they all reflect the same phenomenon. In other words, these descriptions should be independent of coordinates. Differential geometry studies the objects which are independent of coordinates. The coordinatization process had been used by people centuries before mathematicians accepted it as a method. For example, sailors used it to travel from one point to another on Earth. Each point has a latitude and a longitude which completely determines its position on Earth. This coordinatization is not a global one. There exist four domains delimited by the Equator and the Greenwich meridian and each of is then naturally coordinatized. The points on the equator for example admit two different coordinatizations which are smoothly related. The manifolds are precisely those spaces which can be piecewise coordinatized (with smooth correspondence on overlaps) and the intention of this book is to introduce the reader to the problems and the methods which arise in the study of manifolds. The next section is a technical interlude. We will review the implicit function theorem which will be one of the basic tools for detecting manifolds. §1.1.2

The implicit function theorem

We gather here, without proofs, a collection of classical analytical facts. For more details one can consult [22]. Let X and Y be two Banach spaces and denote by L(X, Y) the space of bounded linear operators X —¥ Y. Definition 1.1.1 Let F : [ / c X - » Y b e a continuous function (U is an open subset of X). F is said to be (Frechet) differentiable at u 6 U if there exists T £ L(X,Y) such that ||F(«o + h) - F{u0) - Th\\Y = o(\\h\\x) as h -> 0. Loosely speaking, a continuous function is differentiable at a point if near that point it admits a " best approximation " by a linear map. When F is differentiable at uo £ U, the operator T in the above definition is uniquely determined by Th = — Ifeo F(u0 + th) = lim \ {F{u0 + th) -

F{u0)).

4

Manifolds

We will use the notation T = DUoF and we will call T the Frechet derivative of F at u0. Assume F : U —> y is differentiable at each point w £ U. Then F is said to be of class C 1 if the map u t-t DUF G L(X, Y) is continuous. F is said to be of class C 2 if u >-» D „ F is of class C 1 . One can define inductively Ck and C°° (or smooth) maps. Example 1.1.2 Consider F : U C M™ -> Mm. Using Cartesian coordinates a; = (s 1 , • • •, x n ) in R n and u = (u1, • • •, M"1) in Rm we can think of F as a collection of m functions on U ul = ul(xx, • • •, i " ) , • • •, um = um{x\ • • •, xn). F is differentiable at a point x € U if and only if the functions ul are differentiable at x in the usual sense of calculus. The Frechet derivative of F at x is the linear operator DXF : Rn —> K" given by the Jacobian matrix d(u\---,um) d{x\---,xn)

(ch± \dxi

1
F is smooth if and only if the functions u'(x) are smooth. Exercise 1.1.1 (a) Let U c L(M 7l ,K n ) denote the set of invertible n x n matrices. Show that U is an open set. (b) Let F : U -5- U be defined as A -> A" 1 . Show that DAF(H) = -A^HA-1 for any n x n matrix if. (c) Show the Frechet derivative of the map det : L(»n,K™) -> R ( A n- det A) at 1K» G L(M", K n ) is given by DA\A=1det{H)

= tiH,

VfleL(l",K").

Theorem 1.1.3 (Inverse function theorem) Let X, Y be two Banach spaces and F : U C X —> Y a smooth function. If at a point u0 £ U the derivative DU0F 6 L(X, Y) is invertible, then there exits a neighborhood U\ of u0 in (7 such that F(Ui) is an open neighborhood of v0 = F(u0) in Y and F : f/i —>• F(/7i) is bijective, with smooth inverse. The spirit of the theorem is very clear: the invertibility of the derivative DUQF "propagates" (locally) to F because DU0F is a very good local approximation for F. More formally, if we set T = DU0F, then F(u0 + h} = F{u0) +Th + r{h) where r(h) = o(\\h\\) as h —» 0. The theorem states that for every v sufficiently close to w0 the equation F(u) = v has a unique solution u — u0 + h with h very small. To

5

Preliminaries

prove the theorem one has to show that for \\v — VQ\\Y sufficiently small the equation below v0 + Th + r(h) = v has a unique solution. We can rewrite the above equation as Th = v - v0 - r{h)

T-l{v-vQ-r(h)).

h=

The last equation is a fixed point problem which can be approached successfully via the Banach fixed point theorem. Theorem 1.1.4 (Implicit function theorem) Let X, Y, Z be Banach spaces and F : X x Y —> Z a smooth map. Let (xo,yo) € X xY and set z0 = F(x0,y0). Set F2 :Y -> Z, F2{y) = F(x0,y). Assume DyoF2 e L(Y,Z) is invertible. Then there exist neighborhoods U of x0 £ X, V of y0 € Y and a smooth map G : U —> V such that S = F^izo) n (U x V) is the graph of G i.e. {(x,y)eUxV;

F(x,y)

= z0} = {(x,G{x))

£ Ux V ;

x&U).

Proof Consider the map H : X x Y —> X x Z, £ = (x, y) ^ (x,F(x,y)). if is a smooth map and at £0 = (x 0 , j/o) its derivative D^0H : X x Y —> X x Z has the block decomposition lx D^F,

*V* =

0 DioF2

Above, DF\ (resp. DF2) denotes the derivative o f i H F(x,y0) (resp. the derivative of j/1—> F(xo,y)). D^H is invertible and its inverse has the block decomposition (^oH)-1 =

lx 1

{D^y

0

DM)

(DsBF2yl

Thus, by the inverse function theorem the equation (x,F(x,y))

= (x,zQ)

has a unique solution (x,y) = H~l(x,Zo) in a neighborhood of (x0,y0). It obviously satisfies x = x and F(x,y) = 0. Hence, the set {{x,y) ; F(x,y) = z0} is locally the graph of x >->• H~l(x,z0). Q

6

Manifolds

Figure 1.2: Transition maps

1.2 §1.2.1

S m o o t h manifolds Basic definitions

We now introduce the object which will be the main focus of this book, namely we will define the concept of (smooth) manifold. It formalizes the general principles outlined in Sec. 1.1.1. Definition 1.2.1 A smooth manifold of dimension m is a locally compact Hausdorff space M together with the following collection of data (henceforth called atlas or smooth structure) consisting of: (a) an open cover {Ui]ieI of M; (b) continuous, injective maps $,:£/,—> R m (called charts or local coordinates) such that if [/j n [/j / 0 the transition map $>j o tfr1 : *,.(£/. n Uj) C l M - > %j{Ui n Uj) C R m is smooth. (We say the various charts are compatible; see Figure 1.2). The chart *£; can be viewed as a collection of m functions (x1,..., similarly, we can view ^ as a collection of functions ( y 1 , . . . , ym). map ^j o v?"1 can then be interpreted as a collection of maps ( i 1 , . . . , xm) -> {y\x\

..., xm), •••, y™{x\ ...,

xm) on U{ and The transition

xm)).

The first and the most important example of manifold is R n itself. The natural smooth structure consists of an atlas with a single chart l j j " : R" —» R". To construct more examples we will use the implicit function theorem .

Smooth

manifolds

7

Definition 1.2.2 (a) Let M, N be two smooth manifolds of dimensions m and respectively n. A continuous map / : M —> N is said to be smooth if for any local chart (j> on M and ip on N the composition ^ o / o ^ - 1 (whenever this makes sense) is a smooth map R m -> M". (b) A smooth map / : M —> N is called a diffeomorphism if it is invertible and its inverse is also a smooth map Example 1.2.3 The map t >-» e* is a diffeomorphism ( - c o , oo) —> (0, oo). The map 11-> f3 is a homeomorphism R —> M but it is not a diffeomorphism! If M is a smooth manifold we will denote by C°°(M) the linear space of all smooth functions M -> R. Remark 1.2.4 Let U be an open subset of the smooth manifold M (dim M = m) and tp : U —¥ Mm a smooth, one-to one map with open image and smooth inverse. Then tp defines local coordinates over U compatible with the existing atlas of M. Thus (U, ip) can be added to the original atlas and the new smooth structure is diffeomorphic with the initial one. Using Zermelo's Axiom we can produce a maximal atlas (no more compatible local chart can be added to it). Our next result is a general recipe for producing manifolds. Historically, this is how manifolds entered mathematics. Proposition 1.2.5 Let M be a smooth manifold of dimension m and / i , . . . , /* € C°°(M). Define Z = Z(fu...Jk)

= {pe M ; h(p) = ... = fk(p) = 0}.

Assume the functions / i , . . . , / j t are functionally independent along Z i.e., for each p € Z the matrix ( 1 4 ) has rank k. Here x = (x1,...,xm) denotes a local chart of M near p. Then Z has a natural structure of smooth manifold of dimension m — k. Proof Step 1: Constructing the charts Let po G Z and denote by (x1,..., xm) z local coordinates near p0 such that x (p0) = 0. One of the k x k minors of the matrix ( 1 4 ) is non degenerate. Assume this is determined by the last k columns (and all the k lines). We can now think of the functions / i , • • •, /* as defined on an open subset U of R m . Split R m as R m_ * x R* and set x' = (x1,.. .,xm~k), m k+1 m x" = (x ~ ,... ,x ). We are now in the setting of the implicit function theorem with X = Rm"fc, Y = Mfc, Z = R* and F : X x Y -> Z is x >->• (h{x),.. .,fk(x)). In this case DF2 = ( ^ 7 ) is invertible since it corresponds to our non degenerate minor. Thus, in a neighborhood U of po, the set Z is the graph of some function g : R m -* -> Mk, x' i-> x" = g{x') Zr\U

= {(x',g(x'))

; \x'\ small}.

8

Manifolds

We now define ippo : Z n U -> R" 1- * by

P = {x', g(x')) H-J. x' e Rm-k. ippo is a local chart of Z near poStep2 The transition maps for the charts constructed above are smooth. The details are left to the reader. • Exercise 1.2.1 Complete Step 2 in the proof of Propositionl.2.5. Definition 1.2.6 Let M be a m-dimensional manifold. A codimension k submanifold of M is a subspace N C M locally defined as the common zero locus of functionally independent functions / i , • • •, / t 6 C°°{M). Propositionl.2.5 shows that any submanifold has a natural smooth structure so it becomes a manifold per se. §1.2.2

Partitions of unity

This is a very brief technical subsection describing a trick we will extensively use in this book. Definition 1.2.7 Let M be a smooth manifold and (Ua)aeA an open cover of M. A (smooth) partition of unity subordinated to this cover is a family (fp)psn C C°°{M) satisfying the following conditions. (0 0 < fp < 1 (ii) 30 : B —> A such that supp fp C U^p). (iii) The family (supp/^) is locally finite i.e. any point x € M admits an open neighborhood intersecting only finitely many supports supp/g. (iv)E/j//j(a;) = l f o r a U i e M . We include here for the reader's convenience the basic existence result concerning partitions of unity. For a proof we refer to [73]. Proposition 1.2.8 (a) For any open cover iX = (Ua)aeA) of a smooth manifold M there exists at least one smooth partition of unity {fp)p^s subordinated to i t such that supp fp is compact for any (5. (a) If we do not require compact supports then we can find a partition of unity in which B = A and (j> = 1^. Exercise 1.2.2 Let M be a smooth manifold and S C M a closed submanifold. Prove that the restriction map r : C°°(M) ->• C°°(5) is surjective.

f^f\s

Smooth

manifolds

§1.2.3

Examples

9

Manifolds are everywhere and in fact, to many physical phenomena which can be modeled mathematically one can naturally associate a manifold. On the other hand, many problems in mathematics find their most natural presentation using the language of manifolds. To give the reader an idea of the scope and extent of modern geometry we present here a short list of examples of manifolds. This list will be enlarged as we enter deeper into the study of manifolds. Example 1.2.9 (The round n-dimensional sphere) This is the codimension 1 submanifold of K n + 1 given by the equation \x\2 = f > * ) 2 = r 2 , x = {x°, ...,xn)e

Mn+1.

i=0

One checks that, along the sphere, the differential of \x\2 is nowhere zero so by Proposition 1.2.5 Sn is indeed a smooth manifold. In this case one can explicitly construct an atlas (consisting of two charts) which is useful in many applications. The construction relies on stereographic projections. Let N and S denote the north and resp. south pole of Sn (N = ( 0 , . . . , 0,1) 6 E n + 1 , S = ( 0 , . . . , 0, - 1 ) € K n + 1 ). Consider the open sets UN = Sn\ {N} and Us = Sn \ {S}. They form an open cover of Sn. The stereographic projection from the north pole is the map a^ '• UN —> Kn such that for any P 6 UN, ON(P) is the intersection of the line NP with the hyperplane {xn = 0} = K™. The stereographic projection from the south pole is defined similarly. For P € UN denote by (y1{P), • • -,yn{P)) the coordinates of aN{P) and for Q € Us denote by (z1(Q), • • •, zn(Q)) the coordinates of &s{Q)- A simple argument shows the map (y1(P), • • • ,yn(P)) ^ {z1{P), • • -,zn{P)) ( P e UN (1 UN) is smooth (see the exercise below. Hence {(UN,CTN), (Us,
V'

r,

V« = l , - - - , n .

(£?=i(V')2) Example 1.2.10 (The n-dimensional torus) This is the codimension n submanifold of R2n(xi, j/i; ...; xn, yn) defined as the zero locus A + !/? = ••• = x*n + yl = l. Note that T1 is diffeomorphic with the 1-dimensional sphere S1 (unit circle). As a set T™ is a direct product of n circles T™ = Sl x • • • x S1 (see Figure 1.3) The above example suggests the following general construction.

10

Manifolds

Figure 1.3:

The 2-dimensional torus

Example 1.2.11 Let M and N be smooth manifolds of dimension m and respectively n. Then their topological direct product has a natural structure of smooth manifold of dimension m + n. Example 1.2.12 (The connected sum of two manifolds) Let Mi and M2 be two manifolds of the same dimension m. Pick Pi € Mj (i = 1,2), choose small open neighborhoods [7j of pt in Mj and then local charts ipi identifying each of these neighborhoods with the ball of radius 2 in Mm, #2(0). Let Vj C Ui correspond (via ipi) to the annulus {1/2 < |a:| < 2} c R m . Consider : {1/2 < | Z | < 2} -> {1/2 < \x\ < 2},

ftx)

=

~ .

The action of <j> is clear: it switches the two boundary components of {1/2 < \x\ < 2} and reverses the orientation of the radial directions. Now " glue" V\ to V2 using the "prescription" given by ^ 2 _1 ° ( / ,0, /'i : ^1 ~~• Vi- In this way we obtain a new topological space with a natural smooth structure induced by the smooth structures on Mj. Up to a diffeomeorphism, the new manifold thus obtained is independent of the choices of local coordinates ([16]) and it is called the connected sum of Mi and M 2 and is denoted by Mi#M2 (see Figure 1.4). Example 1.2.13 (The real projective space RP n ) As a topological space HP" is the quotient of R n + 1 modulo the equivalence relation

x ~ y 4 4 3A G H* : x = Xy. The equivalence class of x = ( x ° , . . . , xn) G Rn+l \ {0} is usually denoted by [rr° : . . . : xn}. Alternatively, HP" is the set of all lines (directions) in R" +1 . Traditionally one attaches to each direction in R n + 1 a point at infinity so that HP" can be thought as the collection of all points at infinity along all directions.

Smooth

manifolds

Figure 1.4: Connected sum of tori KP n + 1 has a natural structure of smooth manifold. To describe it consider the sets Uk = {[x° : ... : xn\ e MP" ; xk ^ 0}, k = 0 , . . . , n. Now define i>k : Uk -> K"

[x° : ... : xn] .-> (x°/xk,...,

xk~l/xk,

xk+l/xk,.

..xn).

The maps ipk define local coordinates on the projective space. The transition map on the overlap region Uk C\ Um = {[x° : . . . : xn] ; xkxm =fc 0} can be easily described. Set Mx°

:

• • • : *"]) = (6> • • • . & ) • Mx°

•••••• xn}) = (Vl,... ,r,n).

The equality [x° : ... : xn] = [& : . . . : 6 - 1 : 1 : & : . . . : f„] = [r/i : . . . : 7?m_i : 1 : r?m : . . . : r)n] immediately implies (assume k < m) 6 = Vi/Vk, Zk = Vk+i/m, £m = ?7m%,

•••, &-i = Vk-i/Vk •••, £m-2 = rim-i/Vk •••, £n = W %

6 + i = Vk £m-i = l/»7/t

(1-2-1)

This shows the map T/^ O T/J^1 is smooth and proves that MP" is a smooth manifold. Note that when n = 1, HP 1 is diffeomorphic with S1. One way to see this is to observe that the projective space can be alternatively described as the quotient space of 5 " modulo the equivalence relation which identifies antipodal points .

12

Manifolds

Example 1.2.14 (The complex projective space CP") The definition is formally identical to that of MP". CP" is the quotient space of C + 1 \ {0} modulo the equivalence relation x ~ y 4 4 3A <= C : x = Xy. The open sets Ut are defined similarly and so are the local charts tpk '• £4 —> C n . They satisfy transition rules similar to (1-2.1) so that CP71 is a smooth manifold of dimension In. In the above example we encountered a special (and very pleasant) situation: the gluing maps not only are smooth, they are also holomorphic as maps tl>k0i>^ '• U —> V where U and V are open sets in C n . This type of gluing induces a "rigidity" in the underlying manifold and it is worth distinguishing this situation. Definition 1.2.15 (Complex manifolds)A complex manifold is a smooth, 2ndimensional manifold M which admits an atlas {(Ui,tpi) : Ui -» C"} such that all transition maps are holomorphic. The complex projective space is a complex manifold. Our next example naturally generalizes the projective spaces described above. Example 1.2.16 (The grassmannians Gj;i„(M) and Gk,n{C)) As a set Gfc?n(R) consists of all fc-dimensional linear subspaces of R™. To topologize it introduce (following [41]) the notion of gap between two subspaces U, V C M" as 6{U, V) = sup{dist(M, V) ; u € U, \u\ = 1}. The distance between two subspaces is then defined as 5{U,V)=max{S{U,V),

8{V,U)}.

One can show easily that (Gjtin(M), 5} is a metric space (exercise). Let {ei, • • •, en} denote the standard basis of M". For any subset I c {1, • • •, n} with #1 = k define Ej = span {ei ; i e 7} so that Et G Gktn(R). Now define V, = {V e Gk,n{M) ; n n ^ = 0}. Here Ej- is the orthogonal complement of Ej in W1 with respect to the Euclidian inner product. We leave the reader as an exercise the proofs of the following facts. (a) V is a an open subset of Gfci?1(K). (b) The family (Vj) covers Gfc,„(R). (c) Any U e Vj can be uniquely represented as the graph of a linear operator T = T{U) : Ei -> Ej- i.e. U = {(x,Tx)

€ E, x Ej =* Kn ; x € £ / } .

Smooth

13

manifolds

We obtain continuous (why ?) maps • L(EuEj-) = M{n_k)xk(M.) ^ !*("-*>. One can show that (d) the maps \I// define an atlas on Gjt]71(M). In particular G/t,„(M) has dimension k(n — k). <3fc,n(C) is defined as the space of complex /c-dimensional subspaces of C n . It can be structured as above as a smooth manifold of dimension 2k(n — k). Note that Gi,„(E) = MP" - 1 and Gi,„(C) = OP"" 1 . The grassmannians have important applications in many classification problems. Exercise 1.2.4 Prove the statements (a)-(d) in the above example. G/t,n(C) is a complex manifold (complex dimension k(n — k)).

Show that

Example 1.2.17 (Lie groups) A Lie group is a smooth manifold G together with a group structure on it such that the map GxG-^G

(g,h)^g-h-1

is smooth. These structures provide an excellent way to formalize the notion of symmetry . (a) (Rn, + ) is a commutative Lie group. (b) The unit circle Sl can be alternatively described as the set of complex numbers of norm one and the complex multiplication defines a Lie group structure on it. This is a commutative group. More generally the torus Tn is a Lie group as a direct product of n circles. One can show that any connected commutative Lie group has the from Tn x Mm. (c) The general linear group GL(n, K) defined as the group of invertible nxn matrices with entries in the field DC = R, C is a Lie group. Indeed, GL(n, K) is an open subset (see Exercise 1.1.1) in the linear space o f n x n matrices with entries in K. It has dimension <%n2 where djj is the dimension of K as a linear space over M. (d) The orthogonal group 0(n) is the group of real nxn matrices satisfying T -Tl = id. To describe its smooth structure we will use the Cayley transform trick as in [65] (see also the classical [74]). Set Mn(R)*

= {Te

M n (K) ; det(l + T) / 0}.

The matrices in M n (K)* are called non exceptional. Clearly 1 G 0(n)* = 0(n)C\ M„(M)# so that 0(n)# is a nonempty open subset of 0(n). The Cayley transform is the map # : Mn(R)* -» Mn(U) defined by A^A*

= (l~ A)(l +

The Cayley transform has some very nice properties.

A)-\

14

Manifolds

(i) A* G Mn(R)* for every A G M n ( R ) # . (ii) # is involutory i.e. {A*)* = A for any A G Mn(M)*. (iii) For every T G O(n)* the matrix T * is skew-symmetric and conversely if A e Mn(R)# is skew-symmetric then ,4* G O(n). Thus the Cayley transform is a homeomorphism from 0{n)# to the space of nonexceptional, skew-symmetric, matrices. The latter space is an open subset in the linear space of real n x n skew-symmetric matrices, o(n). Any T G 0(n) defines a self-homeomorphism of 0(n) by left translation in the group LT : 0(n) -> 0(n) S H> LT{S) = T-S. We obtain an open cover of 0(n):

0{n)=

|J

T-0{n)*.

TeO{n)

Define * r : T-0(n)*

->• o{n) by S )-)• ( T - 1 ^ ) * . One can show that (T • O(n)*,
defines a smooth structure on 0(n).

In particular we deduce

dimO(n) = n(n - l ) / 2 . Inside O(n) lies a normal subgroup (the special orthogonal group) SO(n) = {T G 0{n) ; d e t T = l } . SO(n) is a Lie group as well and dimSO(n) = dim0(n). (e) The unitary group U(n) is defined as t/(n) = {Te

GL(n, C ; T • T* = 1}

T* denotes the conjugate transpose (adjoint) of T. To prove that U(n) is a manifold one uses again the Cayley transform trick. This time we coordinatize the group using the space u(n) of skew-adjoint (skew-Hermitian) nxn complex matrices (A = — ^4*). Thus U(n) is a smooth manifold of dimension dimU(n) = dimw(n) = n 2 . Inside U(n) sits the normal subgroup SU(n), the kernel of the group homomorphism det : U(n) —> S 1 . SU(n) is also called the special unitary group. This a smooth manifold of dimension n 2 — 1. In fact the Cayley transform trick allows one to coordinatize SU(n) using the space su(n) = {A€ u{n) ; tr A = 0}.

Smooth

15

manifolds

Exercise 1.2.5 (a)Prove the properties (i)-(iii) of the Cayley transform and then show that [T • 0(n)#, ^T) defines a smooth structure on 0{n). (b) Prove that U(n) and SU(n) are manifolds. (c) Show that 0{n), SO(n), U(n), SU(n) are compact spaces. (d) Prove that 517(2) is diffeomorphic with S3 (Hint: think of S3 as the group of unit quaternions.) Exercise 1.2.6 Let SL(n;K) denote the group o f n x n matrices of determinant 1 with entries in the field K=M, C. Using the Cayley trick show that SL(n; K) is a smooth manifold modeled on the linear space d{n, K) = {A € M n x n (K) ; tr A = 0}. In particular it has dimension d^(n2 — 1), where d = dimjj K.

Exercise 1.2.7 (a)Let G be a connected Lie group and denote by U a neighborhood of 1 € G. If H is the subgroup algebraically generated by U show that H is dense in G. (b) Let G be a compact Lie group and g € G. Show that 1 e G lies in the closure oi{gn;neZ\ {0}}. §1.2.4

How many manifolds are there?

The list of examples in the previous subsection can go on for ever so one may ask whether there is any coherent way to organize the collection of all possible manifolds. This is too general a question to expect a clear cut answer. We have to be more specific. For example we can ask Question 1: Which are the compact manifolds of a given dimension d? For d = 1 the answer is very simple: the only compact 1-dimensional manifold is the circle S1. So we can raise the stakes and try the same problem for d = 2. Already the situation is more elaborate. We know at least two surfaces: the sphere S 2 and the torus T 2 . They clearly look different but we have not yet proved rigorously that they are indeed not diffeomorphic. This is not the end of the story. We can connect sum two tori, three tori or any number g of tori. We obtain doughnut-shaped surface as in Figure 1.5 Again we face the same question: do we get non-diffeomorphic surfaces for different choices of gl Figure 1.5 suggests that this may be the case but this is no rigorous argument. We know another example of compact surface, the projective plane MP2 and we naturally ask whether it looks like one of the surfaces constructed above. Unfortunately, we cannot visualize the real projective plane (one can prove rigorously it

16

Manifolds

Figure 1.5: Connected sum of 3 tori does not have enough room to exist inside our 3-dimensional Universe). We have to decide this question using a little more than the raw geometric intuition provided by a picture. To kill the suspense, we mention that KIP2 does not belong to the family of doughnuts. The reason is that a torus for example has two faces: an inside face and an outside face (think of a car rubber tube). MP2 has a weird behavior: it has "no inside" and "no outside". It has only one side! One says the torus is orientable while the projective plane is not. We can now connect sum any numbers of MP2's to any doughnut an thus obtain more and more surfaces, which we cannot visualize and we have yet no idea if they are pairwise distinct. A classical result in topology says that all compact surfaces can be obtained in this way (see [52]) but in the above list some manifolds are diffeomorphic and we have to describe which. In dimensions > 3 things are not settled and to make things worse in dimension > 4 Question 1 is algorithmically undecidable . We can reconsider our goals and look for all the manifolds with a given property X. In many instances one can give fairly accurate answers. Property X may refer to more than the (differential) topology of a manifold. Real life situations suggest the study of manifolds with additional structure. The following problem may give the reader a taste of the types of problems we will be concerned with in this book. Question 2 Can we wrap a planar piece of canvas around a metal sphere in a one-to-one fashion? (The canvas is flexible but not elastic). A simple do-it-yourself experiment is enough to convince anyone that this is not possible. Naturally, one asks for a rigorous explanation of what goes wrong. The best explanation of this phenomenon is contained the the celebrated Teorema Egregium (Golden Theorem) of Gauss. Canvas surfaces have additional structure (they are made of a special material) and for such objects there is a rigorous way to measure "how curved" are they. One then realizes that the problem in Question 2 is impossible

Smooth

manifolds

17

since a (canvas) sphere is curved in a different way than a plane canvas. There are many other structures Nature forced us into studying them but they may not be so easily described in elementary terms. A word to the reader. The next two chapters are the most arid in geometry but, keep in mind that behind each construction lies a natural motivation and, even if we do not always have the time to show it to the reader, it is there, and it may take a while to reveal itself. Most of the constructions the reader will have to "endure" in the next two chapters constitute not just some difficult to "swallow" formalism but the basic language of geometry. Learning this language may not be the most pleasant thing, but surely enough, it is a very rewarding enterprise.

Chapter 2 Natural Constructions on Manifolds The goal of this chapter is to introduce the basic terminology used in differential geometry. The key concept is that of tangent space at a point which is a first order approximation of the manifold near that point. We will be able to transport many notions in linear analysis to manifolds via the tangent space.

2.1 §2.1.1

The tangent bundle Tangent spaces

We begin with a simple example which will serve as a motivation for the abstract definitions. Example 2.1.1 Consider the sphere (S2):x2

+ y2 + z2 = l

inK 3 .

We want to find the plane passing through the north pole N(0, 0,1) which locally is "closest" to the sphere. The natural candidate for this osculator plane would be a plane given by a linear equation which best approximates the defining equation x2 + y2 + z2 = 1 in a neighborhood of the north pole. The linear approximation of x2 + y2 + z2 near N is x2 + y2 + z2 = 1 + 2(2 - 1) + 0(2) where 0(2) denotes a quadratic error. Hence, the the horizontal plane through the north pole. The called the tangent space to S2 at N. The above construction has one deficiency: it objects "outside" the manifold S2. There is one 18

osculator plane is z = 1 which is linear subspace {z = 0} C R3 is is not intrinsic since it relies on natural way to fix this problem.

19

The tangent bundle

Look at a smooth path j(t) on S 2 passing through N at t = 0. Hence t >-» 7(f) g R3 and |7(<)|2 = 1. (2.1.1) If we derivate (2.1.1) at t = 0 we get (7(0), 7(0)) = 0 i.e. 7(0) _L 7(0) so that 7(0) lies in the linear subspace z = 0. We deduce that the tangent space consists of the tangents to the curves on S 2 passing through N. This is apparently no major conceptual gain since we still regard the tangent space as a subspace of R 3 and this is still an extrinsic description. However, if we use the stereographic projection from the south pole we get local coordinates (u, v) near N and any curve ^{t) as above can be viewed as a curve t H-> (u{t),v(t)) in the (u, v) plane. If (t) is another curve through JV given in local coordinates by 11-» (u(t), vit)) then 7(0) = 0(0) 4=> (u(0),t>(0)) = (u(0U(0)). The right hand side of the above equality defines an equivalence relation ~ on the set of smooth curves passing trough (0,0). Thus, there is a bijective correspondence between the tangents to the curves through N and the equivalence classes of "~". This equivalence relation is now intrinsic modulo one problem: "~" may depend on the choice of the local coordinates. Fortunately, as we are going to see, this is a non-issue. Definition 2.1.2 Let Mm be a smooth manifold and po a point in M. Two smooth paths a, P : {—e,e) —> M such that Q ( 0 ) = /3(0) = po are said to have a 1st order contact at p 0 if there exist local coordinates (x) = (x1,..., xm) near p 0 such that ±Q(0) = xp(Q) where a(t) = (xa(t)) = (a£(t),.. • ,a£(t)) and (5{t) = {x0(t)) = (x}(t),... We write this a~\fi

,xj{t)).

Lemma 2.1.3 ~ i is an equivalence relation. Sketch of proof ~ i is trivially reflexive and symmetric so we only have to check the transitivity. Let a ~ i /? and p ~ i 7. Thus there exist local coordinates (x) = (x\.. .,xm) and {y) = (y\ ...,ym) nearp 0 such that (i a (0)) = {±0(O)) and (yp(0)) = (y 7 (0)). The transitivity follows from the equality

m = m = E !£4(°) = E |§^(o)=»i(o).D Definition 2.1.4 A tangent vector to M at p is an equivalence class of curves through p modulo the the first order contact relation. The equivalence class of a curve a(t) such that a(0) = p will be temporarily denoted by [d(0)]. The set of these equivalence classes is denoted by TPM and is called the tangent space to M at p.

20

Natural constructions on manifolds

Lemma 2.1.5 TPM has a natural structure of vector space. Proof Choose (a; 1 ,..., xm) local coordinates near p such that xl(jp) = 0, Vi and let a, P two smooth curves through p. In the above local coordinates the curves a, /? become {xxa(t)), (arg(i)). Construct a new curve 7 through p by

(4W) = (<(t)+4(i)). Set [d(0)] + [/3(0)] = [7(0)]. For this operation to be well defined one has to check (a) [7(0)] is independent of coordinates. (b) If [d1(0)] = [d2(0)] and [3\(0)} = [ft(0)] then [<*i(0)] + [/MO)] = [ofe(0)] + [&(0)]. We let the reader supply the routine details. • Exercise 2.1.1 Finish the proof of the Lemma 2.1.5. From this point on we will omit the [ , ] in the notation for a tangent vector. Thus [d(0)] will be written simply as d(0). As one expects, all the above notions admit a nice description using local coordinates. Let (xl,..., xm) be coordinates near p G M such that xl(p) = 0, Vi Consider the curves e*(t) = ( * # , . . . , t O k = l,...,m where <5j denotes Kronecker's delta symbol. Equivalently, one can define the e^'s implicitly by x' = 0, Vi / k. We set ^b)=^(0)-

(2.1.2)

Note that these vectors depend on the local coordinates (x1,-• • ,xm). Often, when the point p is clear from the context, we will omit it in the above notation. Lemma 2.1.6 ( g f r b ) )

is a basis of TPM.

Proof It follows from the obvious fact that any curve through the origin in R m has first order contact with a line 11-> (ait,..., amt). • Exercise 2.1.2 Let F : RN -> R* be a smooth map. Assume that (a)M = F-1(O)^0; (b) rankZ? x F = k, for all x € M. Then M is a smooth manifold of dimension N — k and TXM = (DxF)-1(0),

VieM.

The tangent bundle

21

Example 2.1.7 We want to describe TXG where G is one of the Lie groups discussed in Section 1.2.2. (a) G = 0(n). Let T(s) be a smooth path of orthogonal matrices such that T(0) = 1. Then T'^) • T(s) = 1. Differentiating this equality at s = 0 we get

t'(0)+f(0) = 0. T(0) defines a vector in T\0{n) so the above equality says this tangent space lies inside the space of skew-symmetric matrices i.e. TXG C o(n). On the other hand we proved in Section 1.2.2 that dimG = dimo(n) so that

(b) G = SL(n; K). Let T(s) be a smooth path in SL(n; K) such that T(0) = 1. Then detT(s) = 1 and differentiating this equality at s = 0 we get (see Exercise 1.1.1) trT(O) = 0. Thus the tangent space at 1 lies inside the space of traceless matrices i.e. TiSL(n;R) C sl{n;M). Since (according to Exercise 1.2.6) dim5L(n;K) = dim si.(n;M) we deduce TiSL(n;R) = sl{n;R). Exercise 2.1.3 Show that TxU{n) = u(n) and TiSU{n) = su{n). §2.1.2

The tangent bundle

In the previous subsection we have naturally associated to an arbitrary point p on a manifold M a vector space TPM. It is the goal of the present subsection to coherently organize the family of tangent spaces (TPM)P£MIn particular we want to give a rigorous meaning to the intuitive fact that TPM depends smoothly upon p. We will organize the disjoint union of all tangent spaces as a smooth manifold TM. There is a natural surjection •K

: TM = {J TPM -> M,

n{v) = p <(=» v € TPM.

Any local coordinate system x = (x'} defined over an open set U C M produces a natural basis ( £ T ( P ) ) of TPM for any p £ U. Thus, an element v G TU = UPeuTpM is completely determined if we know to which tangent space it belongs (i.e. we know p = ir(v)) and we also know its coordinates in the basis ( ^ r ( p ) ) :

We thus have a bijection * x : TU ->• Ux x R m c Km x R'

22

Natural constructions on manifolds

where Ux is the image of U in R m via the coordinates (xl). We can now transfer the topology on R m x R m to TU via the map ^x. Again we have to make sure this topology is independent of local coordinates. To see this, pick a different coordinate system y = (y1) on U. The coordinate independence referred to above is equivalent to the statement that the transition map *„ o y-1 :UX x R m -> T£/ ->Uy x R m is a homeomorphism. Let A = (x,X) 6 Ux x R m . Then ^(A) = (p,d(0)) where x(p) = x and a(t) C U is a curve through p given in the x coordinates as a{t) = x + tX. Denote by F : Ux —>• Uv the transition map x >-> y. Then * s o * j 1 ( y l ) = (ff(s);y1)...)Km)) where d(0) = {yJa{0)) = Y.Y^-^-ij)). {{ya{t)) is the defining equation for the curve a(t) in the coordinates yi). Applying the chain rule we deduce Yj = vm

= E gz'(O) = E %X-.

(2.1.3)

This proves that ^ j , o ^!~l is actually smooth. The natural topology of TM is obtained by patching together the topologies of TU11 where (C/7, ^ 7 ) 7 is an atlas of M. A set D M is a smooth map. Definition 2.1.8 The smooth manifold TM described above is called the tangent bundle of M. Proposition 2.1.9 A smooth map / : M -> N induces a smooth map Df : TM —> TN such that (a) Df{TpM) C TmN, VpeM (b) The restriction to each tangent space DPF : TPM —> Tf^N is linear. Df is called the differential of / and one often uses the alternate notation / , = Df. Proof Recall that TPM is the space of tangent vectors to curves through p. If a(t) is such a curve (a(0) = p) then /?(£) = f{a(t)) is a smooth curve through q = f(p) and define £>/(d(0))=/?(0). One checks easily that if ax ~ x a2 then / ( a i ) ~ i f(a2) so that Df is well defined. To prove the map Df : TPM —> TqN is linear it suffices to verify it in any particular

The tangent bundle

23

local coordinates (re 1 ,..., xm) near p and (y1,..., yn) near q since any two choices differ (infinitesimally) by a linear substitution. Hence we can regard / as a collection of maps (x\...,xm)^(y\x\...,xm),...,yn(x\...,xm))A basis in TPM is given by {^7} while a basis of TqN is given by {^r}- Then Df is the linear operator given in these bases by the matrix (§j£) this shows that Df is smooth. •

.

• In particular,

Definition 2.1.10 A smooth map / : M —> TV is called immersion (resp. submersion) if for every p 6 M the differential Dpf : TPM -> T/(P)TV is injective (resp. surjective) Exercise 2.1.4 Let / : M —> TV be a submersion. Show that for every q € TV the fiber f~l (q) is either empty or a submanifold of M of dimension dim M — dim TV. §2.1.3

Vector bundles

The tangent bundle TM of a manifold M has some special features which makes it a very particular type of manifold. We list now the special ingredients which enter the special structure of TM since they will occur in many instances. Set for brevity E = TM and F = Km (m = dimM). (a) E is a smooth manifold and there exists a surjective submersion 7r : E —> M. For every U C M set E \v= n"\U). (b) From (2.1.3) we deduce that there exists a trivializing cover i.e. an open cover U of M and for every U €U a diffeomorphism 9u:E\u-*UxF,

V^(P

= TT{V),<S>"(V))

such that (bl) <£p is a diffeomorphism Ep —• F for any p € U. (b2) If Z7, V g W are two trivializing neighborhoods with non empty overlap U (~l V then for any p e U n V the map $y[/(p) = $ ^ o ( $ ^ ) _ 1 : F -> F is a linear isomorphism and moreover, the map pH$ro(p)6i(F,F) is smooth. In our special case the map $vu(p) is explicitly defined by the matrix (2.1.3)

m - (gw)

•

In the above formula (x') are local coordinates on U and (yi) are local coordinates onf.

24

Natural constructions on manifolds

The properties (a) and (b) make no mention of the special relationship between E = TM and M. There are many quadruples {E, n, M, F) with these properties and they deserve a special name. Definition 2.1.11 A quadruple (E,ir,M,F) (where E, M are smooth manifolds, 7r : E —• M is a surjective submersion and F a vector space, real or complex) satisfying conditions (a) and (b) above is called a smooth vector bundle over M. E is called the total space and M is called the base space. The vector space F is called the standard fiber an its dimension (over its field of scalars) is called the rank of the bundle. Roughly speaking, a vector bundle is a smooth family of vector spaces. Note that the properties (ii) and (iii) imply that the fibers 7r_1(p) of a vector bundle have a natural structure of linear space. In particular, one can add elements in the same fiber. The above definition has an "impurity" built in coming from the choices of the open cover (Ua) and trivializations ipa which are somewhat arbitrary. We can get rid of this arbitrariness by introducing an equivalence relation. An open cover (Ua) together with trivializations K • n'l(Ua)

-+UaxF

is said to be equivalent to the open cover (V;) with the trivializations

iPi-.n-'iVj^ViXF if Va, i there exists a smooth map gQ, : UafWi —> GL(F) such that for any x £ UaOVi and any / G F <*°i>il{x,}) = {x,gaif)We will postulate that two equivalent trivializing covers define the same vector bundle structures. There is an equivalent way to define vector bundles. According to Definition 2.1.11, we can find an open cover (Ua) of M such that each of the restrictions Ea = E \Ua is isomorphic to a product \l/a : Ea = Ua x F. Moreover, on the overlaps Ua D Up, the transition maps gap = ^a^!~^1 can be viewed as smooth maps ga/) :UanUp^

GL(F)

where GL (F) denotes the Lie group of linear automorphisms of F. They satisfy the cocycle condition (a) gaa = 1F (b) SQ/)S/37S7a = IF over Ua n Up f~l [77. Conversely, given a cover (Ua) of M and a collection of smooth maps gap : Ua PI Up —> GL (F) satisfying the cocycle condition we can reconstruct a vector bundle by

25

T i e tangent bundle

gluing the product bundles Ea = Ua x F on the overlaps Ua n Up according to the gluing rules (x,v) € Ea is identified with (x,gpa(x)v)

€ Ep Va; e Ua n [//?.

We may say that gpa is the transition from a to f3. In the sequel we will prefer to think of vector bundles in terms of transition maps. A cover (Ua) as above will be called a trivializing cover. The details are carried out in the exercise below. Exercise 2.1.5 Consider a smooth manifold M, a vector space V, an open cover (Ua) and smooth maps ga0 :UanUp-+ GL(V) satisfying the cocycle condition. Set

X=

[)UaxVx{a}. a

We topologize X as the disjoint union of the topological spaces Ua x V. Define a relation ~ C X x X by Ua x V x {a} B (x, u, a) ~ (y, v,f3) eUpxV

x {j3} -&> x = y, v =

g0a(x)u.

(a) Show that ~ is an equivalence relation and E = Xj ~ equipped with the quotient topology has a natural structure of smooth manifold. (b) Show that the projection 7r : X —>• M, (x, u, a) i-> x descends to a submersion E^M. (c) Prove that (E, it, M, V) is naturally a smooth vector bundle. Definition 2.1.12 (a) A section in a vector bundle E A- M defined over the open subset u e M is a smooth map s : U —> E such that s(p) e Ep = 7r -1 (p), Vp G U. Equivalently, this means that JTOS = lv. The space of smooth sections of E over U will be denoted by T(U,E) or C°°(U,E). Note that V(U,E) is naturally a linear space. (b) A section of the tangent bundle is called a vector field. The space of vector fields over U is denoted by Vect(U). In terms of a trivializing cover (Ua) and transition maps (gap) a section can be defined as a collection of smooth maps sa : Ua —» F satisfying a gluing condition on the overlaps sa(x) = ga/}(x)s0(x) Mx G Ua n Up. Definition 2.1.13 Let E1 ^ M; be two smooth vector bundles. A vector bundle map consists of a pair of smooth maps / : Mx —> M2 and F : El -> E2 satisfying the following properties.

26

Natural constructions on manifolds

(a) The diagram below is commutative F

i.e. Vp e Mi, F(££) C Ej{p) (one says that F covers / ) . (b) The induced map F : E* -> £y (p) is linear. The composition of bundle maps is defined in the obvious manner and so is the identity morphism so that one can define the notion of bundle isomorphism in the standard way. If E and F are two vector bundles over the same manifold then we denote by Horn (E, F) the space of bundle maps E —>• F which cover the identity 1MSuch bundle maps are called bundle morphisms. For example the differential Df of a smooth map / : M —> N is a bundle map Df : TM -> TN covering / . Definition 2.1.14 Let E A M be a smooth vector bundle. A bundle endomorphism of E is a morphism F : E —>• E. An automorphism (or gauge transformation) is an invertible endomorphism. Given a trivializing cover (Ua) we can view a bundle map as a collection of smooth maps ha : Ua —> GL(F) such that ha{x) = ga0{x)h0(x)(ga0(x)yl

\fx G Ua n Up.

Exercise 2.1.6 Let V be a vector space, M a smooth manifold, {Ua} an open cover of M and gap, hap : Ua fl Up —> GL(V) two collections of smooth maps satisfying the cocycle conditions. Prove the two collections define isomorphic vector bundles if and only they are cohomologous i.e. there exist smooth maps <j>a : Ua —» GL(V) such that hap

§2.1.4

= aga0't>0 •

Some examples of vector bundles

In this section we would like to present some important examples of vector bundles and then formulate some questions concerning the global structure of a bundle. Example 2.1.15 The tautological line bundle over IIP" and OP" First, let us mention that a rank 1 vector bundle is usually called a line bundle. We consider

27

The tangent bundle

only the complex case. The total space of the tautological line bundle over OP" is the space r n = {(Z,p) g C n + 1 x OP" ; Z belongs to the complex line in OP" determined by p}. Let 7r : rn —> OP" denote the projection onto the second component. Note that for every p G OP" the fiber through p, 7r_1(p) = r n p coincides with the 1-dimensional subspace in C" + 1 defined by p. Example 2.1.16 The tautological vector bundle over a grassmannian We consider here for brevity only complex grassmannian Gjt,n(C). The real case is completely similar. The total space of this bundle is r

jfcTn — {(Z,p) € C" x Gfc,n(C) ; Z belongs to the subspace defined by p}.

If 7r denotes the natural projection it: Tk>n —> Gk,n then for each p G Gk,n the fiber at p coincides with the subspace in C" defined by p. Note that r n = T1>n. Exercise 2.1.7 Prove that r n and Tk X be a rank k (complex) smooth vector bundle over the manifold X. Assume E is ample i.e. there exists a finite family s\,..., s^ of smooth sections of E such that for any i € X the collection {si(x),..., SN(X)} spans Ex. For each x € X set Sx = {v€CN; '£visi{x) = 0} i

Note that dimS x = N - k. We have a map F : X -> GkiN(C) by x - • Sj-. (a) Prove that F is smooth. (b) Prove that E is isomorphic with the pullback F"Tkn.

28

Natural constructions on manifolds

Exercise 2.1.9 Show that any vector bundle over a smooth compact manifold is ample. Thus any vector bundle over a compact manifold is a pullback of some tautological bundle! The notion of vector bundle is trickier than it may look. Its definition may suggest that a vector bundle looks like a direct product -manifold x vector space- since this happens at least locally. Definition 2.1.18 A rank r vector bundle E A M (over the field K= K, C) is called trivial if there exists a bundle isomorphism E = Kr x X. Such an isomorphism is called a trivialization of E. A pair (trivial vector bundle, trivialization) is called a trivialized bundle. A trivial rank n K-bundle over M will be denoted by WM- One can naively ask the following question. Is every vector bundle trivial? We can even limit our search to tangent bundles. Thus we ask the following QUESTION. Is it true that for every smooth manifold M the tangent bundle TM is trivial (as a vector bundle)? Let us look at some positive examples. Example 2.1.19 TS1 = Msi Let 9 denote the angular coordinate on the circle. Then ^ is a globally defined, nowhere vanishing vector field on S1. We thus get a map

Rs^TS1,

(9,s)^(9,s-^)eTgS1

which is easily seen to be a bundle isomorphism. Let us carefully analyze this example. Think of S 1 as a Lie group (group of complex numbers of norm 1). The tangent space at z = 1 (i.e. 9,= 0) coincides with the subspace 9tez = 0 and ^ |i is the unit vertical vector j . Denote by Rg the counterclockwise rotation by an angle 9. Clearly Rg is a diffeomorphism and for thus for each 9 we have a linear isomorphism

De\e=0Re-ToSl^TeSl and moreover ^ = Dg \g=0 Rgj. The existence of the trivializing vector field J^ is due to to our ability to "move freely and coherently" inside S1. One has a similar freedom inside a Lie group as we are going to see in the next example. Example 2.1.20 For any Lie group G the tangent bundle TG is trivial. To see this let n = dimG and consider e\, • • •, e„ a basis of the tangent space at the origin, 7\G. We denote by Rg the right translation (by g) in the group defined by Rg : x H-> x • g, Vrr 6 G.

29

Tiie tangent bundle

Rg is a diffeomorphism with inverse Rg-i so that the differential DRg defines a linear isomorphism DRg : 7\G -> TgG. Set Ei{g) = Dit^e*) € TgG,

i = 1, • • •, n.

Since the multiplication G x G —> G, (g, /i) i-> g • /t is a smooth map we deduce that the vectors -E;( TG,

(g; X\ ... Xn) -»• (ff; £ ^ ( s ) ) .

One checks immediately that $ is a vector bundle isomorphism and this proves the claim. In particular TS3 is trivial since the sphere S3 is a Lie group (unit quaternions). (Using the Cayley numbers one can show that TS7 is also trivial; see [64] for details). We see that the tangent bundle TM of a manifold M is trivial if and only if there exist vector fields X\,- • • ,Xm (m = dimM) such that for each p G M, Xi(p),... ,Xm(p) span TPM. This suggests the following more refined question. Problem For any manifold M compute v(M) the maximum number of pointwise linearly independent vector fields over M. Obviously 0 < v(M) < d i m M and TM is trivial if and only if v(M) = dimM. A special instance of this problem is the celebrated vector field problem: compute v(Sn) for any n > 1 We have seen that v(Sn) = n for n =1,3 and 7. Amazingly, these are the only cases when the above equality holds. This is a highly nontrivial result, first proved by J.F.Adams in [2] using very sophisticated algebraic tools. This fact is related to many other natural questions in algebra. For a nice presentation we refer to [53]. The methods we will develop in this book will not suffice to compute v(Sn) for any n but we will be able to solve "half" of this problem. More precisely we will show that v{S2n) = 0 for any n > 1 so that in particular, TS'in is not trivial. In odd dimensions the situation is far more elaborate (a complete answer can be found in [2]). However one can prove easily the following. Exercise 2.1.10 ^(5 2fc_1 ) > 1 for any k > 1. The quantity v(M) can be viewed as a measure of nontriviality of a tangent bundle. Unfortunately, its computation is highly nontrivial. In the second part of this book we will describe computable ways of measuring the extent of nontriviality of a vector bundle.

30

2.2

Natural constructions on manifolds

A linear algebra interlude

We collect in this section some classical notions of linear algebra. Most of them should be familiar to the reader but we will present them in a form suitable for applications in differential geometry. All vector spaces in this section will be assumed finite dimensional. §2.2.1

Tensor p r o d u c t s

Let E, F be two vector spaces over the field K (K= K, C). Consider the (infinite) direct sum

T(£,F)=e (e , /)6BxF K. Equivalently, T{E, F) can be identified with the space of functions c : E x F —> K with finite support. T(E, F) has a natural basis consisting of "Dirac functions" SeJ:ExF^K,

(X,y)^[l

ij

£ ; ! ; £ #

In particular we have an injection 5:ExF^T(E,F),

(e,f)^6eJ.

(A word of caution: 5 is not linear!) Inside T(£, T) sits the linear subspace TZ(£, T) spanned by \5ej

— 5\ej,

where e, e' G E, f,f

\5ej

— 5e,\f, Se+e'J — $e,f

—

^e'./j ^e,f+f

~ Se,f ~ Se,f'

G F and A € K. Now define E®KFd^f

T(E,F)/1l(£,

F).

and denote by n the canonical projection ir : T(E, F) -> E <S> F. Set e / = n(6ej). We get a natural map i : E x F -> E ® F,

ex/h^e®/.

Obviously i is bilinear. The vector space E ® F is called the t e n s o r product of E and F. Sometimes when we want to emphasize the field of scalars we write E <8>j£ F. The tensor product has the following universality property. P r o p o s i t i o n 2.2.1 For any bilinear map : ExF —> G there exists a unique linear map $ : E F —> G such that the diagram below is commutative:

31

A linear algebra interlude Ex

F-^E®F \ >

$ G

The proof of this result is left to the reader as an exercise. Note that if (e,) is a basis of Sand (fj) is a basis of F then (e* ® fj) is a basis of E ® F and hence dim E <8> F = (dim F ) • (dim F). Exercise 2.2.1 Prove using the universality property of the tensor product that there exists a natural isomorphism E®F = F®E uniquely defined b y e ® / i - » / ® e . The above construction can be iterated. Given three vector spaces E\, E2, E3 we can construct two triple tensor products: (Ei ® E2) ® E3 and F x ® (E2 ® E3). Exercise 2.2.2 Prove there exists a natural isomorphism (E\ <S> E2) ® E3 = Ex {E2 E3).

Thus there exists a unique (up to an isomorphism) triple tensor product which we denote by E\ ® E2 <S> E3. Clearly we can now define multiple tensor products: Ex ® • • • En. Definition 2.2.2 The dual of a K-linear space E is the linear space E* defined as the space of K-linear maps E —• K. For any e* G E* and e e F we set <e*,e>^V(e). Using the universality property of the tensor product one proves easily the following result. Proposition 2.2.3 (a) There is a natural isomorphism F* K. (b) There is a natural isomorphism E*®F^

Horn ( F , F)

uniquely determined by e* ® / H Te.®/ € Horn (2?, F) where T e . 8 /(o:) = (e*,x)f all s e F .

for

32

Natural constructions on manifolds

Exercise 2.2.3 Prove the above proposition. The above constructions are functorial. More precisely, we have the following result. Proposition 2.2.4 (a) Let Ti € Kom(Ei, Ft), i=l,2 be two linear operators. Then they naturally induce a linear operator T = Ti ® T2 • E\ ® E2 —> F\ ® F 2 uniquely defined by T\ ® T 2 (ei ® e2) = (T i e i ) ® (T 2 e 2 ), Ve* e Si. (b) Any linear operator S : E —¥ F induces a linear operator 5 ' : F* —» i?* uniquely defined by

(57*,e) = (/*,5e). S' is called the transpose of S. Let V be a vector space. For r, s > 0

77(10 =; Vs' ® (y*)®s where by definition V 8 0 = (I/*)®0 = K. An element of Tsr is called tensor of type (r,s). Notice that according to Proposition 2.2.3 a tensor of type (1,1) is a linear endomorphism of V. A tensor of type (r, 0) is called contravariant while a tensor of type (0, s) is called covariant. The tensor algebra of V is defined to be

*00 ='07700. r,s

We use the term algebra since the tensor product induces bilinear maps ® : 77 X 77/ -> 77 + + /. The elements of X(V) are called tensors. Exercise 2.2.4 Show that (T(K), +, ®) is an associative algebra. It is often useful to represent tensors using coordinates. To achieve this pick (e^) a basis of V and let (e1) denote the dual basis in V* uniquely defined by

We then obtain a basis of Tsr(V) {eh ® • • • ® eir ® eh ® • • • ® ej'/ 1 < ia, jp < dim V}. Any element T 6 77(1*0

nas a

decomposition

T = Tj^-.^e^ ® • • • ® eir ® e^1 ® • • • ® ej'

33

A linear algebra interlude

where we use Einstein convention to sum over indices which appear twice- once as an upper index and the other time as a lower index. On the tensor algebra there is a natural c o n t r a c t i o n (trace) operation tr : 77 -> V~i uniquely defined by t r ( v i ® - - - ® t v ® u 1 ® - - - ® u s ) = ( u 1 , ^ ) ^ ® - •"u r ®M 2 ® •••&«", Vv; 6 V, v? € V*. In coordinates determined by a basis (ei) of V the contraction can be described as

{trT)%;Z = (l$;;j;) where again we use Einstein's convention. In particular we see that the contraction coincides with the usual trace on 7f*(V) = End (V). §2.2.2

Symmetric and antisymmmetric tensors

Let V be a vector space over K= K, C and set T r ( V ) = TQ(V). group <5r acts naturally on Tr(V) by o(Vi ® ... ® Vr) = ?V(1) ® ••• ® vo(r),

The permutation

O" € © r -

We denote this action of a e <5T on an arbitrary element t E Tr(V) by at. In this subsection we will describe two subspaces invariant under this action. These are special instances of the so called Schur functors. (We refer to [26] for more general constructions.) Define

Sr:T{V)^T(y),

Sr{t) = \

£

at

and

A : T{V) -> T(V), Mt) = { r'! E o e ^

e(ff)at

if r < dim V if r > dim V

When r = 0 we set <50 = {1} so that A0 = S0 = l j j . Above e(a) denotes the signature of the permutation a. The following results are immediate. Their proofs are left to the reader as exercises. L e m m a 2.2.5 Ar and Sr are projectors of Tr{V) i.e. S2T=Sr,

A2r = Ar.

Moreover aSr{t) = ST(at) = Sr(t),

aAr{t) = Ar{at) = e{a)AT{t).

34

Natural constructions on manifolds

Definition 2.2.6 A tensor T G Tr{V) is called symmetric (resp. skew-symmetric) if Sr{T) = T (resp. Ar(T) = T). r is called the degree of the (skew-)symmetric tensor. The space of symmetric tensors (resp. skew-symmetric ones) will be denoted by SrV (resp. KTV). Set S*V = ®r>0STV

and N*V = © r > 0 A r V.

Definition 2.2.7 The exterior product is the bilinear map A : A.rV x ASV ->• A r+ *V defined by

uf A T]S d=f ^-^Ar+Aw r\s\

® v), Vwr e ArV, r)s e A Y .

Proposition 2.2.8 The exterior product has the following properties. (a) associativity: (a A f3) A 7 = a A (/? A 7), Va, fly G A*V. In particular vi A ... vk = k\Ak(vi <8> ... ®vk)=

J2

e{a)va(V) ® ••• ® v^*), Vu; € V.

(b) supercommutativity: r w

Proof

Ar/s = ( - l ) r V A w r .

We first define a new product "Ai" by wr/\ir)s =

Ar+S{u®ri)

which will turn out to be associative and will force A to be associative as well. To prove the associativity of Ai consider the quotient algebra Q* = T*/3* where T* is the algebra (®r>0Tr{V), +, ®) and 3* is the bilateral ideal generated by the set of squares {v ® v/ v G V}. Denote the (obviously associative) multiplication in Q* by U. The natural projection n : X* -¥ Q* induces a linear map n : A*V -> Q*. Step 1: 7r : A*V —>• Q* is a linear isomorphism and moreover ir[aAxp) = TT(Q) U TT(/3).

(2.2.1)

In particular Ai is an associative product. The crucial observation is TT(T) = 7r(A(T)), V t e r t n

(2.2.2)

It suffices to check (2.2.2) on monomials T = ei ® ... ® e r , e* G V. Since (w + v)®2 = 0(mod3*) for any u, v G V we deduce u ® 1; = — 1; ® u(mod3*) so that for any 7r(ex ... ® e r ) = e(a)7r(e«T(i) ® ... ® e„(r))

(2.2.3)

35

A linear algebra interlude

When we sum over a G © r in (2.2.3) we obtain (2.2.2). To prove the injectivity of 7r note that A ( 3 * ) = 0. If n(ui) = 0 for some w £ A*V then a; G kerx = J* n A*K so that

w = A M = 0. The surjectivity of 7r follows immediately from (2.2.2). Indeed any n(T) can be alternatively described as Tt(u) for some u> G A*V. It suffices to take w = A,(T). To prove (2.2.1) it suffices to consider only the special cases when a and /3 are monomials: a = A ( e i ® ... ® e r ), /? = A ( / i ® - ® /»)• We have 7r(aA!/3) = 7T ( A + „ ( A ( e i ® ... ® e r ) ® A ( / i ® - ® /»))) (2 2)

=

de/

7T ( A ( e i ® ... <8» e r ) ® A ( / i ® - ® /,))

7r(A(ei ® ... ® e r )) U T T ( A ( / I ® -

® /»)) = T ( O ) U TT(/3).

Thus At is associative. Step 2: A is associative. Consider a G A r V, /? G ASV and 7 G A'V. We have

(« A /J) A 7 = \(£±?U/j) A 7 = r'.si ^ ^ (r±+^s)\t< (aA^Al7 r!s! J , , ^ ) ! ( a A 1 / 3 ) A l 7 = (r + s + * ) ! Q A i ( / ? A i 7 ) = a A {p A 7 ) r!s!i! r!s!i! The associativity of A is proved. The computation above shows that =

(r+ S

ex A ... A ek = k\Ak(ei ... ® efc). (b) The supercommutativity of A follows from the supercommutativity of Ai (or U). To prove the latter one uses (2.2.2). The details are left to the reader. • Exercise 2.2.5 Finish the proof of part (b) in the proposition above. A*V is called the exterior algebra of V. The exterior algebra is a Z-graded algebra i.e

A is called the exterior product .

A* = 0 A r l / r>0

and ( A T ) A (ASV) C Ar+sV.

In fact ATV = 0 for r > dim V (pigeonhole principle).

Definition 2.2.9 Let V be an n-dimensional K-vector space. The one dimensional vector space AnV is called the determinant line of V and is denoted by det V.

36

Natural constructions on manifolds

There is a natural injection LV : V ^-> A*V such that (LV{X))2 — 0 in A*V for all x (z V. This map enters crucially the formulation of the universality property Proposition 2.2.10 Let V be a vector space over K. For any K-algebra A and any linear map : V —» A such that (<j>(x)2 — 0 there exists an unique morphism of K-algebras $ : A*V —> A such that the diagram below is commutative

i.e. $ o tv = 4>Exercise 2.2.6 Prove Proposition 2.2.10. The space of symmetric tensors S*V can be similarly given a structure of associative algebra with respect to the product a • j3 d=f Sr+S(a ® p), Va E STV, j3 € SSV. The symmetric product "•" is also commutative. Exercise 2.2.7 Formulate and prove the analogue of Proposition 2.2.10 for the algebra S*V. It is often convenient to represent (skew-)symmetric tensors in coordinates. If ei, ..., en is a basis of the vector space V then for any 1 < r < n the family {ejj A ... A eij

1 < »i < ••• < iT < n}

is a basis for ATV so that any degree r skew-symmetric tensor tv can be uniquely represented as u) = 5Z w " ' " 1 ' ^ , A ... Aei,. l
Symmetric tensors can be represented in a similar way. The A* and S* constructions are functorial i.e. any linear map L : V —>• W naturally induces morphisms of algebras A*L : A*V -> A*W, S*L : S*V ->• S*W uniquely defined by their actions on monomials: A*L(w! A ... Atv) = (Lvi) A ... A (Lvr)

A linear algebra interlude

37

and S'Hv!

• ... • vr) = (LVl) • ... • (Lvr).

The functors A* and S* have an exponential like behavior i.e. there exists a natural isomorphism A*(V®W)^A*V®A*W. (2.2.4) S*(V®W)^S*V®S*W.

(2.2.5)

To define the isomorphism in (2.2.4) consider the bilinear map <> / : A*V x A*W —• A*(V © W) uniquely determined by 4>(vi A ... A vr , Wx A ... A ws) = vl A ... A vr A Wi A ... A w s . The universality property of the tensor product implies the existence of a linear map $ : A*V ® A*W -> A*(F © W) such that <3> o t = <j) where t is the inclusion of A*V x A*W in A*V ® A*VK. To construct the inverse of $ note that A*V ® A*W is naturally a K-algebra by (<J ® 77) * ( < / ® 7?') = ( - l ) d e g O d e g a , ' ( w

A

^

^ ^

A

^

V © W is naturally embedded in A*V^ A*W by ^>(w, io) = w ® l + l ® i « € A* V ® A*W. Moreover, for any x E V ®W we have ^(x) * ip(x) = 0. The universality property of the exterior algebra implies the existence of a unique morphism of K-algebras ^ : A*{V®W) ->• A*V®A*W such that ^ o i y ^ = i/>. Note that $ is also a morphism of K-algebras and one verifies easily that ( $ o $ ) o iy9w — W<$w- The uniqueness part in the universality property of the exterior algebra implies $ o vjr = identity. One proves similarly that <]> a $ = identity and this concludes the proof of (2.2.4). We want to mention a few general facts about Z-graded vector spaces i.e. vector spaces equipped with a direct sum decomposition

(We will always assume that each Vn is finite dimensional.) The vectors in Vn are said to be homogeneous, of degree n. For example the ring of polynomials K[x] is a Zgraded vector space. A*V and S* V are Z-graded vector spaces. The direct sum of two def

Z-graded vector spaces V and W is a Z-graded vector space with [V®W)n =J Vn®Wn. The tensor product of two Z-graded vector spaces V and W is a Z-graded vector space with

(v®wo„ =' © vT®ws. r+s=n

To any Z-graded vector space V one can naturally associate a formal series Py(t) € Z[[t,rl}}by JV(i)=£(dimKK)in. regZ

Pv{t) is called the Poincare series of V.

38

Natural constructions on manifolds

Example 2.2.11 The Poincare series of K[x] is %

W

W = i + 1 + 1 2 + ... + 1 " " 1 + ... =

j - ^ .

Exercise 2.2.8 Let V and W be two Z-graded vector spaces. Prove the following statements are true (whenever they make sense). (a.) PVS)W(t) = Pv(t) + Pw{t). (b) PV9W(t) = Pv(t) • Pw(t). (c) dimV = Pv(l). Definition 2.2.12 Let V be a Z-graded vector space. The Euler characteristic of V, denoted by x00> i s defined by (whenever it makes sense) X(V)

^ / V ( - l ) = £ ( - ! ) " dim F n . neZ

Remark 2.2.13 If we try to compute x M z ] ) using the first formula in Definition 2.2.12 we get x(K[a:]) = 1/2 while the second formula makes no sense (divergent series). Proposition 2.2.14 Let V be a vector space over K of dimension n. Then PA.v(t)

= (l + t)n

and

p

-^ = (r^)" = (^T)!(l)nl(r^)-

In particular dimA*V = 2 n and x(A*V) = 0. Proof From (2.2.4) and (2.2.5) we deduce using Exercise 2.2.8 that for any vector spaces V and W we have Pk-(v®w)(t) = P\'v{t)

• P\*w(t)

and Ps-(y®w){t) = Ps'V • Ps'w{t)In particular if V has dimension n then V = K" so that PA-V(*) = (P A -K(*))"

and Ps-v(t)=(Ps.K(*))"The proposition follows using the relations

and

A linear algebra interlude §2.2.3

39

The "super" slang

The aim of this very brief section is to introduce the reader to the "super" terminology. We owe the "super" slang to the physicists. In the quantum world many objects have a special feature not present in the newtonian world. They have parity (or chirality) and objects with different chiralities had to be treated differently. The "super" terminology provides an algebraic background which allows one to deal with different parities on an equal basis. From a strictly syntactic point of view the "super" slang adds the attribute super to most of the commonly used algebraic objects. In this book the prefix "s-" will abbreviate the word "super" Definition 2.2.15 (a) A s-space is a Z^-graded vector space V = Vo © V\(b) A s-algebra over K is a Z2-graded K-algebra i.e. a K-algebra 21 together with a direct sum decomposition 21 = 2t°©2l 1 such that 21' ^ C 2l i + J {mod 2) . The elements in 21' are called homogeneous of degree i. For any o € 21' we denote its degree (mod 2) by |a|. The elements in 21° are said to be even while the elements in 211 are said to be odd. (c) The supercommutator in a s-algebra 21 = 21° ffi 211 is the bilinear map [• , -}s: 2 1 x 2 1 - > 2 l defined on homogeneous elements w' 6 2l ! , rf; e 2F by

A s-algebra is called s-commutative if [ • , • ] s = 0. . Example 2.2.16 Let E = E° © El be a s-space. Any T e End (E) has a block decomposition Too TQ\ where Ttj € End [El, E'). End (E) is naturally a s-algebra. The even endomorphism have the form 'Too 0 0 T„ while the odd endomorphisms have the form 0

T01

Tio

0

Example 2.2.17 Let V be a finite dimensional space. The exterior algebra A*V is naturally a s-algebra. The even elements are gathered in Aeveny

=

Q

^y

40

Natural constructions on manifolds

while the odd elements are gathered in Aoddy =

Q

Ary

r odd

A*V is supercommutative. Definition 2.2.18 Let 21 = 21°©211 be a s-algebra. A s-derivation on 21 is a linear operator on D G End (21) such that for any x G 21 [D,Lx]™M

= LDx

(2.2.6)

where [ , ]End(-A) d e n o t e s the supercommutator in End (21) (with the s-structure defined in Example 2.2.16) while for any z G 21 we denoted by Lz the left multiplication operator a >-> z • a. A s-derivation is called even (resp. odd) if it is even (resp. odd) as an element of the s-algebra End (21). Remark 2.2.19 The relation (2.2.6) is a super version of the usual Leibniz formula. Indeed, assuming D is homogeneous (as an element of the s-algebra End(2l)) then equality (2.2.6) becomes D{xy) = {Dx)y +

{-l)lxmx{Dy)

for any homogeneous elements x, y G 21. Example 2.2.20 Let V be a vector space. Any u* G V* defines an odd s-derivation of A*V denoted by i u . uniquely determined by its action on monomials. T

iu'(v0 A»iA ... AvT) = ^2(—iy{u*,Vi)vo

A vx A ... A vt A • •• A vT.

i=0

As usual, a hat indicates a missing entry. Exercise 2.2.9 Prove the statement in the above example. Definition 2.2.21 Let 21 = (21°ffi211; +; [, ]) be a s-algebra over K, not necessarily associative. For any x G 21 denote by Rx the right multiplication operator a H-> [a,x\. 21 is called s-Lie algebra if it is s-anticommutative i.e. [x, y] + (—l)' 1 " 9 '^, x] = 0, for all homogeneous elements x, y G 21 and Vx G 21, Rx is a s-derivation. When 21 is purely even (i.e. 211 = {0}) then 21 is called simply a Lie algebra. The multiplication in a (s-) Lie algebra is called the (s-)bracket.

41

A linear algebra interlude

The above definition is highly condensed. In more down-to-Earth terms the fact that Rx is a s-derivation for all x 6 21 is equivalent with the super Jacobi identity [\y,z],x] = [[y,x],z] + {-l)^[y,[z,x}}

(2.2.7)

for all homogeneous elements x, y, z € 21.When o is a purely even K-algebra then 21 is a Lie algebra over K if [ , ] is anticommutative and satisfies (2.2.7), which in this case is equivalent with the classical Jacobi identity [[x, y],z] + [{y, z},x} + [[z, x},y} = 0, Vx, y,z e 21.

(2.2.8)

Example 2.2.22 Let E be a vector space (purely even). Then 21 = End (E) is a Lie algebra with bracket given by the usual commutator: [a, 6] = ab — ba. Proposition 2.2.23 Let 21 = 21° e 211 be a s-algebra and denote by £>ers(2l) the space of s-derivations of 21. (a) For any D G £>ers(2l) its homogeneous components D°, D1 £ End (21) are also s-derivations. (b) VD, D' 6 Der s (2l) the s-commutator [D, D']®nd<-A) is again a s-derivation. (c) Vrc £ 21 the bracket Bx : a i-> [o, x]s is a s-derivation called the bracket derivative determined by x. Moreover [£*_ B y]E„d(2l)

=

B [x,j,] %

VX] y

£

gj

Exercise 2.2.10 Prove Proposition 2.2.23. Definition 2.2.24 Let E = E° e E1 and F - F° ® F 1 be two s-spaces. Their stensor product is the s-space E 0 F with the grading (E F)e = © i+J = E ( 2 ) El ® Fj. To emphasize the super-nature of the tensor product we will use the symbol "®" instead of the usual "®". Exercise 2.2.11 Show that there exists a natural isomorphism of s-spaces V*®k*V ^

Ders{A*V)

uniquely determined by v* x u> H-> £)"*»" where Dv~®" is s-derivation defined by DV'®"(V)

= (V\V)UJ,

Vv<= V.

Notice in particular that any s-derivation of A* V is uniquely determined by its action on A1!^. (When a> = 1, D"*®1 coincides with the internal derivation discussed in Example 2.2.20.)

42

Natural constructions on manifolds

Let 21 = 21° © 21 * be a s-algebra over K = M, C. A super/trace on 21 is a K-linear map T:21->&

such that

T([x,y].)=0

Vi,»ea.

If we denote by [21,21]a the linear subspace of 21 spanned by the supercommutators {[x,y}s ; x,j/ e 21} we see that the space of s-traces is isomorphic with the dual of the quotient space 2l/[2l,2l] s . Proposition 2.2.25 Let E = E0 © E\ be a finite dimensional s-space and denote by 21 the s-algebra of endomorphisms of E. Then there exists a canonical s-trace tr s on 21 uniquely defined by t r s l £ = dim£o — d i m i ^ . In fact, if T 6 21 has the block decomposition Too Toi" Tio

then t r , T = tr Toc - t r T u . Exercise 2.2.12 Prove the above proposition. §2.2.4

Duality

Duality is a subtle and fundamental concept which can be detected in all branches of mathematics. This section is devoted to those aspects of the atmosphere called duality which are relevant to differential geometry. In the sequel we will use Einstein's convention without mentioning it. K will denote one of the fields K or C. Definition 2.2.26 A pairing between two finite dimensional K-vector spaces V and W is a bilinear map B : V x W —> K. Any pairing defines two linear maps Dv : V -> W\

v^t

B(v,

-)eW

Dw : W -> V,

v M- B{ • , w) E V.

and A pairing is called a duality if the the maps Dy and Dw are isomorphisms.

A linear algebra interlude

43

Example 2.2.27 The pairing ( , ) : V* x V -» K is a duality. One sees that Dy = l y and Dy = lv- This is called the natural duality. Example 2.2.28 Let V be a finite dimensional real vector space. Any symmetric nondegenerate quadratic form ( , ) : V x V —> M. defines a (self) duality and in particular a natural isomorphism C : V —> V*. When ( , ) is positive definite the the operator £ is called metric duality. This operator can be nicely described in coordinates as follows. Pick (e*) a basis of V and set gtj = (ei,ej). Let (e^) denote the dual basis of V* (defined by (e J , e*) = 8j). The action of C is then

Example 2.2.29 Consider V a real vector space and w : V x V —> M a skewsymmetric bilinear form on V. In particular u> defines a pairing, u is said to be symplectic if this pairing is a duality. In this case the induced operator V —> V* is called symplectic duality. Exercise 2.2.13 Let u : V x V —¥ R define a symplectic duality. Prove the following. (a) V has even dimension. (b) If (e;) is a basis of V and Uij = cj(e;,e.,) then det(wy) 1 R (i = 1, 2) be two pairs of spaces in duality. Then there is a natural duality B = Bi ® B2 : (Vi V2) x (Wx ® W2) -> M uniquely determined by •B(«i ® «2, wi ® W2) = -Bi(vi, Wi) • B2(v2, w2).

Exercise 2.2.14 Prove Proposition 2.2.30. Proposition 2.2.30 implies that given two spaces in duality B : V x f f - > K there is a naturally induced duality B®n

. y®r

x W®r

_^ jg

This defines by restriction a pairing ArB : KrV x KrW -» K uniquely determined by krB (vi A ... A iv, ioi A ... A wr) d= det (B (vt, Wj))l
.

44

Natural constructions on manifolds

Exercise 2.2.15 Prove the above pairing is a duality. In particular, the natural duality ( • , • ) : V* x V —» K induces a duality ( • , • ) : ArV* x A 7 ^ 1 and thus defines a natural isomorphism ArV

^

{ArV)*.

Thus, the elements of ATV* can be viewed as skew-symmetric r-linear forms Vr —> K. Moreover, if n = dimjj V then the natural duality ( • , • ) : AnV x AnV* -> K defines a nontrivial element in AnV ® AnV* = End (AnV) (the identity operator) so that End (AnV) = End (det V) is naturally identified with K. In particular any endomorphism of ^induces a linear map det V —> det V which can be identified with a K-scalar: the classical determinant of a linear operator det T. A duality B :V xW ->K naturally induces a duality Bl :V xW -+Kby B\v\w*)

d

=f

(v^Dy'w*)

where Dy : V —> W* is the linear isomorphism induced by the duality B. Now consider a (real) euclidian vector space V. Denote its inner product by (•, •). The self-duality defined by (•, •) induces a self-duality (still denoted by (•, •)) (

v

):AYxArl/4l

determined by (vi A ... A vr, wl A ... A wr) d= det ((v t , Wj) )1
(2.2.9)

The right hand side of (2.2.9) is a Gramm determinant and in particular the quadratic form in (2.2.9) is positive definite. Thus, we have proved the following result. Corollary 2.2.31 An inner product on a real vector space V naturally induces an inner product on the tensor algebra TV and in the exterior algebra A*V. In an euclidian vector space V the inner product induces the metric duality C : V -> V*. This induces an operator C : TTS(V) -» Ts+l{V) defined by C(vi ® . . . ® vT ® u1 ® . . . ® u5) = (v2 ® ... ®vr) ® ((Cvi ® u1 ® ... ® us). (2.2.10)

A linear algebra interlude

45

The operation defined in (2.2.10) is classically referred to as lowering the indices. The reason for this nomenclature comes from the coordinate description of this operation. Thus if T € Tg(y) is given by T = Til;;%eh ® ... ® eir ® ejl ® ... ®ej' then where V operation TJ{V) —» 7J-\{y) called raising the indices.

induces a linear

Exercise 2.2.16 (Cartan) Let V be an euclidian vector space. For any v € V denote by e„ (resp. iv) the linear endomorphism of h*V defined by evui = vAui (resp. iv = iv. where %v- denotes the interior derivation defined by v* 6 V*-the metric dual of v; see Example 2.2.20). Show that for any u,v &V [ev, iu]s = eviu + iuev = (u, v ) l A . v .

Definition 2.2.32 Let V be a real vector space. A volume form on V is a nontrivial linear form on the determinant line of V: Det : det V -> K. Equivalently, a volume form on V is an element of det V* (n = dim V). Since det V is 1-dimensional a choice of a volume form corresponds to a choice of a basis of det V. Definition 2.2.33 Two volume forms on V, Deti and Det2 are said to be equivalent if there exists a positive constant A such that Det2 = \Det\. (There are only two equivalence classes.) An orientation on a vector space is an equivalence class of volume forms. A pair (vector space + orientation) is called an oriented vector space. There is an equivalent way of looking at orientations. To describe it note first that any volume form Det on V uniquely determines a basis w of det V by the requirement Det{uS) = 1. We say two bases u>\ and CJ2 of det V are equivalent if there exists A > 0 such that ui2 = Awi. An orientation on V selects an equivalence class of bases of det V. A basis is said to be positively oriented if it belongs to the equivalence class defining the orientation. Otherwise, the basis is said to be negatively oriented. To any basis {e 1; ..., e n } of V one can associate a basis ei A • • • A e„ of det V. Note that a permutation of the indices 1 , . . . ,n changes the orientation of the associated basis by a factor equal to the signature of the permutation. Thus to define an orientation on a vector space it suffices to specify a total ordering of a given basis of the

46

Natural constructions on manifolds

space. An ordered basis of V is said to be positively oriented if so is the associated basis of det V. Assume now V is an euclidian space. Denote the euclidian inner product by g( •, •). det V has an induced euclidian structure and in particular there exist only two length-one-vectors in d e t F , ±o>. If we fix one of them - say UJ - as a basis in detV we achieve two things. Firstly, there is a naturally associated volume form Detg defined by Detg(Xu>) = A. Secondly, the equivalence class of this volume form determines an orientation on V. Conversely, an orientation on V uniquely selects a length-one-vector u in det V thus defining a volume form Detg. Thus, we have proved the following result. Proposition 2.2.34 An orientation in an euclidian vector space (V, g) canonically selects a volume form on V, henceforth denoted by Detg Exercise 2.2.17 Let (V, g) be an n-dimensional, oriented, euclidian vector space and denote by Detg the associated volume form. Show that for any basis v\, ..., vn of V we have Detg(viA ... Avn) = e{vl,...,vn)s[{Astg(y^Vj)) where e(vx,..., vn) — +1 if the basis Vi A ... Avn of det V is positively oriented and —1 otherwise. For V = K2 with its standard metric and the orientation given by e\ A e2 prove that \Detg{vi

f\v2)\

is the area of the parallelogram spanned by V\ and v2. Definition 2.2.35 Let (V, g) be an oriented, euclidian space and denote by Detg the associated volume form. The Berezin integral or (berezinian) is the linear form

7

: A*V •

defined on homogeneous elements by

/" = {zM u g

if deg UJ < dim V if deg u> = dim V

Definition 2.2.36 Let u £ A2V', where {V,g) is an oriented, euclidian space. We define its pfaffian as

P/( W )^ / /expw={ 1

0 ± DeUujAn)

if dim V is odd if d i m ^ = 2n

In! expw denotes the exponential in the (nilpotent) algebra A*V: ex

LJk

P w = E IT*:>0

"'•

47

A linear algebra interlude

If (V, g) is as in the above definition (dim V = N) and A : V —> V is a skew-symmetric endomorphism of V we can form LJA e A 2 F by w^ = XJ 9(Aet, e-j)e-i A e,- = - £ p(yle;, e^)e4 A e,where (ei, ..., e#) is a positively oriented orthonormal basis of V. The reader can check that LJA is independent of the choice of basis as above. (Notice that UA(% V) = g(Au, v), Vw, v G V.)We define the pfaffian of A by Pf{A)d^

Pf{u>A).

Example 2.2.37 Let V = M2 denote the standard euclidian space oriented by ex Ae2, where ei, e2 denotes the standard basis. If A = then Wyi = 0e\ A e2 so that Pf{A)

0 9

-0 0

= 9.

Exercise 2.2.18 Let A : V —» V be a skew-symmetric endomorphism of an oriented euclidian space V. Prove that P / ( A ) 2 = det A Exercise 2.2.19 Let (V,g) be an oriented euclidian space of dimension In. Consider A : V —¥ V a skewsymmetric endomorphism and el,...,e2„ a positively oriented orthonormal frame. Prove that (-1)" P A

f( )

= ~^TT

= (-1)"

S

^

e(<7)a
^(17)0^(1)^(2) • • • a
where a^ = <7(ej, ^4ej) is the (i,j) entry in the matrix representing A in the basis (e*) while &2n denotes the set of permutations a e <5 2n satisfying ff(2Jfc-l)<min{<7(2fc),
Let (V, g) be an n-dimensional, oriented, euclidian vector space. The metric duality £ : V —> V* induces both a metric and an orientation on V*. In the sequel we will continue to use the same notation Cg to denote the metric duality

77(v)^77(n=77(n

48

Natural constructions on manifolds

Definition 2.2.38 The Hodge pairing is defined by E:ATVx

An~rV* ->• R,

S(w r , Vn'r) = Detg(ujr A £ 9 r f " r ) .

Exercise 2.2.20 Prove the Hodge pairing is a duality. Definition 2.2.39 The Hodge *-operator is the isomorphism * : A T -» A n - r V induced by the Hodge duality. The above definition obscures the meaning of the ^-operator. We want to spend some time clarifying its significance. Let a e ATV so that *a e An~rV. Denote by ( , ) the standard duality ASV x S A V ->• K and by ( , ) the metric duality ASV x ASV ->• M. Then by definition * satisfies Detg(a A Cg/3) = (*a,/3) d=f (*a,Cgj3) V/J £ A " ' T * . (2.2.11) Let ui denote the unit vector in det V defining the orientation. Then (2.2.11) can be rewritten as aA£g(3 = (*a, CgP)w V/3 e An~TV*. Thus aAj=(*a,j)u> V 7 eA"-7.

(2.2.12)

Equality (2.2.12) uniquely determines the action of *. Example 2.2.40 Let V be the standard euclidian space M3 with standard basis ei, e2, e3 and orientation determined by ex A e2 A e 3 . Then *ei = e2 A e 3 * e2 = e3 A ex * e3 = ei A e2. *1 = ei A e2 A e 3 *(e 2 A e3) = ei

* (ex A e2 A e3) = 1.

* (e3 A ex) = e2

* (ex A e2) = e 3 .

The following result is left to the reader as an exercise. Proposition 2.2.41 The Hodge *-operator satisfies (n = dim V)

*(*w) = (-i) p ("- p) w Vw e Apv. Detg{*\) = 1 a A *f3 = (a, /?) * 1, V a 6 A*V, V/3 e An~kV.

49

A linear algebra interlude

Exercise 2.2.21 Let (V, g) be an n-dimensional, oriented, euclidian space. Denote by ge the rescaled metric ge = e2g. If * is the Hodge operator corresponding to the metric g and *£ is the Hodge operator corresponding to the metric ge show that * e w = s2p-n

* UJ Vw € APV.

We conclude this subsection with a brief discussion of densities. Definition 2.2.42 Let V be a real vector space. For any r € M we define an rdensity to be a function / : det V \ {0} —» R such that /(Aw) = |A| r /(u), Vu G det V \ {0}, VA + 0. The linear space of r-densities on V will be denoted by \A\y. When r = 1 we set |A|y^|A|v. An orientation on V induces a natural isomorphism i: det V -> |A| V w >->• («(w) : det V \ {0} -^ K) where i(u)(u) = sign (Detu)a;(u) and Det is an arbitrary volume form on V defining the orientation. Notice that once we fix a volume form on V, one has identifications: d e t F * ^ |A| V = M. In particular, an orientation in an euclidian vector space canonically identifies |A|y with R. §2.2.5

Some complex linear algebra

In this very short section we want to briefly discuss some aspects specific to linear algebra on complex vector spaces. Thus let V be a complex vector space. Its conjugate is the complex vector space V which coincides with V as a real vector space but in which the multiplication by a scalar A € C is defined by \-v=~\v, Vv E V. V has a complex dual V* which can be identified with the space of complex linear maps V —» C. If we forget the complex structure we obtain a real dual ^consisting of all real-linear maps V —> M. Definition 2.2.43 A Hermitian metric is a complex bilinear map ( , ) : V xV

^C

satisfying the following properties. (a) ( , ) is positive definite i.e.

{v,v) > 0 , (b) (u,v) = (v,u),

Mu,veV.

Vv€V\{0}.

50

Natural constructions on manifolds

A Hermitian metric defines a duality V x V —> C and hence it induces a complex linear isomorphism C : V —¥ V*, v H-> (•, v) £ V*. If V and W are complex Hermitian vector spaces then any complex linear map A : V —*W induces a complex linear map A* : W -> V* A*w

= («H>

(Av, w)) 6 Vc*

where (•, •) denotes the natural duality between a vector space and its dual. We can rewrite the above fact as (Av,w) = (v,A*w). A complex linear map W —>• V* is the same as a complex linear map W -» Vc*. The metric duality defines a complex linear isomorphism Vc* = V so we can view the adjoint A* as a complex linear map W —> V. Let ft = (•, •) be a Hermitian metric on the complex vector space V. If we view (-, •) as an object over K i.e. as a real bilinear map V x V —¥ C then the Hermitian metric decomposes as h = 9te h — iw i = \/^l. The real part is an inner product on the real space V while u is a real, skewsymmetric bilinear form on V and thus can be identified with an element of AJ™ V*. OJ is called the real 2-form associated to the Hermitian metric. Let V be an complex vector space and e\,..., en a basis of V (over C). This is not a real basis of V since d i m ^ F = 2dim(p V. We can however complete this to a real basis. More precisely the vectors e\, iei,..., e„, ien form a real basis of V. Definition 2.2.44 The canonical orientation of a complex vector space is the orientation defined by the basis e\ A \e,\ A ... A en A ie„ above. Exercise 2.2.22 Prove that the orientation defined above is indeed canonical i.e. is independent of the complex basis e\,..., en of V. It is convenient to have a more explicit description of the abstract objects introduced above. Let V be an n-dimensional complex vector space and h a Hermitian metric on it. Pick an unitary basis e\,..., e„ of V i.e. n = dim^ V and h(et, e,-) = &ij. For each j denote by fj the vector \tj. Then e\, / i , ...,e„, /„ is a K-basis of V and C\ A/i A- • - A e n A / n defines the canonical orientation of V. Denote by e 1 , f 1 , . . . , e", / " the dual E-basis in V*. Then SKc ft (e;, e^) = <% = 9^e ft(/;, /,•) and *He ft (e,, /,) = 0 i.e. «Reft = ^(e I (g>e i + / i ® / i ) .

51

A linear algebra interlude Also w(ei, fj) = -3m

h(eu ie,) = <Sy, w(e*, e,) = w(/*> /,•) = 0 Vi, j .

which shows that i

In particular, if we denote by Pfh the pfaffian with respect to the canonical metric SHe h and orientation, we deduce PA(w) = 1. Any complex space V can be also thought of as a real vector space. The multiplication by -\/—T defines a (real) linear operator which we denote by J. Obviously J satisfies J2 = —ly. Conversely, if V is a real vector space then any real operator J : V —> V as above defines a complex structure on V by (a + 6i)v = av + bJv,

V D E F , a + 6i G C.

We will call an operator J as above a complex s t r u c t u r e . Let V be a real vector space with a complex structure J on it. J has no eigenvectors on V. The natural extension of J to the complexification of V, VJQ = V C has two eigenvalues ± i and thus we get a splitting of V£ as a direct sum of complex vector spaces (eigenspaces) Vc = ker ( J - i) © (ker J + i). Exercise 2.2.23 Prove that we have the following isomorphisms of complex vector spaces y = ker(J-i) F = ( k e r J + i). Set V 1 ' 0 = ker(J - i) ^ 10

c

V0'1 = ker(J + i) ^

V

c

V.

01

Thus V c ^ V ' © V ' ^ V © F . We deduce from this an isomorphism of Z-graded complex vector spaces A*VC ^ A*Vlfi ® A'V 0 ' 1 . If we set AP'"V = ApVlfi®£A"V0'1 as

then the above isomorphism can be reformulated AkVc = 0

A"'"V.

(2.2.13)

p+q=k

Note that the complex structure J on V induces by duality a complex structure J* on V* and we have an isomorphism of c o m p l e x vector spaces

v; = (v,jyc^(v;,r).

52

Natural constructions on manifolds

We can define similarly AP'W* as the A p,? construction applied to the real vector space V* equipped with the complex structure J*. Note that Ai,cy.

^ (Ai.
A0-1!/* 9* (A°'V):. and more generally A^v*

^ (A p '«y)*.

If now h is a Hermitian metric on the complex vector space (V, J) then we have a natural isomorphism of complex vector spaces VC*^(V;,J*)=C(V>-J)=CV so that AP,qy*

^

c

AI,PV

The euclidian metric g = SHe h and the associated 2-form u = — 3 m /i are related by
then A ^ V * = s p a n ^ e ' } A0'1 = span c {e J '} and u = i"Y^el

2.3 §2.3.1

At.

Tensor fields Operations with vector bundles

We now return to geometry and more specifically to vector bundles. Let E —>• M be a rank r vector bundle over the smooth manifold M. According to the definition of a vector bundle we can find an open cover (Ua) of M such that each restriction E\Ua is trivial: E\ua = Ua x V where V is an r-dimensional vector space over K=M, C. The bundle E is obtained by gluing these trivial pieces on the overlaps Ua n Up using a collection of transition maps gap : Ua fl Up —> GL(V) satisfying the cocycle condition. Conversely, a collection of gluing maps as above satisfying the cocycle

53

Tensor Gelds

condition uniquely defines a vector bundle. In the sequel we will exclusively think of vector bundles in terms of transition maps. Let E, F be two vector bundles over the smooth manifold M given by a (common) open cover (Ua) and transition maps (gap) and respectively {hal3). Then the collections (ga0®haP), {gals®haP), {(g^))'1, (A*ga/}) satisfy the cocycle condition and therefore define vector bundles which we denote by E®F, E®F, E* and respectively h*E. The reader can check easily that these vector bundles are independent of the choices of transition maps used to characterize E and F (use Exercise 2.1.6). The direct sum E © F is also called the W h i t n e y sum of vector bundles. All functorial constructions discussed in the previous section have a vector bundle correspondent. These constructions are natural in the following sense. Let E' and F' be vector bundles over the same smooth manifold M'. Any bundle maps S : E —> E' and T : F -» F1', both covering the same diffeomorphism : M —> M', induce bundle morphisms S ®T : E ® F ->• E' F', covering <j> and S* : (E1)* ->• E*, covering 0 - 1 . Exercise 2.3.1 Prove the assertion above. Example 2.3.1 Let E, F, E' and F' be vector bundles over a smooth manifold M. Consider bundle isomorphisms S : E —> E' and T : F —> F' covering the same diffeomorphism of the base, 4> : M -> M. Then ( T - 1 ) ' : E* -*• {E'f is a bundle isomorphism covering <j> so that we get an induced map ( S - 1 ) 4 ® T : E* ® F —> {E'Y ® F'. Note that we have a natural identification £*<8>F^Hom(£,F). Definition 2.3.2 Let E —> M be a K-vector bundle over M. A metric on E is a section h of E* ®JJ E* (E — E if K=K) such that for any m € M, /i(m) defines a metric on Em (euclidian if K=M or Hermitian if K= C). §2.3.2

Tensor fields

We now specialize the previous considerations to the special situation when E is the tangent bundle of M, E = TM. The cotangent bundle is then T*Md~f

{TM)\

We define the tensor bundles of M JJ{M)

d

=f TJ{TM)

= (TM)®T ® (T*M)®S.

Note that Tf{M) is naturally a C°°(M)-module; multiplication by a function is multiplication by a smoothly varying scalar in each fiber of TM.

54

Natural constructions on manifolds

Definition 2.3.3 (a) A t e n s o r field of type (r, s) over the open set U C M is a section oiTf(M) over U. (b) A degree r differential form (r-form for brevity) is a section of Ar(T*M). The space of (smooth) r-forms over M is denoted by Qr(M). We set fi*(M) d=

®T>aW{M).

(c) A R i e m a n n i a n m e t r i c on a manifold M is a (0,2), symmetric, (pointwise) positive definite tensor field on M. If we view the tangent bundle as a smooth family of vector spaces then a tensor field can be viewed as a smooth selection of a tensor in each of the tangent spaces. In particular a Riemann metric defines a smoothly varying procedure of measuring lengths of vectors in tangent spaces. Remark 2.3.4 (a) A contravariant tensor field (i.e. a (0, s)-tensor field) S naturally defines a C°°(M)-multilinear map 5 : f[Vect(M)

->• C~(M)

(Xu ... ,X.) - • (p -)• S^X^p),

... ,Xs(p)))

6 C°°(M).

l

Conversely any such map uniquely defines a (0, s)-tensor field. In particular an r-form rj can be identified with a skew-symmetric C°°(M) multilinear map r):f[Vect(M)-±Cco{M). l

Notice that the wedge product in the exterior algebras induces an associative product in 0*(M) which we continue to denote by A. (b) Let / € C°°(M). Its Frechet derivative Df : TM -> TR S R x 1 is naturally a 1-form. Indeed, we get a smooth C°°(M)-linear map df : Vect(M) -> C°°{M) defined by df(X)m

^ Df(X)f{m)

e T / ( r a ) R = R.

In the sequel we will always regard the differential of a smooth function / as a 1-form and to indicate this we will use the notation df (instead of the usual Df). Any diffeomorphism / : M —• M induces a bundle isomorphisms Df : TM —> TM and {Df-1)1 : T*M -> T*M covering / . Thus a diffeomorphism / induces a C°°{M) linear map / . : 77(M) - • 7 J ( M ) (2.3.1) called the push-forward map. Hence the group of diffeomorphisms of M acts naturally (and linearly) on the space of tensor fields.

Tensor fields

55

For contravariant tensor fields a more general result is true. More precisely any smooth map / : M —>• N defines C°°(M)-linear map /* : 7?{N) -> 7 ? ( M ) called the pullback by / . Explicitly, if S is such a tensor defined by a C°°(M)multilinear map 5 : (Vect (N))s -> C°°(N) then f*S is the contravariant tensor field defined by (rS)p(X1{p),

...,Xs(p))

* / Sm(Df

\p (XO, ...,£>/ | p (X s ))

VX 1; ... , X , G ^ ^ ( M ) , p £ M. Note that when / is a diffeomorphism we have

r = f:1 where /„ is the push-forward map defined in (2.3.1). Proposition 2.3.5 Let / : M -» N be a smooth map. The pullback by / defines a morphism of associative algebras /* :fi*(JV)->.fi*(M). Exercise 2.3.2 Prove the above proposition. It is often very useful to have a local description of these objects. If (a;1, ..., xn) are local coordinates on an open set U C M then the vector fields (gfr, •••, gf^r) trivialize TM \tj. We can form a dual trivialization of T*M \v using the 1-forms dxl uniquely determined by <

*

*

•

&

=

*>•

A basis in T*{TXM) is given by < TT-^ ® ••• ® ~zr^®dxil {ox'1 axlr

® ... ®dxi"; 1 < iu ... ,ir < n, 1 < ji, ...,js

Hence any tensor T e 77 (M) has a local description d d T = T£~-y—- 11 ® ... ® 7T— ® dxn ... ® dx3'. •"""•" ox dx'r In particular an r-form looks like w=

Yl l
Wi1...^pda;,1 A •• • Adxlr.

< n

56

Natural constructions on manifolds

Example 2.3.6 Consider the map polar : K2 \ {0} -> R2,

(r,6) H-> (x = rcos6,y

=

rsin0).

The map polar defines the usual polar coordinates. We want to compute the pullback of the volume form dx A dy by polar. We have dx A dy = d(r cos 9) A d(rsin#) = (cos 6dr — rsin#d#) A (sinOdr + r cos 0d#) = r(cos 2 6 + sm29)dr Ad0 = rdr A dO. All operations discussed in the previous section have natural extensions to tensor fields. There is a tensor multiplication, a Riemann metric defines a duality C : Vect (M) —>fl1(M) etc. In particular, there exists a contraction operator

tr : 7J£(M) -> 77(M) defined by tr (X0 ® • • • <8> X r ) (w0 ® • • • ® ws) = w 0 (X 0 )(Xi ® - • • Xr ® wi ® • • • ® w s ). (Xi G Vect (M) Wj € O^M)) In local coordinates the contraction has the form

{to(T£::Z)} = {T$!:::i:}§2.3.3

Fiber bundles

We consider it is useful at this point to bring up the notion of fiber bundle. There are several reasons to do this. On the first hand, they arise naturally in geometry and they impose themselves as worth studying. On the second hand, they provide a very elegant and concise language to describe many phenomena in geometry. We have already met examples of fiber bundles when we discussed vector bundles. These were "smooth families of vector spaces". A fiber bundle wants to be a smooth family of copies of the same manifold. This is a very loose description but may offer a first glimpse at the notion about to be discussed. The model situation is that of direct product X = B x F where B and F are smooth manifolds. It is convenient to regard this as a family of manifolds {Fb)beB. B is called the base, F is called the standard (model) fiber and X is called the total space. This is an example of trivial fiber bundle. In general, a fiber bundle is obtained by gluing a bunch of trivial ones according to a prescribed rule. The gluing may encode a symmetry of the fiber and we would like to spend some time explaining what do we mean by symmetry.

Tensor Gelds

57

Definition 2.3.7 (a) Let M be a smooth manifold and G a Lie group. We say the group G acts on M from the left (resp. right) if there exists a smooth map $ : G x M —> M, (g, m) v-t Tgm such that Tx = 1M and Tg(Thm) = Tghm (resp. Tg{Thm) = Thgm) Vgh 6 G, m € M. In particular, we deduce that V# € G the map Tg is a diffeomorphism of M. For any m G M the set G -m= {Tgm; g € G} is called the orbit of the action through m. (b) Let G act on M. The action is called free if Vp £ G and Vro e M T 9 m ^ m. The action is called effective if Vg e G 3m € M such that Tgm / m. It is useful to think of a Lie group action as defining a symmetry. Example 2.3.8 Let G be a Lie group. A linear representation of G on a vector space V is a left action of G on V such that each Tg is a linear map. One says V is a G-module. Example 2.3.9 Let G be a Lie group. For any g £ G denote by Lg (resp. Rg) the left (resp right) translation by g. In this way we get the tautological left (resp. right) action of G on itself. Definition 2.3.10 Let G be a Lie group. A G-fiber bundle is an object composed of the following: (a) a manifold E called the total space; (b) a manifold B called the base; (c) a manifold F called the standard fiber; (d) a submersion n : E —> B called the natural projection; (e) A Lie group G with a fixed, effective left action on F, g i-> Tg € Diffeo (F). G is called the symmetry group of the bundle; (f) a collection of local trivializations i.e. an open cover (Ua) of the base B and diffeomorphisms ipa : Ua x F —> ir~ 1 (t/ a ) such that

poi,a(b,f)=b, V(b,f)euaxF. We can form the transition (gluing) maps ipQp : Uap x F - > Uap x F (Uap = UaC\ Up) defined by jj)ap = I/J" 1 O ^ . According to (f) these maps can be written as

i>a0(b,f) = (b,Ta0(b)f) where Taf}{b) is a diffeomorphism of F depending smoothly upon b e Uap. The final condition to be imposed is G-compatibility i.e.

58

Natural constructions on manifolds

(g) There exist smooth maps ga/3 : Uap —> G satisfying the cocycle condition 9aa — l

9-ya = 9iP • 9/3a-

and such that Tap{b) = T9al3ib).

We will denote this fiber bundle by (E,p, B, F, G). The choice of an open cover (Ua) in the above definition is a source of arbitrariness since there is no natural prescription on how to perform this choice. We need to describe when two such choices are equivalent. Two open covers (Ua) and (Vi) together with the collections of local trivializations 4>a '• Ua x F —¥ 7r _1 (£/ a ) and fa : Vi x F —» 7r_1(V;) are said to be equivalent if for all a, i there exists a smooth map tai : Ua n VJ —> G such that for any x £ Ua fl Vi and any / € F

CVifa./) =

{x,tai{x)f).

A G-bundle structure is defined by an equivalence class of trivializing covers. As in the case of vector bundles a collection of gluing data determines a G-fiber bundle. Indeed, if we are given a cover (Ua)a€A of the base B and a collection of transition maps gap : Ua fl Up —> G satisfying the cocycle condition, then we can get a bundle by gluing the trivial pieces Ua x F along the overlaps. More precisely if b 6 Ua fl Up then the element (6, / ) e Ua x F is identified with the element (b, 9pa • f) € Up x F where (g, f)>-¥ g • f denotes the left action of G on F. Definition 2.3.11 Let E A B be a G-fiber bundle. A G-automorphism of this bundle is a diffeomorphism T : E —> E such that poT = p (i.e. maps fibers to fibers) and for any trivializing cover (Ua) (as in Definition 2.3.10) there exists a smooth map 9a '• Ua -¥ G such that fa;1Tipa(b,f)

= (b,ga(b)f)),

Vb,g,f.

Definition 2.3.12 (a) A fiber bundle is an object defined by conditions (a)-(d) and (f) in the above definition. (One can think the structure group is the group of diffeomorphisms of the standard fiber). (b) A section of a fiber bundle E A B is a smooth map s : B —> E such that pos = lB, i.e. s(b) 6 p - ' ( 6 ) , V I » e B . Example 2.3.13 A rank r vector bundle (over K= R, C) is a GL(r, K)-fiber bundle with standard fiber Kr and where the group GL(r, K) acts on K r in the natural way.

Tensor fields

59

Example 2.3.14 Let G be a Lie group. A principal G-bundle is a G-fiber bundle with fiber G, where G acts on itself by left translations. Equivalently, a principal G-bundle over a smooth manifold M can be described by an open cover i l of M and a G-cocycle i.e. a collection of smooth maps

guv:UnV^G

such that Wxeunvnw

U,V£il

{u,v,w eU) guv{x)gvw{x)gwu{x)

= 1.

Exercise 2.3.3 (Alternative definition of principal bundle) Let P be a fiber bundle with fiber a Lie group G. Prove the following are equivalent. (a) P is a principal G-bundle. (b) There exists a free, right action of G on G, (P x G —¥ P, {p,g) >-» p- g such that its orbits coincide with the fibers of the bundle P and a trivializing cover {i>a : Ua x G ->

n-\Ua}

such that ipa(u, hg) = ipa{u, h) • g Vg, h G G, u £ Ua. Exercise 2.3.4 (The frame bundle of a manifold) Let Mn be a smooth manifold. Denote by F(M) the set of frames on M i.e. F(M) = {{m;Xu

...,Xn);

meM,

Xt € TmM and span(X 1 , ... ,Xn) = TmM} .

(a) Prove that F(M) can be naturally organized as a smooth manifold such that the natural projection p : F(M) —¥ M, (m; X\, ..., Xn) ^ r n i s a submersion. (b) Show F(M) is a principal GL(n, K)-bundle. F(M) is called the frame bundle of the manifold M. Hint: A matrix T = (Tj) £ GL(n, K) acts on the right on F(M) by (m; Xu ..., Xn) ->• (m; ( T " 1 ) ' ^ , . . . , ( T " 1 ) ^ , ) . Example 2.3.15 (Associated fiber bundles) Let IT : P —> G be a principal Gbundle. Consider (Ua)aeA a trivializing cover and denote by ga/3 : Ua n [/^ —> G a collection of gluing maps determined by this cover. Assume G acts (on the left) on a smooth manifold F T:GXF^F,

(g,f)^T(g)f.

The collection rQ/3 = r(g a/3 ) : [7Q^ —>• Diffeo (F) satisfies the cocycle condition and can be used (exactly as we did for vector bundles) to define a G-fiber bundle with fiber F. This new bundle is independent of the various choices made (cover (Ua) and transition maps gal3). (Prove this!) It is called the bundle associated to P via r and is denoted by P xT F.

60

Natural constructions on manifolds

Exercise 2.3.5 Prove that the tangent bundle of a manifold Mn is associated to F(M) via the natural action of GL(n,M) on Rn. Exercise 2.3.6 (Hopf bundle) If we identify the unit odd dimensional sphere S 2 " - 1 with the submanifold

{(*1( ..,z,)ec"; |z0|2 + --- + kl 2 = i} then we detect an S 1 -action on S2n~l given by ei9-(z1,...,zn)

=

(eiez1,...,eiezn).

The space of orbits of this action is naturally identified with the complex projective space OP" - 1 . (a) Prove that p : 5 2 n _ 1 -> CFn~1 is a principal S1 bundle called Hopf bundle, (p is the obvious projection). Describe one collection of transition maps. (b) Prove that the tautological line bundle over Q P n _ 1 is associated to the Hopf bundle via the natural action of S 1 on C 1 . Exercise 2.3.7 Let E be a vector bundle over the smooth manifold M. Any metric h on E (euclidian or Hermitian) defines a submanifold S(E) C E by S(E) = {veE;

\v\h = 1}.

Prove that S(E) is a fibration over M with standard fiber a sphere of dimension r a n k £ — 1. S(E) is usually called the sphere bundle of E.

Chapter 3 Calculus on Manifolds This chapter describes the "kitchen" of differential geometry. We will discuss how one can operate with the various objects wehave introduced so far. In particular we will introduce several derivations of the various algebras of tensor fields an we will also present the opposite notion of integration.

3.1 §3.1.1

T h e Lie derivative Flows on manifolds

The notion of flow should be familiar to anyone who has had a course in ordinary differential equations. In this section we only want to describe analytic facts in a geometric light. We warmly recommend [4] for more details and excellent examples. A neighborhood M of M x {0} in M x K is called balanced if Mm e M ({m} x ft) n W = {m} x (—r,r),

for some r > 0.

Definition 3.1.1 A local flow is a smooth map $ : M —> M, (m,t) i-> $'(771) (where N is a balanced neighborhood of M x {0} in M x 1) such that (a) $°(m) = m, Vra e M. ( b ) $ ' ( $ ' ( m ) ) = $ i + s ( m ) for alls,* e K, m e M such that (s,m), (s+t,m), (t,$ s (m)) e

N. When Af = M x ft, $ is called a flow or dynamical system. The conditions (a) and (b) above show that a dynamical system is nothing but a left action of the additive (Lie) group (K, +) on M. Example 3.1.2 Let A be a n x n real matrix. It generates a flow $ ^ on ffi" by

61

62

Calculus on manifolds

Definition 3.1.3 Let $ : M —> M be a local flow on M. The infinitesimal generator of $ is the vector field X on M defined by X(m) = X . ( m ) = ' |

| ( = 0 *'(m)

i.e. X(m) is the tangent vector to the smooth curve t >-» $*(m) at £ = 0. This curve is called flow line. Exercise 3.1.1 Show Xj> is a smooth vector field. E x a m p l e 3.1.4 Consider the flow etA on R" generated by an n x n matrix A. Its generator is the vector field X& on W1 defined by XA{U) = — U=o etAu = Au. at Proposition 3.1.5 Let M be a smooth n-dimensional manifold. The map X : {Local flows on M} - » V e c t ( M ) ,

$ i->-A"$

is a surjection. Moreover, if $; : Mi —> N (i=l,2) are two local flows such that A"$j = X$ 2 then $ i = $ 2 on M\ r\J\f2Proof Surjectivity Let X be a vector field on M. An integral curve for A" is a smooth curve 7 : (a, b) —> M such that

7(0 = X( 7 (*)). In local coordinates (xa) over on open subset U C M this condition can be rewritten as ±a(t) = Xa(x1(t),...,xn(t)) (3.1.1) where 7(f) = (x 1 ^), ...,a; n (i)) and X = A " " ^ . (3.1.1) is an ordinary differential equation. Classical existence results (see [4]) can be used to show that for any precompact open subset K C U there exists £ > 0 such that for all x g K there exists a unique integral curve for X, yx : {—£,e) —> M satisfying 7,(0) = x.

(3.1.2)

Moreover, as a consequence of the smooth dependence upon initial data we deduce that the map $ K : MK = K x ( - e , e) -> M, (a;, i) H- 7 x (i) is smooth.

The Lie derivative

63

Now we can cover M by open, precompact local coordinate neighborhoods (Ki)iei and as above, we get smooth maps <3>; : A/i = Ki x (—£{,£;) —¥ M solving the initial value problem (3.1.1-2). Moreover, by uniqueness we deduce $; = $ j on Mi nMj. Define M = U^/A/"; and set $ : M -¥ M, $ = $j on A/;. Clearly, $ satisfies all the conditions in the definition of a local flow. Tautologically, X is the infinitesimal generator of $. The second part of proposition follows from the uniqueness in initial value problems. CI The family of local flows on M with the same infinitesimal generator X G Vect(M) is naturally ordered according to their domains: ( $ ! : A/i -> M) < ($ 2

:M2-*M)

iff M\ C A/2. This family has a unique maximal element which is called the local flow generated by X and is denoted by $ x • §3.1.2

The Lie derivative

Let X be a vector field on the smooth n-dimensional manifold M and denote by $ = <J>x the local flow it generates. We assume $ is actually a flow so its domain is actually M x K. The local flow situation is conceptually identical but unpleasantly lengthens the presentation of the facts to come. For each t E R, $ ' is a diffeomorphism of M and so it induces a push-forward map on the space of tensor fields. If 5 is a tensor field on M we define its Lie derivative along the direction given by M in a natural way Lx5m'^/-limi(($iS)m-Sra)

Vm&M.

(3.1.3)

Intuitively LxS measures how fast is the flow $ changing the tensor 5. Clearly the limit in (3.1.3) exists and one sees that LxS is a tensor of the same type as S. We want provide more explicit descriptions of this operation. Lemma 3.1.6 For any X G Vect (M) and / G Cco{M) we have Xf "=fLxf

= (df,X)

=

df(X).

Above, ( , ) denotes the natural duality between T*M and TM, (,):

C°°(r*M) x C°°{TM) -+ C°°(M).

In particular, Lx is a derivation of

Coa(M).

64

Calculus on manifolds

Proof Let $ ' = $x be the local flow generated by X and assume for simplicity it is defined for all t. $ ' acts on C°°(M) by the pullback of its inverse i.e. $ t ( / ) = / ° $ ~ f Hence Lxf(m)

= lim l-{f{m) - f^m))

= - |

| ( = 0 / ( * - « r o ) = {df, X).

D

Exercise 3.1.2 Prove that any derivation of the algebra C°°(M) is of the form Lx for some X G Vect(M). Thus L>er(C°°(M)) ^ V e c t ( M ) .

Lemma 3.1.7 Let X, Y G Vect(M). Then the Lie derivative of F along X is a new vector field LxY which, viewed as a derivation of Cco(M), coincides with the commutator of the two derivations of Cca{M) defined by X and Y i.e. LxYf

= [X,Y]f,

V/€C°°(M).

The vector field [X, Y] = LXY is called the Lie bracket of X and Y. In particular the Lie bracket induces a Lie algebra structure on Vect (M). Proof We will work in local coordinates {xl) near a point m G M so that X = X 1 ^ and Y = Yj £.. We first describe the commutator [X, Y]. If / G C^iM) then

[x,y]/ = ( x - A ) ( y ^ ) - ( F ^ ) ( x ^ ) dx'dxi

dxl dxJ J

\

dx'dxi

dxJ dxl

so that the commutator of the two derivations is the derivation defined by the vector field

Note in particular that [-£r, -g^-} = 0 i.e. the basic vectors -^- commute as derivations. So far we have not proved the vector field in (3.1.4) is independent of coordinates. We will achieve this by identifying it with the intrinsically defined vector field LXY. Set y(t) = $ e m so that we have a parametrization j(t) = (xl(t)) with x' = X1. Then $ - f m = 7 ( _ t ) = 7 (0) - 7 (0)i + 0(t2) = ( 4 - tXl + 0(t2)) and BY*

Y](_t) = Yi-tX<—

+ 0(e).

(3.1.5)

65

The Lie derivative

Note that $ 7 ' : T7(0)M -> T^-^M is the linearization of the map (1*) >->• (xj - tX' + 0(t2)) so it has a matrix representation

*-'= i d -'(£) +o<'!>\

/ z,j

In particular, using the Neumann's series ( 1 - . 4 ) - 1 = 1 + A + A2 + --(where A is a matrix of norm strictly less than 1) we deduce that $ ' = ( $ ^ ! ) _ 1 T7(-t)M —> T 7 ( 0 )M has the matrix form ^=id

+ t(^\

+ 0(t2).

:

(3.1.6)

Using (3.1.6) in (3.1.5) we deduce

This concludes the proof of the lemma. • We can now completely describe the Lie derivative on the algebra of tensor fields. Proposition 3.1.8 Let X be a vector field on the smooth manifold M. Lx is the unique derivation of T*{M) with the following properties. (a) Lxf = {df,X) = Xf, V/ 6 C"»(M). (b) LXY = [X, Y], VX, Y e Vect (M). Lx commutes with the contraction tr : 7J+~i(M) —> T^{M). Moreover, Lx is a natural operation i.e., for any diffeomorphism 4>: M —>• N we have *oLx = Llp,x°4>*> VX e Vect (M) i.e. 0.(1*) = L ^ x Proof

The fact that Lx is a derivation i.e. LX(S

® T) = L x 5 ® T + S <8> L X T

follows easily from the definition (3.1.3). Properties (a) and (b) were proved above. As for part (c) this is Leibniz' rule in disguise. In its simplest form when T — Y ®ui where Y <E Vect (M) and u £ fix(M) we have t r T = w(y) and (c) is equivalent to Lx MY))

= (LXUJ)(Y)

+ u(Lx(Y).

(3.1.7)

66

Calculus on manifolds

(3.1.7) follows immediately from (3.1.3) mimicking the proof of the usual product rule. In particular (3.1.7) uniquely determines the Lie derivative of a 1-form u> by (Lxu)(Y)

= Lx ( w (y)) -

U([X,

Y}).

(3.1.8)

Since Lx is a derivation of the algebra of tensor fields, its restriction to C°°{M) © Vect [M)®Q}(M) uniquely determines the action on the entire algebra of tensor fields which generated by the above subspace. The reader can check easily that property (c) is satisfied in its entire generality. The naturality of Lx is another way of phrasing the coordinate independence of this operation. We leave the reader fill in the routine details. • Corollary 3.1.9 For any X,Y

e Vect (M) we have [LX,LY]

= L[X,Y]

as derivations of the algebra of tensor fields on M. In particular this says that Vect (M) as a space of derivations of T*(M) is a Lie subalgebra of the Lie algebra of derivations. Proof [Lx, Ly] is a derivation (as a commutator of derivations). By Lemma 3.1.7, [LX,LY] = i[x,r] on C°°(M). Also, a simple computation shows that [Lx, LY}Z = L[XX]Z,

VZ 6 Vect (M)

so that [Lx,Ly] — L[X]Y] on Vect(M). Finally, since the contraction commutes with both Lx and Ly it obviously commutes with LxLy — LyLx. The corollary is proved.

• Exercise 3.1.3 Prove that the map V : Vect (M) © End (TM) ->

Der(T;{M))

given by V(X, S) = Lx + S is well defined and is a linear isomorphism. Moreover [V(XuSl),V(X2,S2)]

=

V([Xl,X2],[S1,S2}).

Lx is a derivation of T* with the remarkable property LX{Q.*{M)) C

fi'(M).

The wedge product makes fT (M) a s-algebra and it is natural to ask whether Lx is a s-derivation with respect to this product. Proposition 3.1.10 The Lie derivative along a vector field X is an even s-derivation of n*(M) i.e. LX(CJ A rj) = (Lxu>) A 7)+ io A {Lxrj),

Vw, 77 € Q*(M).

Proof Denote (as in Section 2.2.2) by A the anti-symmetrization operator A : (r*M) 0 A : —¥ Qk(M). The statement in the proposition follows immediately from the straightforward observation that the Lie derivative commutes with this operator (which is a projector). We leave the reader to fill in the details. •

67

The Lie derivative §3.1.3

Examples

Example 3.1.11 Let u = u)idx' be a 1-form o n l " . If X = X^-^j on R" then Lxu> = (Lxu))kdxk is defined by

is a vector field

(^) t = (^)(^) = x w( A)_ w(Ljf |.) (dxi d = X -Uk+u

-^-r^\oxK ox,

Hence (vidwk

dX>\

,

k

Example 3.1.12 Consider X ~ F^ + G%-y + H-§~z a smooth vector field on R 3 . We want to compute Lxdv where dv is the volume form on R 3 , dv = dx Ady A dz. Since Lx is an even s-derivation of fi*(M) we deduce Lx(dx A dy A dz) = (Lxdx)

A dy A dz + dx A (Lxdy) Adz + dx Ady A (Lxdz).

Using the computation in the previous example we get Lx(dx)

dF = dF :=—-dx dx

OF OF +—-dy +—-dz, dy dz

Lx(dy)

= dG, Lx(dz)

= dH

so that

In particular, we deduce that if div X = 0, the local flow generated by X preserves the form dv. We will get a better understanding of this statement once we learn integration on manifolds, later in this chapter. Example 3.1.13 The exponential map on Lie groups Consider G a Lie group. Any element g € G defines two diffeomorphisms of G: the left (Lg) and the right translation (Rg)on G. A tensor field T on G is called left (resp. right) invariant if for any g G G (Lg),T = T (resp. {Rg)*T = T). The set of left invariant vector fields on G is denoted by £ G . The naturality of the Lie bracket implies (Lg),[X,Y}

=

{(Lg)tX,(Lg)tY]

so that VX, Y € £ G , [X, Y] € £ G . Hence £ G is a Lie subalgebra of Vect (G). £ G is called the Lie algebra of the group G. FACT 1. d i m £ G = dimG. Indeed, the left invariance implies that the restriction map £ G —> T\G, X M- Xi is an isomorphism (Exercise). We will often find it

68

Calculus on manifolds

convenient to identify the Lie algebra of G with the tangent space at 1. Any X e £G defines a local flow §x on G. FACT 2. $' x is defined for all i € M so that it is a flow. (Exercise) Set exp(tX) d= $x(l). We thus get a map exp : 7\G ^ £ G ->• G, X H-> exp(X) called the exponential map of the group G. FACT 3. &x(g) = g • exp(iX) i.e. $x

=

Rexp(tX)-

Indeed, it suffices to check that d dt We can write (gex.p(tX)) |-

(= o

((/exp(iX)) = Xg.

= Lgexp(tX)

so that

| (= o (L 9 exp(iX)) = (L ( Lg),(, ) . ( | \t=0 exp(<X)) = (Lg)tX

= X 9 (by left invariance).

The reason for the notation exp(iX) is that when G = GL(n, K) the Lie algebra of G is the Lie algebra gl(n, K) of all n x n matrices with the bracket given by the commutator of two matrices (Exercise) and for any X £ £G, exp(X) = ex = E*>o hXk (Exercise). Exercise 3.1.4 Prove the statements left as exercises in the example above. Exercise 3.1.5 Let G be a matrix Lie group i.e. a Lie subgroup of some general linear group GL(N,K). This means the tangent space T^G can be identified with a linear space of matrices. Let X, Y £ TiG and denote by exp(tX) and exp(iF) the 1-parameter groups with they generate and set g{s,t) = exp(sX) exp(ty) exp(—sX) exp(—tY). (a) Show that gStt = l + [X,Y\algst

+ 0({s2 + t2f2)

as s , i - > 0

where the bracket [X, Y]aig (temporarily) denotes the commutator of the two matrices X and Y.

Derivations

ofQ*(M)

69

(b) Denote (temporarily) by [X, Y]geom the Lie bracket of X and Y viewed as left invariant vector fields on G. Show that at 1 6 G [X, Y\alg = [X, Y]ge0m. (c) Show that o(n) C gl(n, R) (defined in Section 1.2.2) is a Lie subalgebra with respect to the commutator [•, •]. Similarly, show that u(n), su(n) C gUn,C) are real Lie subalgebras of gl(n,C), while su(n, C) is even a complex Lie subalgebra of g/(n,C). ~ (d) Prove that we have the following isomorphisms of real Lie algebras. £o(n) — Q_{n), £u(n) = u{n), £Su(n) = su{n) and £ S L ( n ] Q - §l{n,C)Remark 3.1.14 In general in a non-commutative matrix Lie group G the traditional equality exp(iX) exp(tY) = exp(t{X + Y)) no longer holds. Instead, one has the Campbell-Hausdorff formula exp(iX) • exp(ir) = exp ( tdx(X, Y) + t2d2{X, Y) + t3d3{X, Y) + • • •) where d^ are homogeneous polynomials of degree k in X and Y where the multiplication between X and Y is given by their bracket. The dk's are usually known as Dynkin polynomials. For example d,(X,Y)

= X+Y,

d2{X,Y)

= \[X,Y],

d3(X,Y)

= i ( [ X , [X, Y]]+[Y, \Y,X]]) etc.

For more details we refer to [65].

3.2 §3.2.1

Derivations of Q*(M) T h e exterior derivative

The super-algebra of exterior forms on a smooth manifold M has additional structure and in particular its space of derivations has special features. This section is devoted precisely to these new features. The Lie derivative along a vector field X defines an even derivation in Q.*{M). The vector field X also defines an odd derivation ix (called the interior derivative along X or contraction) via the contraction map ixu

d

= U{X<S:u}),

Va;enr(M).

More precisely ix<*> is the (r — l)-form determined by (»xw)(*i. - . X r - i ) =u{X,Xu

. . . , X r _ i ) , VXi, ...,X r _i € Vect(M).

The fact that ix is an odd s-derivation is equivalent to ix(u

A ri) = (ixw) A 7? + (-l) d e s w w A {ixr)),

Vw, f] e fi*(M).

70

Calculus on manifolds

Exercise 3.2.1 Prove that the interior derivative along a vector field is a s-derivation. Proposition 3.2.1 (a) [ix,iy]s = ixiy + iyix = 0. (b) The super-commutator of Lx and iy as s-derivations of fi*(M) is given by [Lx, iy]s = Lxiy

— iyLx

= i[x,Y]-

The proof uses the fact that the Lie derivative commutes with the contraction operator and it is left to the reader as an exercise. The above s-derivations by no means exhaust the space of s-derivations of Q.*(M). In fact we have the following fundamental result. Proposition 3.2.2 There exists an universal odd s-derivation on the s-algebra of differential forms f2*( • ) uniquely characterized by the following conditions. (a) For any smooth function / 6 Q.°{M), df coincides with the differential of / . (b) d2 = 0 (c) d is natural (universal) i.e. for any smooth function : N —> M and for any form u on M we have dcjfuj = 4>*duj.

d is called the exterior derivative. Proof We first prove uniqueness. Let U be a local coordinate chart on M " with local coordinates (x1, ... ,xn). Then, over U, any r-form ui can be described as CJ =

tJi1...tTdxil A...

Yl

Adxir.

l
Since d is a s-derivation and d^x1) duj =

= 0 we deduce that over U

^T

(dwii...!,.) A (dx11 A • * - A dx%r)

l
= E Kii

^ d z M A (da;*1 A---Adx{').

(3.2.1)

<~-
Thus the form dm is uniquely determined on any coordinate neighborhood and this completes the proof of the uniqueness of d. Existence To prove existence consider an r-form u> and for each coordinate neighborhood U we define du\u as in (3.2.1). To prove this is a well defined operation we must show that if U, V are two coordinate neighborhoods then duj\u= dui\v

onUnV.

71

Derivations of fi*(M)

Denote by (a;1, ... ,xn) the local coordinates on U and by (j/ 1 , ... ,yn) the local coordinates along V so that on the overlap U fl V we can describe the y's as functions of the x's. Over [/ we have u) =

^h...irdxH A ... A dx'r

5Z l
tL,

=

(^^dx*\A(dxhA...Adxi')

E l
"X'

\

J

while over V we have w

=

^h-jrdy31

E

A • • • A d2/Jr

l<ji<-<3r
E f^r1^) w A • • •A dyjry

^=

l<ji<-<jr
\

"V

)

The components u)^...^ and C)^...^ are skew-symmetric i.e. Vcr G &r

jl

:

dy U C\V v/e deduce and similarly for the Q's. Since u \v=dy LJ \V over Wil lr &* &* Hence

ax*

£-; v ax** ''' dxidx** ' ax*' WJ'1"JV dx* ax*' ax1

fc=i

where in the above equality we also sum over the indices j \ , ...,jr according to Einstein's convention. We deduce duJ

^-i'dxi

y l
=

£ £ ^ . . . *f*

i tTi^a;*' Y^Tj^x"!

. 2 ^ ,r^A^A...A^

a ^ JI-Jr

ax-ax"

+ V r ? C . . .

Adxh A...A dxir

U X

9 T d

"il-:iT dx{ A dx" A ... A da*.

/

1

ax -- ax

1

(3.2.2)

v y

Notice that . f * ; . = _ ?* , while dx' A d x " = — dx'k A dx1 so that the first term in ox1 ox *

oxfcax*

the right hand side of (3.2.2) vanishes. Consequently d

^dx
=

^ lrd

^ . - . dx'1 dx

onUC\V

^ A d x ox'

i

A d x " . . - d x -

72

Calculus on manifolds

dxl

)

\dx'i

J

\dx'*

1

= (du)jl...jT) A dy^ A • • • A dy** d&h~JT

-dyj Adyh A ••• Adyir. dyi This proves duj \u= dw \v over U(~)V. We have thus constructed a well defined linear map d:Q*(M) ->fi* + 1 (M). To prove that d is an odd s-derivation it suffices to work in local coordinates and show the (super)product rule on monomials. Thus, let 6 = fdx11 A • • • A dxlr and w = gdx*1 A • • • A dxi'. We set for simplicity dx1 = dxn A • • • A dx1' and dxJ = dxjl A---Adxj'. Then d{9 A w) = d{fgdxl

A dxJ) = d{fg) A dx1 A dxJ

= {df -g + f -dg)A dx1 A dx3 = df A dx1 A dx3 + ( - l ) r ( / A dx1) A (dg A dx3) = d0Aw + (-l) d e g "0Adu;. 2

We now prove d = 0. We check this on monomials fdx1 as above. d2{fdx')

= d(df A dx1) = {d2f) A dx1.

Thus it suffices to show d2f = 0 for all smooth functions / . We have d2f =

df2 "' ox1 ox3

.dx'Adx3.

The desired conclusion follows from the fact that gJaxJ = gJdxi while dx1 A dxj = —dx? A dx1. Finally, let i ^ b e a smooth map N —• M and u = Y,i widx1 be an r-form on M. Here / runs through all ordered multi-indices 1 < «i < • • • 'u) = £ (dAr(<^*W/) A cf>*(dx') + (jfu1 A d{4>*dx')) i

For functions, the usual chain rule gives d^ffl^i) = (jf^M^1). I n terms of local coordinates (x') the map ^> looks like a collection of n functions <j>1 € C°°(N) and we get ^(cfe 7 ) = dtp1 = dNh A • • • A d w f ' • In particular, d^(d0 7 ) = 0. We put all the above together and we deduce dN{4>*w) = *(dMu/) A d<j)! = ^{dMUi1)

The proposition is proved. •

A 4>*dx3 = 4>*{dMoj).

Derivations

ofQ*(M)

73

Proposition 3.2.3 The exterior derivative satisfies the following relations ([ , ]3 denotes the super-commutator in the s-algebra of real endomorphisms of fi*(M)). (a) [d, d]s = 2d2 = 0. (b) (Homotopy formula) [d, ix]s = dix + ixd = Lx VX € Vect (M). (c) [d, Lx]s = dLx - Lxd = 0, VX e Vect (M). An immediate consequence of the homotopy formula is the following invariant description of the exterior derivative:

(du>)(X0,Xu ... ,Xr) = J2(-l)iXi(u>(X0,...,Xi,...,Xr)) t=0

+

£

(-l) i + ^([X i ,X : ,-] ) X 0 ,... ) X i ,...,^ 1 -,-X-r).

(3.2.3)

0
Above, the hat indicates that the corresponding entry is missing. Proof To prove the homotopy formula set V = [d, ix]s = dix + ixd. T> is an even s-derivation of Q*(M). It is a local s-derivation i.e. if w e Q*(M) vanishes on some open set U then VUJ vanishes on that open set as well. The reader can check easily by direct computation that VUJ = Lxu>, Mu € Q°(M) © fix(M). The homotopy formula is now a consequence of the following technical result left to the reader as an exercise. Lemma 3.2.4 Let 2>, V be two local s-derivations of Q*(M) which have the same parity (i.e. they are either both even or both odd). If T> = V on Q°(M) © £2X(M) thenX> = X>'onfi*(M). Part (c) of the proposition is proved in a similar way. Equality (3.2.3) is a simple consequence of the homotopy formula. We prove it in two special case r = 1 and r = 2. T h e case r = l . Let u be an 1-form and let X, Y G Vect (M). We deduce from the homotopy formula du,(X,Y)

= (ixdu)(y)

= (Lxu)(Y)

- (dw(X))(Y).

On the other hand, since Lx commutes with the contraction operator, we deduce Xui{Y) = Lx(u{Y))

=

{Lxu>){Y)+u{[X,Y]).

Hence dui{X, Y) = Xw{Y) - w([X, Y}) - {du){X)){Y) = Xu>{Y) - YUJ{X) - u([X, Y\). This proves (3.2.3) in the case r = 1. T h e case r = 2 . Consider a 2-form w and three vector fields X, Y and Z. We deduce from the homotopy formula {du)(X, Y, Z) = (ixduj)(Y, Z) = (Lx - dix)u>{Y, Z).

(3.2.4)

74

Calculus on manifolds

Since Lx commutes with contractions we deduce {Lxu){Y, X) = X{UJ{Y, Z)) - U([X, Y], Z) - u,(Y, [X, Z\).

(3.2.5)

We substitute (3.2.5) into (3.2.4) and we get (CLJ)(X, Y, Z) = X(u{Y, Z)) - U{[X, Y], Z) - u{Y, [X, Z\) - d{ixu>){Y, X).

(3.2.6)

We apply now (3.2.3) for r = 1 to the 1-form ixw. We get d{ixu){Y,X)

= Y(ixuj(Z))

= YLJ{X,

- Z{ixu>{Y)) -

(iXLj)([Y,Z})

Z) - ZUJ(X, Y) - u{X, [Y, Z]).

(3.2.7)

If we use (3.2.7) in (3.2.6) we deduce (du)(X, Y, Z) = Xu{Y,

Z) - YUJ{X, Z) + ZUJ{X,

-UJ([X, Y], Z) + UJ([X, Z], Y) -

W([Y,

Y)

Z], X).

(3.2.8)

The general case in (3.2.3) can be proved by induction. The proof of the proposition is complete. • Exercise 3.2.2 Prove Lemma 3.2.4. Exercise 3.2.3 Finish the proof of (3.2.3) in the general case. §3.2.2

Examples

Example 3.2.5 (The exterior derivative in K 3 ) (a) Let / 6 C°°(R 3 ) then df = —dx + — * / + -w-dz. ox oy oz df looks like the gradient of / . (b) Let w e n ^ K 3 ) , w = Pdx + Qdy + Rdz. Then duj = dP A dx + dQ A dy + dR A dz fdQ

dP\

J

J

(dR

dQ\ ,

J

(dP

so that dui looks very much like a curl. (c) Let w = Pdy Adz + Qdz Adx + Rdx A dy <E Q2(M3). Then (dP dQ dR\ , , , du) = ——h ——h -r- \dx A dy A dz. \ ox dy oz j This looks very much like a divergence.

dR\ ,

A

J

75

Connections on vector bundles

Example 3.2.6 Let G be a connected Lie group. In Example 3.1.11 we defined the Lie algebra £<j of G as the space of left invariant vector fields on G. Set QTleft(G) = left invariant r-forms on G. In particular £*G = Q}eft(G). If we identify £*G = TfG then we get a natural isomorphism Qruft{G) = Ar2G. The exterior derivative of a form in tyeft can be described only in terms of the algebraic structure of £Q. Indeed, let w e £G = n}eft{G). For X,Y £ £*G we have (see (3.2.3) ) dw{X, Y) = Xu>(Y) - Yu(X)

- w{[X, Y]).

Since w, X and Y are left invariant UJ(X) and u(Y) are constants. Thus, the first two terms in the above equality vanish so that du)(X,Y) nen tne

More generally, if w € ^Lft t for all X0,..., XT e £ G we have

Mx0,xu...,xr)=

=

-u{[X,Y}).

same arguments applied to (3.2.3) imply that

J2 {-iY+M[xi,xi],x1,...,xi,-,xj,...,xT).

(3.2.9)

0
3.3 §3.3.1

Connections on vector bundles Covariant derivatives

We learned several methods of differentiating tensor objects on manifolds. The tensor bundles are not the only vector bundles arising in geometry and very often one is interested in measuring the "oscillations" of sections of vector bundles. Let E be a. K vector bundle over the smooth manifold M (K= M, C). For such an arbitrary E we encounter a problem which was not present in the case of tensor bundles. Namely, the local flow generated by a vector field X on M no longer induces bundle homomorphisms. For tensor fields the transport along a flow was a method of comparing objects in different fibers which otherwise are abstract linear spaces with no natural relationship between them. To obtain something that looks like a derivation we need to formulate clearly what properties we should expect from such an operation. (a) It should measure how fast is a given section changing along a direction given by a vector field X. Hence it has to be an operator V : Vect (M) x C°°(.E) -> C°°(E),

{X, u) H-> V X U .

76

Calculus on manifolds

(b) If we think of the usual directional derivative, we expect that after "rescaling" the direction X the derivative along X should only rescale by the same factor i.e. V/€C°°(M):

VJxu =

fVxu.

(c) Since V is to be a derivation it has to satisfy a sort of (Leibniz) product rule. The only product that exists on an abstract vector bundle is the multiplication of a section with a smooth function. Hence we require V * ( / u ) = (Xf)u

+ fVxu,

V/ e C°°(M), u e C°°(E).

Conditions (a) and (b) can be rephrased as follows: for any u G Cco(E) the map

Vu: Vect(M)->C°°(E),

I ^ V ^

is C°°(M) linear so that it defines a bundle map Vw e C°°(Hom {TM, E)) =* C^iT'M

® E).

Summarizing, we can formulate the following definition. Definition 3.3.1 A covariant derivative (or linear connection) on £ is a EClinear map V : C°°{E) -> C°°{T*M E) such that V/ € C°°(M) and Vu G C°°{E).

V(/u) = df®u + fVu. Example 3.3.2 Let K M = M x W be the rank r trivial bundle over M. The space C°°(KM) of smooth sections coincides with C°°(M,Kr). We can define V° : C°°{M, Kr -> C^iM, T*M ® K r ) V ° ( / i , . . . , / r ) = (d/i,...,d/ r ). One checks easily that V is a connection. This is called the trivial connection. Remark 3.3.3 Let V°, V 1 be two connections on a vector bundle E —>• M. Then for any a e Cco{M) the map V = a V 1 + (1 - Q ) V ° : C°°(S) -> C°°(T* ® E) is again a connection. Notation For any vector bundle F over M we set Qk(F)

d

=f C°°(A*T*M <8> F).

77

Connections on vector bundles

Proposition 3.3.4 Let E be a vector bundle. The space %l(E) of linear connections on E is an affine space modeled on fi^End (E)). Proof We first show 21 (E) is not empty. To see this, choose {Ua} an open cover of M such that E \ua is trivial, Va. Next, pick (fip) a smooth partition of unity subordinated to this cover. Since E \ua is trivial it admits at least one connection, the trivial one, as in the above example. Denote such a connection by V a . Now define

a,/3

One checks easily that V is a connection so that 21(E) is nonempty. To check that 21 (E) is an affine space consider two connections V° and V 1 . Their difference A = V 1 — V° is an operator A : C°°(£) -> C^iT'M

® E)

satisfying A{fu) = fA(u), Vw € C°°{E). Thus A e C°°(Hom (E,T*M ® E)) ^ C°°{T*M E* ® E) ^ Ql{E* ® E) = Q}(EndE). Conversely, given V° € 2l(.E) and A e f l 1 (End-E) one can verify that the operator VA = V° + A : C°°{E) -»

rt(E).

is a linear connection. This concludes the proof of the proposition.

•

The tensorial operations on vector bundles extend naturally to vector bundles with connections. The key principle behind this fact is the Leibniz' formula. More precisely if E{ (i = 1,2) are two bundles with connections V1 then E\ ® E2 has a naturally induced connection V E l ® E 2 K u2) = ( W ) ® u2 + ux ® V2u2. The dual bundle E{ has a natural connection V* defined by the identity X(v, u) = (V*xv, u) + (v, V^w),

VM € C^iEx),

v 6 C0°(£1*)>

x

£

Ve

ct (M)

where ( • , • ) : C°°{El) x C°°{E{) ->•

C^iM)

is the pairing induced by the natural duality between the fibers of E{ and E\. In particular, any connection V B on a vector bundle E induces a connection yEnd(E) o n End (E) ^ E* <8> E by (VEnd{E)T){u)

= VE(Tu) - T(V B u) = [ V £ , 7 >

(3.3.1)

78

Calculus on manifolds

VTeEnd(£) ueC°°(£). It is often useful to have a local description of a covariant derivative. This can be obtained using Cartan's moving frame method. Let E —> M be a rank r K-vector bundle over the smooth manifold M. Pick a coordinate neighborhood U such E \u is trivial. A moving frame is a a bundle isomorphism : U x Kr —> E \v. (A moving frame is what physicists call a choice of local gauge). Consider the sections ea = (j>(5a), a = 1,..., r, where 8a are the natural basic sections of U x K r . As x moves in U, the collection (ei(x),..., er(x)) describes a basis of the moving fiber Ex and thus the terminology moving frame. A section u € C°°(E\u) can be written as a linear combination u = uaea

uaeC°°({/,K).

Hence if V is a covariant derivative in E we have VM = dua ®ea +

uaVea.

Thus, the covariant derivative is completely described by its action on a moving frame. To get a more concrete description pick local coordinates (x ! ) over U. Vea € Ql(E \v) so that we can write Vea = Tldx1 ® ep, I t e C°°{U, K). Thus, for any section uaea of E\u we have Vu = dua ®ea + r?auadx{ ® ep.

(3.3.2)

It is convenient to view f rftt 1 as an r x r-matrix valued 1-form and we write this as

(rftt)=dxi®ri. The form T = dx' ® T; is called the connection 1-form associated to the choice of local gauge. A moving frame allows one to identify sections of E \u with K r -valued functions on U and we can rewrite (3.3.2) as Vu = du + Tu.

(3.3.3)

A natural question arises: how does the connection 1-form changes with the change of the local gauge? Let f = (fa) be another moving frame of E \u- The correspondence ea i-> fa defines an automorphism of E \u- Using the local frame e we can identify it with a smooth map g : U -» GL(r;K). g is called the local gauge transformation relating e tof. Let f denote the connection 1-form corresponding to the new moving frame i.e.

v/0 = r%-

79

Connections on vector bundles

Consider a a section of E\u. With respect to the local frame (e a ) it has the decomposition a = uaea while with respect to (fp) it has a decomposition

a = vPfp. The two decompositions are related by u = gu.

(3.3.4)

Now, we can identify the ^-valued 1-form Va with a K r -valued 1-form in two ways: either using the frame e or using the frame f. In the first case Va is identified with the K r -valued 1-form du + Tu while in the second case it is identified with du + Tu. These two identifications are related by the same rule as in (3.3.4): du + Tu = g(du + Tu). Using (3.3.4) in the above equality we get (dg)u + gdu + Tgu = gdu + gTu. Hence f = g-ldg + g-1Tg. The above relation is the transition rule relating two local gauge descriptions of the same connection. The above argument can be reversed producing the following global result. Proposition 3.3.5 Let E —> M be a rank r smooth vector bundle and (Ua) a trivializing cover with transition maps gap : UanUp —> GL(r; K). Then any collection of matrix valued 1-forms Ta 6 fi1(End,Ky ) satisfying r

/? = [9Zld9an) + 9ZlTagafj

= -(dg0a)g^

uniquely defines a covariant derivative on E. Exercise 3.3.1 Prove the above proposition.

+ gpaTag^

over Ua D Ug

Calculus on manifolds

80

A word of warning. The identification {moving frames}

= {local

trivialization}

should be treated carefully. These are like an object and its image in a mirror and there is a great chance of confusing the right hand with the left hand. More concretely, if ta : Ea = Ua x K r (resp. tp : Ep = Up x K r ) is a trivialization of a bundle E over an open set Ua (resp. Up) then the transition map over Ua n Up "from a to @" is 5/3a = tp °t~l. The standard basis in W, denoted by (<5;) induces two local moving frames on E: e a = t^iS.) and ep = tp1^.). On the overlap UaC\Up these two frames are related by the local gauge transformation (tpep)

=g^(taea).

This is precisely the opposite way the two trivializations are identified. Example 3.3.6 (Complex line bundles) Let L —> M be a complex line bundle over the smooth manifold M. Let {Ua} be a trivializing cover with transition maps zap : UaC\Up -> C* = GL{\, C). The bundle of endomorphisms of L, End (L) = L*®L is trivial since it can be defined by transition maps (zap)~l ®zap = 1. Thus the space of connections on L, 21 (L) is an affine space modeled by the linear space of complex valued 1-forms. A connection on L is simply a collection of C-valued 1-forms u>a on Ua related on overlaps by ^

= ^ - + oja =

d\nzap+iJa.

Za/3

§3.3.2

Parallel transport

As we have already pointed out, one reason we could not construct natural derivations on the space of sections of a vector bundle was the lack of a canonical procedure of identifying fibers at different points. We will see in this subsection that such a procedure is all we need to define covariant derivatives. More precisely, we will show that once a covariant derivative is chosen, it offers a simple way of identifying different fibers. Let E —> M be a rank r K-vector bundle and V a covariant derivative on E. For any smooth path 7 : [0,1] —> M we will define a linear isomorphism T 7 : i?7(o) —> £7(1) called the parallel transport along 7. More exactly, we will construct an entire family of linear isomorphisms Tt : .Eyfo) —>•

Ey(t).

81

Connections on vector bundles

One should think of this Tt as identifying different fibers. In particular, if u0 G £7(o) then 11-» ut = T(«o € £7(() should be thought of as a "constant" path. The rigorous way of stating this "constancy" is via derivations: a quantity is "constant" if its derivatives are identically 0. Now, the only way we know how to derivate sections is via V i.e. ut should satisfy V d u, = 0,

where — = 7.

The above equation offers a way of defining Tt. For each u0 G S7(o) define Ttuo as the value at t of the initial value problem

an

d t 6 [0,1]

(3.3.5)

dt

u(0)

=

u0

The equation (3.3.5) is a linear ordinary differential equation in disguise. To see this let us make the simplifying assumption that j(t) lies entirely in some coordinate neighborhood U with coordinates {xl, ...,xn), such that E\u is trivial. This is always happening at least on every small portion of 7. Denote by (e Q )i< Q < r a local moving frame trivializing E \u so that u = uaea. The connection 1-form corresponding to this moving frame will be denoted by F G O x (End (K r )). Equation (3.3.5) becomes rf"° it a u {0)

4. rQ ifi + ^u"

— nu = «J

where Tt = i±T e fi°(End (K r )) = End (K r ). This is obviously a linear ordinary differential equation whose solutions exist for any t. We deduce «(0) = - r t u 0 . (3.3.6) This gives a geometric interpretation of the connection 1-form T: for any vector field X the contraction —ix? = —r(X) G End (E) describes the infinitesimal parallel transport along the direction prescribed by the vector field X, in the non-canonical identification of nearby fibers via a local moving frame. In more intuitive terms, if 7(2) is an integral curve for X and Tt denotes the parallel transport along 7 from E7(0) to E^t) then, given a local moving frame for E in a neighborhood of 7(0), Tt is identified with a t-dependent matrix which has a Taylor expansion of the form Tt = id - I V + 0{t2), withT 0 = (i*r)| 7 ( 0 ).

t small

(3.3.7)

82 §3.3.3

Calculus on manifolds The curvature of a connection

Consider E —> M a rank r smooth vector bundle over the smooth manifold M and let V be a covariant derivative on E: V

:Q0(E)^Q}(E).

Proposition 3.3.7 V has an extension to an operator dv

:Qr(E)-^nr+\E)

uniquely defined by the requirements: (a) d v | n 0 ( B ) = V;

(b)Vw e n r (M), T] e ns(E) dv{w A r}) = duj A 77 + ( - l ) r w A dvr?. Brief outline of the proof

Existence For w G fT(M), u G fi°(£} set

dv(w(8)M) = d o ; ® u + ( - l ) r w V u .

(3.3.8)

Using a partition of unity one shows that any 77 G fir (£) is a locally finite combination of monomials as above so the above definition induces an operator Qr(E) —> QT+1(E). We let the reader check that this extension satisfies conditions (a) and (b) above. Uniqueness Any operator with the properties (a) and (b) acts on monomials as in (3.3.8) so it has to coincide with the operator described above using a given partition of unity. • Example 3.3.8 The trivial bundle KM has a natural connection V°- the trivial connection. This coincides with the usual differential d : Q°(M) ® K -> Q1(M) ®K. d v ° is the usual exterior derivative. There is a major difference between the usual exterior derivative d and an arbitrary d?. In the former case we have d2 = 0 which is a consequence of the commutativity III"' af~] = 0' where (xl) are local coordinates on M. In the second case, the equality (d v ) 2 = 0 does not hold in general. Still, something very interesting happens. Lemma 3.3.9 For any smooth function / G C°°(M) and any w G QT(E) we have

(d^(H Hence {dvf

= /{(dv)2M-

is a bundle morphism A T * M ® E ^ Ar+2T*M E.

Connections on vector bundles Proof

83

We compute (dv)2(/u) = dv(rf/Au + / d v u ) = -df

A (IVLO + dfA dvu

+ f{dv)2u

= /(dv) V

a

As a map Q°(E) —» £12{E), (rf v ) 2 can be identified with a section of Horn (E, k2T*M ® E) =* E* ® A2T"M 8 £ = A 2 T*M ® End (E). Thus ( d v ) 2 is an End (£)-valued 2-form. Definition 3.3.10 For any connection V on a smooth vector bundle E —>• M the object ( d v ) 2 G fl2(End (E)) is called the c u r v a t u r e of V and is usually denoted by F(V). Example 3.3.11 Consider the trivial bundle KTM- The sections of this bundle are smooth IfC-valued functions on M. The exterior derivative d defines the trivial connection on K ^ and any other connection differs from d by a M r (K)-valued 1-form on M. If A is such a form then the curvature of the connection d + A denoted by F(A) is F{A)u ={d + Afu = (dA + A A A)u, Vu G C^iM, Kr). The A operation above is defined for any vector bundle E as the bilinear map fir(End (E)) x O s (End (E)) -> ftr+s(End {E)) uniquely defined by (wr ® A) A (r/s <8> B) = ur A r)s ® AB,

A,B € End (E).

We conclude this subsection with an alternate description of the curvature which hopefully will shed some light on its analytical significance. Let E —> M be a smooth vector bundle on M and V a connection on it. Denote its curvature by F = F ( V ) G fi2(Bnd (E)). For any X, Y G Vect (M) the quantity F(X, Y) is an endomorphism of E. In the remaining part of this section we will give a different description of this endomorphism. For any vector field Z denote by iz : Q,r{E) -> CT^iE) the C°°{M) - linear operator defined by iz(w ® u) = (izw) ® u, V u e O r (M), « G Q 0 ^ ) . The covariant derivative Vz extends naturally to elements of Q.r(E) by V z (oi ® w) = (L^oi) ® u + w ® Vz«.

84

Calculus on manifolds

The operators rfv, iz, Vz satisfy the usual super-commutation identities. izdv + dviz

= Vz.

(3.3.9)

ixiY+iYix=Q.

(3.3.10)

VxiY-ir'Vx=i[x,Y]-

(3.3.11)

For any u G Q°(E) we compute using (3.3.9)-(3.3.11) F(X, Y)u = iYix{dv)2u = «y(Vx - dvix)Vu

=

= (iYVx)Vu

= (V x «y - i[x,K])Vu -

iY{ixdv)Vu -

{iYdv)Vxu

VyVxu

= ( V X V y - V y V X - V[X,y])M. Hence F(X, Y) = [ V x , Vy] - V [x ,y].

(3.3.12)

If in the above formula we take X = -~ and y = g|- where (a;1) are local coordinates on M we deduce (V, = V_g_, Vj = V_a_)

^

= -Ffi = F(~,

A ) = [ Vi , V,].

(3.3.13)

Thus Fij measures the extent to which the partial derivatives Vj, V,- fail to commute. This is in sharp contrast with the classical calculus and an analytically oriented reader may object to this by saying we were careless when we picked the connection. Maybe an intelligent choice will restore the classical commutativity of partial derivatives so we should concentrate from the very beginning to covariant derivatives V such that F(V) = 0. Definition 3.3.12 A connection V such that F(V) = 0 is called flat. A natural question arises: given an arbitrary vector bundle £ - > M d o there exist flat connections on El If E is trivial then the answer is obviously positive. In general the answer is negative and this has to do with the global s t r u c t u r e of the bundle. In the second half of this book we will understand the motivation behind this fact.

85

Connections on vector bundles §3.3.4

Holonomy

The reader may ask a very legitimate question: why have we chosen to name curvature, the deviation from commutativity of a given connection. In this subsection we describe the geometric meaning of curvature and maybe this will explain the terminology. Let E —>• M be a smooth vector bundle and V a connection on it. Consider (x 1 , ...,xn) local coordinates on an open subset U C M such that E\u is trivial. Pick (ei,..., e r ) (r = rankiJ) a moving frame over U. The connection 1-form associated to this moving frame is T = r W = {Tfp)dx\

l
It is defined by the equalities (V; = V a ) Vie/, = I > a .

(3.3.14)

Using (3.3.13) we compute Fijep = (ViV,- - V j V i )e / 3 =

V1(rje/j)-Vj{rieli)

- (?£k _ ^Ji.\ P + (r 7 r Q - r 7 r a ^ P '[dx* dx> ) e a + {lil>L* l i " i ^ C o

Though the above equation looks very complicated it will be the clue to understanding the geometric significance of curvature. Assume for simplicity the point of coordinates (0, ...,0) lies in U. Denote by T' the parallel transport (using the connection V) from (a;1, ...,xn) to (x1 + s,x2, ...,xn) along the curve r 1-4 (X 1 + T , X 2 , . . . , xn). Define T\ in a similar way using the coordinate x2 instead of x1. Look at the parallelogram PSit in the "plane" (x J ,x 2 ) described in Figure 3.1. We now perform the parallel transport along the boundary of P Si( with the counterclockwise orientation. The outcome is a linear map T s>t : E0 —> E0, where E0 is the fiber of E over Po = (0,...,0). Set F12 = F ^ ^ ) |(o o)- Fl2 is an endomorphism of Ea. Proposition 3.3.13 for any u € EQ we have

FuU =

-jk^

86

Calculus on manifolds

s,t

Figure 3.1: Parallel transport along a coordinate parallelogram. We see that the parallel transport of an element u G E0 along a closed path may not return it to itself. The curvature is an infinitesimal measure of this deviation. P r o o f The parallel transport along dPSjt can be described as

Label the vertices of Pst in counterclockwise order starting at p0, by pi, p2, ps- Fix M0 £ E0. The parallel transport T' : E0 —> EP1 can be approximated using (3.3.6) «i = Ui(s, t) = T°UQ = M0 — sr!(po)Mo + C\S2 + 0(s3

(3.3.16)

Ci is a constant vector in E0 whose exact form is not relevant to our computations. In the sequel the letter C (eventually indexed) will denote constants. u2 = u2{s, t) = Tffiu

= Tl2ux = «! - tT2(Pl)ui

+ C2t2 + 0{t3)

= MO - sr!(p0)Mo - tr 2 (pi)(« 0 - s I M p o H ) + Cis2 + C2t2 + 0 ( 3 )

= {i - srx(po) - tr 2 ( P l ) + <sr2(p1)r1(p0)}M0 + clS2 + c2t2 + 0(3). 0(k) denotes an error that can be estimated from above by C(s2 + i 2 )*/ 2 as s, t —> 0. Now use dV2 r 2 (Pi) = r 2 (p 0 ) + s ^ ( p 0 ) + o ( 2 ) to deduce M2

= < 1 - sTi - tT2

st

dT2 dx1

r2rr) 11

+ ( V + C2i2 + 0 ( 3 ) .

M0

(3.3.17)

87

Connections on vector bundles Similarly we have u3 = u3(s,t) = Tr s T 2 *7> 0 = T{su2 = u2 + S r 1 (p 2 )w 2 + C3s2 + 0(3). The T-term in the right-hand-side can be approximated as Tfa)

= T^po) + s g i ( P o ) + t | j ( P o ) + 0(2).

Using « 2 described as in (3.3.17) we get after an elementary computation

«3 = u3(s, t) = {i - tr 2 + st (|^i - ~ + r 2 r x - r^r, J J \po u0 +C 4 s 2 + C 5 t 2 + 0(3).

(3.3.18)

Finally we have w4 = u4(s, t) = T2"' = u3 + tT2(p3)u3 + C6t2 + 0(3) with

r2(p3) = r 2 ( Po ) + tg^fa)

+ c7t2 + 0(3).

Using (3.3.18) we get

w4(s, t) = w0 + st ( ^ - j - ^-j- + r 2 rx - r ^ j |po M0 +C 8 s 2 + C9t2 + 0(3) = « 0 - stFu{p0)u0 + Css2 + C9t2 + 0 ( 3 ) . Clearly | J J = -F i2 (po)wo as claimed. • Remark 3.3.14 If we had kept track of the various constants in the above computation we would have arrived at the conclusion that C& = C9 = 0 i.e. T,, t = i d - s t F 1 2 + 0 ( 3 ) . Alternatively, the constant C$ is the second order correction in the Taylor expansion of s i-»- TSio = id so it has to be 0. The same goes for C g . Thus we have rfareaFs,t

ds

The result in the above proposition is usually formulated in terms of holonomy. Definition 3.3.15 Let E —> M be a vector bundle with a connection V. The holonomy of V along a closed path 7 is the parallel transport along 7. We see that the curvature measures the holonomy along infinitesimal parallelograms. A connection can be viewed as an analytic way of trivializing a bundle. We can do so along paths starting at a fixed point, using the parallel transport, but using different paths ending at the same point we may wind up with trivializations which differ by a twist. The curvature provides an infinitesimal measure of that twist.

88

§3.3.5

Calculus on manifolds Bianchi identities

Consider E —¥ M a smooth vector bundle with a connection V : Q°(E) —> Q1(E). We have seen that the associated exterior derivative dv : Q,P(E) —¥ £lp+1(E) does not satisfy the usual {dv)2 = 0 and the curvature is to blame for this. The Bianchi identity describes one remarkable algebraic feature of the curvature. Recall that V induces a connection in any tensor bundle constructed from E. In particular it induces a connection in E* ® E = End (E) which we continue to denote by V. This extends to an "exterior derivative" D : fip(End (E)) -> H p+1 (End {E)). Proposition 3.3.16 (Bianchi identity) Let E —> M and V as above. Then £>F(V) = 0. Roughly speaking, the Bianchi identity states that (dv)3 is almost 0. Proof have

We will use the identities (3.3.9) -(3.3.11). For any vector fields X, Y, Z we ixD = Vx

-Dix-

Hence (DF)(X, Y, Z) = iziYixDF

= iziY{Wx

= «z(Vjf«V - i[x,Y))F - iz(Vy = (Vxiziy - i[x,z]iy - izi[x,Y})F - {Vyizix

-

Dix)F

DiY)ixF - i[Y,z\ix -

^ziyix)F

= {i[x,Y]iz + i[Y,z\ix + i[z,x]iY)F -(Vxiyiz

+ ^yizix

+

^zixiy)F.

We compute immediately i[x,Y]izF = F{Z, [X,Y\) = [ v z , V ( x , y ] ] -

V[z,[x,n-

Also for any u e £l°(E) we have (VxirizF)u

= VX(F(Z, Y)u) - F(Z, Y)Vxu = [VX,F(Z, = [VX, V[y,zl] U - [V X , [Vy, V Z ]] U.

Y)] u

The Bianchi identity now follows from the classical Jacobi identity for commutators. D Example 3.3.17 Let K be the trivial line bundle over a smooth manifold M. As we have seen any connection on K has the form V w = d + u>

where d is the trivial connection and a; is a K-valued 1-form on M. The curvature of this connection is F(u) = du. The Bianchi identity is in this case precisely the equality d2ui = 0.

89

Connections on vector bundles §3.3.6

Connections on tangent bundles

The tangent bundles are very special cases of vector bundles so the general theory of connections and parallel transport is applicable in this situation as well. However, the tangent bundles have some peculiar features which enrich the structure of a connection. Recall that when looking for a local description for a connection on a vector bundle we have to first choose local coordinates and a moving frame and this is a very arbitrary decision. For tangent bundles it happens that once local coordinates (xl) are chosen these automatically define a moving frame of the tangent bundle ( ^ j and it is thus very natural to work with this frame. Hence, let V be a connection on TM. With the above notations we set V

^ =r ^ T

<*< = * * >

The coefficients T^ are usually known as the Christoffel symbols of the connection. As usual we construct the curvature tensor F(X, Y) = [V x , Vy] - V{xy]

€ End {TM).

Still, this is not the only tensor naturally associated to V. Lemma 3.3.18 For X,Y

e Vect (M) consider

T{X, Y) = VXY Then V/ e

- VYX

- [X, Y] G Vect (M).

CX{M)

T(fX,Y)=T(XJY)

=

fT(X,Y).

so that T(-, •) is a tensor T e D.2(TM). T is called the torsion of the connection V. The proof of this lemma is left to the reader as an exercise. In terms of Christoffel symbols the torsion has the description K

dxi'dxj'

{ y

n,

dxk

Definition 3.3.19 A connection on TM is said to be symmetric if T = 0. We guess by now the reader is wondering how the mathematicians came up with this object called torsion. In the remaining of this subsection we will try to sketch the geometrical meaning of torsion. To seek such an interpretation we have to look at the finer structure of the tangent space at a point x € M. It will be convenient to regard TXM as an affine space modeled by M", n = dimM. Thus, we will no longer think of the elements of TXM as vectors but instead we will treat them as points. TXM can be coordinatized using

90

Calculus on manifolds

affine frames. These are pairs (p;e) where p is a point in TXM and e is a basis of the underlying vector space. A frame allows one to identify TXM with M" where p is thought of as the origin. If A, B are two affine spaces, both modeled by R™, and (p;e) , (p;f) are affine frames of A and respectively B. Denote by (xl) the coordinates in A induced by the frame (p; e) and by (yi) the coordinates in B induced by the frame (g;f). An affine map T : A-¥ B can then be described using these coordinates as T : KJ -> W^ x>->y = Sx + v where v is a vector in R" and S is an invertible nxn real matrix. Thus an affine map is described by a "rotation" S followed by a translation v. This vector measures the "drift" of the origin. If now (x1) are local coordinates on M then they define an affine frame 2t x at each x € M: (Qlx = (0; (gfr))- Given a connection V on TM and 7 : / —> M a smooth path we will construct a family of affine isomorphisms Tt : T7(0) —t T7(t) called the affine transport of V along 7. In fact we will determine Tt by imposing the initial condition T0 = id and then describing Tt. This is equivalent to describing the infinitesimal affine transport at a given point x0 € M along a direction given by a vector X = X1-^- £ TxaM. The affine frame of

TX0M is mxo = (0; ( £ ) ) . If xt is a point along the integral curve of X, close to x0 then its coordinates satisfy x\ = xi0 + tXt + O{t2). This shows the origin xQ of 2l l 0 "drifts" by tX + 0(t2). The frame (^-) suffers a parallel transport measured as usual by id — ii^T + 0(t2). The total affine transport will be

Tt = id + t(-ixr

X)+0(t2).

+

The holonomy of V along a closed path will be an affine transformation and as such it has two components: a "rotation"" and a translation. As in Proposition 3.3.13 one can show the torsion measures the translation component of the holonomy along an infinitesimal parallelogram. Since we will not need this fact we will not include a proof of it. Exercise 3.3.2 Consider the vector valued 1-form w £ £ll{TM)

u){X) = 1

V I 6 Vect{M).

Show that if V is a linear connection on TM then dVLJ =

where T v denotes the torsion of V.

T

V

defined by

Integration on manifolds

91

Exercise 3.3.3 Consider a smooth vector bundle E -» M over the smooth manifold M. We assume that both E and TM are equipped with connections and moreover the connection on TM is torsionless. Denote by V the induced connection on A 2 T*M End (E). Prove that \/X, Y,Z e Vect (M)

V X j F(y, Z) + V y ^ ( Z , X) + VZF{X, Y) = 0.

3.4

Integration on manifolds

§3.4.1 Integration of 1-densities We spent a lot of time learning to derivate geometrical objects but. as in classical analysis, the story is only half complete without the reverse operation, integration. Classically, integration requires a background measure and in this subsection we will describe the differential geometric analogue of a measure, namely the notion of 1-density on a manifold. Let E —¥ M be a rank k, smooth real vector bundle over a manifold M defined by an open cover (Ua) and transition maps gap : Uap —> GL{k, E) satisfying the cocycle condition. For any r e R we can form the real line bundle |A| r (£) defined by the same open cover and transition maps ta/} = | detga/}\~r. The fiber at p E M of this bundle consists of r-densities on Ep (see Section 2.2.4). Definition 3.4.1 Let M be a smooth manifold. The bundle of r-densities on M is |A| r (TM).

|A|k ^

When r = l w e will use the notation |A|M = |A|^. We call |A| M the density bundle of M. Denote by C ° ° ( | A | M ) the space of smooth sections of |A| M and by C£°(|A| M ) its subspace consisting of compactly supported densities. It helps to have local descriptions of densities. To this aim, pick an open cover of M consisting of coordinate neighborhoods, (Ua). Denote the local coordinates on Ua by {xla)- This choice of a cover produces a trivializing cover of TM with transition maps

\

^ / l
where n is the dimension of M. Set 5ap = |det# Q/3 |. A 1-density on M is then a collection of functions /iQ 6 C°°((7Q) related by ^ a — bap^p-

92

Calculus on manifolds

It may help to think that for each point p e Ua the basis ^-,..., g^- of TPM spans an infinitesimal parallelepiped and na(p) is its "volume". A change in coordinates should be thought as a change in the measuring units. The gluing rules describe how the numerical value of the volume changes from one choice of units to another. The densities on a manifold resemble in many respects the differential forms of maximal degree. A density can be viewed as a map fj, : C°°(det(TM)) ->

C^iM)

such that jJ.{foS) = \f\fi{uj) for all smooth functions / and all ui € C°°(detTM). In particular, any smooth map <j> : M —>• N between manifolds of the same dimension induces a pullback transformation 4>* : C°°(\A\N) -> C°°(|A| M ) defined by {4>*n){u)) = det(Z>0)/*(w). Example 3.4.2 Consider the special case M = R". Denote by ei,..., en the canonical basis. This extends to a trivialization of TK" and in particular the bundle of densities comes with a natural trivialization. It has a nowhere vanishing section \dvn\ defined by |du„|(ei A ... Ae„) = 1. In this case any smooth density on K" takes the form \i = f\dvn\ where / is some smooth function on Mn. The reader should think of |dti n | as the standard Lebesgue measure on M". If <j> : Kn —)• ffin is a smooth map, viewed as a collection of n smooth functions 0i = n(x\...,xn) then

«<§>

The importance of densities comes from the fact that they are precisely the objects that can be integrated. More precisely, we have the following abstract result. Proposition 3.4.3 There exists a natural way to associate to each smooth manifold M a linear map / : C0°°(|A|M) -»• K JM

uniquely defined by the following conditions: (a) JM is invariant under diffeomorphisms i.e. for any smooth manifolds M, N of the same dimension n, any diffeomorphism (f> : M —> N and for every fjt € CQ°(|A|A/) have / 0 V = / M; JM

JN

we

93

Integration on manifolds (b) / M is a local operation i.e. for any open set U C M and any p € C^{\K\M) supp p c U we have

with

/ M= / Mn

(c) For any p e C£°(R ) we have /

plduJ = /

p(x)dx

where in the right-hand-side stands the Lebesgue integral of the compactly supported function p. JM is called the integral on M. Proof To establish the existence of an integral we associate to each manifold M a collection of data as follows. (i) A smooth partition of unity A C C%°(M) such that Va € A the support supp a lies entirely in some precompact coordinate neighborhood Ua and such that the cover (Ua) is locally finite. (ii) For each Ua we pick a collection of local coordinates (xla) and we denote by \dx'a\ (n = dim M) the density on Ua defined by

\d<\ (A A - A -£-} =1For any p € C°°(|A|) a\i is a density supported in Ua and can be written as a p = fia\dxla\ where p Q is some smooth function compactly supported on Ua. The local coordinates allow us to interpret p Q as a function on M™. Under this identification \dx'a\ corresponds to the Lebesgue measure \dvn\ on R" and p a is a compactly supported, smooth function. We set Finally define

/ M =JM/ M *-^ Y,Jv I a^-

JM

A

a

The above sum contains only finitely many nonzero terms since suppp is compact and thus it intersects only finitely many of the U'as which form a locally finite cover. To prove property (a) we will first prove that the integral defined as above is independent of the various choices, the partition of unity A C C™(M) and the local coordinates (x'a)a£A. Independence of coordinates Fix the partition of unity A and consider a new collection of local coordinates (y'a) on each Ua. These determine two densities \dx'a\ and respectively \dyJa\. For each p e C£°(|A| M ) we have ap, = ap,l\dxia\ = ap?a\dyja\

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Calculus on manifolds

where nl, nya e C™(Ua) are related by ny = det ,

' dx /V

dyi The equality

is the classical change in variables formula for the Lebesgue integral. (b) Independence of the partition of unity. Let A,B C C$°(M) two partitions of unity on M. We will show that rA

rB

JM

JM

Form the partition of unity A * B = {ap ; {a,(3)eAxB}C

C0°°(Af).

Note that suppa/? C Uap = Ua{MJfj. We will prove rA JM

rA*B

rB

JM

JM

Let n € C Q ° ( | A | M ) . We can view afi as a compactly supported function on Rn. We have / ajj, = y f pan = T [ afifi. (3.4.1) Similarly J

Uf,

a

Juaff

Summing (3.4.1) over a and (3.4.2) over /? we get the desired conclusion. To prove property (a) for a diffeomorphism : M —> N consider a partition of unity A C C™(N). From the classical change in variables formula we deduce that for any coordinate neighborhood Ua containing the support of a e A and any H € C^(\A\N) we have / , a(cj))*n = / U'l(ua) Jva

a/i.

(a())aeA forms a partition of unity on M. Property (a) now follows by summing over a the above equality and using the independence of the integral on partitions of unity. To prove property (b) on the local character of the integral pick U C M and then choose a partition of unity B C CQ°(U) subordinated to the open cover (V/})/)eB. For any partition of unity A C C£°(M) with associated cover (Va)ae^ we can form a new partition of unity A * B of U with associated cover Vag — Va n Vg. We use

Integration on manifolds

95

Figure 3.2: The Mobius strip. this partition of unity to compute integrals over U. For any distribution ( i o n W supported on U we have

Property (c) is clear since for M = Kn we can assume that all the local coordinates chosen are cartesian. The uniqueness of the integral is immediate and we leave the reader fill in the details. • §3.4.2

Orientability and integration of differential forms

Operating with densities on a smooth manifold is not always a very pleasant thing to do. However, under some mild restrictions on the manifold, the calculus with densities can be reduced to the friendlier calculus with differential forms. The mild restrictions referred to above have a global nature. We have to restrict our attention to oriented manifolds. Roughly speaking these are "2-sided manifolds" i.e. one can distinguish between an "inside face" and an "outside face" of the manifold. ( Think of a 2-sphere in M3 (a soccer ball) which is naturally a "2-faced" surface.) The 2-sidedness feature is such a frequent occurrence in the real world that for many years it was taken for granted. This explains the "big surprise" produced by the famous counter-example due to Mobius in the first half of the 19th century. He produced a 1-sided surface nowadays known as the Mobius strip using paper and glue. More precisely, he glued the opposite sides of a paper rectangle attaching arrow to arrow as in Figure 3.2. The 2-sidedness can be formulated rigorously as follows. Definition 3.4.4 A smooth manifold M is said to be orientable if the determinant line bundle det TM (or equivalently det T*M) is trivial.

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Calculus on manifolds

We see that det T*M is trivial iff it admits a nowhere vanishing section. Such a section is called a volume form on M. We say that two volume forms wx and ui2 are equivalent if there exists / 6 C°°(M) such that u)2 =

e'uj\.

This is indeed an equivalence relation and an equivalence class of volume forms will be called an orientation of the manifold. An orientable connected manifold can have two orientations. A pair [orientable manifold, orientation) is called an oriented manifold. A natural question arises: how can one decide whether a given manifold is orientable or not. We see this is just a special instance of the more general question we addressed in Chapter 2: how can one decide whether a given vector bundle is trivial or not. The orientability question can be given a very satisfactory answer using topological techniques. However, it is often convenient to decide the orientability issue using ad-hoc arguments. In the remaining part of this section we will describe several simple ways to detect orientability. E x a m p l e 3.4.5 If the tangent bundle of a manifold M is trivial then clearly TM is orientable. In particular, all Lie groups are orientable. Example 3.4.6 Suppose M is a manifold such that the Whitney sum RkM © TM is trivial. Then M is orientable. Indeed, we have detQR* © TM) = detKfc det TM. Both detKfc and detQR* © TM) are trivial. We deduce det TM is trivial since det TM ^ d e t ( l * © TM) (det!*)*. This trick works for example when M = Sn. Indeed, let v denote the normal line bundle. The fiber of v at a point p e Sn is the 1-dimensional space spanned by the position vector of p as a point in W1; (see Figure 3.3). This is clearly a trivial line bundle since it has a tautological nowhere vanishing section p i-> p G vp. v has a remarkable feature: v®TSn = l n + 1 . Hence all spheres are orientable. Important convention The canonical orientation on Kn is the orientation defined by the volume form dx1 A • • • A dxn where x1,..., xn are the canonical cartesian coordinates. The unit sphere Sn C M"+1 is orientable. In the sequel we will exclusively deal with its canonical orientation. To describe this orientation it suffices to describe

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Integration on manifolds

Figure 3.3: The normal line bundle to the round sphere. a positively oriented basis of det TPM for some p e Sn. To this aim we will use the relation K n + 1 ^ vv © TPM. An element ui € det TpM defines the canonical orientation if pAuj € detK n + 1 defines the canonical orientation of R" +1 . Above, by p we denoted the position vector of p as a point inside the euclidian space K n+1 . We can think of p a s the "outer" normal to the round sphere. We call this orientation outer normal first. When n = 1 it coincides with the counterclockwise orientation of the unit circle S 1 .

Lemma 3.4.7 A smooth manifold M is orientable if and only if there exists an open cover (Ua)aeA and local coordinates (:cj,, ...,a:™) on Ua such that det

dx\ —f dx'a

> 0 on Ua n Ua

(3.4.3)

Proof 1. det T*M is trivial iff there exists a volume form. Assume there exists an open cover with the properties in the lemma. Consider B C C™{M) a partition of unity subordinated to the cover {Ua)a<=A i-e. there exits a map ip : B —> A such that

suppleu^p)

ypSB.

Consider 0 where for all a e A we define ua = dxla A • • • A dx1^. The form u is nowhere vanishing since condition (3.4.3) implies that on an overlap (7 0 i n- • -C\Uam the forms uai, ...,wQm differ by a positive multiplicative factor.

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Calculus on manifolds

2. Conversely, let w be a volume form on M and consider an atlas {Ua\ (xla)). Then to\ua= t^adxla A • • • A dxna where the smooth functions fia are nowhere vanishing and on the overlaps they satisfy the gluing condition

*--'«(§)-£ A permutation tp of the variables ^ , . . . , 1 ^ will change dxxa A • • • A dx™ by a factor e(tp) so we can always arrange these variables in such an order so that fia > 0. This will insure the positivity condition Aa0

> 0.

The lemma is proved. • From the above lemma we deduce immediately the following consequence. Proposition 3.4.8 The connected sum of two orientable manifolds is an orientable manifold Exercise 3.4.1 Prove the above result. Exercise 3.4.2 Prove that a complex manifold is orientable. In particular, the complex grassmannians Gfc,n( |A|y. Similarly, an orientation on a smooth manifold M defines an isomorphism iM • C°°(detT*M) -> C°°(|A| M )For any compactly supported differential form ui on M of maximal degree we define its integral by f /

JM

def f LU =

/

lMU).

JM

We want to emphasize that this definition depends on the choice of orientation.

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Integration on manifolds

Figure 3.4: Spherical coordinates. Example 3.4.9 Consider the 2-form on K3 oj = xdy A dz 2

and let S denote the unit sphere. We want to compute

where S2 has the canonical orientation. To compute this integral we will use spherical coordinates (r,
(

x = r sin

At the point p = (1,0,0) we have d dr

d dx

89

d dy

d dip

dz

so that the standard orientation on S2 is given by d(p A d9. On S 2 we have r = 1 and dr = 0 so that xdy A dz \S2= sin ip cos 9 (cos 9 sin ipd9 + sin 9 cos

dip)) A (— sin (p)d(p = sin 3 ip cos2 9dip A d9. Finally, we compute rlt

Js

2

r2lT

sin3 ip cos2 9dip A d9 = / sin3 ipdip • / J [0,7r]x[0,27r] it Jo Jo

cos2 9d9

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Calculus on manifolds A.'K

= — = volume of the unit ball B 3 C I 3 . As we will see in the next subsection the above equality is no accident. Example 3.4.10 (Invariant integration on compact Lie groups) Let G be a compact Lie group. Fix once and for all an orientation on £Q- Consider u> G det £,*G a positively oriented volume element. By left translation we can extend a; to a leftinvariant volume form on G which we continue to denote by u>. This defines an orientation and in particular, by integration, we get a positive scalar c=

I oi. JG

Set dg = \u> so that / dg = 1.

(3.4.4)

JG

dg is the unique left-invariant n-form (n = dimG) on G satisfying (3.4.4) (assuming a fixed orientation on G). We claim dg is also right invariant. Assume for simplicity that G is connected. To prove this consider the modular function G 3 / I H A ( / I ) 6 1 defined by R*h(dg) = A(h)dg. A(h) is a constant because R*hdg is a left invariant form so it has to be a scalar multiple oi dg. Since (R^^)* = {RhiRh,)* = R^R*^ w e deduce A{hxh2) = A(/n)A(/i 2 )

V/H, h2 € G.

Hence h i-> A/i is a smooth morphism G - > ( R \ {()},•)• Since G is connected A(G) C M+ and since G is compact A(G) is bounded. If there exists x G G such that A(z) / 1 then either A(x) > 1 or A(a; _1 ) > 1 and in particular we would deduce the set (A(xn))nez is unbounded. Thus A = 1 which establishes the right invariance of dg. The invariant measure dg provides a very simple way of producing invariant objects on G. More precisely, if T is tensor field on G, then for each x G G define

T[ = [ ({Lg).T)xdg. JG

Then x i-¥ Tx defines a smooth tensor field on G. We claim T is left invariant. Indeed, for any ft G G we have

(L^T1 = J{Lh).{{Lg).T)dg = fG((Lhg)tT)dg

101

Integration on manifolds "=fts f (Lu)*Td{h-lu)

= Te

(d(/i- 1 «)) = Ll-,du = du).

JG

If we average once more on the right we get a tensor jG{{R9)tTl)xdg

x H+

which is both left and right invariant. Exercise 3.4.3 Let G be a Lie group. For any X € £ G denote by ad{X) the linear map £(j —> £ G defined by

£G^Y^[X,Y]eSlG(a) If u> denotes a left invariant volume form prove that VX £ £Q LXUJ =

trad(X)u}.

(b) Prove that if G is a compact Lie group then trad(X) §3.4.3

= 0 for any X € £ Q .

Stokes formula

The Stokes' formula is the higher dimensional version of the fundamental theorem of calculus (Leibniz-Newton formula)

f"df = f(b)-f(a) Jo.

where / : [a, b] —• R is a smooth function and df = f'(i)dt. In fact, the higher dimensional formula will follow from the simplest 1-dimensional situation. We will spend most of the time finding the correct formulation of the general version and this requires the concept of manifold with boundary. The standard example is the upper half-space Hl = {{x1,...,xn) e l " ; ! 1 >0}. Definition 3.4.11 A smooth manifold with boundary is a closed subset M of a smooth manifold M (dim M = n) such that (a) M° = int M + 0. (b) For each point p G dM = M\M° there exist local coordinates (x 1 , ...,xn) defined on an open neighborhood 91 of p in M such that

(6i) M° n m = {{x\..., xn) • xl > o}. 1 (62) dMnm = {x = o}.

dM is called the boundary of M. orientable if M° is orientable.

A manifold with boundary (M, dM) is called

Example 3.4.12 A closed interval / = [a,b] is a smooth 1-dimensional manifold with boundary dl = {a, b}.

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Calculus on manifolds

Example 3.4.13 The closed unit ball B3 C R 3 is an orientable manifold with boundary dB3 = S2. Proposition 3.4.14 Let (M, dM) be a smooth manifold with boundary. Then dM is also a smooth manifold of dimension dimdM = d i m M — 1. Moreover if M is orientable then so is its boundary. The proof is left to the reader as an exercise. Important convention Let (M, dM) be an orientable manifold with boundary. There is a (non-canonical) way to associate to an orientation on M° an orientation on the boundary. This will be the only way in which we will orient boundaries throughout this book. If we do not pay attention to this convention then our results may be off by a sign. We now proceed to described this induced orientation on dM. For any p £ dM choose local coordinates (a;1, ...,xn) as in Definition 3.4.11. Then the induced orientation of TpdM is defined by edx2 A---AdxnedetTpdM,

e = ±1

1

where e is chosen so that for x > 0 (i.e. inside M ) the form ei-dx1)

A dx2 A • • • A dxn

is positively oriented, dx1 is usually called an inner conormal since x1 increases as we go towards the interior of M. —dx1 is then the outer conormal for analogous reasons. The rule by which we get the induced orientation on the boundary can be rephrased as {outer conormal}A{induced orientation on boundary} = {orientation in the interior}. We may call this rule "outer (co)normal first" for obvious reasons. Example 3.4.15 The canonical orientation on Sn C M n+1 coincides with the induced orientation of S" + 1 as the boundary of the unit ball Bn+1. Exercise 3.4.4 Consider the hyperplane Hi C K" defined by the equation {xl = 0}. Prove that the induced orientation of Hi as the boundary of the half-space H"' 1 = {x' > 0} is given by the (n — l)-form (—lj'dx 1 A • • • A dx1 A • • • dxn where as usual the hat indicates a missing term. Theorem 3.4.16 (Stokes formula)Let M be an oriented n-dimensional manifold with boundary dM and u € fln_1(M) a compactly supported form. Then / JM°

du = /

u.

JdM

In the above formula d denotes the exterior derivative and dM has the induced orientation.

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Integration on manifolds

Proof Via partitions of unity the verification is reduced to the following two situations. Case 1. w is a compactly supported (n — 1) form in W1. We have to show / du> = 0. JW It suffices to consider only the special case u) = f(x)dx2

A • • • A dxn

where f(x) is a compactly supported smooth function. The general case is a linear combination of these special situations. We compute 1 / du= f ^/rdx JW JW dxl

A f ^-rdx1)

= f

l

1

Jw~ \m dx

A • • • A dxn

dx 2 A - • • A dx n = 0

)

since L &dxl=/(o°'x2' •••• xn) ~ / ( _ o ° ' x 2 ' •••• xn) and / has compact support. Case 2. u is a compactly supported (n — 1) form on H " . Let = ] T fi^dx1

w

A • • • A dx1 A • • • A dxn.

i

Then ^=(E(-l)

i + 1

^)^1A--Ado;".

One verifies as in Case! 11 that that /

-J-dx1 A • • • A dxn = 0 for i ^ 1.

For i = 1 we have /

l^^

1

JH^ OX1

A • • • A dx" = / f / ° ° l ^ 1 | dx 2 A • • • A dx" yR"-i Vyo a x 1 J

= f ^ (/(oo, a;2, ..., x") - / ( 0 , x 2 ,..., xn)) dx2 A • • • A dx" = ~ /

^ / ( 0 , x 2 , ...,x")dx 2 A • • • A dxn = | a H n w.

The last equality follows from the fact that the induced orientation on 3 H " is given by —dx2 A • • • A dx n . This concludes the proof of the Stokes formula. •

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Calculus on manifolds

Remark 3.4.17 Stokes formula illustrates an interesting global phenomenon. It shows that the integral JM duj is independent of the behavior of ui inside M. It only depends on the behavior of w on the boundary. Example 3.4.18 / xdy/\dz=

dx Ady Adz = vol (B 3 ) — — .

/

Js2

3

JB*

Remark 3.4.19 The above considerations extend easily to more singular situations. For example, when M is the cube [0, l ] n its topological boundary is no longer a smooth manifold. However, its singularities are inessential as far as integration is concerned. The Stokes formula continues to hold

f du= f OJ Vw6 Q:n-l(In).

Ji"

Jdi

The boundary is smooth outside a set of measure zero and is given the induced orientation: " outer (co)normal first". The above equality can be used to give an explanation for the terminology "exterior derivative" we use to call d. Indeed if w 6 n n _ 1 (M n ) and Ih = [0, h] then we deduce du) L = 0 = lim hTn [ h->0

w.

(3.4.5)

Jdl"

When n = 1 this is the usual definition of the derivative. Example 3.4.20 We now have sufficient technical background to describe an example of vector bundle which admits no flat connections thus answering the question raised at the end of Section 3.3.3. Let M be the unit sphere in M3. As in Chapter 1, we have a distinguished cover of M, {[/„, Us} where U„ = S2 \ {south pole} and U, = S2 \ {north pole}. Each of the two sets can be identified with the complex plane C and we get coordinates zn on Un and zs on Us. On the overlap U, D U„ the two sets of coordinates are related by _ 1 Consider the complex line bundle L obtained by gluing two trivial line bundles Ln —> U„ and Ls —> Us via the transition map gsn :UnnUs^

{zn ± 0} -»• C*, gSn{Zn) = zn.

A connection on L is a collection of two complex valued forms ujn G n 1 (f/„) ® C, u>s G O^iUs) ® C satisfying a gluing relation on the overlap (see Example 3.3.6) dz

Vn(z) = — + W,(z), z

{Z = Zn).

Integration on manifolds

105

If the connection is flat then duin = 0 on Un and dujs = 0 on u>s. Let E+ be the equator equipped with the induced orientation as the boundary of the northern hemisphere and E~ the equator with the opposite orientation (as the boundary of the southern hemisphere). The orientation of E+ coincides with the orientation given by the form dQ where zn = exp(i#). We deduce from the Stokes formula (which works for complex valued forms as well) that /

u)n = 0

JE+

/

u>s = -

JE-

us = 0. JE+

On the other hand over the equator we have uin — u>s = — = ido

z

from which we deduce 0= /

JE+

un - ws = [

JE+

id0 = 27ri !!!

Thus there exist no flat connections on the line bundle L and at fault is the gluing cocycle defining L. In a future chapter we will quantify the measure in which the gluing data obstruct the existence of flat connections. §3.4.4

Representations and characters of compact Lie groups

The invariant integration on compact Lie groups is a very powerful tool with many uses. Undoubtedly, one of the most spectacular application is Hermann Weyl's computation of the characters of representations of compact semi-simple Lie groups. The invariant integration occupies a central place in his solution to this problem. We devote this subsection to the description of the most elementary aspects of the representation theory of Lie groups. Let G be a Lie group. Recall that a (linear) representation of G is a left action on a (finite dimensional) vector space V

GxV^V

(g,v)^T(g)veV

such that the map T(g) is linear for any g. One also says V has a structure of Gmodule. If V is a real (resp. complex) vector space then it is said to be a real (resp. complex) G-module. Example 3.4.21 Let V - C n . Then G = GL(n, C) acts linearly on V in the tautological manner. Moreover V , V®k, A.mV and SeV are complex G-modules.

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Calculus on manifolds

Definition 3.4.22 A morphism of G-modules V\ and V2 is a linear map L : V\ —> V2 such that for any g € G the diagram below is commutative i.e. T2{g)L = LT\(g)

The space of morphisms of G-modules is denoted by Hom<3(Vi, V2). The collection of isomorphisms classes of complex G-modules is denoted by G-Mod. If V is a G-module than an invariant subspace (or submodule) is a subspace U C V such that T(g)(U) C U, \/g G G. A G-module is said to be irreducible if it has no invariant subspaces other than {0} and V itself. Proposition 3.4.23 The direct sum "©" and the tensor product "®" define a structure of semi-ring with 1 on G-Mod. 0 is represented by the null representation {0} while 1 is represented by the trivial module G —> Aut (C), g i-+ 1. The proof of this proposition is left to the reader. Example 3.4.24 Let Ti : G -+ Aut ([/,) (i = 1, 2) be two complex G-modules. Then U{ is a G-module by

(s,u*).->i7GrVHence Horn {U\, U2) is also a G-module. Explicitly, the action of g € G is given by (g,L)^T2(g)LTl(g-1). We see that H o r n e d , [72) can be identified with the linear subspace in Horn (Ui, t/ 2 ) consisting of the linear maps U\ —¥ U2 unchanged by the above action of G. Proposition 3.4.25 (Weyl's unitary trick) Let G be a compact Lie group and V a complex G-module. Then there exists a hermitian metric h on V which is Ginvariant i.e. h(gv1,gv2) = h{v1,v2) Vui, v2 G V. Proof

Let h be an arbitrary hermitian metric on V. Define its G-average by h(u, v)=

h(gu, gv)dg JG

where dg denotes the normalized bi-invariant measure on G. One can now check easily that h is G-invariant. • In the sequel, G will always denote a compact Lie group.

Integration on manifolds

107

Proposition 3.4.26 Let V be a complex G-module and h a G-invariant hermitian metric. If U is an invariant subspace of V then so is Ux, where "_l_" denotes the orthogonal complement with respect to h. Proof Since h is G-invariant it follows that Vg e G, T(g) is an unitary operator, T(g)T*(g) = \ v . Hence, T*(g) = T " 1 ^ ) = TO?" 1 ), V
= 0.

Thus T{g)x e C1- so that [71- is G-invariant. D Corollary 3.4.27 Every G-module V can be decomposed as a direct sum of irreducible ones. If we denoted by Irr(G) the collection of isomorphism classes of irreducible G-modules then we can rephrase the above corollary by saying that Irr(G) generates the semigroup (G-Mod, ©). To gain a little more insight we need to use the following remarkable trick due to Isaac Schur. Lemma 3.4.28 (Schur lemma)Let Vi, V2 be two irreducible complex G-modules. Then

dimcHomG(y1,^2) = | j ' £ ~ £ * Proof Let L G HomG(Vi, V2). Then kerL C VY is an invariant subspace of V\. Similarly, Range (L) C V2 is an invariant subspace of V2. Thus, either kerL = 0 or k e r i = Vi. The first situation forces range L 7^ 0 and since V2 is irreducible Range L = V2. Hence L has to be in isomorphism of G-modules. We deduce that if V\ and V2 are not isomorphic as G-modules H o r n e d , V2) = {0}Assume now that Vx = V2 and S : V\ —> V2 is an isomorphism of G-modules. According to the previous discussion, any other nontrivial G-morphism L : V\ —> V2 has to be an isomorphism. Consider the automorphism T = S^L : V\ —>• Vx. Since V\ is a complex vector space T admits at least one (non-zero) eigenvalue A. The map Alv^ — T is an endomorphism of G-modules and ker (XlVl — T) / 0. Invoking again the above discussion we deduce T = A l ^ i.e. L = XS. This shows dimHom G (V 1 ,V 2 ) = 1. • Schur's lemma is powerful enough to completely characterize S ^ M o d .

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Calculus on manifolds

Example 3.4.29 The irreducible (complex)representations of S 1 .) Let V be a complex irreducible 5 1 -module {eie, v) i-»- T9v

where In particular this implies that each Tg is an 5 1 -automorphism since it obviously commutes with the action of this group. Hence Tg = \(8)ly. In particular dim V = 1 since any 1-dimensional subspace of V is S'-invariant. We have thus obtained a smooth map A : S 1 ->• C* such that \{e'9 • eiT) =

\{eie)\{eie).

Hence A : S 1 —» C* is a group morphism. As in the discussion of the modular function we deduce that |A| = 1. Thus A looks like an exponential {verify!) A(e i S )=exp(ia0). Moreover, when exp(27ria) = 1 so that a £ Z . Conversely, for any integer n e Z w e have a representation 5x^Aut(C)

{eie,z)^einez.

The exponentials exp(m#) are called the characters of the representations pn. Exercise 3.4.5 Describe the irreducible representations of T"-the n-dimensional torus. Definition 3.4.30 (a) Let V be a complex G-module, g i-> T(g) e Aut (V). character of V is the smooth function Xv--G->C

Xv{g) =

The

trT(g).

(b) A class function is a continuous function / : G —> C such that f{hgh-l)

= f(g)

Vg,heG.

(The character of a representation is an example of class function). Theorem 3.4.31 Let G be a compact Lie group, U\, U2 complex G-modules and Xut their characters. Then the following hold. ( a ) x ^ e c / 2 = XUi + Xu2, XUi<su2 = XUi • XUi-

(b)xc/ i (l) = dim[/ i . ( c ) Xu" = Xt/;-^ 6 c o m p l e x conjugate of xu{-

Integration on manifolds

109

(d) f Xui{g)dg = AimUf JG

where Up denotes the space of G-invariant elements of Ui U? = {xeUi;x

= Ti(g)x V 5 e G}.

e / XuAd) • Xu2{9)d9 = dim H o r n e d , ^ i ) . Jc P r o o f (a) and (b) are left to the reader. To prove (c) fix an invariant hermitian metric on U = Ui. Thus each T(g) is an unitary operator on U. The action of G on U* is given by T f [g~ l ). Since T(g) is unitary we have Tf(gl) = T(g). This proves (c). P r o o f of ( d ) . Consider P:U

->U

Pu=

f

T(g)udg.

Note that PT(h) = T(h)P, Vft 6 G i.e. P e RomG{U, U). We now compute T(h)Pu

= JGT(hg)udg I T(j)udj

=

jj^uRUd-f

= Pu.

JG

Thus, each Pu is G-invariant. Conversely, if x G U is G-invariant then Px = / T(g)xdg = / xdg = x i.e. UG = Range P . Note also that P is a projector i.e. P2 = P. Indeed P2u = J T(g)Pudg = f Pudg = Pu. Hence P is a projection onto UG and in particular dime UG = t r P = fGtrT{g)dg

=

jGXu{g)dg.

P r o o f of (e). J XUi • Xu2d9 = J Xc/i • Xv;dg = J Xui®u;dg = / Xnom(u2,Ui)

= dime (Horn {U2, Ux)f

= dim c Hom G (i7 2 , U{)

since HoniG coincides with the space of G-invariant morphisms. •

Calculus on manifolds

no Corollary 3.4.32 Let U, V be irreducible G-modules. Then {Xu, Xv) = I Xv Xvd9 = <W = JG

Proof Follows from Theorem 3.4.31 using Schur lemma. D Corollary 3.4.33 Let U, V be two G-modules. Then U = V if and only if xu = XvProof

Decompose U and V as direct sums of irreducible G-modules U = ®?{mlUt)

V = ®[{njVj).

Hence xu = Y,miXVi and xv = Y,njXvr The equivalence "representation" •*=>• "characters" stated by this corollary now follows immediately from Schur lemma and the previous corollary. D Thus, the problem of describing the representations of a compact Lie group boils down to describing the characters of its irreducible representations. This problem was completely solved by Hermann Weyl. Its solution requires a lot more work and goes beyond the scope of this book. We will spend the remaining part of this subsection analyzing the equality (d) in Theorem 3.4.31. Describing the invariants of a group action was a very fashionable problem in the second half of the nineteenth century. Formula (d) mentioned above is a truly remarkable result. It allows (in principle) to compute the maximum number of linearly independent invariant elements. Let V be a complex G-module and denote by xv its character. The complex exterior algebra A*V* is a complex G-module, as the space of complex multi-linear skew-symmetric maps V x • • • x V -> C. Denote by bck(y) the complex dimension of the space of G-invariant elements in A^V*. One has the equality bck(V) = /

XMv-dg.

These facts can be presented coherently by considering the Z-graded vector space

3e(V) = ©*ALV. Its Poincare polynomial is

^W*) = Y,tkK(v) = I/xw-dg. To obtain a more concentrated formulation of the above equality we need to recall some elementary facts of linear algebra.

Integration on manifolds

111

For each endomorphism A of V denote by ak (A) the coefficient of tk in the characteristic polynomial at{A) =det(lv + tA). Explicitly,
°k(.A) =

det a

J2

( >«v)

(n = dimV).

l
Equivalently ak(A) =

tr(AkA)

k

where A M is the endomorphism of A V induced by A. If g e G acts on V by T(g) then g acts on A*V* by A ^ T ' ^ " 1 ) = A*T(ff). (We implicitly assumed that each T(g) is unitary with respect to some G-invariant metric on V). Hence XAJV = ak{f{g)). (3.4.6) We conclude that P3{v)(t)

= ^^(TigVdg

= fGdet(lv

+ tf(g))dg.

(3.4.7)

Consider now the following situation. Let V be a c o m p l e x V-module. Denote by A*V the space of R-multi-linear, skew-symmetric maps V x • • • x V ->• K. A* V* is a real G-module. We complexify it. A*V ® C is the space of M-multi-linear, skew-symmetric maps V x ••• x V - > C and as such it is a complex G-module. The real dimension of the subspace 3k of G invariant elements in AkV* will be denoted by brk(V) so that the Poincare polynomial of TT{V)=®k3kis On the other hand bTk(V) is equal to the complex dimension of AkV* ® C. Using the results of Subsection 2.2.5 we deduce A*TV C s A:V* ® c A^T

= 0

(ei+,-=*AiV ® A ' F ) .

(3.4.8)

k

Each of the above summands is a G-invariant subspace. Using (3.4.6) and (3.4.8) we deduce P

x(v) = E / E it

JG

^(TGrtK^cfo,))fdfl

i+j=k

f d e t ( l v + tT(s)) d e t ( l F + tT( f f ))d 5 = / | d e t ( l v + *T(ff))|2dff.

(3.4.9)

We will have the chance to use this result in computing topological invariants of manifolds with a "high degree of symmetry" like e.g. the grassmannians.

112 §3.4.5

Calculus on manifolds Fibered calculus

In the previous section we have described the calculus associated to objects defined on a single manifold. The aim of this subsection is to discuss what happens when we deal with an entire family of objects parameterized by some smooth manifold. We will discuss only the fibered version of integration. The exterior derivative also has a fibered version but its true meaning can only be grasped by referring to Leray's spectral sequence of a fibration and so we will not deal with it. The interested reader can learn more about this operation from [32]. Assume now that instead of a single manifold F we have an entire (smooth) family of them (Fj,)(,€S. In more rigorous terms this means we are given a smooth fiber bundle p : E —> B with standard fiber F. On the total space E we will always work with split coordinates (xl; yi) where (x1) are local coordinates on the standard fiber F and (yi) are local coordinates on the base B (the parameter space). The model situation is the bundle

E = Rk xRm^Rm

=B

{x,y)&y.

We will first define a fibered version of integration. This requires a fibered version of orientability. Definition 3.4.34 Let p : E -» B be a smooth bundle with standard fiber F. The bundle is said to be orientable if (a) F is oriented; (b) there exists an open cover (Ua) and trivializations

p-\Ua)

*AFxUa

such that the gluing maps ipp o V" 1 : F x Ua0 -> F x Ua0

{Ua/3 = Uan Up)

are orientation preserving in the sense that for each y € Uap, the diffeomorphism

is orientation preserving. Exercise 3.4.6 If the the base B of an orientable bundle p : E —• B is orientable, then so is the total space E (as an abstract smooth manifold). Important convention Let p : E —> B be an orientable bundle with oriented basis B. The natural orientation of the total space E is defined as follows.

Integration on manifolds

113

If E = F x B then the orientation of the tangent space Ty,b)E is given by QF x LJB where up S det T/F (resp. wg € detT(,B) defines the orientation oiT/F (resp. TbB). The general case reduces to this one since any bundle is locally a product and the gluing maps are orientation preserving. This convention can be briefly described as orientation total space = orientation fiber A orientation base. The natural orientation can thus be called the fiber-first orientation. In the sequel all orientable bundles will be given the fiber-first orientation. Let p : E —> B be an orientable fiber bundle with standard fiber F. Proposition 3.4.35 The integration along fibers is an operation P

* = /,,„ : ^ P i (£) -> 0£{B

(r = dimF).

uniquely defined by its action on forms supported on domains D of split coordinates D ^ l

r

If <j = fdx1 A dyJ (/ G C?(W+m)) r JE/B

x E

m

4 l

m

(x; y) M- y.

then

f 0 , \I\^r \ (/ R r fdx1) dyJ , | / | = r

The proof goes exactly as in the non-parametric case (i.e. when B is a point). One shows using partitions of unity that these local definitions can be patched together to produce a well defined map

/ • « ; * ( £ ) ^:P7(B). I E/B

The details are left to the reader. Proposition 3.4.36 Let p : E —> B be an orientable bundle with an r-dimensional standard fiber F. Then for any w € tt*cpt(E) and 77 e w*p((B) such that degw+degr/ = dim E we have / dEto = {-l)rdB I u. JE/B

JE/B

If B is oriented then

Js">*P'W = fB(fE/B<>>)*V-

(F»bini)

114

Calculus on manifolds

The last equality implies immediately p„(u> Ap*rf) = ptLj A T / .

Proof

It suffices to consider only the model case p:£ = l ' x I 1

and ui = fdx

m

4 l

m

= B

{x;y)Ay

J

A dy . Then

dEuJ = Y, J~idxi

A dx

Ad J

V + (- 1 )' 7 ' E J~jdxI

'

Ad

^

A

V-

The above integrals are defined to be zero if the corresponding forms do not have degree r. The Stokes formula shows that the first integral is always zero. Hence

/

JE/B

djw=(-l)l^ dyi

/

£ d x ' ) *dyiAdyJ j

WR' J

= (-iydB

[

JE/B

u.

The second equality is left to the reader as a practice exercise. • Definition 3.4.37 A d-bundle is a collection (E, dE,p, B) consisting of the following. (i) A smooth manifold E with boundary dE. (ii) A smooth map p : E —>• B such that the restrictions p : Int E —> B and p : dE —• B are smooth bundles. The standard fiber of p : IntE —¥ B is called the interior fiber. One can think of a 9-bundle as a smooth family of manifolds with boundary. Example 3.4.38 The projection p : [0,1] x M -> M

(t;m) >-> m

defines a 9-bundle. The interior fiber is the open interval (0,1). The fiber of p : d(I x M) —> M is the disjoint union of two points. Standard Models

A d-bundle is obtained by gluing two types of local models.

Interior models I ' x C - t Km Boundary models H ^ x l " 1 - ) Mm where Kr+ = {{x\---,xr)e&r

; x1 > 0 } .

Integration on manifolds

115

Remark 3.4.39 Let p : (E,dE) - > B b e a d-bundle. If p : Int E -> B is orientable and the basis B is oriented as well, then on dE one can define two orientations, (i) The fiber-first orientation as the total space of an oriented bundle dE —> B. (ii) The induced orientation as the boundary of E. Exercise 3.4.7 Prove that the above orientations on dE coincide. Theorem 3.4.40 Let p : (E, dE) —>• B be an orientable 9-bundle with an r-dimensional interior fiber. Then for any u> 6 Q.*cpt(E) we have / u> = dsu — (—Wd.B I JdE/B JE/B JE/B

u

(Homotopy formula).

The last equality can be formulated as

I JdE/B

= l

dE- (~iydB [

JE/B

.

JE/B

This is "the mother of all homotopy formulae". It will play a crucial part in Chapter 7 when we embark on the study of DeRham cohomology. Exercise 3.4.8 Prove the above theorem.

Chapter 4 Riemannian Geometry Now we can finally put to work the abstract notions discussed in the previous chapters. Loosely speaking the Riemannian geometry studies the properties of surfaces (manifolds) "made of canvas". These are manifolds with an extra structure arising naturally in many instances. In particular we will study the problem formulated in Chapter 1: why a plane (flat) canvas disk cannot be wrapped in an one-to-one fashinon around the unit sphere in R3. Answering this requires the notion of Riemann curvature which will be the central theme of this chapter.

4.1 §4.1.1

Metric properties Definitions and examples

To motivate our definition we will first try to formulate rigorously what do we mean by a "canvas surface". A "canvas surface" can be deformed in many ways but with some limitations: it cannot be stretched as a rubber surface and this is because the fibers of the canvas are flexible but not elastic. Alternatively, this means that the only operations we can perform are those which do not change the lengths of curves on the surface. Thus one can think of "canvas surfaces" as those surfaces on which any "reasonable" curve has a well defined length. Adapting a more constructive point of view one can say that such surfaces are endowed with a clear procedure of measuring lengths of piecewise smooth curves. Classical vector analysis describes one method of measuring lengths of curves in M3. If 7 : [0,1] ->• K3 is such a curve then its length is given by

length (7) =

Jo

f\iit)\dt

where |7(£)| is the euclidian length of the tangent vector j(t). We want to do the same thing on an abstract manifold and we are clearly faced with one problem: how do we make sense of |7(£)|? Obviously, this problem can be 116

117

Metric properties

solved if we assume that there is a procedure of measuring lengths of tangent vectors at any point on our manifold. The simplest way to do achieve this is to assume that each tangent space is endowed with an inner product (which can vary from point to point in a smooth way). Definition 4.1.1 (a) A Riemann manifold is a pair (M, g) consisting of a smooth manifold M and a metric g on the tangent bundle i.e. a smooth, symmetric positive definite (0,2) tensor field on M. g is called a Riemann metric on M. (a) Two Riemann manifolds (M;, &) (i — 1,2) are said to be isometric if there exists a diffeomorphism cj>: M\ -> M2 such that <£*g2 = 9iIf (M, g) is a Riemann manifold then for any x € M the restriction gx : TXM xTxM —>• R is an inner product on the tangent space TXM. We will frequently use the alternative notation (•, -)x = gx(-, •). The length of a tangent vector v € TXM is defined as usual as

K=VM) 1 / 2 If 7 : [a, b] —> M is a piecewise smooth curve then we define its length by

1(7) = fm)\,(t)dt. If we choose local coordinates (a;1,..., xn) onM then we get a local description of g as g ~ gijdx'dx3,

g{j = g{-g~, j - ) •

Proposition 4.1.2 Let M be a smooth manifold and denote by SHM the set of Riemann metrics on M. Then 9 1 ^ is a non-empty convex cone in the linear space of symmetric (0,2) tensors. Proof The only thing we have to prove is that SH^ is non-empty. We will use again partitions of unity. Cover M by coordinate neighborhoods (Ua)aeA- Let (xla) be a collection of local coordinates on Ua. Using these local coordinates we can construct by hand the metric ga on Ua by ga = {dx\)2 + ••• +

[dxlf.

Now pick a partition of unity B C C™(M) subordinated to the cover {Ua)ae^ there exits a map <j>: B —> A such that V/3 £ B supp/5 C U^p). Then define

i.e.

The reader can check easily that g is well defined and is indeed a Riemann metric on M. D

118

Riemannian

Figure 4.1: The unit sphere and an ellipsoid look

geometry

"different".

Example 4.1.3 (Euclidian space) The space H" has a natural Riemann metric g0 = (dx1)2

+ ••• +

(dxn)2.

The geometry of (Rn,g0) is the classical Euclidian geometry. Example 4.1.4 (Induced metrics on submanifolds)Let (M,g) be a Riemann manifold and S C M a submanifold. If i: S —» M denotes the natural inclusion then we obtain by pull back a metric on S 9s = i*9 = 9\s • For example, any invertible symmetric nx n matrix defines a quadratic hypersurface in Kn by •HA = {xeRn; (Ax,x) = l} where (•, •) denotes the euclidian inner product on W. %A has a natural metric induced by the Euclidian metric on M". For example when A = In then 'Hin is the unit sphere in H" the induced metric is called the round metric of 5 n _ 1 .

Remark 4.1.5 On any manifold there exist many Riemann metrics and there is no natural way of selecting one of them. One can visualize a Riemann structure as denning a "shape" of the manifold. For example the unit sphere x2 + y2 + z2 = 1 is diffeomorphic to the ellipsoid y? + f? + W = ^ ^ut they look "different " (see Figure 4.1). However, appearances may be deceiving. In Figure 4.2 it is illustrated the deformation of a sheet of paper to a half cylinder. They look different but the metric structures are the same since we have not changed the lengths of curves on our sheet. The conclusion to be drawn from these two examples is that we have to be very careful when we use the attribute "different".

119

Metric properties

Figure 4.2: A plane sheet and a half cylinder are "not so different". Example 4.1.6 (The hyperbolic plane) The Poincare model of the hyperbolic plane is the Riemann manifold (D, g) where D is the unit open disk in the plane R2 and the metric g is given by 9= i -2 j ( ^ 2 + dy2). 1 — xl — yi Exercise 4.1.1 Let S) denote the upper half-plane

Sj = {(u,v) e l 2 ; v> 0} endowed with the metric h = ^-{du2

+ dv2).

Show that the Cayley transform z = x + iyi-)w

.z + i = —l =u+ w 2 — 1

establishes an isometry (D,
hg(x,Y) = (x,Y)9 = <(V0.*>(V'W vs e G, x,y eT9G where (Lg-i), : TgG —> T\G is the differential at g G G of the left translation map Lg-i. One checks easily that g i-> (•, -)g is a smooth tensor field and is left invariant i.e. L*gh = h MgeG. If G is also compact one then can use the averaging technique of Subsection 3.4.2 to produce metrics which are both left and right invariant.

120

Riemannian

geometry

Exercise 4.1.2 Let M = Gk,n be the grassmannian of complex k-planes in C". For each such A;-plane V denote by Py the orthogonal projection onto V. Thus Pv = Pv = Pv and Range (Pv) = V so that Py can be viewed as a complex, selfadjoint n x n matrix. Denote by Sn the linear space of such matrices and by h0 the inner product h0{A, B) = tr (AB*) \/A, B G Sn. (a) Prove that the map Pk : GkiH —» Sn defined by V >-» Pv is an embedding i.e. it is a one-to-one immersion. (b) Let k = 1 so that Glt„ = a P n _ 1 . Describe the induced metric P{h0 on OP"" 1 using the homogeneous coordinates introduced in Section 1.2.2. §4.1.2

The Levi-Civita connection

To continue our study of Riemann manifolds we will try to follow a close parallel with classical euclidian geometry. The first question one may ask is whether there is a notion of "straight line" on a Riemann manifold. In the euclidian space M3 there are at least two ways to define a line segment. (i) A line segment is the shortest path connecting two given points. (ii) A line segment is a smooth path 7 : [0,1] —> M3 satisfying 7(0=0.

(4.1.1)

Since we have not said anything about the calculus of variations (which deals precisely with problems of type (i) ) we will use the second interpretation as our starting point. We will see however that both points of view yield the same conclusion. Let us first reformulate (4.1.1). As we know the tangent bundle of K3 is equipped with a natural trivialization and as such it has a natural trivial connection V° defined by V?9,-=0 Vi,j ( $ = £ ; , V i

=

Vaj)

i.e. all the Christoffel symbols vanish. Moreover if go denotes the euclidian metric then

(v°ff0) (a,-, 9*) = v°i53k - so(v?a,-, ak) - 9o(dv v%) = o i.e. the connection is compatible with the metric. Condition (4.1.1) can be rephrased as V°(t)7W = 0 (4.1.2) so that the problem of defining "lines" in a Riemann manifold reduces to choosing a "natural" connection on the tangent bundle. Of course we would like this connection to be compatible with the metric but even so, there are infinitely many connections to choose from. The following fundamental result will solve this dilemma.

121

Metric properties

Proposition 4.1.8 Consider a Riemann manifold (M, g). Then there exists a unique symmetric connection V on TM compatible with the metric g i.e. T(V) = 0

Vg = 0.

V is usually called the Levi-Civita connection associated to the metric g. Proof We first prove that there exists at most one connection with the required properties. We will achieve this by producing an explicit description of it. Let V be a connection with the desired properties i.e. Vg = 0 and For any X,Y,Z

VXY

- VYX

= [X, Y] VZ, Y G Vect (M).

£ Vect (M) we have Zg(X, Y) = g(VzX, Y) + g(X,

VZY)

since Vg = 0. Using the symmetry of the connection we compute Zg(X, Y) - Yg(Z, X) + Xg(Y, X) = g(VzX, Y) - g(VYZ, +g(X, VZY)

- g(Z, VYX)

= g([Z, n X) + g([X, Y],Z)+

+ g(Y,

X) + g(VxY,

Z)

VXZ)

g([Z, X], Y) + 2g(VxZ,

Y).

We conclude g(VxZ,

Y) = l-{Xg{Y, Z) - Yg(Z, X) + Zg(X,

-g([X,Y],Z)

Y)

+ g([Y, Z],X) - g([Z,X],Y)}.

(4.1.3)

The above equality establishes the uniqueness of V. Using local coordinates {xl,...,xn) on M we deduce from (4.1.3) (with X = foTi'Y=

d^'

Z

=

a!~)

that

g{Vidj, dk) = guYlj = - (digjk - dkg{j +

djgik).

If (g'e) denotes the inverse of (g^) we deduce

4 = \gu (di93k - dk9i] + dm).

(4.1.4)

Existence It boils down to showing that (4.1.3) indeed defines a connection with the required properties. The routine details are left to the reader. • We can now define the notion of "straight line" on a Riemann manifold.

122

Riemannian

geometry

Definition 4.1.9 A geodesic on a Riemann manifold (M, g) is a smooth curve 7 : (a, b) —> M satisfying V W ,7(*) = 0 (4.1.5) where V is the Levi-Civita connection. Using local coordinates (x1, ...,xn) with respect to which the Christoffel symbols are (Tkj) and j(t) = (x1^), ...,xn(t)) we can reformulate the geodesic equation as a second order nonlinear system of ordinary differential equations. Set £ = -y(i) = xldi. Then V±Mt) = £% + x'VjLdi dt

w

dt

= xkdk + rk]lxiijdk

(r£- = rj4)

so that the geodesic equation is equivalent to xk + rkjxixj

= 0

VJfc = 1,..., n.

(4.1.6)

k

Since the coefficients r^- = T j(x) depend smoothly upon x we can use the classical Banach-Picard theorem on existence in initial value problems (see e.g. [4]). We deduce the following local existence result. Proposition 4.1.10 Let (M, g) be a Riemann manifold. For any compact subset K C TM there exists e > 0 such that for any (a;, X) 6 K there exists a unique geodesic 7 = jx<x • (—s,e) —>• M such that 7(0) = x, 7(0) = X. One can think of a geodesic as defining a path in the tangent bundle t i-t (7(i),7(i)). The above proposition shows that the geodesies define a local flow $ on TM by $ t (z,A-) = (7(i),7(t)) 7 = 7*,*. Definition 4.1.11 The local flow defined above is called the geodesic flow of the Riemann manifold (M,g). When the geodesic glow is a global flow i.e. any j X t X is defined at each moment of time t for any (x, X) 6 TM then the Riemann manifold is called geodesically c o m p l e t e . The geodesic flow has some remarkable properties. C o n s e r v a t i o n of energy: If j(t) is a geodesic then the length of j(t) is independent of time. Indeed ^|7(«)|2 = ^(7(*),7(*)) = 2g{Vi(thj(t))

= 0.

Thus if we consider the sphere bundles ST{M) = {XeTM

; \X\ = r}

we deduce that Sr(M) are invariant subsets of the geodesic flow.

123

Metric properties Exercise 4.1.3 Describe the infinitesimal generator of the geodesic flow. Example 4.1.12 Let G be a connected Lie group and £ G its Lie algebra. X 6 £ G defines an endomorphism ad (X) of £ G by ad{X)Y

Any

= [X,Y\.

The Jacobi identity implies that ad([X,Y])

=

[ad{X),ad(Y)}

where the bracket in the right hand side is the usual commutator of two endomorphisms. Assume there exists an inner product (•, •) on £ G such that for any X 6 £ G , ad{X) is skew-adjoint i.e. {[X,Y],Z)

= ~(Y,[X,Z})

(4.1.7)

We can now extend this inner product to a left invariant metric h on G. We want to describe its geodesies. First, we have to determine the associated Levi-Civita connection. Using (4.1.3) we get h{VxZ,

Y) = \{Xh{Y,

Z) - Y(Z, X) + Zh(X,

Y)

-h([X, Y],Z) + h([Y, Z], X) - h([Z, X], Y)}. If we take X, Y, Z e £ G i-e. these vector fields are left invariant then h(Y, Z) = constant, h(Z,X) — const., h(X,Y) = const, so that the first three terms in the above formula vanish. We obtain the following equality (at l e G ) {VXZ,Y)

= i{-([X,Y],Z) + <[Y,Z\,X)-

([Z,X],Y)}.

Using the skew-symmetry of ad(X) and ad(Z) we deduce (VXZ,Y)

=

~([X,Z},Y)

so that at 1 € G VxZ=l-[X,Z]

VX,Ze£G.

(4.1.8)

This formula correctly defines a connection since any X £ Vect (G) can be written as a linear combination X = YJ(*iXi

aiEC^iG)

XiEZo.

124

Riemannian

geometry

Now, if j(t) is a geodesic we can write j(t) = Y,liXi so that

0 = V W 7(t) = E 7 ^ +

\Y,wAxi>Xi]i,j

i

Since psT^X,] = —[Xj,Xi\ we deduce 7* = 0 i.e. 7W = £ 7 i ( 0 L Y i = X. This means 7 is an integral curve of the left invariant vector field X so that the geodesies through the origin with initial direction X e TiG are 7*(i) =

exp{tX).

Exercise 4.1.4 Let G be a Lie group and h a bi-invariant metric on G. Prove that its restriction to £Q satisfies (4.1.7). In particular, on any compact Lie groups there exist metrics satisfying (4.1.7). Definition 4.1.13 Let £ be a finite dimensional real Lie algebra. pairing or form is the bilinear map K : £ x £ —> K, K(X, y) = —tr (ad(x)ad(y))

The Killing

x, y G £ .

The Lie algebra £ is said to be semisimple if the Killing pairing is a duality. A Lie group G is called semisimple if its Lie algebra is semisimple. Exercise 4.1.5 Prove that SO(n) and SU(n) and SL{n, M) are semisimple Lie groups but U(n) is not. Exercise 4.1.6 Let G be a compact Lie group. Prove that the Killing form is positive definite and satisfies (4.1.7). N o t e The converse of the above exercise is also true i.e. any semisimple Lie group with positive definite Killing form is compact. This is known as Weyl's theorem. Its proof, which will be given later in the book, requires substantially more work. Exercise 4.1.7 Show that the parallel transport of X along exp(iF) is ( i exp(fy))*(^exp(|y))*^-

Example 4.1.14 (Geodesies on flat tori and SU(2)) The n-dimensional torus Tn = S1 x • • • x S1 is an abelian, compact Lie group. If (9l, •••,6n) are the natural angular coordinates on T™ then the flat metric is defined by g

= {de1)2 + •••

[denf.

125

Metric properties

g is a bi-invariant metric on on Tn and obviously its restriction to the origin satisfies the skew-symmetry condition (4.1.7) since the bracket is 0. The geodesies through 1 will be the exponentials 701

an{t)

t^(eta»t,...,ete-')

ak€R.

If the numbers ak are linearly dependent over the rationals Q then obviously 7Q1,...,Q„(t) is a closed curve. On the contrary, when the a's are linearly independent over Q then a classical result of Kronecker (see e.g. [33]) states that the image of 7Ql,...lQn is dense in Tn\\\ (see also Section 7.4 to come) The special unitary group SU(2) can also be identified with the group of unit quaternions {a + bi + cj + dk; a2 + b2 + c2 + d2 = 1} so that SU(2) is diffeomorphic with the unit sphere S3 C K4. The round metric on S 3 is bi-invariant with respect to left and right (unit) quaternionic multiplication (verify this) and its restriction to (1,0,0,0) satisfies (4.1.7). The geodesies of this metric are the 1-parameter subgroups of S3 and we let the reader verify that these are in fact the great circles of S3 i.e. the circles of maximal diameter on S3. Thus, all geodesies on S3 are closed. §4.1.3

The exponential map and normal coordinates

We have already seen that there are many differences between the classical euclidian geometry and the the general riemannian geometry in the large. In particular we have seen examples in which one of the basic axioms of euclidian geometry no longer holds: two distinct geodesic (read lines) may intersect in more than one point. The global topology of the manifold is responsible for this "failure". Locally however, things are not "as bad". Local riemannian geometry is similar in many respects with the euclidian geometry. For example, locally, all of the classical incidence axioms hold. In this section we will define using the metric some special collections of local coordinates in which things are very close to being euclidian. Let (M, g) be a Riemann manifold and U an open coordinate neighborhood with coordinates (x 1 , ...,xn). We will try to find a local change in coordinates (a;*) >-> (yj) in which the expression of the metric is as close as possible to the euclidian metric So = Sijdtfdyi. Let q g U be the point with coordinates (0,...,0). Via a linear change in coordinates we may as well assume that 9iM) -

S

'j-

We can formulate this by saying that (gi3) is euclidian up to order zero. We would like to "spread" the above equality to an entire neighborhood of q. To achieve this we try to find local coordinates (yJ) near q such that in these new

126

Riemannian

geometry

coordinates the metric is euclidian up to order one i.e.

We now describe a geometric way of producing such coordinates using the geodesic flow. Denote as usual the geodesic from q with initial direction X G TqM by 7,,x(i). Note the following simple fact Vs > 0 jq,sx(t)

= -yq,x(st).

Hence, there exists a small neighborhood V of 0 € TqM such that for any X EV the geodesic 7,,x(*) is defined for all \t\ < 1. We define the exponential map at q by exp, : V C TqM -+ M

X^

7,,x(l)-

TqM is an euclidian space and we can define D ? (r) C TqM - the open "disk" of radius r centered at the origin. We have the following result. Proposition 4.1.15 Let (M,g) and q € M as above. Then there exists r > 0 such that the exponential map exp g : D,(r) ->• M is a diffeomorphism onto. The supremum of all radii r with this property is denoted by PM{q)Definition 4.1.16 PM{I) is called the injectivity r a d i u s of M at q. The infimum pM =

'mipM(q)

is called the injectivity radius of M. The proof relies on the following key fact. Lemma 4.1.17 The Frechet differential at 0 6 TqM of the exponential map Do exp, : TqM -> TexPq{a)M = TqM is the identity TqM ->• TqM. Proof of the lemma. Consider X G TqM. It defines a line in TqM by t >-* tX which is mapped via the exponential map to the geodesic 7,,x(i)- By definition (D0expq)X

= j9,x(0)=X.

O

Proposition 4.1.15 follows immediately from the above lemma using the inverse function theorem.

Metric properties

127

Now choose an orthonormal frame (ei,...,e n ) of TqM and denote by ( x 1 , . . . ^ " ) the resulting cartesian coordinates. For 0 < r < PM{Q) any point p 6 exp g (D g (r)) can be uniquely written as V = exp^x^i) so the collection (x 1 ,..., x") provides a coordinatization of the open set exp g (D ? (r)) c M. The coordinates thus obtained are called normal coordinates at q , the open set exp 9 (D,(r)) is called a normal neighborhood and will be denoted by B r (g) for reasons that will become apparent a little later. Proposition 4.1.18 Let (x1) be normal coordinates at q € M and denote by gy the expression of the metric tensor in these coordinates. Then we have dgigij(. expg fem'glrj so that

€ Kn the curve x* = mlt is the geodesic

rjt(x(t)K"m* = 0. In particular T)k(Q)m?mk = 0 Vmj 6 » " from which we deduce the lemma. • The result in the above lemma can be formulated as ff(V^--,—I) \

=0

Vt.j.fc

a*J OX* OX* J

so that V ^ — - = 0 at q Vt,j. 8xJ OTC

Using Vp = 0 we deduce

g^) = (^g,)l,= o. *

(4.1.9)

128

Riemannian

geometry

The reader may ask whether we can go one step further and find local coordinates which produce a second order contact with the euclidian metric. At this step we are in for a big surprise. This thing is in general not possible and in fact there is a geometric way of measuring the "second order distance" between an arbitrary metric and the euclidian metric. This is where the curvature of the Levi-Civita connection comes in and we will devote an entire section to this subject. §4.1.4

The minimizing property of geodesies

We defined geodesies via a 2nd order equation imitating the 2nd order equation defining lines in an euclidian space. As we have already mentioned, this is not the unique way of extending the notion of Euclidian straight line to arbitrary Riemann manifolds. One may try to look for curves of minimal length joining two given points. We will prove that the geodesies defined as in the previous subsection do just that, at least locally. We begin with a technical result which is interesting in its own. Let (M, g) be a Riemann manifold. Lemma 4.1.20 For each } E M there exists 0 < r < PM(Q) and e > 0 such that Vm G B r (g) we have e < pM{m) and B e (m) 3 B r (g). In particular, any two points of Gr(q) can be joined by a unique geodesic of length < e. We must warn the reader that the above result does not guarantee that the postulated connecting geodesic lies entirely in B r (g). This is a different ball game. Proof of the lemma Using the smooth dependence upon initial data in ordinary differential equations we deduce that there exists an open neighborhood V of (q, 0) € TM such that exp m X is well defined for all (m, X) G V. We get a smooth map F:V

^ M xM

(m,X)^{m,

exp m X).

We compute the differential of F at (q, 0). First, using normal coordinates (x1) near q we get coordinates (x*; ~XP) near (q, 0) G TM. The partial derivatives of F at (q, 0) are

Thus the matrix defining D^i0)F has the form id *

0 id

and in particular it is nonsingular. It follows from the implicit function theorem that F maps some neighborhood V of (9, 0) G TM diffeomorphically onto some neighborhood U of (q, q) G M x M. We

129

Metric properties

can choose V to have the form {(m, X) ; \X\m < e, m 6 Bs(q)} for some sufficiently small e and 5. Choose 0 < r < min(e,pAf (g)) such that Vm1,m2€Br(g)

(m 1 ,m 2 ) € [/.

In particular, it follows that for any m G Br( U is injective. D We can now formulate the main result of this subsection. Theorem 4.1.21 Let q, r and e as in the previous lemma and consider 7 : [0,1] —• M the unique geodesic of length < e joining two points of B r (g). If tv : [0,1] —>• M is a piecewise smooth path with the same endpoints as 7 then

f W)\dt < f \u{t)\dt

Jo

Jo

with equality if and only if w([0,1]) = 7([0,1]). Thus 7 is the shortest path joining its endpoints. The proof relies on two lemmata. Let m € M be an arbitrary point and assume 0 < R < pM{m). Lemma 4.1.22 (Gauss) In Bfi(m) the geodesies through m are orthogonal to the hypersurfaces E, = exp,(5 4 (0)) = {exp m (X) ; |X| = &} 0 < 6 < R. Proof Let t i-t X(t), 0 < t < 1 denote an arbitrary smooth curve in TmM such that |X(t)| m = 1 i.e. X(t) is a curve on the unit sphere Si(0) C TmM. We have to prove that the curves t >-¥ exp m (5X(i)) are orthogonal to the radial geodesic s i-» exp m (sX(0)), 0 < s < R. Consider the parameterized surface f(s, t) = expm(sX(t)) Set

(a, t) e [0, R) x [0,1].

130

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geometry

Define -£ similarly. | £ and | £ are not vector fields in the usual sense since the surface / may have double points and at such point /» may associate two tangent vectors. One way around this problem is to pullback everything (metric, connection etc.) to the rectangle [0, R] x [0,1] via / and everything will be correctly defined. We will denote the pulled-back objects by the same symbols as the originals. It is convenient to view these "vector fields" as the tangent vectors to the "coordinate lattice" f(t = const.), f(s = const.). We have to show

,df df.

, d d,

w/

.

where (•, •} denotes the inner product defined by g. The first equality is tautological. Using the metric compatibility of the (pulled-back) Levi-Civita connection we compute d , d d, , _ d d. ,d _ d, os os ot 9* ds ot os o> ot Since the curves f(t = const.) are geodesies we deduce 9' OS

On the other hand, since | Jj, JU = 0 we deduce (using the symmetry of the LeviCivita connection) d_

W

d_ _ } {

d_

%dt ~ ds'

d_

mds'~

2 _ld_ d_ 2dt ds

0

since | ^ = \X(t)\ = 1. We conclude that the quantity (Jj, ^ ) is independent of s. For s = 0 we have / ( 0 , t) = exp m (0) so that | £ = 0 and therefore

as needed. • Now consider any continuous, piecewise smooth curve w: [a,6]->BR(m)\{m}. Each uj(t) can be uniquely expressed in the form cj(t)=exPm(p(t)X(t))

\X(t)\ = l

0<\p(t)\
L e m m a 4.1.23 The length of the curve u>(t) is > \p(b) — p(a)\. The equality holds iff X(t) = const and p(t) > 0. In other words, the shortest path joining two concentrical shells Tig is a radial geodesic.

131

Metric properties Proof

Let f{p,t)

= expm(pX(t))

so that w{t) =

f(p{t),t).

Since the vectors §^ and | £ are mutually orthogonal and since = \X(t)\ = 1 we get 2

H2 = | / f + | ; >|P|2The equality holds if and only if §£ = 0 i.e. X = 0. Thus / |w|dt> /" | p | d t > | / 9 ( 6 ) - p ( a ) | . Ja

Ja

Equality holds if and only if p(t) is monotone and X(i) is constant. This completes the proof of the lemma. • The proof of Theorem 4.1.21 is now immediate. Let m0,mi e B r (g) and 7 : [0,1] —¥ M a geodesic of length < e such that 7(1) = rrij, i = 0,1. We can write j(t) = expmo(tX)

l £ D

£

( 4

Set .R = |X|. Consider any other piecewise smooth path u> : [a, b] —> M joining m 0 to rri!. For any 8 > 0 this path must contain a portion joining the shell T,j(m0) to the shell E R (m 0 ) and lying between them. By the previous lemma the length of this segment will be > R — 6. Letting 5 —> 0 we deduce l(u>)>R=l{j). If ui([a, b}) does not coincide with 7([0,1]) then we obtain a strict inequality. • Any Riemann manifold has a natural structure of metric space. More precisely we set d(p, q) = inf {l(w) ui : [0,1] —> M piecewise smooth path joining p to q} A piecewise smooth path w connecting two points p, q such that 1(LJ) = d(p, q) is said to be minimal. From Theorem 4.1.21 we deduce immediately the following consequence.

132

Riemannian

geometry

Corollary 4.1.24 The image of any minimal path coincides with the image of a geodesic. In other words, any minimal path can be reparameterized such that it satisfies the geodesic equation. Exercise 4.1.8 Prove the above corollary. Theorem 4.1.21 also implies that any two nearby points can be joined by a unique minimal geodesic. In particular we have the following consequence. Corollary 4.1.25 Let q € M. Then for all r > 0 sufficiently small exp,(D r (0))(= Br(q)) = {peM;

d{p,q) < r}

(4.1.10)

Corollary 4.1.26 For any q € M we have the equality PM(Q) = sup{r ; r satisfies (4.1.10)}. Proof The same argument as in the proof of Theorem 4.1.21 shows that Vr < PM(Q) the radial geodesies expJtX) are minimal. • Definition 4.1.27 A subset U C M is said to be convex if any two points in U can be joined by a unique minimal geodesic which lies entirely inside U. Proposition 4.1.28 For any q € M there exists 0 < R < LM(q) such that for any r < R the ball B r (g) is convex. Proof Choose a small e > 0 (0 < 2e < PM(Q)) a n d 0 < R < e such any two points m 0 , m-i in BR(q) can be joined by a unique minimal geodesic 7 m o , m i (t) (0 < t < 1) of length < e (not necessarily contained in BR(q)). We will prove that Vm0, m1 6 B fl (g) the map t >-> d{q, 7 m 0 j m i (t)) is convex and thus it achieves its maxima at the endpoints t = 0,1. Note that

d(q,j{t)) mi (t) can be uniquely expressed as 7mo,mi(*) = exp,(r(t)X(t))

X{t)eTqM

with r{t) =

d{q,ym(hrni{t))-

It suffices to show J^(r 2 ) > 0 for t G [0,1] if d(q,m0) and d(q,mi) are sufficiently small. At this moment it is convenient to use normal coordinates (x1) near q. The geodesic 7 mo>mi takes the form (x ! (i)) and we have r2 = (x 1 ) 2 + • • • + (x") 2 . We compute easily ^ ( r 2 ) = 2r 2 (x 1 + --- + x") + |x| 2

(4.1.11)

133

Metric properties where x(t) = X) x ' e i e TqM. 7 satisfies the equation x i + r;.fc(x)xJ'x* = 0

Since 1^(0) = 0 (normal coordinates) we deduce that there exists a constant C > 0 (depending only on the magnitude of the second derivatives of the metric at q) such that

|r;*(x)| < c\x\. Using the geodesic equation we obtain x* > - C | x | | x | 2 . We substitute the above inequality in (4.1.11) to get ^(r2)>2|x|2(l-nC|x|3).

(4.1.12)

If we choose from the very beginning

then because along the geodesic \x\ < R + e the right-hand-side of (4.1.12) is nonnegative. This establishes the convexity of 11-> r2(t) and concludes the proof of the proposition. • In the last result of this subsection we return to the concept of geodesic completeness. We will see that this can be described in terms of the metric space structure alone. Theorem 4.1.29 (Hopf-Rinow)Let M be a Riemann manifold and q 6 M. The following assertions are equivalent: (a) exp„ is defined on all of TqM. (b) The closed and bounded (with respect to the metric structure) sets of M are compact. (c) M is complete as a metric space. (d) M is geodesically complete. (e) There exists a sequence of compact sets Kn C M, Kn C Kn+X and (Jn Kn = M such that if pn $ Kn then d(q,pn) —» 00. Moreover, on a (geodesically) complete manifold any two points can be joined by a minimal geodesic. Remark 4.1.30 On a complete manifold there could exist points (sufficiently far apart) which can be joined by more than one minimal geodesic. Think for example of a manifold where there exist closed geodesic (e.g the tori Tn).

134

Riemannian

geometry

Exercise 4.1.9 Prove the Hopf-Rinow theorem. Exercise 4.1.10 Let (M, g) be a Riemann manifold and let (Ua) an open cover consisting of bounded geodesically convex open sets. Set da = (diameter (Ua))2. Denote by ga the metric on Ua defined by ga = d~*g so that the diameter of Ua in the new metric is 1. Using a partition of unity (ipi) subordinated to this cover we can form a new metric 9 = Y, ViMi)

(SUPP W

c

U

«{i))-

i

Prove that g is a complete Riemann metric. Hence, on any manifold there exist complete Riemann metrics. §4.1.5

Calculus on Riemann manifolds

The classical vector analysis extends nicely to Riemann manifolds. We devote this subsection to describing this more general "vector analysis". Let (M,g) be an oriented Riemann manifold. We now have two structures at our disposal: a Riemann metric and an orientation and we will use both of them to construct a plethora of operations on tensors. First, using the metric we can construct by duality an isomorphism C : Vect (M) —>• &}{M) (called lowering the indices). Example 4.1.31 Let M = R 3 with the euclidian metric. A vector field V on M has the form ox

ay

az

Then W = CV = Pdx + Qdy + Rdz. If we think of V as a field of forces in the space then W is the infinitesimal work of V. On a Riemann manifold there is an equivalent way of describing the exterior derivative. Proposition 4.1.32 Let e : C^iTM

® A*T*M) ->

Cx'(A*+1T*M)

denote the exterior multiplication operator e{a®P)=a/\p,

VQ G Qr{M), /3 €

Qk(M).

Then d = eo V where d denotes the exterior derivative and V is the connection induced on AkT*M by the Levi-Civita connection.

135

Metric properties Proof We will use a strategy useful in many other situations. normal coordinates will payoff. Denote temporarily by D the equality d = D is a local statement and it suffices to prove neighborhood. Choose (x 1 ) normal coordinates at an arbitrary di = ^ T . Note that

Our discussion about operator e o V. The it in any coordinate point p € M and set

i k

Let u) e £l (M). Near p it can be written as u) — ^2 ujjdx1 i

where as usual, for any ordered multi-index / : (1 < ix < • • • < ik < n) we set dx1 = dx' 1 A • • • A dxlk. In normal coordinates a t p we have (Vj3,) | p = 0 from which we get the equalities (VidxO | p (dk) = -(dx.J(Vidk))

| p = 0.

Thus a t p Dw = ^ dx" A V i (w / dx / ) = J2

dx

'

A

(djUidx1 + w/Vi(dx f )) = ^ dx* A 3jW/ = dw.

i

i

Since the point p was chosen arbitrarily this completes the proof of Proposition 4.1.32. D

Exercise 4.1.11 Show that for any fc-form u> on the Riemann manifold (M,g) the exterior derivative dui can be expressed by

Mx°,Xi,---,,xk) = ^2(-i)i(vXiu)(x0,---,xi,---,xk) for all Xo, • • • ,Xk € Vect (M). (V denotes the Levi-Civita connection.) The Riemann metric defines by duality a metric in any tensor bundle TJ~(M) which we continue to denote by g. Thus, given two tensor fields 1\, T2 of the same type (r, s) we can form their pointwise scalar product MBp^

g(T, S)p = ^ ( ^ ( p ) , T 2 (p)).

In particular, any such tensor has a pointwise norm

M3p^\T\g,p

=

{T,T)y\

Using the orientation we can construct (using the results in subsection 2.2.4) a natural volume form on M which we denote by dvg and we call it the metric volume .

136

-Riemannian geometry

This is the positively oriented volume form of pointwise norm = 1. If (x 1 ,..., i") are local coordinates such that dx1 A • • • A dxn is positively oriented then dvg = \f\g\dxl A • • • A dx11 where \g\ = det(^y). In particular we can integrate (compactly supported) functions on M by

/

f=f[

fdvg

V/ G C0°°(M).

We have the following (not so surprising) result. Proposition 4.1.33 Vxdvg = 0

Proof

V i e Vect (M).

We have to show that for any p G M (Vxdvg)(eu...,en)

=0

(4.1.13)

where basis of TPM. Choose (x1) normal coordinates near p. Set d{ = -g~, Sij — g{di,dk) and e* = d^ \p. Since the expression in (4.1.13) is linear in X we may as well assume X = dk for some k = 1,..., n. We compute (Vxdv9)(ei,...,

en) = X{dvg{du ..., 3n)) | p - £ dv9(eu ..., (Vxd{) \p,..., dn).

(4.1.14)

z

We consider each term separately. Note first that dvg(d1, ...,dn) = (det(gy)) 1 / 2 so that X(det(gij))1^2 | p = 9/t(det(gy)) 1 / 2 | p is a linear combination of products in which each product has a factor of the form dkgij \p- Such a factor is zero since we are working in normal coordinates. Thus the first term in(4.1.14) is zero. The other terms are zero as well since in normal coordinates p

Vxdt = V^di = 0. Proposition 4.1.33 is proved. • . Once we have an orientation we also have the Hodge *-operator * : Qk(M) -+

Sln'k(M)

uniquely determined by a A *j3 = (a, p)dvg

Vaj3eQk{M).

In particular *1 = dvg.

(4.1.15)

Metric properties

137

Example 4.1.34 To any vector field F = Pdx + Qdy + Rdz on K3 we associated its infinitesimal work WF = C{F) = Pdx + Qdy + Rdz. The infinitesimal energy flux of F is the 2-form §F = *WF = Pdy Adz + Qdz Adx + Rdx A dy. The exterior derivative of WF is the infinitesimal flux of the vector field curl F dWF = {dyR - dzQ)dy Adz + (dzP - dxR)dz Adx + (dxQ - dyP)dx A dy = $cur\F = *WcurlF-

The divergence of F is the scalar defined as div F = *d*WF = * d $ F = * {(dxP + dyQ + dzR)dx AdyA dz} = dxP + dyQ + dzR. If / is a function on R3 then we compute easily *d*df = d2j + %f + d2j = Af. Definition 4.1.35 (a) For any smooth function / on M we denote by g r a d / the vector field dual to the 1-form df. In other words (grad / , X) = df{X) = X • f

V i e Vect (M).

(b) For any X € Vect (M) we denote by div X the smooth function defined by the equality Lxdvg = (div X)dvg. Proposition 4.1.36 Let A" be a vector field on M and denote by a the 1-form dual to X. Then (a) d i v X = t r ( V X ) where we view V X as an element of C°°(End (TM)) via the identifications V X 6 Q}{TM) =* CX(T*M

® TM) ^ C°°(End (TM)).

(b) d i v X = *d * a. (c) If (x1,..., xn) are local coordinates such that dx1 A • • • A dxn is positively oriented then

divx = ^Ld i (^jx') y\g\ where X = X^d*. The proof will rely on the following technical result which is interesting in its own.

138

Riemannian

geometry

Lemma 4.1.37 Denote by S the operator S = *d* : nk(M)

->

tf-^M).

Let a be a (k — l)-form and fi a &-form such that at least one of them is compactly supported. Then I {da,P)dvg = {-iy-» J {a,5P)dvg JM

JM

where vn£ = nk + n + 1. Definition 4.1.38 Define d* : Q,k{M) -> Qk~l{M)

by

,

d* = (-l)"»-*5 = ( - l ) ' » ' » * d * . Proof of the lemma

We have

/ (da, p)dv„ = f da A */3 = [ d(a A *j3) + (-1)* / a Ad* /}.

JM

JM

JM

JM

The first integral in the right-hand-side vanishes by the Stokes formula since a A *j3 has compact support. Since d * (3 € fin-*+1(M) and * 2 = (_i)(»-*+i)(*-i) 0 n

fi"-*+1(Af)

we deduce f (da,P)dvq = (~l)k+^-k+l^-k^

JM JM

f

aA*6

JM

This establishes the assertion in the lemma since (n-

k + l ) ( f c - 1) + k = j/„ i t (mod2). D

Proof of the proposition Set 0 = di>9 and let (Xi,...,Xn) frame of TM in a neighborhood of some point. Then

be a local moving

(Lxsi)(xu..., xn) = x{n(xu..., xn)) - Y, n(*i> -. [*> xl -. *»)•

(4-L16)

i

Since VQ. = 0 we get X • (fl(Xi,..., X n )) = £ to{Xu - , VxXi,...,

Xn).

i

Using the above equality in (4.1.16) we deduce from VXY {LxQ)(Xlt....

Xn) = Y,n{Xlt...,

— [X,Y] = VyX

VXiX,..., Xn).

that (4.1.17)

i

Over the neighborhood where the local moving frame is defined we can find smooth functions / / such that VXiX

= f?XJ=>tr(VX)

= fi.

139

Metric properties

Part (a) of the proposition follows after we substitute the above equality in (4.1.17). Proof of (b) From (a) we deduce that for any / € C™(M) we have

Lx(fuj) = (xm + fti(vx)n. On the other hand LxifSl)

= (ixd + dix){fQ)

=

dix{fQ).

Hence

{(Xf) + ftv(vx)}dv9 = d(ixfn). Since the form fQ is compactly supported we deduce from Stokes formula

[ d{ixfU) = 0. We have thus proved that for any compactly supported function / we have the equality - J ftr(VX)dvg JM

= [ Xfdv9 JM

= J (grad f,X)dvg=

= /

df{X)dvg

JM

f

{df,a)dvg.

Using Lemma 4.1.37 we deduce - J ftr{VX)dvg

= - [ f5advg

V/ € C0°°(M).

This completes the proof of (b). Proof of (c) We use the equality Lxiy/lgldx1

A • • • A dxn) = div {X^^dx1

A • • • A dxn).

The desired formula follows derivating in the left-hand-side. One uses the fact that Lx is an even s-derivation and the equalities Lxdxl

= diX'dx1

(no summation)

proved in Subsection 3.1.3. • Exercise 4.1.12 Let {M,g) be a Riemann manifold and X € Vect (M). Show that the following conditions on X are equivalent. (a) Lxg = 0. (b) g{VYX, Z) + g{Y, VZX) = 0 for all Y, Z e Vect (M). (A vector field X satisfying the above equivalent conditions is called a Killing vector field).

140

Riemannian

geometry

Exercise 4.1.13 Consider a Killing vector field X on the Riemann manifold (M, g) and denote by 77 the 1-form dual to X. Show that Srj — 0 i.e. in other words div (X) = 0. Definition 4.1.39 Let (M, g) be an oriented Riemann manifold (possibly with boundary). For any /c-forms a, @ define {a,/3) = {a,0)M=

J {a,/3)dvg=

JM

f a A */3

JM

whenever the integrals in the right-hand-side are finite. Let (M, g) be an oriented Riemann manifold with boundary dM. By definition, M is a closed subset of a boundary-less manifold M of the same dimension. Along dM we have a vector bundle decomposition

(TM)\9M=T(8M)®n where n = (TdM)1- is the orthogonal complement of TdM in (TM) \QM- Since both M and dM are oriented manifolds it follows that tt is a trivial line bundle. Indeed, over the boundary det TM = det {TdM) ® n so that n^detTM®det(TdM)*. In particular n admits nowhere vanishing sections and each such section defines an orientation in the fibers of n. An outer normal is a nowhere vanishing section a of n such that for each x £ dM and any positively oriented wx e det TxdM ax A w j i s a positively oriented element of det TXM. Since n carries a fiber metric we can select a unique outer normal of pointwise length = 1. This will be called the unit outer normal and will be denoted by v. Using partitions of unity we can extend v to a vector field defined on M. Proposition 4.1.40 (Integration by parts)Let (M, g) be a compact, oriented Riemann manifold with boundary, a G nk~l{M) and fi € ttk{M). Then / (da, P)dvg = J

(a A */?) \9M + [ (a, d*0)dvg

JdM

JM

= /

JdM

a\9M Ai(ip/3)\dM

JM

+ JM

[a,d*(3)dvg

where * denotes the Hodge ^-operator on dM with the induced metric g and orientation.

Metric properties

141

Using the (•, •) notation of Definition 4.1.39 we can rephrase the above equality as (da, /3)M = (a, ip/3)dM + (a, d*P)M. Proof of the proposition

As in the proof of Lemma 4.1.37 we have

(da, P)dvg = daA*P

= d(a A */?) + ( - l ) f c a Ad* p.

The first part of the proposition follows from Stokes formula arguing precisely as in Lemma 4.1.37. To prove the second part we have to check that (a A *P) \9M= a \9M Ai(ipP) \

m

.

This is a local (even a pointwise) assertion so we may as well assume M = H " = {(a;1,...,a;") G Mn ; x1 > 0} and the metric is the euclidian metric. Note that v = —d\. Let I be an ordered (k — l)-index and J an ordered k- index. Denote by Jc the ordered (n — fc)-index complementary to J so that (as sets) J U Jc = {1, ...,n}. By linearity, it suffices to consider only the cases a = dx1, P = dxJ. We have *dx] = ejdxJC

(ej = ±1)

(4.1.18)

and ivax

- j _dxj,

l e J

where J ' = J \ {1}. Note that if 1 £ J then 1 € Jc so that (a A *p) \BM= 0 = a \aM Ai(ipP) \gM so the only nontrivial situation left to be discussed is 1 €
= -*{dxJ')

= ~e'jdxJC

(e'j = ±1).

(4.1.19)

We have to compare the two signs ej and e'j. in (4.1.18) and (4.1.19). ej is the signature of the permutation JUJ C of {1,..., n} obtained by writing the two increasing multi-indices one after the other, first J and then Jc. Similarly, since the boundary dM has the orientation — dx2 A- • -Adxn, e'j is (—l)x(the signature of the permutation J ' U J c o f {2,...,n}). Obviously sign (JUJ c ) = sign (J'QJ c ) so that ej = —e'j. The proposition now follows from (4.1.18) and (4.1.19). • Corollary 4.1.41 (Gauss)Let {M,g) be a compact, oriented Riemann manifold with boundary and X a vector field on M. Then I div (X)dv9 = f JM

where9 9 = 5 | a M .

JdM

(X, u)dv9a

142 Proof

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geometry

Denote by a the 1-form dual to X. We have / div (X)dv„ = / 1 A *d * adv„

JM

JM

= / (l,*d*a)dv9

= — /

JM

=

(l,d*a)dvg

JM

L„awdv*>= Ljx^dvn»- ° JoM

J oM

Remark 4.1.42 The compactness assumption on M can be replaced with an integrability condition on the forms a, 0 so that the previous results hold for noncompact manifolds as well provided all the integrals are finite. Definition 4.1.43 Let (M, g) be an oriented Riemann manifold. The geometric Laplacian is the linear operator A M : Cco(M) -> C°°(M) defined by A M = d*df = — *d*df

= - d i v (grad / ) .

A smooth function / on M satisfying the equation A M / = 0 is called harmonic. Using Proposition 4.1.36 we deduce that in local coordinates (a;1,..., xn) the geometric laplacian takes the form

AM = —)=di

(y/Wtdj)

where (g ,J ) denotes as usual the matrix inverse to (
A0 = -(a? + ... + a2) which differs from the physicists Laplacian by a sign. Corollary 4.1.44 (Green)Let (M,g) as in Proposition 4.1.40 and f,g€ Then (/, A M S ) M = (df, dg)M

C°°(M).

- (/, -~Z)BM-

{f, A M 5 ) M - ( A M / , S ) M = (^p, g)dM - (/,

^>aw-

Proof The first equality follows immediately from the integration by parts formula (Proposition 4.1.40) with a = f and /? = dg. The second identity is now obvious. •

143

Metric properties

Exercise 4.1.14 (a) Prove that the only harmonic functions on a compact oriented Riemann manifold M are the constant ones. (b) If u, f e C°°{M) are such that A M w = / show that

/ = o.

JM

Exercise 4.1.15 Denote by (w 1 ,... ,un) the coordinates on the round sphere Sn ' M n+1 obtained via the stereographic projection from the south pole. (a) Show that the round metric go on Sn is given in these coordinates by 00 =

1+ r

{(du1)2 + ••• +

(dunY}

where r 2 = (u 1 ) 2 H h (un)2. (b) Show that the n-dimensional "area" of Sn is r , 27r("+1>/2 n = n dv„90 Js r(a±i) where T is Euler's gamma function TOO

T(s) = / *s-1e~*1dt. Jo Hint: Need to know the "doubling formula"

7r1/2r(2s) = 2 2s - 1 r(s)r(s +1/2) and (see [31]) -dr = Vo (1 + r )" ' ~ Jo 2

(r(n/2)) 2 2r(n)

Exercise 4.1.16 Consider the Killing form on su(2) (the Lie algebra of SU(2)) defined by

(X,Y) =

-trX-Y.

(a) Show that the Killing form defines a bi-invariant metric on SU(2) and then compute the volume of the group with respect to this metric. SU(2) is given the orientation defined by ex A e2 A e$ £ A 3 SM(2) where e* € su{2) are the Pauli matrices ex

[ i

01

0 -i

[ 0 1 1 -1 0

e2 =

e3 =

[Oil i 0

(b) Show that the trilinear form on su(2) defined by

B(X,Y,Z)

= ([X,Y},Z) 3

is skew-symmetric. In particular B e A su(2)*. (c) B has a unique extension as a left-invariant 3-form on SU(2) known as the Cartan form on SU{2)) which we continue to denote by B. Compute !su^2) B.

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Hint: Use the natural diffeomorphism SU(2) = S 3 and the computations in the previous exercise.

4.2

The Riemann curvature

A Riemann metric on a manifold has roughly speaking the effect of "giving a shape" to the manifold. Thus a very short (in diameter) manifold is different from a very long one. A large (in volume) manifold is different from a small one. However there is a lot more information encoded in the Riemann manifold than just its size. To recover it we need to look deeper in the structure and go beyond the first order approximations we have used so far. The Riemann curvature tensor achieves just that. It is an object which is very rich in informations about the "shape" of a manifold and loosely speaking provides a second order approximation to the geometry of the manifold. As Riemann himself observed we do not need to go beyond this order of approximation to recover all the informations. In this section we introduce the reader to the Riemann curvature tensor and its associates. We will describe some special examples and we will conclude with the Gauss-Bonnet theorem which shows that the local object which is the Riemann curvature has global effects. §4.2.1

Definitions and properties

Let (M, g) be a Riemann manifold and denote by V the Levi-Civita connection. Definition 4.2.1 The Riemann curvature denoted by R = R(g) is defined as R{g) = F(V) where F(V) is the curvature of the Levi-Civita connection. The Riemann curvature is a tensor R e fi2(End (TM)) explicitly defined by R{X, Y)Z = [ V x , Vy]Z -

V[XtY]Z.

In local coordinates (a;1, ...,xn) we have the description

4 ^ =^,3^. In terms of the Christoffel symbols we have r>i _ % -pi

n

ijk

— ujL

ik

a r ' x r ' r m _ r^

u L

k

ij "+" L mjL ik

L

rm

mkL ij •

Lowering the indices we get a new tensor Rijkt = 9imRfke = {R{dk, dt)dj, d{).

The Riemann

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Theorem 4.2.2 (The symmetries of the curvature tensor) The Riemann curvature tensor R satisfies the following identities {X, Y, Z,U,V G Vect (M)). (a) g(R(X, Y)U, V) = -g(R{(Y, X), U, V). (b) g(R(X,Y)U,V) = -g{R{X,Y)V,U). (c) (The 1st Bianchi identity) R{X, Y)Z + R{Z, X)Y + R(Y, Z)X = 0. (d) g{R{X, Y)U, V) = g(R(U, V)X,

Y).

(e) (The 2nd Bianchi identity) (VXR)(Y,Z)

+ (VYR)(Z,X)

+ (VZR){X,Y)

= 0

In local coordinates the above identities take the form Rijkl = —Rjikt — —RijikRijkt — Rklij)kt + R\jk + R'ktj = 0-

R

(ViR)Le + {ViR)U + (VtR)L = 0. Proof (a) It follows immediately from the definition of R as an End(TM)-valued skew-symmetric bilinear map (X, Y) i-> R(X,Y). (b) We have to show the symmetric bilinear form Q(U, V) = g(R(X, Y)U, V) + g(R(X, Y)V, U) is trivial. Thus it suffices to check Q(U, U) = 0. We may as well assume that [X, Y] = 0 since (locally) X, Y can be written as linear combinations (over C°°(M)) of commuting vector fields. (E.g. X = Xldi). Then Q(U, U) =

ff((VxVy

- VYVX)U,

U).

We compute Y(Xg(U,U))

=

2Yg(VxU,U)

= 2
VYU)

and similarly X(Yg(U, U)) = 2g(VxVYU,

U) + 2g(VxU,

VyU).

Subtracting the two equalities we deduce (b). (c) As before, we can assume the vector fields X, Y, Z pairwise commute. The 1st Bianchi identity is then equivalent to VXVYZ

- VY\7XZ

+ VZVXY

- VXVZY

+ VYVZX

- VZVYX

= 0.

The identity now follows from the symmetry of the connection: V x ^ = VYX (d) We will use the following algebraic lemma ([44], Chapter 5).

etc.

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Lemma 4.2.3 Let R : E x E x E x E ->• 1R be a quadrilinear map on a real vector space E. Define S(Xi, X2, X3, X4) = R{X\, X2, X3, X 4 ) + R(X2, X3, Xi, X4) +

R(X3,Xi,X2,X4).

If R satisfies the symmetry conditions R{X\, Xi-, X3, X4) = —R(X2, X\, X3, Xi) R(Xi, X2, X3, X^ — —R(Xi,X2,

X$, X3)

then R(Xi, X2, X3, X4) — R(X3, X^ X\, X2) =

7s \S(Xi, X2, X3, Xi) — S(X2, X3, Xi, X\) — S(X3, Xi, X\, X2) + S(Xi, X3, X\, X2)} •

The proof of the lemma is a straightforward (but tedious ) computation which is left to the reader. The Riemann curvature R = g{R{Xx, X2)X3, Xi) satisfies the symmetries required in the lemma and moreover, the 1st Bianchi identity shows the associated form 5 is identically zero. This concludes the proof of (d). (e) This is the Bianchi identity we established for any linear connection (see Exercise 3.3.3). • The Riemann curvature tensor is the source of many important invariants associated to a Riemann manifold. We begin by presenting the simplest ones. Definition 4.2.4 Let (M, g) be a Riemann manifold with curvature tensor R. Any two vector fields X, Y on M define an endomorphism of TM by U M- R(U, X)Y. The Ricci curvature is the trace of this endomorphism i.e. Ric {X, Y) = tr {U

H->

R(U,

We view it as a (0,2)-tensor (X, Y) H* Ric {X, Y) e

X)Y). CX(M).

If (x1,..., xn) are local coordinates on M and the curvature R has the local expression R = (Rekij) then the Ricci curvature has the local description Ric=(i*y) = £ ^ « I

The symmetries of the Riemann curvature imply that Ric is a symmetric (0,2)-tensor (as the metric).

The Riemann

147

curvature

Definition 4.2.5 The scalar curvature s of a Riemann manifold is the trace of the Ricci tensor i.e. in local coordinates s = gijRio = gijR^j where as usual (g*i) is the inverse matrix of (
X,Y

Let (M, g) be a Riemann manifold and p e M. e TPM set KP{X,Y)-

For any linearly independent

| X A y |

where \X A Y\ denotes the Gramm determinant

\XAY\

(X,X) (Y,X)

(X,Y) (Y,Y)

and is non-zero since X and Y are linearly independent. (\X A Y\ll2 measures the area of the parallelogram in TPM spanned by X and Y.) Exercise 4.2.1 Let X, Y,Z,W 6 TPM such that span(X,Y)= span(Z,W) is a 2dimensional subspace of TPM prove that KP(X, Y) = KP(Z, W). According to the above exercise the quantity KP{X, Y) depends only upon the 2plane in TPM generated by X and Y. Thus Kp is in fact a function on Gr2(p) the grassmannian of 2-dimensional subspaces of TPM. Definition 4.2.6 The function Kp : Gr2{p) —> M defined above is called the sectional curvature of M at p. Exercise 4.2.2 Prove that Gr2(M) = disjoint union of Gr2{p)

p GM

can be organized as a smooth fiber bundle over M with standard fiber Gr2^n(R), n = d i m M such that if M is a Riemann manifold Gr2{M) 3 (p;7r) i-> Kp(n) is a smooth map. §4.2.2

Examples

Example 4.2.7 Consider again the situation discussed in Example 4.1.12. Thus G is a Lie group and (•, •) is a metric on £ G satisfying (od{X)Y, Z) = -(Y,

ad(X)Z)

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geometry

In other words, (•, •) is the restriction of a bi-invariant metric m on G. We have shown that the Levi-Civita connection of this metric is

VXY = ±[X,Y] vx,re£ G . We can now easily compute the curvature R(X, Y)Z = i {[X, [Y, Z}} - [Y, [X, Z}}} - ±[[X, Y], Z] (Jacobi identity) = ±[[X, Y], Z] + hjY, [X, Z\\ - ~{Y, [X, Z}} - ±][X, Y], Z\ =

-\[lX,Y},Z}.

We deduce (R(X,Y)Z,W)

= -\{[[X,Y),Z],W)

= -\{{X,Y],ad{Z)W)

=

=

\(ad(Z)[X,Y},W)

-\{[X,Y],[Z,W}).

Now let 7r € Gr2(g) for some g £ G. If (X, Y) is an orthonormal basis of TT (viewed as left invariant vector fields on G) then the sectional curvature along 7r is Kg(rr) =

^{[X,Y},{X,Y})>0

Denote the Killing form by K(X,Y) — —tr (ad(X)ad(Y)). To compute the Ricci curvature we pick an orthonormal basis E\,...,En of £ Q . For any X = X'Ei,Y = YjEj e £ G we have Ric(X,Y)

=

±ti(Z^[[X,Z\,Y])

= \Yt([[X,Ei],Y],Ei)) =

—ZiadWfrEhEi)

i

i

= - £ < [ X , Et\, [Y, Ei]) = \ Y,{ad{X)Eu i

ad(Y)Et)

i

= -^(ad(Y)ad{X)Ei,Ei)

= -^tr

(ad(Y)ad(X))

i

=

\K{X,Y).

In particular, on a compact semisimple Lie group the Ricci curvature is a symmetric positive definite (0,2)-tensor (in fact it is a scalar multiple of the Killing metric.)

The Riemann

curvature

149

We can now easily compute the scalar curvature. Using the same notations as above we get s

= \ E

Ric

($> Ei) = \ £ <Ei> Ei)-

In particular, if G is a compact semisimple group and the metric is given by the Killing form then the scalar curvature is S(K) = - dimG. 4

Remark 4.2.8 Many problems in topology lead to a slightly more general situation than the one discussed in the above example namely to metrics on Lie groups which are only left invariant. Although the results are not as "crisp" as in the bi-invariant case many nice things do happen. For details we refer to [57]. Example 4.2.9 Let M be a 2-dimensional Riemann manifold (surface) and consider local coordinates on M, (x 1 ,^ 2 ). Due to the symmetries of R Rijkl = —Rijlk = Rklij

we deduce that the only nontrivial component of the Riemann tensor is R = #1212The sectional curvature is simply a function on M K = y-R12n

= rs(s)

where \g\ = d e t a -

in this case K is known as the total curvature or the Gauss curvature of the surface. If in particular M is oriented and the form dx1 A dx2 defines the orientation, we can construct a 2-form e

(ff) = —Kdvg 9 = — s(g)dv g9 = v; 27r"""' ~47r"

"

j^Ruudx1

A dx2

2TTJ|5|

e(g) is called the Euler form associated to the metric g. We want to emphasize that this form is defined only when M is oriented. We can rewrite this using the pfaffian construction of Subsection 2.2.4. The curvature R is a 2-form with coefficients in the bundle of skew-symmetric endomorphisms of TM so we can write R = A dvg

A =

0 #2112

i?1212 0

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Riemannian

geometry

Assume for simplicity that (x 1 ,^ 2 ) are normal coordinates at a point q € M. Thus at q, \g\ = 1 since d\, d2 is an orthonormal basis of TqM. Hence at q, dvg = dx1 A dx2 and ^-g(R(di,d2)d2,d1)dx1Adx2.

e{g) = Z7T

Hence we can write
*!

^Pfa{-R).

The Euler form has a very nice interpretation in terms of holonomy. Assume as before that (x1 ,x2) are normal coordinates at q and consider the square St = [0, \/i\ x [0, y/i\ in the (xx,x2) plane. Denote the (counterclockwise) holonomy along dSt by %t- This is an orthogonal transformation of TqM and with respect to the orthogonal basis (d\, 82) of TqM it has a matrix description as _ _ cos9{t) l ~ [ sin0(i)

—sm0{t) cos 9(t)

'

The result in Subsection 3.3.4 can be rephrased as sin 9{t) = -tg(R{dud2)d2,

di) + 0{t2)

so that #1212 =
Cartan's moving frame method

This method was introduced by Elie Cartan at the beginning of this century. Cartan's insight was that the local properties of a manifold equipped with a geometric structure can be very well understood if one knows how the frames of the tangent bundle (compatible with the geometric structure) vary from one point of the manifold to another. We will begin our discussion with the model case of K n . Example 4.2.10 Consider Xa = X^-^, a = 1, ...,n an orthonormal moving frame on K" where (x1, ...,xn) are the usual cartesian coordinates. Denote by (0°) the dual coframe i.e. the moving frame of T*K™ defined by

e°(x0) = si. The 1-forms measure the infinitesimal displacement of a point P with respect to the frame (X Q ). Note that the TM-valued 1-form 9 = 9aXa is the differential of the identity map id : M™ —> R" expressed using the given moving frame.

The Riemann

curvature

151

Introduce the 1-forms wjj defined by dX& = w%Xa

(4.2.1)

where we set

We can form the matrix valued 1-form (u>f) which measures the infinitesimal rotation suffered by the moving frame (Xa) following the infinitesimal displacement x i-> x + dx. In particular u) = (u/g) is a skew-symmetric matrix since 0 = d(Xa,Xf,)

= (uXa,Xp)

+

(Xa,uXi,).

Since 9 = did we deduce d& = 0 and we can rewrite this as dOa = 90 A ul or dO = -w A 9.

(4.2.2)

Using d/'Xp = 0 in (4.2.1) we deduce dw'p = — w" A UJJ or equivalently dw = - w A w. (4.2.3) The equations (4.2.2)-(4.2.3) are called the structural equations of the Euclidian space. The significance of these structural equations will become evident in a little while. We now try to perform the same computations on an arbitrary Riemann manifold M. We choose a local orthonormal moving frame (Xa) and construct similarly its dual coframe (9°). Unfortunately, there is no natural way to define dXa to produce the forms w2 entering the structural equations. We will find them using a different (dual) search strategy. Proposition 4.2.11 (E. Cartan)There exists a collection of 1-forms a/g uniquely defined by the requirements (a)wjj = -wg. {b)d9a = 0? A 0$, Va. Proof Uniqueness Suppose a/g satisfy the conditions (a)&(b) above. Then there exist functions / ^ and gpy such that

<4 = ftJ1

152

Riemannian

geometry

Then the condition (a) is equivalent to

(al) fa = - / £ while (b) gives

(bl) fa ~ f°p = 9$, The above two relations uniquely determine the / ' s in terms of the g's via a cyclic permutation of the indices a, j3, 7 #7 = ^ Existence

7

+

«&-*&)

(4-2.4)

Consider the functions g^ defined by dea = \ g ^ ^

{g% =

-galfi).

Next define CJ2 = fS^O1 where the / ' s are given by (4.2.4). We let the reader check that the forms wjj satisfy both (a) and (b). • The reader may now ask why go through all this trouble. What have we gained by constructing the forms w and after all, what is their significance? To answer these questions consider the Levi-Civita connection V. Define u>0 by VX^ = u}pXa. Hence VXlX0 = w|(X 7 )X t t . Since V is compatible with the Riemann metric we deduce in standard manner that (a) C$ = -<%.

The differential of 9" can be computed in terms of the Levi-Civita connection (see Subsection 4.1.5) and we have d9a(X0,X7)

= X09a{X7)

- X^iXp)

- 9a(VXpX7)

(use 9a{Xp) = 6% = const) = -9a(v67{X0)Xs)

+ +

9a(VXiX0) 9a{^(X7)Xs)

= tf(X7) - &°{XP) = (P A tfXXf,, X7). Thus the w's satisfy both conditions (a) and (b) of Proposition 4.2.11 so that we must have u>$ = w$.

In other words, the matrix valued 1-form (ujf) is the 1-form associated to the LeviCivita connection in this local moving frame. In particular, the 2-form Q = (did + u A u )

The Riemann

153

curvature

is the Riemannian curvature (see the computation in Example 3.3.11). The structural equations of a Riemann manifold take the form d9 = -u> A 9 du) + w A u) = fl. Comparing these with the euclidian structural equations we deduce another interpretation of the Riemann curvature: it measures "the distance" between the given Riemann metric and the euclidian one". We refer to [70] for more details on this aspect of the Riemann tensor. The technique of orthonormal frames is extremely versatile in concrete computations. Example 4.2.12 We will use the moving frame method to compute the curvature of the hyperbolic plane i.e. the upper half space H + = {(!,»);

y>0}

endowed with the metric g = —(dx2 + dy2). {ydx, ydy) is an orthonormal moving frame and (6X = -dx, 0y = -dy) is its dual coframe. We compute easily d0x = d(-dx) y

= ~dx Ady = (-dx) A 6V y2 y

dOy = d(-dy) = 0 = (--dx) y

A 0*.

y

Thus the connection 1-form in this local moving frame is ' 0 1.

.v

•

_ i

y

0

dx.

The Riemann curvature is

0

Q = du> =

i

i y

--V o

dy A dx :

0 1

-1 0

y2

The Gauss curvature is 1

K = y-Mto(dx,dy)dy,dx) = y4{-

=-1.

154

Riemannian

§4.2.4

geometry

The geometry of submanifolds

We now want to apply Cartan's method of moving frames to discuss the the local geometry of submanifolds of a Riemann manifold. Let (M, g) be a Riemann manifold of dimension m and S a ^-dimensional submanifold in M. The restriction of g to S induces a Riemann metric on S. We want to analyze the relationship between the Riemann geometry of M (assumed to be known) and the geometry of S with the induced metric. Denote by V M (resp. V s ) the Levi-Civita connection on (M, g) (resp on (S,g|s). The metric g produces an orthogonal splitting of vector bundles (TM)\S=TS@NS. Ns is called the normal bundle of S <-> M. Thus, a section of (TM) \s, that is a vector field X of M along S, splits into two components: a tangential component XT and a normal component X". Now choose a local orthonormal moving frame (Xi, ...,Xk;Xk+i, •••,Xm) such that the first k vectors (Xi,...,Xk) are tangent to S. Denote the dual coframe by (6a)i k. Denote by (nf), (1 < a, f5 < m) the connection 1-forms associated to V M in this frame and let crfi, (1 < a, ft < k) be the connection 1-forms of V 5 . We will analyze the structural equations of M restricted to S <-> M. d0a = 0" A [fp 1 < a, 0 < m.

(4.2.5)

We distinguish two situations. A . 1 < a < k. Since 9P | s = 0 for /? > k the equality (4.2.5) yields

^Q = E ^ A ^ ,

I$ = -I4

l
0=1

The uniqueness part of Proposition 4.2.11 implies that along S aap=Hap

l
This can be equivalently rephrased as VSXY = ( V ^ F ) T B.

VX, Y e Vect (5).

k < a < m. We deduce 0=i

At this point we want to use the following elementary result.

(4.2.6)

The Riemann

155

curvature

Exercise 4.2.3 (Cartan's Lemma) Let V be a d-dimensional real vector space and consider p linearly independent elements u^, ...,ui p € AlV, p < d. If #1, ..., 8p 6 AlV are such that

then there exist scalars Aij, 1 < i,j f c , 1 < (3, 7 < fc satisfying //37

=

A/3-

^ = /^7Now form We can view 01 as a symmetric bilinear map Vect (5) x Vect (5) -> C°°(Ars). If[/,VeVect(S)

then

U = Ul3Xl3=efi{U)Xp

\
V = VXJ=8"<(V)XJ

l<-y
W. n = EIE fe /^7(*o) ^ 1 *A A>A: (_ 0

\ 1

1

)

= E(E/W'W A>k \ £

/

The last term is precisely the normal component of VyU so that we have established (v£ft/)" = W(U, V) = *tt(V, £/) = ( v £ v ) " .

(4.2.7)

9t is called the 2nd fundamental form of S <-» M. (The first fundamental form is the induced metric). There is an alternative way of looking at 9t. Choose U, V G Vect(S), N G C°°(NS). We have (g(-, •) = <-,•»

<9i(y, v), iv) = <(v£f v)", JV) = «

v, TV>

156

Riemannian =

geometry

V%{V,N)-{V,V}fN) =

-(V,(V^N)T).

We have thus established -(V, (V2?JV)T> = (Vl(U, V), N) = («rt(V, U), N) = -(U, (vjf JV)T>.

(4.2.8)

The 2nd fundamental form can be used to determine a relationship between the curvature of M and that of S. More precisely we have the following celebrated result. T H E O R E M A E G R E G I U M (Gauss) Let RM (resp. Rs) denote the Riemann curvature of (M, g) (resp. (S, g | s ) . Then for any X, Y,Z,Te Vect (5) we have (RM{X,Y)Z,T)

=

(RS(X,Y)Z,T)

+(w(x, z), ^(y; T)) - {m(x, r), m(Y, z)). Proof

(4.2.9)

Note that \>%Y = VsxY +

Vl(X,Y).

We have RM{X, Y)X = [V^, V f )Z -

V$X]Z

= V^f ( V y Z + *rt(y, Z)) - V f ( V | Z + OT(X, Z)) - Vf x , y ] Z - 9t([X, Y], Z). Take the inner product with T of both sides above. Since 9fl(-, •) is iVs-valued we deduce using (4.2.6)-(4.2.8) (RM(X, Y)Z, T) = ( V ^ V y Z , T) + (V$f*tt(Y, Z), T)

-(vf

V^Z.T)

-

= ([v|, v y ]z, T) - <«n(y, z),m(x, T » +(9fi(A-,z),«n(y;T)) - (vfx,y]z,T). This is precisely the equality (4.2.9). • . The above result is especially interesting when 5 is a transversally oriented hypersurface i.e. a codimension 1 submanifold such that the normal bundle Ns is trivial. (Locally, all hypersurfaces are transversally oriented since Ns is locally trivial by definition). Pick n an orthonormal frame of Ns (i.e. a length 1 section of Ns) and pick an orthonormal moving frame (Xi,..., X m _i) of TS- Then (Xi,..., Xm-i, n) is an orthonormal moving frame of (TM) \s and the second fundamental form is completely described by

mfi(x,Y) = {m(x,Y),n).

The Riemann

157

curvature

?&n is a bona-fide symmetric bilinear form and moreover according to (4.2.8) we have 9t*(X,Y) = - < V £ n , Y ) =

-(VPn,X).

Gauss formula becomes in this case (RS(X,Y)Z,T)

= {RM(X,Y)Z,T)

-

OT«(X,Z) mg{Y,Z)

*fln(X,T) 9fla(y,T)

Let us further specialize and assume M = K m . Then (RS{X,Y)Z,T)

=

Ol«(y,T)

(4.2.10)

*Hn(F,^)

In particular, the sectional curvature along the plane spanned by X, Y is {Rs(X,Y)Y,X)=Vln(X,X)-Vlii(Y,Y)-\Vln(X,Y)\2. This is a truly remarkable result. On the right-hand-side we have an extrinsic term (it depends on the "space surrounding S ") while in the left-hand-side we have a purely intrinsic term (which is defined entirely in terms of the internal geometry of 5). Historically, the extrinsic term was discovered first (by Gauss) and very much to Gauss surprise (?!?) one does not need to look outside S to compute it. This marked the beginning of a new era in geometry. It changed dramatically the way people looked at manifolds and thus it fully deserves the name of The Golden (egregium) Theorem of Geometrv. Example 4.2.13 (Quadrics in M3) Let A be a selfadjoint, invertible —> linear operator with at least one positive eigenvalue. This implies the quadric QA = {ueR3

; {Au,u) = 1}

is nonempty and smooth (use implicit function theorem to check this). Let MO € QAThen TU0QA = {x e R 3 ; (Au0,x) = 0} = {AUQ)2-. QA is a transversally oriented hypersurface in R3 since the map QA B U >-» Au defines a nowhere vanishing section of the normal bundle. Set n = rfiAu. Consider °

\Au\

(eo,ei,e2) an orthonormal frame of R 3 such that e0 = n(uo). Denote the cartesian coordinates in R3 with respect to this frame by (a: 0 ,! 1 ,^ 2 ) and set <% = ^ - . Extend (ei, e2) to a local moving frame of TQA near MoThe second fundamental form of QA at u0 is ^h{di,dj)

=

(difi,dj)\

158

Riemannian

geometry

We compute dtti = di ( 4 ^ 7 ) = 9i((Au, Au)-V2)Au \\Au\J _

(diAu, Au) -Au\Au\V2

Hence (

yin(di,dj)\uo 1

+

~Adti \Au\

" mdiAu-

1 {AdiU, ej) |„ 0 \Au0\ e

(diu,Aej)\V0=

iy

Aej).

(4.2.11)

We can now compute the Gaussian curvature at uQ. Kun =

1 \Au0\*

(i4e!,ei) (Ae2,e1)

(Aeue2) (Ae2,e2)

In particular, when A = r 2I so that QA is the round sphere of radius r we deduce K

" = ^ Vlul

Thus, the round sphere has constant positive curvature. We can now explain rigorously why we cannot wrap a plane canvas around the sphere. Notice that when we deform a plane canvas the only thing that changes is the extrinsic geometry while the intrinsic geometry is not changed since the lengths of the "fibers" stays the same. Thus, any intrinsic quantity is invariant under "bending". In particular, no matter how we deform the plane canvas we will always get a surface with Gauss curvature 0 which cannot be wrapped on a surface of constant positive curvature! Gauss himself called the total curvature a "bending invariant". Example 4.2.14 (Gauss)Let E be a transversally oriented, compact surface in M3. (E.g. a connected sum of g tori). Note that the Whitney sum N^ © T E is the trivial bundle R | . We orient N% such that orientation JVS A orientation T E = orientation M3. Let n be the unit section of N-^ defining the above orientation. We obtain in this way a map <J5 : E -> S2 = {u e K3 ; |u| = 1} E 3 i 4 n(x) € S2. & is called the G a u s s map of E •->• S2. It really depends on how E is embedded in M3 so it is an extrinsic object. Denote by 91^ the second fundamental form of E •-> M3 and let (xl,x2) be normal coordinates at q 6 E such that orientationTgE = Si A 62-

The Riemann

159

curvature

Consider en the Euler form on E with the metric induced by the euclidian metric in M3. Then, taking into account our orientation conventions, we have 2irex(d1,d2) = Rnn «a(9i,0i)

Wn(di,di)

(4.2.12)

Now notice that din = -«tta(Qj, dx)di - Vlgidi, d2)d2. We can think of n,di \q and d2 \q as defining a (positively oriented) K3. The last equality can be rephrased by saying that the derivative of the Gauss map (8>t : T,E —> Ts(,)5 acts according to $ ^ -9ta(Qj, 9i)9i - 0 ^ ( 9 i ; 9 2 )9 2 . In particular, we deduce <25, preserves (reverses) orientations •<=>• i?i2i2 > 0 (< 0)

(4.2.13)

because the orientability issue is decided by the sign of the determinant 1 0 0 -Vflnidudi) 0 -*fln(d2,di)

0 -mn(dl,d2) -QM^.ft)

At q, di _L 9 2 so that (a ; n, 8,-fi) = gna(ft, 9001^(9,, 9X) + 5 ^ ^ , ^ S t ^ a , - , 9 2 ). We can rephrase this coherently as an equality of matrices {d\fi, d\n) (d2n, dxn) Vln(di,di) ^(92,90

(din, d2n) (d2n, d2n)

Qflfl(9i,92) Vln(d2,d2)

*>Mdi,ai) OTa^.ft)

^ln(9i,9 2 ) 5rin(9 2 ,9 2 )

Hence <

yin(dudl) ^n(9i,92)

VU(dud2) ^-(92,92)

( 9 ^ , d\n) (din, d2n)

(din, d2n) (d2n, d2n)

(4.2.14)

If we denote by dv0 the metric volume form on S2 induced by the restriction of the euclidian metric on M3 we see that (4.2.12) and (4.2.14) put together yield 27:\es(di,d2)\

= \dv0(din,d2n)\

=

\dv0{&,{di),&,{d2))\.

160

Riemannian

Using (4.2.13) we get

e = & Edv =

* h * ° h&^£s2-

geometry

(4 2 15)

--

This is one form of the celebrated Gauss-Bonnet theorem . We will have more to say about it in the next subsection. Note that the last equality offers yet another interpretation of the Gauss curvature. Prom this point of view the curvature is a "distortion factor". The Gauss map "stretches" an infinitesimal parallelogram to some infinitesimal region on the unit sphere. The Gauss curvature describes by what factor the area of this parallelogram was changed. §4.2.5

The Gauss-Bonnet theorem

We conclude this chapter with one of the most beautiful results in geometry. Its meaning reaches deep inside the structure of a manifold and can be viewed as the origin of many fertile ideas. Recall one of the questions we formulated at the beginning of our study: explain unambiguously why a sphere is "different" from a torus. This may sound like forcing our way in through an open door since everybody can "see" they are different. Unfortunately this is not a conclusive explanation since we can see only 3-dimensional things and possibly there are many ways to deform a surface outside our tight 3D Universe. The elements of Riemann geometry we discussed so far will allow us to produce an invariant powerful enough to distinguish a sphere from a torus. But it will do more than that. Theorem 4.2.15 (Gauss-Bonnet: preliminary version)Let S be a compact oriented surface without boundary. If po a n d
Jse90(S) = Jse91(S). Hence the quantity fs £g(S) is independent of the Riemann metric g so that it really depends only on the topology of S!!! The idea behind the proof is very natural. Denote by gt the metric gt = go + t{gi — So)- We will show

! / / * = ov* e [ o,i}. It is convenient to consider a more general problem. Definition 4.2.16 Let M be a compact oriented manifold. For any Riemann metric g on E define <£(
s{g)dvg

The Riemann

161

curvature

where s(g) denotes the scalar curvature of {M,g). Einstein functional.

<£(g) is called the Hilbert-

We have the following remarkable result. Lemma 4.2.17 Let M be a compact oriented manifold without boundary and gl = {g\j) be a 1-parameter family of Riemann metrics on M depending smoothly upon t e R. Then

-£<*(
JM

I

Vt.

In the above formula (•, -)t denotes the inner product induced by gl on the space of symmetric (0,2)-tensors while the dot denotes the i-derivative. The (0, 2)-tensor -s{x)gij{x) is called the Einstein tensor of (M, g). Definition 4.2.18 A Riemann manifold (M,g) is said to be Einstein if the metric g satisfies Einstein's equation Ric 9 =

^s(g)g.

Remark 4.2.19 The attribute Einstein usually refers to a larger class of Riemann manifolds satisfying the condition Ric9(:r) =

\(x)g(x)

for some smooth function A € Cca(M). We refer to [12] for a comprehensive presentation of this subject. In this book we reserve the name Einstein to the special case described in the above theorem. Exercise 4.2.4 Consider a 3-dimensional Riemann manifold {M,g). Show that Rijkt = £ik9ji - £ie9jk + £ji9ik — £jk9u + 7}{9ie9jk — 9ik9ji)In particular, this shows that on a Riemann 3-manifold the full Riemann tensor is completely determined by the Einstein tensor. Exercise 4.2.5 (Schouten-Struik, [67])Prove that the scalar curvature of an Einstein manifold of dimension > 3 is constant.

162

Riemannian

geometry

Hint Use the 2nd Bianchi identity. Notice that when (S, g) is a compact oriented Riemann surface two very nice things happen. (i) (S, g) is Einstein. (Recall that only #1212 is nontrivial). (ii)C(p) = 2 / s £ 8 . Theorem 4.2.15 is thus an immediate consequence of Lemma 4.2.17. Proof of the lemma We will produce a very explicit description of the integrand (*(»')*¥ = = (-(s(
(4.2.16)

of ^<£(
dk9ij = o.

Vidj = 0 f} fc = 0.

(4.2.17) (4.2.18)

If a = a,dx' is a 1-form then at q 5tja = Yldiai.

{5 = *d*)

(4.2.19)

i

In particular, for any smooth function u (AMiSu)(q)

= -J29i2u. i

Set h = (hi,) = (5) = (<)„). his a, symmetric (0,2)-tensor. Its ^-trace is the scalar tTgh =

giihij=tiC-1(h)

(4.2.20)

The Riemann curvature

163

where C is the lowering the indices operator defined by g. In particular a t q

%x~ah = Y,h«-

(4-2-21)

The curvature of g is given by R-ikj

=

~^Hjk

=

d>^ij — dfiik + ^mk^Tj ~ r m j T ^ .

The Ricci tensor is T> "ij

T>k o pA: o -pk . pfc T^m pA: r m — "ikj — °ki ij " °jL ik + l mkL ij ~ l mjl ik-

Finally the scalar curvature is s = tr 9 Rtj = gl3Rij

= j* (^r* - art + r^r^ - r^.rs). Derivating s at t = 0 and then evaluating a t q we obtain

s = gij (&f£ - djt 1k) + s» (fci* - dji*) = g^Rij + Yl{dktki-ditkk).

(4.2.22)

i

The term
(4.2.23)

To evaluate the derivatives T's we use the known formula? Fv = \
+ djgik)

which upon derivation at t = 0 yield II? = \ {di9ik ~ dkgij + dj9ik) + ±gkm (dihjk - dkh{j + djhik) = ^ (dihim ~ dmhij + djhim). We substitute (4.2.23) -(4.2.24) in (4.2.22) and we get a t q s = - X) hijRij + - Y,(dkdihik ij

i,k

- dk2hu + dkdihik)

(4.2.24)

164

Riemannian ~ o Jl(di2hkk

-

geometry

didkhik)

= - X) hijRij - X di2hkk + X didkhik i,j

i,k

i,k

= -(Ric,
+ Y,didkhik.

(4.2.25)

i,k

To get a coordinate free description of the last term note that a t q {Vkh)(di,dm)

= dkhim.

The total covariant derivative V/i is a (0,3)-tensor. Using the g-trace we can construct a (O.l)-tensor trj(V/t) = t r ( £ r ! v / i ) where C^1 is the raising the indices operator defined by g. In the local coordinates (x') we have Using (4.2.17) and (4.2.19) we deduce that the last term in (4.2.25) can be rewritten (at q ) as 5trg(Vh) = Strg{Vg). We have thus established that s = -(Ric,g)g

+ AM,gtrgg

+ 6trg(Vg).

(4.2.26)

The second term of the integrand (4.2.16) is a lot easier to compute. dvg = ±yj\g\dxx

• ••dxn

so that

dv9 =

±\\g\-^jt\g\dxl.-.dx\

A t q the metric is Euclidian, gy = 6ij and jt\g\

= Y,9u = \g\-tTS(g) =

\g\{g,g)g\g\.

Hence £(ff) = /

{~s(9)9-Ric{g),g)gdvg

+ / AM^gttg g +

5tig(Vg)dvg.

JM

The Green formula shows the last two terms vanish and the proof of the Lemma is concluded. D

The Riemann

curvature

165

Definition 4.2.20 Let S be a compact, oriented surface without boundary. define its Euler characteristic as the number

We

where g is an arbitrary Riemann metric on S. The number l

9(S) =

-(2-X(S))

is called the genus of the surface. Remark 4.2.21 According to the theorem we have just proved the Euler characteristic is independent of the metric used to define it. Hence, the Euler characteristic is a topological invariant of the surface. The reason for this terminology will become apparent when we discuss DeRham cohomology, a Z-graded vector space naturally associated to a surface whose Euler characteristic coincides with the number defined above. So far we have no idea whether x(S) is even an integer. Proposition 4.2.22 X (S

Proof

2

) = 2 and

X (T

2

) = 0.

To compute x{S2) we use the round metric 50 for which K = 1 so that x(5 2 ) = ^ /

s 2

^ o = ^ a r e a 9 0 ( 5 2 ) = 2.

To compute the Euler characteristic of the torus we think of it as an abelian Lie group with a bi-invariant metric. Since the Lie bracket is trivial we deduce from the computations in Subsection 4.2.2 that its curvature is zero. This concludes the proof of the proposition. D Proposition 4.2.23 If 5; (i=l,2) are two compact oriented surfaces without boundary then x(S1#S2)=X(5i) + x(52)-2. Thus upon iteration we get

x(5i#---#5 t ) = E x ( S i ) - 2 ( * - l ) i=l

for any compact oriented surfaces S i , . . . , 5 * . In terms of genera, the last equality can be rephrased as i=l

166

Riemannian

f

geometry

f

Figure 4.3: Generating a hot-dog-shaped

surface

In the proof of this proposition we will use special metrics on connected sums of surfaces which require a preliminary analytical discussion. Consider / : (—4,4) —> (0, oo) a smooth, even function such that (i) f{x) = l for \x\ < 2. (ii) f(x) = yjl-(x + 3) 2 for x € [-4, -3.5]. (iii) f(x) = yjl-{x3) 2 for x € [3.5,4]. (iv) / is non-decreasing on [-4,0]. One such function is graphed in Figure 4.3 Denote by Sf the surface inside R3 obtained by rotating the graph of / about the x-axis. Because of properties (i)-(iv) 5 / is a smooth surface diffeomorphic to S2. (One such diffeomorphism can be explicitly constructed projecting along radii starting at the origin). We denote by g the metric on Sf induced by the euclidian metric in R 3 . Since Sf is diffeomorphic to a sphere f Kgdvg =

2-KX(S2)=4ir.

Set

sf = sfn {±x > o} sf = sf n {±x > i} Since / is even we deduce

j

Kgdvg = '-[ K.gdvg 2Jsf

!

2%.

(4.2.27)

The Riemann curvature

167

Figure 4.4: Special metric on a connected sum On the other hand, on the neck C = {\x\ < 2} the metric g is locally euclidian g = dx2 + d62 so that over this region Kg = 0. Hence f Kgdvg = 0.

(4.2.28)

Proof of the Proposition 4.2.23 Let A C Si (i = 1,2) be a local coordinate neighborhood diffeomorphic with a disk in the plane. Pick a metric & on Si such that (Di,gi) is isometric with Sj!" and (1)2,52) is isometric to Sj. The connected sum 5 i # 5 2 is obtained by chopping off the regions Sj from D\ and SJ1 from Di and (isometrically) identifying the remaining cylinders 5 * n {|x| < 1} = C and call O the overlap created by gluing (see Figure 4.4). Denote the metric thus obtained on S\#S2 by g. We can now compute

+

hfs2K*dv°>-hLK°dv°

168

Riemannian (4

geometry

= 7 ) x(5 1 )+x(5 2 )-2.

The proposition is proved. • Corollary 4.2.24 (Gauss-Bonnet)Let E 9 denote the connected sum of p-tori. (By definition E 0 = S 2 . Then X(E 9 ) = 2 - 2 5 and g(Eg) = g. In particular a sphere is not diffeomorphic to a torus. Remark 4.2.25 It is a classical result that the only compact oriented surfaces are the connected sums of g-tori (see [52]) so that the genus of a compact oriented surface is a complete topological invariant.

Chapter 5 Elements of the calculus of variations This is a very exciting subject lieing at the frontier between mathematics and physics. The limited space we will devote to this subject will hardly do it justice and we will barely touch its physical significance. We recommend to anyone looking for an intellectual feast the Chapter 16 in vol.2 of [27] which in our opinion is the most eloquent argument for the raison d'etre of the calculus of variations.

5.1 §5.1.1

The least action principle l-dimensional Euler-Lagrange equations

From a very "dry" point of view the fundamental problem of the calculus of variations can be easily formulated as follows. Consider a smooth manifold M and denote L : t x TM —> R be a smooth function called the lagrangian. Fix two points po, p\ 6 M. For any piecewise smooth path 7 : [0,1] —> M connecting these points we define its action by 5(7) = 5 L ( 7 ) = / 1 L ( t , 7 ( t ) , 7 ( t ) ) d * . Jo

In the calculus of variations one is interested in those paths as above with minimal action. Example 5.1.1 Given U : K3 —> R a smooth function called the potential we can form the lagrangian L(q, q) : K3 x R 3 ^ TR 3 -» R as L = Q — U = kinetic energy — potential energy = -m|<j| 2 — U(q). (m is the mass). The action of a path (trajectory) 7 : [0,1] —> R3 is a quantity called the newtonian action. 169

170

Calculus of variations

Example 5.1.2 To any Riemann manifold (M, g) one can naturally associate two lagrangians L\, L2 : TM -» M defined by Li(q,v)=gq{v,v)1'2

(veTqM)

L2{q,v) =

-g(v,v).

We see that the action defined by L\ coincides with the length of a path. The action defined by L2 is called the energy of a path. Before we present the main result of this subsection we need to introduce a bit of notation. Tangent bundles are very peculiar manifolds. Any collection (ql,...,qn) of local coordinates on a smooth manifold M automatically induces local coordinates on TM. Any point in TM can be described by a pair (q, v) where q £ M, v £ TqM. Furthermore, v has a decomposition v = v>di

(* = £ ) .

We set q' = v' so that The collection (q1,... ,qn; q1,... ,qn) defines local coordinates on TM. These are said to be holonomic local coordinates on TM. This will be the only type of local coordinates we will ever use. Theorem 5.1.3 (The least action principle) Let L : RxTM —> K be a lagrangian and p0, pi € M two fixed points. If 7 : [0,1] —> M is a smooth path such that (1)7(1) = Pi, i = 0 , l .

(ii) S.L(7) < £x(7) for any smooth path 7 : [0,1] —> M joining p0 to pi. Then 7 satisfies the Euler-Lagrange equations d d

r

,

L(

.,

7 7) =

5*F *' '

dL

^-

More precisely, if (q';^) are holonomic local coordinates on TM such that "/(t) = {ql{t)) and 7 = {
JtW{t,qi'^)

=

dq~k{t'9i'4J)

fc = 1

'-'n=

dimM

-

Definition 5.1.4 A path 7 : [0,1] —>• M satisfying the Euler-Lagrange equations with respect to some lagrangian L is said to be an extremal of L.

171

The least action principle

To get a better feeling of these equations consider the special case discussed in Example 5.1.1 L = \m\q\2 - U(q). The Euler-Lagrange equations become mq = -VU(q).

(5.1.1)

These are precisely Newton's equation of the motion of a particle of mass m in the force field —VU generated by the potential U. In the proof of the least action principle we will use the notion of variation of a path. Definition 5.1.5 Let 7 : [0,1] —> M be a smooth path. A variation of 7 is a smooth map a = as(t) : (-e,e) x [0,1] ->• M such that «o(t) = 7(i). If moreover, as(i) = Pi Vs (i=0,l) then we say that a is a variation rel endpoints. Proof of Theorem 5.1.3

Let as be a variation of 7 rel endpoints. Then SL(CHO) < SL(as)

Vs

so that -H s =o SL(as)

= 0.

Assume for simplicity that the image of 7 is entirely contained in some open coordinate neighborhood U with coordinates (q1,..., qn). Then for very small \s\ we can write a s (t) = (9 I (s,i)) and

_ | = (,f ( s ,*)).

Following the tradition, we set da

9q'

d

das

dq3 d

5a is a vector field along 7 called infinitesimal variation (see Figure 5.1). In fact, the pair (5a;6a) € T{TM) is a vector field along t H-> {j{t),j{t)) £ TM. Note that 6a = jt5a and at endpoints 6a = 0. Exercise 5.1.1 Prove that if t >-> X(t) e T^t)M is a smooth vector field along 7 such that X[t) = 0 for t = 0,1 then there exists at lest one variation rel endpoints a such that 6a = X.

172

Calculus of variations

Figure 5.1: Deforming a path rel endpoints (Hint: Use the exponential map of some Riemann metric on M.) We compute (at s = 0)

Jo dql

Jo dqi

s

Integrating by parts in the second term we deduce

The last equality holds for a n y variation a. From Exercise 5.1.1 deduce it holds for any vector field Sa'di along 7. At this point we use the following classical result of analysis. " If f(i) is a continuous function on [0,1] such that f f{t)g(t)dt Jo

= 0 V 5 eC 0 °°(0,l)

then / is identically zero." Using this result in (5.1.2) we deduce the desired conclusion. • Remark 5.1.6 (a) In the proof of the least action principle we used a simplifying assumption i.e. the image of 7 lies in a coordinate neighborhood. This is true locally and for the above arguments to work it suffices to choose only a special type of variations, localized on small intervals of [0,1]. In terms of infinitesimal variations

173

The least action principle

this means we need to look only at vector fields along 7 supported in local coordinate neighborhoods. We leave the reader fill in the details. (b) The Euler-Lagrange equations were described using holonomic local coordinates. The minimizers of the action (if any) are objects independent of any choice of local coordinates so that the Euler-Lagrange equations have to be independent of such choices. We check this directly. If (xl) is another collection of local coordinates on M and (xl\£i) are the coordinates induced on TM then we have the transition rules rM .

so that d _dxi__d_ 'dq~i ~~ a ^ & r J

+

d V .k d dqkdqi9 dxi

_d__dx^_d__ dxj d dqi ~ dqi dxi ~~ dqi dxi Then

aL_a^aL dq* ~ dq{ dxi d (dL\

ay .kdL dqkdq'9

_ d (dxi

dxi ' dL\

Jt yWj ~ dl {dq* OH J d2xi .kdL dxi_d_ (dL\ ~~ dqkdqiq 'd±i + 'dq7dl \d&) ' We now see that the Euler-Lagrange equations in the ^-variables imply the EulerLagrange in the x-variable i.e. these equations are independent of coordinates. The aesthetically conscious reader may object to the way we chose to present the Euler-Lagrange equations. These are intrinsic equations we formulated in a coordinate dependent fashion. Is there any way of writing these equation so that the intrinsic nature is visible "on the nose" ? If the lagrangian L satisfies certain nondegeneracy conditions there are two ways of achieving this goal. One method is to consider a natural nonlinear connection V L on TM as in [59] . The Euler-Lagrange equations for an extremal y(t) can then be rewritten as a "geodesies equation" V^7The example below will illustrate this approach on a very special case when L is the lagrangian L^ defined in Example 5.1.2 in which the extremals are precisely the geodesies on a Riemann manifold. Another far reaching method of globalizing the formulation of the Euler-Lagrange equation is through the Legendre transform which again requires a nondegeneracy

174

Calculus of variations

condition on the lagrangian. Via the Legendre transform the Euler-Lagrange equations become a system of first order equations on the cotangent bundle T*M known as the Hamilton equations. These equations have the advantage that can be formulated on manifolds more general than the cotangent bundles namely on symplectic manifolds. These are manifolds carrying a closed 2-form whose restriction to each tangent space defines a symplectic duality (see §2.2.4.) Much like the geodesies equations on a Riemann manifold the Hamilton equations carry many informations about the structure of symplectic manifolds and are currently the focus of very intense research. For more details and examples we refer to the monographs [5] or [21]. E x a m p l e 5.1.7 Let [M,g) be a Riemann manifold. We will compute the EulerLagrange equations for the lagrangians L\, L2 in Example 5.1.2. L2(q,q) =

^gaiqWq3

so that dL2

W

., = 9jkq

dL2

W

=

1 dgij .t., qq

2W -

The Euler-Lagrange equations are

^+w^

= ^q3-

(5L3)

Since gkmgjm = 5™ w e get

^""(tN§0«'^°-

5

< -">

When we derivate with respect to t the equality gikQ' = gjkq3

we deduce km^gj^-i-i

9n

qq

_

1

km f%*

w -2

9n

_, dgjk\

Ai

w+wiqq

We substitute this equality in (5.1.4) and we get

*-+WiM?-|?)«=°-

<5i5»

The coefficient of qlq* in (5.1.5) is none other than the Christoffel symbol r™ so this equation is precisely the geodesic equation.

175

The least action principle Consider now the lagrangian L\{q,q) = {gijCfqi)1!2- Note that the action

[Pl L{q,q)dt •>po

is independent of the parameterization t >-» q(t) since it computes the length of the path. Thus, when we express the Euler-Lagrange equations for a minimizer 70 of this action we may as well assume it is parameterized by arclength i.e.

I-Vol = 1 . The Euler-Lagrange equations for L\ are d dt

gkjgj

\l9itfq?

_

% W

2\l9irfq>'

Along the extremal we have gutftf = 1 (arclength parameterization) so that the previous equations can be rewritten as

Jt^q)

= 2Wqq-

We recognize here the equation (5.1.3) which, as we have seen, is the geodesic equation in disguise. This fact almost explains why the geodesies are the shortest paths between two nearby points. §5.1.2

Noether's conservation principle

This subsection is intended to offer the reader a glimpse at a fascinating subject touching both physics and geometry. We first need to introduce a bit of traditional terminology commonly used by physicists. Consider a smooth manifold M. The tangent bundle TM is usually referred to as the space of states or the lagrangian phase space. A point in TM is said to be a state. A lagrangian L : R x TM —> K associates to each state several meaningful quantities. (a) The generalized momenta: pi = jpj. (b) The energy: H = p,g' — L. (c) The generalized force: F = f t . This terminology can be justified by looking at the lagrangian of a classical particle in a potential force field F = —VU

L = lm\q\2-U(q). The momenta associated to this lagrangian are the usual kinetic momenta of the Newtonian mechanics Pi = mql

176

Calculus of variations

while H is simply the total energy H=1-m\q\2

+ U(q).

It will be convenient to think of an extremal for an arbitrary lagrangian L(t, q, q) as describing the motion of a particle under the influence of the generalized force. Proposition 5.1.8 (Conservation of energy) Let -y(t) be an extremal of a time independent lagrangian L = L(q, q). Then the energy is conserved along 7 i.e.

Proof

By direct computation we get ! * ( 7 , 7 ) = ! ( * * * - L)

dt \dqi)q = { l ( i ) - | }

dqit} ?

= °

dqiq

dqiQ

( ^ Euler-Lagrange).

•

At the beginning of this century (1918) Emmy Noether discovered that many of the conservation laws of the classical mechanics had a geometric origin: they were, most of them, reflecting a symmetry of the lagrangian!!! This became a driving principle in the search for conservation laws and in fact, conservation became synonymous with symmetry. It eased the leap from classical to quantum mechanics and one can say it is a very important building block of quantum physics in general. In the few instances of conservation laws where the symmetry was not apparent the conservation was always "blamed" on a "hidden symmetry". What is this Noether principle? To answer this question we need to make some simple observations. Let X be a vector field on a smooth manifold M defining a global flow $ ' . This flow induces a flow \l>' on the tangent bundle TM defined by *'(z,i>) = ($'(*).*'.(«))• One can think of \P' as defining an action of the group ffi on TM and thus defining a symmetry of M. Example 5.1.9 Let M be the unit round sphere S2 C ffi3. The rotations about the 2-axis define a 1-parameter group of isometries of S2 generated by Jg (9 is the longitude on S2).

177

The least action principle

Definition 5.1.10 Let L be a lagrangian on TM and X a vector field on M. The lagrangian L is said to be X-invariant if L o $ ' = L Vt. Denote by X G Vect (TM) the infinitesimal generator of * ' and by Lx the Lie derivative on TM along A". We see that L is X-invariant iff CXL = 0. We describe this derivative using the local coordinates (ql,(f).

Set (ql(t), q>(t)) =

jt\t=0q-(t)=XkSl To compute Jj | (= o ${t)-£j

we use the definition of the Lie derivative on M d

q

-7

d

T

-Jt ^ 'X

dqk

=

d

t-i

Lx{q

\

]

W

^ dqk J dqqj ~

-4{

dXi d dq1 dqi

since dqj/dqi = 0 on TM. Hence dqi

dXi d dqk dip'

Corollary 5.1.11 L is X-invariant iff vi3L

x

.kdXi

0L

+q

w ww~

Theorem 5.1.12 (E. Noether) If the lagrangian L is X-invariant then the quantity

* = *£-*» is conserved along the extremals of L. Proof

Consider an extremal 7 = 7(9'(i)) of L. We compute d

o /

M

d

\ v i , u^dL\

dX> .k0L

k = (Euler - Lagrange) -7ri-q -7rrr + X{—~=0 oqk dq1 oq*

vld

(dL\

by Corollarv 5.1.11.

•

The classical conservation of momentum law is a special consequence of Noether's theorem.

178

Calculus of variations z

r

J

—

/

—/

—^22

y

*

Figure 5.2: A surface of revolution Corollary 5.1.13 Consider a lagrangian L = L(t,q,q) component of the force is zero) then ^ of the momentum is conserved).

on K n . If f^ = 0 (the z-th

= 0 along any extremal (the i-th component

To prove this it suffices to take X — A in Noether's conservation law.. The conservation of momentum has an interesting application in the study of geodesies. Example 5.1.14 (Geodesies on surfaces of revolution) Consider a surface of revolution S in E 3 obtained by rotating about the 2-axis the curve y = f(z) situated in the yz plane. If we use cylindrical coordinates (r, 8, z) we can describe S as r = f(z). In these coordinates the euclidian metric in M3 has the form ds2 = dr2 + dz2 + r2d92. We can choose (z, 9) as local coordinates on S then the induced metric has the form 9s = {1 + (f'(z))2}dz2

+ f2(z)d92

= A(z)dz2 + r2d92. (r =

f(z))

The lagrangian defining the geodesies on 5 is L=

2 2 2 \(A z + r 9 )

We see that L is independent of 9: ^ = 0 so that the generalized momentum d L

2n

—- = r 9 39 is conserved along the geodesies.

179

The variational theory of geodesies

This fact can be given a nice geometric interpretation. Consider a geodesic 7 = (z(t),8(t)) and compute the angle between 7 and ^ . We get cos,= C0S

<7,a/w>

=

\i\-\d/d6\

i!j r| 7 |

i.e. rcos = r 2 # | 7 | _ 1 . The conservation of energy implies that |7| 2 = 2L = H is constant along the geodesies. We deduce the following classical result. Theorem 5.1.15 (Clairaut)On a a surface of revolution the quantity r cos <> / is constant along any geodesic. cf> € (—7r, n) is the oriented angle the geodesic makes with the parallels z = const. Exercise 5.1.2 Describe the geodesies on the round sphere S2 and on the cylinder {x2 + y2 = 1} C t t 3 .

5.2

The variational theory of geodesies

We have seen that the paths of minimal length between two points on a Riemann manifold are necessarily geodesies. Conversely, given a geodesic joining two points qo, qi it may happen it is not a minimal path. This should be compared with the situation in calculus when a critical point of a function / may not be a minimum or a maximum. To decide this issue one has to look at the second derivative. This is precisely what we intend to do in the case of geodesies. This situation is a bit more complicated since the action functional

S = \jWdt is not defined on a finite dimensional manifold. It is a function defined on the "space of all paths" joining the two given points. With some extra effort this space can be organized as an infinite dimensional manifold. We will not attempt to formalize these prescriptions but rather follow the ad-hoc, intuitive approach of [55]. §5.2.1

Variational formulae

Let M be a connected Riemann manifold and consider p,q G M. Denote by fiM = fiM(M) the space of all continuous, piecewise smooth paths 7 : [0,1] —> M connecting p to q. An infinitesimal variation of a path 7 e fiM is a continuous, piecewise smooth vector field V along 7 such that ^(0) = 0 and V(l) = 0. The space of infinitesimal variations of 7 is an infinite dimensional linear space denoted by T =T n

180

Calculus of variations

Definition 5.2.1 Let 7 G QP:q. A variation of 7 is a continuous map a = as(t) : ( - e , e ) x [0,1] ->• M such that (i) Vs € (-e,e), as e Qp>,. (ii) There exists a partition 0 = ta < t\ • • • < tk-i < tk = 1 of [0,1] such that the restriction of a to each (—£,£•) x (£;_!,£;) is a smooth map. Every variation a of 7 defines an infinitesimal r

variation

def das

Exercise 5.2.1 Given 7 e T T construct a variation a such that 5a = V. Consider now the energy functional

E-.nM->R

1 r1.

E(1) = -Jg I^WI2(ii-

Fix 7 6 QPig and let a be a variation of 7. The velocity j(t) has a finite number of discontinuities so that the quantity A t 7 = lira (j(t + h)-

j(t - h))

ft-»0+

is nonzero only for finitely many t's. Theorem 5.2.2 (The first variation formula) E.(Sa)

=' 4- l»=o E{a.) = - £<(<*")(*). ^ 7 ) - l\&a, V±i)dt. (5.2.1) as ( Vo <" V denotes the Levi-Civita connection. (Note that the right-hand-side depends on a only through 5a so it is really a linear function on T7.) Proof

Set as = ^-.

We derivate under the integral sign using the equality — |d s | 2 = 2 ( V B Ads,ds) us >

and we get

d r1 — \s=oE(as)= / (VjLas,as)\s=0dt. as Jo a" Since the vector fields Jj and J^ commute we have V j _ ^ = V a . f j - Let 0 = to < t2 < • • • < tk = 1 be a partition of [0,1] as in Definition 5.2.1. Since as = 7 for s = 0 we conclude

^(H = E£ i (Vafo, 7).

181

The variational theory of geodesies We use the equality — (<5a,7) = (Va_5a,j)

+ (5a,V

±j)

to integrate by parts and we obtain

E.(5a) = J2(6a,i) |*;_x " £ f {6<*,V±i)dt. i=iy«i-i

i=i

*'

This is precisely equality (5.2.1). D Definition 5.2.3 A path 7 € Qp,q is critical if

£,(10 =0 w e r 7 . Corollary 5.2.4 A path 7 G fiP), is critical if and only if it is a geodesic. Exercise 5.2.2 Prove the above corollary. Note that a priori a critical path may have a discontinuous first derivative. The above corollary shows that this is not the case; the criticality also implies smoothness. This is a manifestation of a more general analytical phenomenon called elliptic regularity. We will have more to say about it in Chapter 9. The map E, : T 7 —> E, 5a H-» E„(5a) is called the first derivative of E at 7 e f2p>?. We want to define a second derivative of E in order to address the issue raised at the beginning of this section. We will imitate the finite dimensional case which we now briefly analyze. Let / : X —> R be a smooth function on the finite dimensional smooth manifold X. If x0 is a critical point of / i.e. df(xo) = 0 then we can define the hessian at x0 / „ : TX0X x TXoX -> 1 as follows. Given Vi, V2 6 TX0X consider a smooth map (si, S2) •-> a(si, s 2 ) € X such that dc*. a(0,0) = zo and — (0,0) = VI, i = 1,2. (5.2.2) Now set h,{VuV2)-

dsids2

-|(o,o).

Note that since x 0 is a critical point of / the hessian /**(Vi, V2) is independent of the function a satisfying (5.2.2). We now return to our energy functional E : i1p^q —¥ M. Let 7 G Qpq be a critical path. Consider a 2-parameter variation of 7 aSUS2{t)

: U x [0,1] ->• M.

U is the tiny square (—e, e) x (—e,e) C R2 and a0,o = 7- a is continuous and has second derivatives everywhere except maybe on finitely many "coordinate planes Sj = const or t = const. Set ^ a = ~ |(0,o), 2 = 1,2. Note that 5ia 6 T 7 .

182

Calculus of variations

Exercise 5.2.3 Given Vi,V2 € T 7 construct a 2-parameter variation a such that Vi = Sta. We can now define the hessian of E at 7 by E„(6ia,8,a)

=

d g

^

|(o.o) -

Theorem 5.2.5 (The second variation formula) £„(o"ia, 52a) = - ^ ( f c a , A(<5lQ> - ^ V 2 a , V 2 J^i<* - #(7,5 i a )j)dt

(5.2.3)

where i? denotes the Riemann curvature. In particular, E„ is a bilinear functional onT7. Proof

According to the first variation formula we have 9E

v-^.,.

rllr

. da.

_

9a. ,

Hence _d2E ds fl - / {V_a_5 2a,Va_j)dtds st 7o

i

r1 da / {52a,V_a_Va_—)dt. Bt Bs

Jo

i

ot

(5.2.4)

Since 7 is a geodesic A t 7 = 0 and V.a.7 = 0. Using the commutativity of Jj with ^ we deduce

Finally, the definition of the curvature implies V A V I = V 3 V_a_ + R(Sxa, 7).

aj!

at

at dax

Putting all the above together we deduce immediately the equality (5.2.3). • Corollary 5.2.6 Ett(Vi, V2) = E„{V2, Vi) W i , V2 e TT

183

The variational theory of geodesies §5.2.2

Jacobi fields

In this subsection we will put to work the elements of calculus of variations presented so far. Let (M,g) be a Riemann manifold and p,q € M. Definition 5.2.7 Let 7 € fip,9 be a geodesic. A geodesic variation of 7 is a smooth map a3(t) : ( - e , e ) x [0,1] -> M such that a0 = 7 and 11-> a,(t) is a geodesic for all s. We set as usual 5a = | ^ | s = 0 . Proposition 5.2.8 Let 7 € fiPi, be a geodesic and (as) a geodesic variation of 7. Then the infinitesimal variation 5a satisfies the Jacobi equation V2t5a = R(i,6a)j

(V t = V J L ) .

Proof

^.-v.(v.|)-v.(v.|) Definition 5.2.9 A smooth vector field J along a geodesic 7 is called a Jacobi field if it satisfies the Jacobi equation

Vp =

R(j,J)j.

Exercise 5.2.4 Show that if J is a Jacobi field along a geodesic 7 then there exists a geodesic variation as of 7 such that J = 5a. Exercise 5.2.5 Let 7 € fiPi, and J a vector field along 7. (a) Prove that J is a Jacobi field if and only if

E„(J,V} = 0 W 6 T

r

(b) Show that J € T 7 (i.e. J is a vector field along 7 vanishing at endpoints) is a Jacobi field if any only if S*,(J, W) = 0 for all vector fields W along 7. (It is important to emphasize the point that not all vector fields W along 7 belong to T 7 .)

Exercise 5.2.6 Let 7 6 fiPi9 be a geodesic. Define 3 P to be the space of Jacobi fields V along 7 such that V(p) = 0. Show that d i m 3 P = d i m M and moreover, the evaluation map

ev, :ZP^TPM is a linear isomorphism.

V^VtV{p)

Calculus of variations

Figure 5.3: The poles are conjugate along meridians. Definition 5.2.10 Let j(t) be a geodesic. Two points 7(^1) and 7^2) on 7 are said to be conjugate along 7 if there exists a nontrivial Jacobi field J along 7 such that

Jfc) = 0, i = l,2. Example 5.2.11 Consider 7 : [0, 2n] -> S2 a meridian on the round sphere connection the poles. One can verify easily (using Clairaut's theorem) that 7 is a geodesic. The counterclockwise rotation by an angle 9 about the 2-axis will produce a new meridian, hence a new geodesic jg. Thus (79) is a geodesic variation of 7 with fixed endpoints. £7 is a Jacobi field vanishing at the poles. We conclude that the poles are conjugate along any meridian (see Figure 5.3). Definition 5.2.12 Let 7 e Qpq be a geodesic. 7 is said to be nondegenerate if q is not conjugated to p along 7. The following result (partially) explains the geometric significance of conjugate points. Theorem 5.2.13 Let 7 G Qp,g be a nondegenerate, minimal geodesic. Then p is conjugate with no point on 7 other than itself. In particular, a geodesic segment containing conjugate points cannot be minimal ! Proof We argue by contradiction. Let pi = 7(^1) be a point on 7 conjugate with p. Denote by Jt a Jacobi field along 71[0 tt] such that J 0 = 0 and Jtl = 0. Define V e T-, by f Jt , t e [o,t!] \ 0 , t>h We will prove that Vt is a Jacobi field along 7 which contradicts the nondegeneracy of 7. In the sequel I denotes the length.

185

The variational theory of geodesies Step 1 E„(U,U)>0

V£/GT 7 .

(5.2.5)

Indeed, let as denote a variation of 7 such that 5a = U. One computes easily that j^E{a.*)

=

2E„{U,V).

Since 7 is minimal for any small s we have l(as2) > l{a0) so that

E(as>) > \ (jf1 |ds,|dt)2 = i[(a s2 ) 2 > |l(«o)2 = |l(T)2 =

E(aa).

Hence — | J=0 £ ( « . » ) > 0. This proves (5.2.5). Step 2 Ett(V, V) = 0. This follows immediately from the second variation formula and the fact that the nontrivial portion of V is a Jacobi field. Step 3 E„{U,V) = 0 V l / € T 7 . From (5.2.5) and Step 2 we deduce 0 = E„{V, V) < £U(V + TU,V

+ TU)

= fu(r)

Vr.

Thus, T = 0 is a global minimum of /c/(r) so that fu(0) = 0. Step 3 follows from the above equality using the bilinearity and the symmetry of E„. The final conclusion (that V is a Jacobi field) follows from Exercise 5.2.5. • Exercise 5.2.7 Let 7 : M -> M be a geodesic. Prove that the set {t € R ; 7(£) is conjugate to 7(0)} is discrete. Definition 5.2.14 Let 7 G f2p>, be a geodesic. We define its index, denoted by ind (7), as the cardinality of the set C 7 = {t e (0,1) ; is conjugate to 7(0)} which by Exercise 5.2.7 is finite.

186

Calculus of variations

Theorem 5.2.13 can be reformulated as follows: the index of a nondegenerate minimal geodesic is zero. The index of a geodesic obviously depends on the curvature of the manifold. Often, this dependence is very powerful. Theorem 5.2.15 Let M be a Riemann manifold with non-positive sectional curvature i.e. {R(X, Y)Y, X) < 0 \/X, Y € TXM Vz e M. (5.2.6) Then for any p,q e M and any geodesic 7 G n P i „ ind(7) = 0. Proof It suffices to show that for any geodesic 7 : [0,1] —• M the point 7(1) is not conjugated to 7(0). Let Jt be a Jacobi field along 7 vanishing at the endpoints. Thus

V2tJ =

R(j,J)j

so that

[\v2J,J)dt

Jo

= f\R{j,J)j,J)dt Jo

= - f1 {R( J, 7)7, J)dt. Jo

We integrate by parts the left-hand-side of the above equality and we deduce

(v,J,J)\l - f \vtj\2dt = Jo

[\R{J,J)J,J)dt. Jo

Since J{T) = 0 for r = 0,1 we deduce using (5.2.6)

f \VtJ?dt
Jo

This implies V ( J = 0 which coupled with the condition J(0) = 0 implies J = 0. The proof is complete. • The notion of conjugacy is intimately related to the behavior of the exponential map. Definition 5.2.16 Let / : X —> Y be a smooth map between the smooth manifolds x and Y. (a) A point i G X i s said to be critical if r a n k ( D / x : TXX -»• Tf{x)Y)

< minfdimT.X,

dimTf(x)Y}.

(b) A point y G Y is called a critical value for / if / _1 (2/) contains at least one critical point of / . (c) y € Y is a regular value if it is not a critical value.

187

The variational theory of geodesies

Theorem 5.2.17 Let (M, g) be a connected, complete, Riemann manifold and g0 € M. A point q G M is conjugated to go along some geodesic if and only if it is a critical value for the exponential map exp go : TqoM -> M. Proof Let q = exp w t ( « € TqoM). Assume first that q is a critical value for exp9o and v is a critical point. Then Dvexpqo(X) = 0 for some X € Tv(TqoM). Let v(s) be a path in TqoM such that v(0) = v and v(0) = X. The map (s, t) *-¥ expqo(tv(s)) is a geodesic variation of the radial geodesic -yv : 1i-+ exp 9o (tv). Hence the vector field W = — | s = 0 exp go (to(s)) is a Jacobi field along 7„. Obviously W(0) = 0 and moreover d_

W{1) = — | s = 0 expw(t>(a)) = Dv expJX)

= 0.

ds

On the other hand this is a nontrivial field since VtW

d_ = V 5 |,=o ^ exp go (to(s)) = V,w(a) | s = 0 / 0. dt

This proves
Tv(TqoM)

(5.2.7)

The existence of such a Jacobi field follows from Exercise 5.2.6. As in that exercise denote by 3 , 0 the space of Jacobi fields J along j v such that J(qo) = 0. The map T,(rwM)->3w

X^tJx

is a linear isomorphism. Thus, a Jacobi field along 7„ vanishing at both q0 and q must have the form Jx where X € Tv(TqoM) satisfies Dvexpqo(X) = 0. Since v is not a critical point this means X — 0 so that Jx = 0. • Corollary 5.2.18 On a complete Riemann manifold M with non-positive sectional curvature the exponential map exp g has no critical values for any q € M. We will see in the next chapter that this corollary has a lot to say about the topology of M. Consider now the following experiment. Stretch the round two-dimensional sphere of radius 1 until it becomes "very long". A possible shape one can obtain may look like in Figure 5.4. The long tube is very similar to a piece of cylinder so that the total

188

Calculus of variations

Figure 5.4: Lengthening a sphere. (= scalar) curvature is very close to zero, in other words is very small. The lesson to learn from this intuitive experiment is that the price we have to pay for lengthening the sphere is decreasing the curvature. Equivalently, a highly curved surface cannot have a large diameter. Our next result offers a more quantitative description of this phenomenon. Theorem 5.2.19 (Myers)Let M be an n-dimensional complete Riemann manifold. If for all X € Vect (M) Ric(X,X)>^-=i!|X|2 then every geodesic of length > irr has conjugate points and thus is not minimal. Hence diam (M) = sup{dist (p, q) ; p,q € M} < irr and in particular the Hopf-Rinow theorem implies that M must be compact. Proof Fix a minimal geodesic 7 : [0, £} —> M of length t and let e,(i) be an orthonormal basis of vector fields along 7 such that en(t) = j(t). Set q0 = 7(0) and qi = "/(£)• Since 7 is minimal we deduce £«(V,V)>0 Set W{ = sm(irt/£)ei.

WeT7.

Then

E„(Wi7 W{) = - j\Wi, Jo = j

sin2(irt/e) [fit2

VtW, +

R{Wi,i)j)dt

- (R(ei,j)-y,ei))

dt.

189

The variational theory of geodesies We sum over i = 1 , . . . , n — 1 and we obtain J2 E„{Wh Wi) = / sin2 Trt/e {[n - 1)TT 2 /£ 2 - Ric (7,7)) dt > 0. i=l

''0

If I > 7ir then (n - 1)TT 2 /£ 2 - Ric (7,7)

< 0

so that

£fi„(Wi)Wi)<0. Hence for at least for some W{ we have Ett(Wi,Wi) minimality of 7. The proof is complete. •

< 0. This contradicts the

Corollary 5.2.20 A semisimple Lie group G with positive definite Killing pairing is compact. Proof The Killing form defines in this case a bi-invariant Riemann metric on G. Its geodesies through the origin 1 € G are the 1-parameter subgroups exp(tX) which are defined for all t € R. Hence, by Hopf-Rinow theorem G has to be complete. On the other hand we have computed the Ricci curvature of the Killing metric and we found Ric (X, Y) = ^K{X, Y) VX, Y e £ G . The corollary now follows from Myers' theorem. • Exercise 5.2.8 Let M be a Riemann manifold and q £ M. For the unitary vectors X, Y £ TqM consider the family of geodesies ~/s{t)=expqt{X

+ sY).

Denote by Wt = Sjs the associated Jacobi field along 70(t). Form /(£) = \Wt\2. Prove the following. (a) Wt = Dtx exp g (y) = Frechet derivative of D H expq(v) (b) f(t) = t 2 - \(R(Y,X)X, Y)qt* + 0(f). (c) Denote by x 1 a collection of normal coordinates at q. Prove that 9u(x) =

6u-±Rkijixix?+0{3l).

detpy(x) = 1 - ^ x V

+0(3).

(e) Let D r («) = {x e TqM ; |x| < r } .

190

Calculus of variations

Prove that if the Ricci curvature is negative definite at q then volo (D r (q)) < vols (exp„(D r (g)) for all r sufficiently small, volo denotes the euclidian volume in TgM while vol9 denotes the volume on the Riemann manifold M. Remark 5.2.21 The interdependence "curvature-topology" on a Riemann manifold has deep reaching ramifications which stimulate the curiosity of many researchers. We refer to [23] or [55] and the extensive references therein for a presentation of some of the most attractive results in this direction.

Chapter 6 The fundamental group and covering spaces In the previous chapters we almost exclusively studied local properties of manifolds. This study is interesting only if some additional structure is present since otherwise all manifolds are locally alike. We noticed an interesting phenomenon: the global "shape" (topology) of a manifold restricts the types of structures that can exist on a manifold. For example, the Gauss-Bonnet theorem implies that on a connected sum of two tori there cannot exist metrics with curvature everywhere positive because the integral of the curvature is a negative universal constant. We used Gauss-Bonnet theorem in the opposite direction and we deduced-the intuitively obvious fact- that a sphere is not diffeomorphic to a torus because they have distinct genera. The Gauss-Bonnet theorem involves a heavy analytical machinery which obscures the intuition. Notice that S2 has a remarkable property which distinguishes it from T 2 : on the sphere any closed curve can be shrunk to a point while on the torus there exist at least two "independent" unshrinkable curves (see Figure 6.1). In particular this means the sphere is not diffeomorphic to a torus. This chapter will set the above observations on a rigorous foundation.

Figure 6.1: Looking for unshrinkable loops.

191

192

6.1

The Fundamental group and covering spaces

The fundamental group

§6.1.1

Basic notions

In the sequel all topological spaces will be locally path connected spaces. Definition 6.1.1 (a)Let X and Y be two topological spaces and / o , / i : X —¥ Y continuous maps, /o and / i are said to be homotopic if there exists a continuous map F:[0,l]xX^Y (t,x)^Ft{x) such that Fi = fa for i = 0,1. We write this as f0 ^h / i (b) Two topological spaces X, Y are said to be homotopically equivalent if there exist maps / : X —> Y and g : Y —> X such that / o j ~ f t 1Y and g o / ~ h lx. We write this X ~^ Y. (c) A topological space is said to be contractible if it is homotopically equivalent to a point. Example 6.1.2 The unit disk in the plane is contractible. The annulus {1 < \z\ < 2} is homotopically equivalent with the unit circle. Definition 6.1.3 (a) Let X be a topological space and x0 £ X. A loop based at x0 is a continuous map 7 : [0,1] ->• X such that 7(0) = 7(1) = x0. The space of loops in X based at x0 is denoted by Q(X,x0). (b) Two loops 70,71 : I -¥ X based at x0 are said to be homotopic rel x0 if there exists a continuous map r:[0,l]x/->A-

(i,s)^r((s)

such that ri(s)=7i(a)

t = 0,l

and

{s^rt(s))eQ{x,x0) Vie [0,1]. We write this as 70 —xo 7iNote that a loop is more than a closed curve; it is a closed curve + a description of a motion of a point around the closed curve. Example 6.1.4 The two loops 71,72 : / -> C, jk(t) different though they have the same image.

= exp(2knt),

k = 1,2 are

The fundamental

193

group

Definition 6.1.5 (a) Let 71,72 be two loops based at x0 G X. The product of 71 and 72 is the loop 7l *7 2 (s)

7i *l2(s)

={

^2s) > °^sl/2

|72(2s_i)

, 1/2 < s < 1 •

The inverse of a based loop 7 is the based loop 7 " defined by 7-(s)

=7(l-s).

(c) The identity loop is the constant loop e l 0 (s) = xa. Intuitively, the product of two loops 71 and 72 is the loop obtained by first following 71 (twice faster) and then 72 (twice faster). The following result is left to the reader as an exercise. Lemma 6.1.6 Let a 0 c^l0 OJI, Po—XoPi based loops.. Then (a) a0*p0~xoa1 * fa.

an

d 7o—x07i be three pairs of homotopic

(b) a0 * QQ

(c) a0 * cxo~xoaa. (d) (a0 * Po) * 7o-x 0 Q o * (Po * 7o)Hence the product operation descends to an operation "•" on tt(X, Xo)/—x0~ t n e set of homotopy classes of based loops. Moreover the induced operation is associative, it has a unit and each element has an inverse. Hence (Q,(X,xo)/^xo, •) is a group. Definition 6.1.7 The group (£1{X, Zo)/—xa, •) is called the f u n d a m e n t a l g r o u p (or the Poincare group) of the topological space X and is denoted by TTI(X,X0). The image of a based loop 7 in ^ ( X , XQ) is denoted by [7]. The elements of ni(X, x0) are the "unshrinkable loops" discussed at the beginning of this chapter. The fundamental group ni(X,x0) "sees" only the connected component of X which contains a;0- To get more information about X one should study all the groups {ni(X,x)}xeXProposition 6.1.8 Let X and Y be two topological spaces x 0 € X and yo £ Y. Then any continuous map / : X —>• Y such that f(x0) = y0 induces a morphism of groups / , ITTip^Zo) ->7Ti(Y,2/ 0 )

satisfying the following functoriality properties. ( a ) ( l x ) , = l^(x,io)-

194

The Fundamental group and covering spaces

Figure 6.2: Connecting base points. (b)If

(X,x0)A(Y,y0)^(Z,z0) are continuous maps (such that f{x0) = y0 and g{yo) = ZQ) then (g o /)„ = gt o /„. (c) Let /o, / i : (X, x0) —> (Y, y0) be two base-point-preserving continuous maps. Assume f0 is nomotopic to f\ rel xo i.e. there exists a continuous map F : I x X —¥ Y, (t,x) i-> Ft(x) such that Fi(x) = fi(x) for i = 0,1 and Ft(x0) = yo- Then (/o). = ( / i ) . -

Proof

Let 7 G 0.(X, x0) Then 7(7) G fl(Y, j/ 0 ) and one can check immediately that 7^xoV => / ( 7 ) - W /(V)-

Hence the correspondence

n{X,x0)By^f{y)eQ{Y,yo) descends to a map / : ni(X,x0) —• ^ ( Y , j/ 0 )- This is clearly a group morphism. (a) and (b) are now obvious. We prove (c). Let / o , / i : {X,x0) -¥ {Y,yQ) be two continuous maps and Ft a homotopy rel x0 connecting them. For any 7 G fi(X, x0) A = /o(7)-M/i(7)=AThe above homotopy is realized by Bt = Ft(j).

•

A priori, the fundamental group of a topological space X may change as the base point varies and it almost certainly does if X has several connected components. However, if X is connected (and thus path connected since it is locally so) all fundamental groups ITI{X,x), x G X are isomorphic. Proposition 6.1.9 Let X be a connected topological space. Any continuous path a : [0,1] —> X joining x 0 to X\ induces an isomorphism a , :7ri(X,xo)

-^-K\(X,xx)

defined by a»([7]) = \pT * 7 * a] (see Figure 6.2).

The fundamental

195

group

Exercise 6.1.1 Prove Proposition 6.1.9. Thus, the fundamental group of a connected space X is independent of the base point modulo some isomorphism. We will write ni(X, pt) to underscore this weak dependence on the base point. Corollary 6.1.10 Two homotopically equivalent connected spaces have isomorphic fundamental groups. Example 6.1.11 (a) 7Ti(M",pi) ~ ^i{pt,pt) (b) 7Ti(annulus) ~ x^S1).

= {1}.

Definition 6.1.12 A connected space X such that iri(X,pt) simply connected.

= {1} is said to be

Exercise 6.1.2 Prove that the spheres of dimension > 2 are simply connected. Exercise 6.1.3 Let G be a connected Lie group. Define a new operation "* " on

n(G,l)by (o*/?)(a) = a(s)-/J(s) where • denotes the group multiplication. (a) Prove that a* ft ~i a * f3. (b) Prove that 7Ti(G, 1) is abelian. Exercise 6.1.4 Let E —> X be a rank r complex vector bundle over the smooth manifold X and let V be a flat connection on E i.e. F ( V ) = 0. Pick x0 € X and identify the fiber Exo with C . For any continuous, piecewise smooth 7 e il(X,xo) denote by T 7 = T 7 (V) the parallel transport along 7 so that T 7 £ GL (r, C). (a) Prove that a~Xo/3 =>• Ta =Tp. (b) Tpm =TaoTp. Thus, any fiat connection induces a group morphism T:7r1(X,a;o)^GL(7-,C)

J^T~\

(This morphism (representation) is called the monodromy of the connection). Example 6.1.13 We want to compute the fundamental group of the complex projective space CP". More precisely, we want to show it is simply connected. We will establish this by induction. For n = 1, CP 1 = S2 and by Exercise 6.1.2 the sphere S 2 is simply connected. We next assume CPfc is simply connected for k < n and prove the same is true for n. Notice first that the natural embedding C fc+1 "—> C" + 1 induces an embedding CP* ^-» CP". More precisely, in terms of homogeneous coordinates this embedding is given by [z0 : ... : zk] .-> [zo : ... : zk : 0 : . . . : 0] € C P n .

196

The Fundamental group and covering spaces

Choose as base point p t = [1 : 0 : . . . : 0] e OP" and let 7 € fi(OPn,pt). We may assume 7 avoids the point P = [0 : . . . : 0 : 1] since we can homotop it out of any neighborhood of P. We now use a classical construction of projective geometry. We project 7 from P to the "hyperplane" H = Q P " - 1 ^ OP". More precisely, if C = [-zo : • • • : zn] e OP" we denote by n(Q the intersection of the line P £ vvith the hyperplane %. In homogeneous coordinates "•(0 = [zo(l - zn) : ...: z„_i(l - zn) : 0] (= [z0 : . . . : z n _i : 0] when zn ^ 1). Clearly 7r is continuous. For t € [0,1] define ^(C) = [2o(l - tzn) : ... : z„_i(l - tzn) : (1 - t)zn]. Geometrically, 7rt flows the point C, along the line P(, until it reaches the hyperplane %. Note that irt(() = C Vt and V£ e W. Clearly 7r( is a homotopy rel p t connecting 7 = 71-0(7) to a loop 71 in % = O P " - 1 based at p t . Our induction hypothesis shows that 7x can be shrunk to p t inside % . This proves QP" is simply connected. §6.1.2

Of categories and functors

The considerations in the previous subsection can be very elegantly presented using the language of categories and functors. This brief subsection is a minimal introduction to this language. The interested reader can learn more about it from the monograph [51]. A c a t e g o r y is a triple C=(Ob, Horn, o) where (i) O b is a collection of elements called the objects of the category, (ii) Horn is a family of sets Horn (X, Y), one for each pair of objects X and Y. The elements of Horn (X, Y) are called the morphisms (or arrows) from X to Y. (iii) o is a collection of maps o : Horn (X, Y) x Horn (Y, Z) -» Horn (X, Z)

(f,9)^9°fwhich satisfies the following conditions. (Cl) For any object X there exists a unique element lx 6 Horn (X, X) such that f°lx

=f

9°lx=9

V/eHom(X,y)V5GHom(Z,X).

(C2) V/ € Horn {X, y ) j g Horn (Y, Z) h £ Horn (Z, W) ho(gof)

=

(hog)of.

The fundamental

group

197

R e m a r k 6.1.14 We note in passing that we were deliberately sloppy when we used the term "collection" when we introduced Ob. It may happen O b is not a set and one has to deal with thorny foundational issues which are beyond the scope of this book E x a m p l e 6.1.15 Top is the category of topological spaces. The objects are topological spaces and the morphisms are the continuous maps. (Top, *) is the category of marked topological spaces, the objects are pairs (X, *) where X is a topological space and * is a distinguished point of X. The morphisms

(A»4(y,*) are the continuous maps / : X —>Y such that /(*) = Jfr. pVect is the category of vector spaces over the field F. The morphisms are the F-linear maps. G r is the category of groups, while A b denotes the category of abelian groups. The morphisms are the obvious ones. iiMod denotes the category of left .R-modules, where R is some ring. Definition 6.1.16 Let €.\ and
con-

T : O b ( d ) x Horn (Ci) - • O b (C 2 ) x Horn (C 2 )

(X,f)»{F{X),F{f)) such that if X -4 Y then T(X) (i) T(lx) = 1HX)

^W F{Y) (resp.^(X) ^W ?{Y))

and

(ii) Ha) o HI) = Hg ° /) (resP. ?{f) ° r{9) = T{9 O /)). E x a m p l e 6.1.17 Let V be a real vector space. Then right tensoring with V defines a covariant functor ®V : ]g> Vect —> jgVect defined by U^U^V

L

%v U2 ® V).

{U1AU2)^{U1®V

On the other hand, taking the dual * : jjVect —¥ RVect V

H->

V

(U A V)

H>

(V* %• U")

defines a contravariant functor. The fundamental group construction of the previous is a covariant functor 7T! : (Top, *) ->• Gr. In Chapter 7 we will introduce other functors very important in geometry.

198

6.2 §6.2.1

The Fundamental group and covering spaces

Covering spaces Definitions and examples

As in the previous section we will assume that all topological spaces are locally path connected. Definition 6.2.1 (a) A continuous map ir : X —> Y is said to be a covering map if for any y £Y there exists an open neighborhood U such that 7r _1 ([/) is a disjoint union of open sets V, C X each of which is mapped homeomorphically onto U by n. Such an U is said to be an evenly covered neighborhood. The sets V, are called the sheets over U. (b) Let Y be a topological space. A covering space of Y is a topological space X together with a covering map 7r : X —> Y. If 7r : X —> Y is a covering map then for any y £ Y the set 7r_1(y) is called the fiber over y. Example 6.2.2 Let D be a discrete set. Then for any topological space X the product X x D is a covering of X with covering projection ir(x, d) = x. This type of covering space is said to be trivial. Exercise 6.2.1 Show that a fibration with standard fiber a discrete space is a covering. Example 6.2.3 The exponential map exp : t -> S 1 , t H exp(27ri<) is a covering map. However its restriction to (0, oo) is no longer a a covering map. (Prove this!). Exercise 6.2.2 Let (M, g) and (M, g) be two Riemann manifolds of the same dimension such that (M, g) is complete. Let <j> : M —> M a surjective local isometry i.e. $ is smooth and \v\g = \D(j>{v)\-g VveTM. Prove that 4> is a covering map. The above exercise has a particularly nice consequence. Theorem 6.2.4 (Cartan-Hadamard) Let (M, g) be a complete Riemann manifold with non-positive sectional curvature. Then for every point q € M the exponential map exp, : T„M -+ M is a covering map.

Covering spaces

199

Proof Consider the pull-back h = exp*(p). h is a symmetric non-negative definite (0,2)-tensor field on TqM. It is in fact a metric (i.e. it is positive definite) since the map exp. has no critical points due to the non-positivity of the sectional curvature. The lines 11-» tv through the origin oiTqM are geodesies of h and they are defined for all t € ffi. By Hopf-Rinow theorem we deduce that (TqM, h) is complete. The theorem now follows from Exercise 6.2.2. • Exercise 6.2.3 Let G and G two Lie groups of the same dimension and 4>: G —> G a smooth, surjective group morphism. Prove that 0 is a covering map. In particular, this explains why exp : R —> S1 is a covering map. Exercise 6.2.4 Identify S 3 C K4 with the group of unit quaternions

S3 = { ? e i ; M = i}. The linear space R3 can be identified with the space of purely imaginary quaternions

l 3 = a m l = {xi + yj + zk}. (a) Prove that Vq G S 3 qxq~l e J m H . (b) Prove that for any q G S3 the linear map x H-> qxq'1

T , : 3 m i - > 3mH

is an isometry so that Tq G S O ( 3 ) . Moreover the map S3Bq^Tqe

50(3)

is a group morphism. (c) Prove the above group morphism is a covering map. Example 6.2.5 Let M be a smooth manifold. A Riemann metric on M induces a metric on the determinant line bundle det TM. The sphere bundle of det TM (with respect to this metric) is a covering space of M called the orientation cover of M. Definition 6.2.6 Let X\ ^ Y and X2 ^ Y be two covering spaces of Y. A morphism of covering spaces is a continuous map F : X\ -¥ X2 such that 7r2 ° F = 7Ti i.e. the diagram below is commutative.

Y

200

T i e Fundamental group and covering spaces

If F is also a homeomorphism we say F is an isomorphism of covering spaces. Finally, if X A Y is a covering space then its automorphisms are called deck transformations. The deck transformations form a group denoted by Deck (X, 7r). Exercise 6.2.5 Show that Deck(Re-^? S 1 ) =* Z. Exercise 6.2.6 (a) Prove that the natural projection Sn —> is a covering map. (b) Denote by r i the tautological (real) line bundle over KP n . Using a metric on this line bundle form the associated sphere bundle S^Tik) —> MP". Prove that this defines a covering space isomorphic with the one described at part (a). §6.2.2

Unique lifting property

Definition 6.2.7 Let X A Y be a covering space and F : Z - > Y a continuous map. A lift of / is a continuous map F : Z —> X such that IT O F = f i.e. the diagram below is commutative. X

Y Proposition 6.2.8 (Unique P a t h Lifting)Let X A Y be a covering map, 7 : [0,1] —»• Y a path in Y and XQ a point in the fiber over j/o = 7(0), X0 € 7r_1(y0)- Then there exists at most one lift of 7, T : [0,1] —• Y such that T(0) = Xsy. Proof We argue by contradiction. Assume there exist two such lifts, I Y r ^ : [0,1] ->• X. Set

5 = {te[o,i] ; r!(t) = r2(t)}. S / 0 since 0 € 5. Obviously S is closed so it suffices to prove that it is also open. We will prove that there exists ro > 0 such that [0, r 0 ] C S. The general situation is entirely similar. Pick a small open neighborhood U of x0 such that n restricts to a homeomorphism onto w{U). There exists r0 > 0 such that 7t([0,ro] C U, i = 1,2. Since 7roFi = 7ror 2 we deduce r \ |[o,r0]= ^2 l[o,r0]- The proposition is proved. • Theorem 6.2.9 Let X 4 V be a covering space and / : Z - > F a continuous map, where Z is a connected space. Fix z0 € Z and XQ a point in 7T-1(?/o)i where y0 = f(z0). Then there exists at most one lift F : Z —>• X of / such that F(z0) = x0.

201

Covering' spaces

Proof For each z G Z let az be a continuous path connecting z0 to z. If Fi,F2 are two lifts of / such that Fi(z 0 ) = F2(z0) = xo then for any z G Z the paths i \ = F,(a z ) and T2 = F2{az) are two lifts of 7 = f(az) starting at the same point. From Proposition 6.2.8 we deduce that Tx = T2 i.e. Fi(z) = F2(z) for any z G Z. • §6.2.3

Homotopy lifting property

Theorem 6.2.10 (Homotopy lifting property)Let X 4 7 be a covering space, / : Z - > F a continuous map and F : Z —¥ X a lift of / . If ft: [0,1] x Z->Y

(i,z)>->ft ( (z)

is a homotopy of / (fto(z) = f{z)) then there exists a unique lift of ft H : [0,1] x Z -> X

(t, z) >-> Ht{z)

such that #0(2) = F(z). Proof For each z G Z we can find an open neighborhood C/z of z G Z and a partition 0 = t0 < £1 < • • • < tn = 1 (depending on z) such that ft maps [£;_i, ij] x Uz into an evenly covered neighborhood of ft^^z). Following this partition can now succesively lift ft \jxu, to a continuous map H = Hz : I x Uz —> X such that #„(£) = F(C), VC G Uz. By unique lifting property the liftings on / x UZl and / x UZ2 agree on / x (UZ1 n C/Z2) for any Zi, Z2 € Z and hence we can glue all these local lifts together to obtain the desired lift H on I x Z. • Corollary 6.2.11 (Path lifting property)Let X A Y be a covering space and 7 : [0,1] —• y a continuous path starting at y0 G Y. Then for every xo 6 7r_1 (j'o) there exists a unique lift r : [0,1] —¥ X of 7 starting at a;0. Proof 7(i).

Use the previous theorem with / : {pt} —»• F , / ( p t ) = 7(0) and ft((pt) = •

Corollary 6.2.12 Let X A y be a covering space, y0 G Y and 70,71 G fi(y, y0) If 70 and 7! are homotopic rel yo then any lifts r 0 , Ti which start at the same point also end at the same point i.e. F 0 (l) = T ^ l ) . Proof Lift the homotopy connecting 70 to 71 to a homotopy in X. By unique lifting property this lift connects r 0 to IY We thus get a continuous path r ( ( l ) inside the fiber 7r~1(y0) which connects r 0 ( l ) to T ^ l ) . Since the fibers are discrete this path must be constant. • Let X -^ y be a covering space and y0 G Y. Then for every x G ir~l{ya) and any 7(5) G fi'y, yo) denote by Tx(s) the unique lift of 7 starting at x. Set d

x-7=

4rx(i).

202

The Fundamental group and covering spaces

By Corollary 6.2.12 if w G Q,(Y,y0) is homotopic to 7 rel yo then x • 7 = x • u>.

Hence x • 7 depends only upon the equivalence class [7] € ^i(Y,y0).

Clearly

*-([7]-M) = ( * - [ 7 ] ) - M and X ' CyQ

X

so that the correspondence Tr~l(y0) x 7Ti(Y,2/o) 3 (a;,7) ^

x

• 7 e x • 7 G 7r_1(y0)

defines a right action of ^ ( F , 2/0) on the fiber 7r_1(yo)- This action is called the monodromy . The map x H-> X • 7 is called the monodromy along 7. Note that when Y is simply connected the monodromy is trivial. The map 7r induces a group morphism 7rt :7r x (X,a; 0 )->7ri(r,j/o) ^o G 7r_1(j/0)Proposition 6.2.13 TT» is injective. Proof Indeed, let 7 G tt(X, xo) such that 7r(7) is trivial in 7rj (F, yo)- The homotopy connecting n(j) to cOT lifts to a homotopy connecting 7 to the unique lift of eyo at xo which is txo. • §6.2.4

On the existence of lifts

Theorem 6.2.14 Let X A Y be a covering space, x0 € X, y0 = ir(x0) G Y, / : 2 4 V a continuous map and z0 € Z such that /(zo) = 2/o- Assume the spaces Y and .Z are connected (and thus path connected). / admits a lift F : Z —> X such that F(z0) = x0 if and only if /.MZ.^CTr.fa^.zo))-

(6.2.1)

Proof Necessity. If F is such a lift then using the functoriality of the fundamental group construction we deduce / , = 7r« o Ft. This implies the inclusion (6.2.1). Sufficiency For any z € Z choose a path j z from zo t o z- Then az = f(ryz) is a path from y0 to y = f(z). Denote by Az the unique lift of az starting at x0 and set F(z) = AZ{1). We claim that F is a well defined map. Indeed, let uz be another path in Z connecting z0 to z. Set \ z = }{LJZ) and denote by Az its unique lift in X starting at XQ. We have to show that A z (l) = Az{\). Construct the loop based at z0 /}z = uiz*~(z.

203

Covering spaces

f(/3z) is a loop in Y based at y^. From (6.2.1) we deduce that the lift Bz of /(/9Z) at x0 € X is a closed path (i.e. the monodromy along /(/3 2 ) is trivial). We now have A,(l) = B , ( l / 2 ) = A,(0) =

Az(l).

This proves F is a well defined map. We still have to show this map is also continuous. Pick z € Z. Since / is continuous, for every arbitrarily small, evenly covered neighborhood U of f(z) € Y there exists a path connected neighborhood V of z S Z such that f(V) C U. For any ( £ V pick a path a = a^ in V connecting z to £. Let UJ denote the path w = j z * a^ (go from zo to z along yz and then from z to C along ) starting at xQ. Since (/(C) G t/ we deduce that Q(l) belongs to the local sheet E, containing F(z), which homeomorphically covers [/. We have thus proved z £ V C F - 1 ( E ) . The proof is complete since the local sheets E form a basis of neighborhoods of F(z). • Definition 6.2.15 Let Y be a connected space. A covering space X A Y is said to be universal if X is simply connected. Corollary 6.2.16 Let X\ -^ Y (i=0,l) be two covering spaces of Y. Fix xt G Xt such that po(x0) = p(ii) = y0 € F . If X 0 is universal there exists a unique covering morphism F : Xo —> Xi such that F ( s 0 ) = ^iProof A bundle morphism F : X 0 —> Xi can be viewed as a lift of the map Po : XQ —> Y to the total space of the covering space defined by p\. The corollary follows immediately from Theorem 6.2.14 and the unique lifting property. D Corollary 6.2.17 Every space admits at most one universal covering space (up to isomorphism). Theorem 6.2.18 Let Y be a connected, locally path connected space such that each of its points admits a simply connected neighborhood. Then Y admits an (essentially unique) universal covering space. Sketch of proof Assume for simplicity that Y is a metric space. Fix y0 6 Y. Let Vyo denote the collection of continuous paths in Y starting at y0. It can be topologized using the metric of uniform convergence in Y. Two paths in Vyo are said to be homotopic rel endpoints if they can be deformed from one to another keeping the endpoints fixed. This defines an equivalence relation on Vyo. We denote the space of equivalence classes by Y and we endow it with the quotient topology. Define p : Y -> Y by P(W)=7(1)

V7G7V

Then (Y,p) is an universal covering space of Y.

•

Exercise 6.2.7 Finish the proof of the above theorem.

204

The Fundamental group and covering spaces

Example 6.2.19 K ^? 5 1 is the universal cover of Sl. More generally, Exp : Mn -> (*i, - - -, *«) •—> (exp(27rit 1 ),..., exp(27rztn)) is the universal cover of Tn. The natural projection p : Sn -> I P " is the universal cover of MP". Example 6.2.20 Let (M, g) be a complete Riemann manifold with non-positive sectional curvature. By Cartan-Hadamard theorem the exponential map exp 9 : TqM —> M is a covering map. Thus the universal cover of such a manifold is a linear space of the same dimension. In particular the universal covering space is contractible!!! We now have another explanation why Exp : Kn —• Tn is a universal covering space of the torus: the sectional curvature of the (flat) torus is zero. Exercise 6.2.8 Let (M,g) be a complete Riemann manifold and p : M —>• M its universal covering space. (a) Prove that M has a natural structure of smooth manifold such that p is a local diffeomorphism. (b) Prove that the pullback p*g defines a complete Riemann metric on M locally isometric with g. Example 6.2.21 Let (M,g) be a complete Riemann manifold such that Ric {X, X) > const.\X\2g

(6.2.2)

where const denotes a strictly positive constant. By Myers theorem M is compact. Using the previous exercise we deduce that the universal cover M is a complete Riemann manifold locally isometric with (M, g). Hence the inequality (6.2.2) continues to hold on the covering M. Myers theorem implies again that the universal cover M is compact!! In particular, the universal cover of a semisimple, compact Lie group is compact!!! §6.2.5

The universal cover and the fundamental group

Theorem 6.2.22 Let i ^ X b e the universal cover of a space X. Then ni{X,pt) Proof

^Deck(X^X).

Fix (,o € X and set x0 = p(£o)- There exists a bijection Ev:Deck(X) ->p_1(i0)

205

Covering spaces given by the evaluation

Ev(F)=F(£o). _1

For any £ 6 7r (xo) let 7^ be a path connecting £0 to £. Any two such paths are homotopic rel endpoints since X is simply connected (check this). Their projections on the base X determine identical elements in it\{X, xo). We thus have a natural map tfiDeckW-^psT.Xo)

F ^

p{-yF{io)).

$ is clearly a group morphism. (Think monodromy!). The injectivity and the surjectivity of \£ are consequences of the lifting properties of the universal cover. Q Corollary 6.2.23 If the space X has a compact universal cover then iri(X,pt) finite.

is

Proof Indeed the fibers of the universal cover have to be both discrete and compact. Hence they must be finite. The map E v in the above proof maps the fibers bijectively onto Deck (X). O Corollary 6.2.24 (Weyl) The fundamental group of a compact semisimple group is finite. Indeed from we deduce from Example 6.2.21 that the universal cover of such a group is compact. Example 6.2.25 From Example 6.2.5 we deduce that 7ri(5x) = (Z, + ) . Exercise 6.2.9 (a)Prove that 7r1(MP",pt) S Z 2 , Vn > 2. (b) Prove that T T ^ T ^ S Z " . Exercise 6.2.10 Show that the natural inclusion U(n — 1) •-+ U(n) induces an isomorphism between the fundamental groups. Conclude that the map det : U{n) -> S 1 induces an isomorphism

Chapter 7 Cohomology 7.1 §7.1.1

D e R h a m cohomology Speculations around the Poincare lemma

To start off, consider the following partial differential equation in the plane: given two smooth functions P and Q find a smooth function u such that £

= '•

%">•

<"•*>

As it is, the formulation is still fuzzy since we have not specified the domains of the functions u, P and Q. As it will turn out, this aspect has an incredible relevance in geometry. Equation (7.1.1) has another interesting feature: it is overdetermined i.e. it imposes too many conditions on too few unknowns. It is therefore quite natural to impose some additional restrictions on the data P, Q just like the zero determinant condition when solving overdetermined linear systems. To see what restrictions one should add it is convenient to introduce the 1-form a = Pdx + Qdy. (7.1.1) can be rewritten as du = a.

(7.1.2)

If (7.1.2) has at least one solution u then 0 = d2u = da so that a necessary condition for existence is da = 0 i.e. dP__dQ^ dy dx' 206

(7.1.3)

DeRham

207

cohomology

A form satisfying (7.1.3) is said to be closed. Thus, if the equation du = a has a solution then a is necessarily closed. Is the converse also true? Let us introduce a bit more terminology. A form a such that the equation (7.1.2) has a solution is said to be exact. The motivation for this terminology comes from the fact that sometimes du is called the exact differential of u. We thus have an inclusion of vector spaces {exact forms} C {closed forms}. Is it true the opposite inclusion also holds? Amazingly, the answer to this question depends on the domain on which we study (7.1.2). The Poincare lemma comes to raise our hopes. It says that this is always true at least locally. Lemma 7.1.1 (Poincare lemma) Let C be an open convex set in R" and a € Qfc(C). Then the equation du = a (7.1.4) has a solution w € Q* -1 (C) if and only if a is closed, da = 0. Proof Necessity is clear. We prove sufficiency. We may as well assume 0 £ C. Consider the radial vector field on C d

r- = xli — OXi

and denote by <3>( the flow it generates. $ t is the linear flow $ t (x) = e'x, i G B " . Note that since C is convex the flow lines of $ s , which are the straight lines through the origin, intersect C along connected segments originating at 0. We begin with an a priori study of (7.1.4). Let u satisfy du = a. Using the homotopy formula Lf — dif + ifd we get dL/u = d(dif + ifd)u = difCt i.e. d{LfU — ifOt) = 0. This suggests looking for solutions of LfU = if a.

(7.1.5)

208

Cohomology

If u is a solution of this equation then Lfdu = dLfU — difCc = LfOt — igda. = Lfa. Hence u also satisfies Lf(du — a) = 0. 1

Set u) = du — a = Y,i i^idx . Using the computations in Subsection 3.1.3 we deduce Lfdx' = dx% so that LfOj = y^XLftu^dx1 = 0. / We deduce that Lftjj = 0 and consequently the coefficients uij are constants along the the flow lines of $ ( which all converge at 0. Thus u>\ = ci = constant. Each monomial cidx1 is exact i.e. there exits r\i 6 Q,k~l(C) such that dr]j = Cjdx1. For example, when I =\ <2 < •••
u= J

mifct)dt.

(7.1.6)

J —CO

Here the convexity assumption on C enters essentially since it implies that *,(C)CC

Vi<0

so that if the above integral is convergent then u is a form on C. If we write

then u{x) = Y,(f

rfj{x)dt\ dx1.

(7.1.7)

DeRham

209

cohomology

We have to check two things. A. The integral in (7.1.7) is well denned. To see this we first write ajdxJ

a = ^ \j\=k

and set A(x) = max v

'

laj(r:r)|. v

J;0
"

Then $*(va) = e ' v f ^2

aJ(etx)dxJ

so that 1^/(^)1 < C*e*|a:tA(x) Vt < 0. This proves the integral in (7.1.7) converges. B . u defined by (7.1.6) is a solution of (7.1.5). Indeed, on differential forms LfU = lim - (0

lixa-U'f

S

&{ipa)dt-

s->0 s \

[

J-oo

= lim f f

$*t{ifa)dt) I

J-oo

$*t(i?a)dt - f

s->0 \J-oo

$*t{i?a)dt) J

J-oo

= lim / $*t{ifa)dt = %{ifa)

= ifO.-

Poincare lemma is proved. • Local solvability does not in any way implies global solvability. Something happens when one tries to go from local to global. Example 7.1.2 Consider the form d9 on K2 \ {0} where (r, 6) denote the polar coordinates in the punctured plane. To write it in cartesian coordinates (x, y) we use the equality V tan0 = x so that d Jfl__ (1 + tan, 22 m 0)d0 = — y\2djx _ +, v x x

(-i^y2\M ,.

-ydx + xdy

-ydx + xdy x2 + y2

•• a .

210

Cohomology

Obviously, da = d?9 = 0 on I 2 \ {0} so that a is closed on the punctured plane. Can we find a smooth function u on I 2 \ {0} such that du — a? We know that we can always do this locally. However, we cannot achieve this globally. Indeed, if this was possible then

f du= f 1 a= f l d6 = 2-K.

Js1

Js

Js

On the other hand, using polar coordinates u — u(r, 9) we get

f du=f 1 %d9 Js1 Js 89 = JQ ^ d 0 = u(l,2ir)-u(l,O) = O. Hence on M2 \ {0} {exact forms} / {closed forms}. We see what a dramatic difference a point can make: K2 \ {point} is structurally very different from K2. The artifice in the previous example simply increases the mystery. It is still not clear what makes it impossible to patch-up local solutions. The next subsection describes two ways to deal with this issue. §7.1.2

Cech vs. D e R h a m

Let us try to analyze what prevents the "spreading" of local solvability of (7.1.4) to global solvability. We will stay in the low degree range. Cech approach Consider w a closed 1-form on a smooth manifold. To solve the equation du = u, u 6 C°°(M) we first cover M by geodesically convex open sets (Ua) (with respect to some fixed Riemann metric). By Poincare lemma we can solve du = u on each open set Ua so that we can find a smooth function fa € C°°(Ua) such that dfa — w. We get a global solution if and only if

fa/) = fa - h = °

on each U

"P = UanU0^
For fixed a the solutions of the equation du = uo on Ua differ only by additive constants (i.e. closed 0-forms). The quantities /Q/? satisfy dfap = 0 on the (connected) overlaps Ua/3 so they are constants. Clearly they satisfy the conditions /Q/3 + hi + ha = 0 on (7Q/37 7^ 0.

(7.1.8)

DeRham

211

cohomology

On each Ua we have as we have seen several choices of solution. Altering a choice is tantamount to adding a constant fa —> fa + ca. The quantities fap change according to faB —> faB +Ca— Cp.

Thus the global solvability issue leads to the following situation. Pick a collection of local solutions fa. The equation du = u> is globally solvable if we can alter each fa by a constant ca such that faB = {cp - ca)

Va, /? such that Ua3 ± 0.

(7.1.9)

We can start the alteration at some open set Ua and work our way up from one such open set to its neighbors, always trying to implement (7.1.9). It may happen that in the process we might have to return to an open set whose solution was already altered. Now we are in trouble. (Try this on S1 and w = dQ.) After several attempts one can point the finger to the culprit: the global topology of the manifold may force us to always return to some already altered local solution. Notice that we replaced the partial differential equation du = ui with a system of linear equations (7.1.9), where the constants fap are subject to the constraints (7.1.8). This is no computational progress since the combinatorics of this system makes it impossible solve in most cases. The above considerations extend to higher degree and one can imagine the complexity increases considerably. This is however the approach Cech adopted in order to study the topology of manifolds and although it may seem computationally hopeless, its theoretical insights are invaluable. D e R h a m Approach This time we postpone asking why the global solvability is not always possible. Instead, for each smooth manifold M one considers the Z-graded vector spaces B*(M) = ®k>0Bk(M),

(B°(M)

d

=f {0})

Z*(M} =

®k>0Zk(M)

where Bk(M)

= {du; 6 Qk(M) ; u) € Q,k~l(M)} = exact k - forms

and Zk(M = {rie nk{M) k

; dq = 0} = closed k - forms.

k

Clearly B C Z . Form the quotients Hk{M) =

Zk{M)/Bk{M).

Intuitively, this space consists of those closed fc-forms w for which the equation du =

UJ

has no global solution u € fl*-1(M). Thus if we can somehow describe these spaces we may get an idea " who" is responsible for the global nonsolvability.

212

Cohomology

Definition 7.1.3 For any smooth manifold M the vector space Hk(M) fc-th DeRham cohomology group.

is called the

Clearly Hk{M) = 0 for k > dimM. Example 7.1.4 The Poincare lemma shows that Hk(Rn) sion in Example 7.1.2 shows that H^M2 \ {0}) # 0.

= 0 for k > 0. The discus-

Proposition 7.1.5 For any smooth manifold M dim H°(M) = number of connected components of M. Proof

Indeed H°(M) = Z°(M) = {/ € C^iM)

; df = 0}

Thus H°(M) coincides with the linear space of locally constant functions. These are constant on the connected components of M. O Already H°(M) contains an important topological information. Obviously the groups Hk are diffeomorphism invariants and its is reasonably to suspect the higher cohomology groups may contain more topological information. Thus, to any manifold we can now associate the graded vector space H*{M)"=f

($Hk{M).

A priori, the spaces Hk may be infinite dimensional. The Poincare polynomial, denoted by PM{t) is defined by P M (t) = ^ i f c d i m F A : ( M ) k>0

every time the right-hand-side expression makes sense. The number dim Hk (M) is usually denoted by 6^(M) and is called the k-th Betti number of M. Hence PM(t) = ] > > ( M ) t * . k

Exercise 7.1.1 Show that Psi(t) =

l+t.

We will spend the remaining of this chapter trying to understand what is that these groups do and which (if any) is the connection between the two approaches outlined above.

DeRham §7.1.3

cohomology

213

Very little homological algebra

At this point it is important to isolate the common algebraic skeleton on which both DeRham and Cech approaches are built. This requires a little terminology from homological algebra. In the sequel all rings will be assumed commutative with 1. Definition 7.1.6 (a) Let R be a ring, C = © „ 6 z C n and D = © n 6 z A i two Z-graded left .R-modules. A degree fc-morphism : C -» D is an fl-module morphism such that 4>{Cn)cDn+k VneZ. (b) Let C = ®nez b e a Z-graded fl-module. A boundary (resp. coboundary) operator is a degree -1 (resp. a degree 1) endomorphism d : C —» C such that d2 = 0. A chain (resp. cochain) complex over R is a pair (C, d) where C is a Z-graded i?-module and d is a boundary operator (resp. a coboundary operator). In this book we will be interested mainly in cochain complexes so in the remaining part of this subsection we will stick to this situation. In this case cochain complexes are usually described as (C = © n 6 z C " , d ) . Moreover we will consider only the case C" = 0 for n < 0. Traditionally, a cochain complex is represented as a long sequence of -R-modules and morphisms of .R-modules > Cn~l

(C, d) :

d

^z\Cn

such that range [dn-i) C ker (dn) i.e. dndn-i

-^

Cn+l -> • • •

= 0.

Definition 7.1.7 Let >

£„_!

d^cn

_d^

cn+1

_^

be a cochain complex. Set Zn(C) = kerd„

Bn{C) = range

(dn^).

The elements of Zn(C) are called cocycles, while the elements of Bn(C) are called coboundaries. Two cocycles c, d £ Zn(C) are said to be cohomologous if c — d G Bn(C). The quotient module Hn{C)

d

=

Zn{C)/Bn(C)

is called the n-th cohomology group (module) of C. It can be identified with the set of equivalence classes of cohomologous cocycles. A cochain complex complex C is said to be acyclic if Hn(C) = 0 for all n > 0.

214

Cohomology For a cochain complex C one usually writes H'(C) =

@Hn{C). n>0

Example 7.1.8 ( D e R h a m complex) Let M be an m-dimensional smooth manifold. Then the sequence

o -> n°(M) -4 9} -4 • • • 4 nm(M) -> o (where d is the exterior derivative) is a cochain complex called the DeRham complex. Its cohomology is the DeRham cohomology of the manifold. Example 7.1.9 Let (g, [•, •]) be a real Lie algebra. Define d : A*g* -> A*+1fl* by

(<M(*o, Xlt...,Xk)=

£

( - W K . **]. *o, • • •, Xit..., Xit..., X,)

0
where as usual, the hat indicates a missing argument. According to the computations in Example 3.2.6 d is a coboundary operator so that (A*g*, d) is a cochain complex. Its cohomology is called the Lie algebra cohomology and is denoted by H*(g). Exercise 7.1.2 (a) Let g be a real Lie algebra. Show that H1(g)^(g/[g,g}T where [g, g] = span {[X, Y] ; X,Y e g}. (b) Compute Hl{gl{n, K)) where gUn, R) denotes the Lie algebra of nxn real matrices with the bracket given by the commutator. (c) (Whitehead) Let g be a semisimple Lie algebra (i.e. nondegenerate Killing pairing). Prove that H1^) = {0}. (Hint: Prove that [fl^]"1" = 0 where _L denotes the orthogonal complement with respect to the Killing pairing.) Proposition 7.1.10 Let (C,d):

• C"-1 ^ C " - ^ > C n + 1 -> • • •

be a cochain complex of /^-modules. Assume moreover that C is also a Z-graded .R-algebra i.e. there exists an associative multiplication such that Cn-CmcCm+n

Vm,n.

If d is a quasi-derivation i.e. d(x • y) = ±(dx) • y ± x • (dy)

\/x, y G C

then H*(C) inherits a structure of Z-graded .R-algebra.

DeRham

cohomology

215

A cochain complex as in the above proposition is called a differential graded algebra or D G A . Proof It suffices to show Z*{C) • Z*{C) C Z*{C) and B*{C) • B*{C) C B*{C). If dx = dy = 0 then d(xy) = ±(dx)y ± x{dy) = 0. If x = dx' and y = dy' then since d2 = 0 we deduce xy = ±(dx'dy'). D Corollary 7.1.11 The DeRham cohomology of a smooth manifold has an K-algebra structure induced by the exterior multiplication of differential forms. Definition 7.1.12 Let (A,d) and {B,S) be two cochain complexes. (a) A cochain map is a degree 0 morphism <j> : A —¥ B such that <j> o d = 5 o cj> i.e. the diagram below is commutative for any n.

An

_*!_

^n+1

Bn

_i2_^

Bn+1

(b) Two cochain maps 4>,ip : A —> B are said to be cochain nomotopic and we write this ~h ip if there exists a degree -1 morphism x : {An —> Bn~1} such that 4>{a) - tp{a) = ±do x(a) ±x° d(a). (c) Two cochain complexes (^4, d) and {B, 5) are said to be homotopic if there exist cochain maps 4>: A-*

such that tp o ~h 1A and ^ o ^ ~

B and ip : B -¥ A k

1B.

Example 7.1.13 The commutation rules in Subsection 3.2.1, namely [Lx,d] = 0 and [ i x i ^ s = Lx show that for each vector field X on a smooth manifold M the Lie derivative along X, Lx : fl*(M) —> £l*(M) is a cochain map homotopic with the trivial map (= 0). The interior derivative ix is the cochain homotopy achieving this. Proposition 7.1.14 (a) Any cochain map <j> : (A, d) -> (B,5) induces a degree zero morphism in cohomology 0 , : H*{A) -»• f T ( 5 ) . (b) If the cochain maps cj),ip : A —• B are cochain homotopic then they induce identical morphisms in cohomology, » = i/>,. (c) (1A)» = 1H'(,A) and if

(A ) d)4(A , ,d')4(A",d") are cochain m a p s t h e n (ip o <[>)t = %j)t o <^>„.

216 Proof

Cohomology (a) It boils down to checking the inclusions
and 4>{Bn{A)) c

Bn{B).

These follow immediately from the definition of a cochain map. (b) We have to show that (cocycle) — ^(cocycle) = coboundary. Let da = 0. Then 4>{a) — ip(a) = ±5(x{a) ± x{da) = • {A, dA) 4 (B, dB) 4 (C, dc) -> 0 be a short exact sequence of cochain complexes. This means we have a commutative diagram

dA

0

dB

An+1

•

An

• dA

0

An~l

• dA

t l * Bn+1

Wn V'n+l.

+\

Qn

dB

dA

0

dc

B

n

ll>n

C

dB

dp

Bn

C

dB

dc

n

(7.1.10)

n

- 0

DeRham cohomology

217

in which the rows are exact. Then there exists a long exact sequence > Hn-\C)

V

Hn{A) h Hn(B) % Hn{C) % Hn+1(A)

->••••.

(7.1.11)

We will not include a proof of this proposition. We believe this is one proof in homological algebra the reader should try it on its own. We will just indicate the construction of the connecting maps dn. This construction and in fact the entire proof relies on a simple technique called diagram chasing. Start with x e Hn(C). x can be represented by some cocycle c e Zn(C). Since tp„ is surjective there exists b G Bn such that c = i>n{b). From the commutativity of the diagram (7.1.10) we deduce 0 = dc^n{b) = ipn+idB(b) i.e. dB(b) € keri/'n+i = rangen+i- In other words, there exists a € An+1 such that n+i(a) = dBb. We claim o is a cocycle. Indeed n+ia = dB+1dBb

= 0.

Since nll°dB

o^(c)

=

we can

^+1odBb.

This is not entirely rigorous since a depends on various choices. We let the reader check that the correspondence Zn(C) 3 c >-> a e Zn+1(A) above induces a well defined map in cohomology, dn : Hn{C) —>• Hn+1(A} and moreover, the sequence (7.1.11) is exact. Exercise 7.1.3 (Abstract Morse inequalities) Let C = (&n>0Cn be a cochain complex over the field F . Assume each of the vector spaces Cn is finite dimensional. Form the Poincare series Y,tn
Pc{t) = 71>0

and J^tndimFHn(C).

PH'(C)(t) = n>0

Prove that there exists a formal series R(i) (E Z[[t]] with non-negative coefficients such that Pc(t) = PH.{c)(t) + (1 +t)R(t). In particular, whenever it makes sense, the graded spaces C* and H* have identical Euler characteristics X(C*) = P C ( - 1 ) = P H . ( C ) ( - 1 ) =

X(H*(C)).

218

Cohomology

Exercise 7.1.4 (Additivity of Euler characteristic) Let 0-+yl->B->C->0 be a short exact sequence of cochain complexes over the field F. Prove that if at least two of the cohomology modules H*(A), H*{B) and H*(C) have finite dimension over F then the same is true about the third one and moreover x(H*(B))=x(H*(A})

+

x(H*(C)).

Exercise 7.1.5 (Finite dimensional Hodge theory) Let (©„>oV™, dn) be a cochain complex over the reals such that dim Vn < oo for all n. Assume each Vn is an euclidian space and denote by d*n : Vn+1 —> V" the adjoint of dn. We can now form the Laplacians A„ : Vn -)• Vn An = d*ndn + <*„_!<_!. (a) Prove that Anc = 0 iff dnc = 0 and d'.jC = 0. (b) Let c € Zn(C). Prove that there exists a unique c G Zn(C) such that \c\ = min{|c'| ; c-c' e Bn(C)}

cohomologous to c

where | • | denotes the euclidian norm in V". (c) Prove that c determined in part (b) satisfies A n c = 0. Deduce from all the above the Hodge theorem Hn(V) = kevAn. Exercise 7.1.6 Let V be a real vector space and v0 G V. Define dk : AkV —• Ak+1V by LJ t-> Vo A u).

(a) Prove that . . . " V A*v 4 Ak+1V %' ••• is a cochain complex (known as Koszul complex). (b) Use the finite dimensional Hodge theory described in previous exercise to prove that the Koszul complex is acyclic if v0 ^ 0 i.e. Hk{A*V,d) = 0 V/fc>0 §7.1.4

Functorial properties of D e R h a m cohomology

Let M and N be two smooth manifolds and <j>: M —> N a smooth map. The pullback * : Q*{N) -*•

is a cochain map i.e. 4>*dN = dM*.

Q*(M)

DeRham

219

cohomology

Thus <j>* induces a morphism in cohomology which we continue to denote by *; * :H*{N) -> H*{M). In fact we have a more precise statement. Proposition 7.1.17 The DeRham cohomology construction is a contravariant functor from the category of smooth manifolds and smooth maps to the category of Z-graded vector spaces with degree zero morphisms. Note that the pull-back is an algebra morphism * : Q*(iV) —> fl*(M) and the exterior differentiation is a quasi-derivation so that the map it induces in cohomology will also be a ring morphism. Definition 7.1.18 (a) Two smooth maps o,4>\ '• M —>• N are said to be (smoothly) homotopic (and we write this <j>o ~ s /j 4>\) if there exists a smooth map $ : I x M ->• N

(£, m) i-> $ ( (m)

such that i for i = 0,1. (b) A smooth map cj>: M —• N is said to be a (smooth) homotopy equivalence if there exists a smooth map ip : N —>• M such that cj>otp ~ s h 1^ and ip o

: M —• N. Proposition 7.1.19 Let <j>Q,<j>\ : M -* N be two homotopic smooth maps. Then they induce identical maps in cohomology <% = $* : H*(N) -»

H*(M).

Proof According to the general results in homological algebra it suffices to show the pullbacks #J,fi*(M) are cochain homotopic. Thus, we have to produce a map X:W{N)

-¥Q*~1{M)

such that

^!M - $S(w) = ±x(<M ± dxw. At this point, our discussion on the fibered calculus of Subsection 3.4.5 will pay off. The projection $ : / x A f - t t f defines an oriented 9-bundle (with standard fiber I). For any u> G Q*(N) we have the equality

*0{u) = $ * M IIXM - $ * ( w ) | 0 xM

220

Cohomology

JldIxM)/M J(dIxM)/M

We now use the homotopy formula of Theorem 3.4.40 of Subsection 3.4.5. We get J(dIxM)/M

J(IxM)/M

= f

J(IxM)/M

&(dNU>)-dU[

J(IxM)/M

*».

J(IxM)/M

7(/xM)/M

is the sought for cochain homotopy. • Corollary 7.1.20 Two homotopically equivalent spaces have isomorphic cohomology rings. Consider now a smooth manifold and U, V two open subsets such that M = ULiV. Denote by ly (resp. iy) the inclusions U *-» M (resp. V •-» M). These induce the restriction maps i\j : Q*(M)

->• Q*(U)

u)^u\u

and ^ : fi*(M) -> fi*(V) u H u |

F

.

We get a cochain map r :fi*(M) -»•«*(£/) ©fi*(V)

There exists another cochain map

5 •. n*(u) efl'(v) -> n*(u nv) {w,rf) H-> -w|t/nv +v\unv • Lemma 7.1.21 The short Mayer-Vietoris sequence

o -+ n*(M) A n*(i7) © n*(v) 4o*(c/nv)->o is exact. Proof Obviously r is injective. The proof of the equality Range r = ker 6 can be safely left to the reader. The surjectivity of 5 requires a little more effort. {U, V} is an open cover of M so we can find a partition of unity {
DeRham

221

cohomology

subordinated to this cover i.e. supp ipu C U, supp ipv C V 0 <
¥>t/ + W = 1-

Note that for any u> 6 Q*([/ n V) supp ^ w C supp ipv

cV

and thus, upon extending ipuui by 0 outside V we can view it as a form on M. Restricting to U we get a form ipvu> £ Q*(C/). Similarly, ipvuj £ fT(K). Note that 5(-Pv + H>u)u = w. This establishes the surjectivity of 6. • Using the abstract results in homological algebra we deduce from the above lemma the following fundamental result. Theorem 7.1.22 (Mayer-Vietoris)Let M = UliV be an open cover of the smooth manifold M. Then there exists a long exact sequence > Hk{M) A Hk{U) © Hk(V) -4 Hk{U f\V)A

Hk+\M)

-> • • •

called the long Mayer-Vietoris sequence. The connecting morphisms d can be explicitly described using the prescriptions following Proposition 7.1.16 in the previous subsection. Start with ui € Qk(U D V) such that dui = 0. Writing as before

we deduce d(
Thus we can find r\ € Ct

on U n V.

(M) such that r] \u= d(ipvtu)

T)\v=

d(-ipuu).

Then duj =

TJ.

The reader can prove directly that the above definition is independent of the various choices. The Mayer-Vietoris sequence has the following functorial property.

222

Cohomology

P r o p o s i t i o n 7.1.23 Let <j> : M —> TV be a smooth map and {U, V} an open cover of N. Then U' = _1(C/), V = ^> _1 (^) f ° r m a n ° P e n cover of M and moreover, the diagram below is commutative. Hk{N)

Hk{U)®Hk{V)

Hk{UCiV)

Hk+\N)

Hk{M)

Hk{U')®Hk{V)

Hk{U' n V)

Hk+1(M)

Exercise 7.1.7 Prove the above proposition. §7.1.5

Some simple e x a m p l e s

The Mayer-Vietoris theorem established in the previous subsection is a very powerful tool for computing the cohomology of manifolds. In principle, it allows one to recover the cohomology of a manifold decomposed into simpler parts, knowing the cohomologies of its constituents. In this subsection we will illustrate this principle on some simple examples. E x a m p l e 7.1.24 ( T h e cohomology of spheres)The cohomology of S 1 can be easily computed using the definition of DeRham cohomology. We have /f°(S' 1 ) = R since S 1 is connected. The closed 1-forms on S 1 have the form const.dO so that H^S^^M. Thus Psi(t) = l + t. To compute the cohomology of higher dimensional spheres we use Mayer-Vietoris theorem. The (n + l)-dimensional sphere Sn+1 can be covered by two open sets Us = Sn+1 \ {north pole} and Un = Sn+1 \ {south pole}. Each is diffeomorphic to M n+1 . Note that the overlap Un D Us is homotopically equivalent with the "Equator" 5". The Mayer-Vietoris sequence gives Hk{Un) 0 Hk{Us) -> Hk(Un (~l Us) -> Hk+1(Sn+1)

-> Hk+\Un)

® Hk+\US)

^ 0.

The last isomorphism is precisely Poincare lemma. For k > 0 the group on the left is also trivial so that we have the isomorphisms Hk(Sn)^Hk(UnDUs)^Hk+1{Sn+1)

k>0.

Denote by Pn{i) the Poincare polynomial of Sn and set Qn(t) = Pn{t) — Pn(0) = P„{t) — 1. We can rewrite the above equality as Qn+i(t) = tQn(t)

n>0.

DeRham

223

cohomology

Since Q\{t) = t we deduce Qn(t) = tn i.e. Psn{t) =

l+tn.

Example 7.1.25 Let {U, V} be an open cover of the smooth manifold M. We assume that all the Betti numbers of U, V and U C\V are finite. Using the Mayer-Vietoris short exact sequence and Exercise 7.1.4 in Subsection 7.1.3 we deduce that all the Betti numbers of M are finite and moreover we have x(M) = x(U) + x(V)-X(UnV).

(7.1.12)

This resembles very much the classical inclusion-exclusion principle in combinatorics. We will use this simple observation to prove that the Betti numbers of a connected sum of p-tori is finite and then compute its Euler characteristic. Let E be a surface with finite Betti numbers. From the decomposition E = (E \ disk) U disk we deduce (using again Exercise 7.1.4) x(E) = X ( S \ disk) + x(disk) -

X

((E \ disk) n (disk)).

Since (E \ disk) n disk is homotopically a circle and x(disk) = 1 we deduce X (E)

= x(E \ disk) + 1 - xiS1) = x ( S \ disk) + 1.

If now Ej and £2 are two surfaces with finite Betti numbers then E X # E 2 = (E x \ disk) U (E 2 \ disk) where the two holed surfaces intersect over an entire annulus, which is homotopically a circle. Thus X(Ei#E2)

= X (Ei \ disk) + X (E 2 \ disk) -

l

X(S

)

= x(2i) + X(E2) - 2. This equality is identical with the one proved in Proposition 4.2.23 of Subsection 4.2.5. We can decompose a torus as a union of two cylinders. The intersection of these cylinders is the disjoint union of two annuli so homotopically, this overlap is a disjoint union of two circles. In particular, the Euler characteristic of the intersection is zero. Hence x(torus) = 2x(cylinder) = 2x(circle) = 0. We conclude as in Proposition 4.2.23 that X (connected sum of g tori) = 2 — 2g. This is a pleasant, surprising connection with the Gauss-Bonnet theorem. And the story is not over.

224 §7.1.6

Cohomology The Mayer-Vietoris principle

We describe in this subsection a "patching" technique which is extremely versatile in establishing general homological results about arbitrary manifolds building up from elementary ones. Definition 7.1.26 A smooth manifold M is said to be of finite type if it can be covered by finitely many open sets U\,..., Um such that any nonempty intersection f/ij n • • • n Uik (k > 1) is diffeomorphic to R d l m M . Such a cover is said to be a good cover. Example 7.1.27 All compact manifolds are of finite type. To see this it suffices to cover such a manifold by finitely many open sets which are geodesically convex with respect to some Riemann metric. If M is a finite type manifold and U C M is a closed subset homeomorphic with the closed unit ball in R d l m M them M \ U is a finite type non-compact manifold. (It suffices to see that M" \ closed ball is of finite type). The connected sums and the direct products of finite type manifolds are finite type manifolds. Proposition 7.1.28 Let p : E —> B be a smooth vector bundle. If the base B is of finite type then so is the total space E. In the proof of this proposition we will use the following fundamental result. Lemma 7.1.29 Let p : E —> B be a smooth vector bundle such that B is diffeomorphic to K". Then p : E ->• B is a trivial bundle !!! Proof of Proposition 7.1.28. Denote by F the standard fiber of E. F is a vector space. Let (C/j) be a MV-cover of B. For each ordered multi-index I: {i\ < • • • < i^} denote by Uj the multiple overlap U^ D • • • D Uik. Using the previous lemma we deduce that each Ej = E \Vl is a product FxUi and thus it is diffeomorphic with some vector space. Hence (£*) is a MV-cover. D. Exercise 7.1.8 Prove Lemma 7.1.29. Hint Assume E is a vector bundle over the unit open ball B e l " . Use V-parallel transport along f= —x'-^ where V is a connection on E. We organize the family of n-dimensional, finite type smooth manifolds as a category 9Jln. The morphisms of this category are the embeddings (i.e. 1-1 immersions) Afi -> M 2 (Mi G mn).

DeRham

225

cohomology

Definition 7.1.30 Let R be a commutative ring with 1. A contravariant MayerVietoris functor (or MV-functor for brevity) is a contravariant functor from the category 9Jt n to the category of Z-graded .R-modules T = ®n^Tn

-> G r a d R-MoA

M .->

®Pi{M)

with the following property. If {U, V} is a MV-cover of M G SD?„ i.e. U, V, UC\ V G 97t„ then there exist morphisms of .R-modules dn:Fl{Ur\V)^Tn+1{M) such that the sequence below is exact.

—>^"(M) 4 J^{U)® J^{V) - 4 ^ ( u n v ) ^

J^+1(M)

->••••

where r* is defined by r* = F(iv) ® T{iv) while 8 is defined by 5(x ®y)=

T{iunv)(y)

-

T(tunv)(x).

(The maps i. denote natural embeddings.) Moreover, if TV G 97tn is an open submanifold of TV and {U, V) is a MV cover of M such that {U f~l TV, V D TV} is a MV-cover of TV then the diagram below is commutative.

r(f/nV) —^— ^"+1(M)

^"(c/nvnTV) —^-

^"+1(TV)

The vertical arrows are the morphisms ^ ( J . ) induced by inclusions. The covariant MV-functors are defined in the dual way, by reversing the orientation of all the arrows in the above definition. Definition 7.1.31 Let T, Q be two contravariant MV-functors 97tn -> G r a d ijMod. A c o r r e s p o n d e n c e between these functors is a collection of .R-module morphisms $M = © nM •• ®Jrn{M)

-»• © e n ( M )

226

Cohomology

(one collection for each M e 97ln) such that for any embedding Mx A M 2 the diagram bellow is commutative ^ " ( M 2 ) -?&L

^»(Mi)

0M,

en(M2)

g(v).

^(MO

and for any M G !J7tn and any MV cover {U, V} of M the diagram below is commutative. F((/(17) ~^- r'+^M) <}>UC\V

gn{unv)

<$>M

- ^ — gn+1(M)

The correspondence is said to be a natural equivalence if all the morphisms <J>M are isomorphisms. Theorem 7.1.32 (Mayer-Vietoris principle)Let T, Q be two (contravariant) MayerVietoris functors on 2DIn and <j>: T —> Q & correspondence. If <j)kw : Fk{Rn)

-> 0*(R")

is an isomorphism for any k € Z then 0 is a natural equivalence. Proof

The family of finite type manifolds SDTn has a natural filtration

tml c m2n c • • • c an; c • • • where 9JTn is the collection of all smooth manifolds which admit a good cover consisting of at most r open sets. We will prove the theorem using an induction over r. The theorem is clearly true for r = 1 by hypothesis. Assume 4>kM is an isomorphism for all M € SD^ -1 . Let M € WlTn and consider a good cover {Uu..., Ur} of M. Then {U =

U1U---UUr-1,UT}

is a MV-cover of M. We thus get a commutative diagram .^"(E/) © ^ " ( t / r ) — • J ^ ( * y n t / r )

Gn{U) ® Gn{Ur)

gn{unur)

^"+1(M)

— ^" +1 (c/)©a n+1 (c/r

e n+1 (M) — ^ i + 1 (c/)©e n + 1 (^-)

DeRham

227

cohomology

The vertical arrows are defined by the correspondence . Note the inductive assumption implies that in the above infinite sequence only the morphisms M may not be isomorphisms. At this point we invoke the following technical result. Lemma 7.1.33 (The five lemma)Consider the following commutative diagram of .R-modules. A-2 — - -4-i — - A0 Ax • A2 /o

/-l

B-2 —>• B - i — * Bo

h " B\

h

*- Bi

If fi is an isomorphism for any i ^ 0 then so is / 0 . Exercise 7.1.9 Prove the five lemma. The five lemma applied to our situation shows that the morphisms 4>M must be isomorphisms. D Remark 7.1.34 (a) The Mayer-Vietoris principle is true for covariant MV-functors as well. The proof is obtained by reversing the orientation of the horizontal arrows in the above proof. (b) The Mayer-Vietoris principle can be refined a little bit. Assume that !F and Q are functors from £DT„ to the category of Z-graded .R-algebras and : T —>• Q is a correspondence compatible with the multiplicative structures i.e. each of the .R-module morphisms (J>M are in fact morphisms of .R-algebras. Then if -^ are isomorphisms of Z-graded .R-algebras then so are the 4>M'S for any M e Wln. (c) Assume R is a field. The proof of the Mayer-Vietoris principle shows that if T is a MV-functor and dim R T*(9.n) < oo then dim ^ * ( M ) < oo for all M e 9DTn. Corollary 7.1.35 Any finite type manifold has finite Betti numbers. §7.1.7

The Kiinneth formula

We learned in principle how to compute the cohomology of an "union of manifolds". We will now use the Mayer-Vietoris principle to compute the cohomology of products of manifolds. Theorem 7.1.36 (Kiinneth formula) Let M G 971m and N € SDTn. Then there exists a natural isomorphism of graded M-algebras H'{M xN)

= H'(M)

® H*{N) = 0

(®p+q=nH>>(M) ®

In particular, we deduce PMxN{t)

= PU{t) • PN(t).

H"{N)).

228 Proof

Cohomology We construct two functors JF,S:S!7tm->GradRAlg T : M •->• © ^ ( M ) = 0 r>0

{®p+q=rH"{M)

® H"(N)}

r>0

© r > 0 £(M) = © r > 0 e r ( M x N).

G:M^

) V/ : M -> M 2 $ ( / ) = 0 ( / x l w ) * U'(M2xiV)

V/ : Mi ^

M2.

r>0

We let the reader check the following elementary fact. Exercise 7.1.10 T and Q are contravariant MV-functors. For M € WlM define 4>M • T{M) -> Q(N) by def

<^M(W ® 7?) = W X TJ = 7T^o; A n*Nr] (w € H*(M),T] e

H*(M))

where 7rM (resp. TTN) are the canonical projections M xN —• M (resp. M x TV —» TV). The operation x:H*(M)®H*(N)^H*{MxN)

(w ® r?) >->• W X 77

is called the cross product. The Kiinneth formula is a consequence of the following lemma. Lemma 7.1.37 (a) cj) is a correspondence of MV-functors. (b) ]]jm is an isomorphism. Proof of the lemma The only nontrivial thing to prove is that for any MV-cover {U, V} of M G 93tm the diagram below is commutative. ®p+q=rH"(U

f)V)®

H"{N) - i - ® p + 9 = r i F + 1 ( M ) ® #«(/V)

^(/nv

Hr({U xN)n(Vx

<£M

N))

Hr+1{M

xN)

We briefly recall the construction of the connecting morphisms d and &'. One considers a partition of unity {tpu,
DeRham

229

cohomology

and ipv — TMVV form a partition of unity subordinated to the cover {U x N, V x N} of M x N. If w ® 7? € # * ( [ / n V) #*(JV) then 9 ( w ® r?) = Qi 77

where ^ \u= — d((pvw)

& = d(ipuuj).

On the other hand 0tmv(w® 77) = w x 77 and ^(w x 77) = w x 77. This proves (a). To establish (b) note that the inclusion j : TV M- E m x N

ii4(0,i)

is a homotopy equivalence with ir^ a homotopy inverse. Hence, by the homotopy invariance of the DeRham cohomology we deduce S(R m ) =* H*{N). Using the Poincare lemma and the above isomorphism we can identify the morphism <%» with % - . D Example 7.1.38 Consider the n-dimensional torus, Tn. By writing it as a direct product of n circles we deduce from Kunneth formula that Pm{t) = {Psl(t)}n

= (l + t)n.

Thus &*(Tn)=Q,

dimH*(Tn)

= 2n

and

xcn = 0. One can easily describe a basis of H*(Tn). Choose angular coordinates (01,... ,6n) on Tn. For each ordered multi-index I : 1 < i\ < • • • < j * < n we have a closed, nonexact form dO1. These monomials are linearly independent (over K) and there are 2" of them. Thus, they form a basis of H*{Tn). In fact, one can read the multiplicative structure using this basis. We have an isomorphism of K-algebras H*{Tn) S* A*R".

230

Cohomology

Exercise 7.1.11 Let M e 9Jtm and N € QJI„. Show that for any w* e rjj e H*(N) (i,j=0,l) the following equality holds. (wo x no) A (Wl x Vl) = {-l)de^de^(tj0

H*(M),

A W l ) x (% A tfc).

Exercise 7.1.12 (Leray-Hirsch)Let p : £ -> M be smooth bundle with standard fiber F. We assume the following: (a) Both M and F are of finite type. (b) There exists cohomology classes e j . , . . . , e r € H*(E) such that their restrictions to any fiber generate the cohomology algebra of that fiber. The projection p induces a H* (M)-modu\e structure on H*(E) by U-TI = P*LJAT1

VW € H*(M)r) e # * ( £ ) .

Show that H*(E) is a free i?*(M)-module with generators e i , . . . , e r .

7.2 §7.2.1

The Poincare duality Cohomology with compact supports

Let M be a smooth n-dimensional manifold. Denote by f2^,t(M) the space of smooth compactly supported &-forms. Then

0^fi° p t (M)4--.4fi; t (M)->0 is a cochain complex. Its cohomology is denoted by H*t(M) and is called the DeRham cohomology with compact supports. Note that when M is compact this cohomology coincides with the usual DeRham cohomology. Although it looks very similar to the usual DeRham cohomology, there are many important differences. The most visible one is that if <j> : M —> N and u> G fl*p((Af) is a smooth map then the pull-back <jfu> may not have compact support so this new construction is no longer a contravariant functor from the category of smooth manifolds and smooth maps to the category of graded vector spaces. On the other hand if dim M = dim N and cj> is an embedding we can identify M with an open subset of N and then any 77 e fi*p((M) can be extended with 0 outside M C N. This extension by zero defines a push-forward map , •. n ^ ( M ) -»• n ; t ( A 0 . One can verify easily that , is a cochain map so that it induces a morphism

4>.: #; t (M) -> #; ( (w). In terms of our category SDT„ we see that H*pt is a covariant functor from the category SDT„ to the category of graded real vector spaces. As we will see, it is a rather nice functor.

Poincare duality

231

Theorem 7.2.1 H^t is a covariant MV-functor and moreover

<((Kn) = { E0

, k < n , k= n

The last assertion of this theorem is usually called the Poincare lemma for compact supports. We first prove the Poincare lemma for compact supports. The crucial step is the following technical result we borrowed from [10]. L e m m a 7.2.2 Let E A B be a rank r real vector bundle, orientable in the sense described in Subsection 3.4.5. Denote by p„ the integration-along-fibers map

p. : ^t(E)

-> U£{B).

Then there exists a smooth bilinear map -> n ^ T " 1 ^ )

m : n%t(E) x Q^iE) such that p'p.a A 0 - a A p'p,/3 = {-l)rd{m(a,

Proof

/?)) - m{da, /?) + ( - l ) d e g a m ( a , dp).

Consider the 9-bundle n:£ •K

= I x(E®

E) ^ E

: (t; v0, vi) i-» (t- v0 + t(vi - v0)).

Note that d£ = ({0} x(E®

E)) U ({1} x {E © E)).

Define 7r*: E ® E -> E as the composition E © E 9* {<} x £ © E ^ I x (E © E) A £\ Observe that d7r = 7r|a £ = ( - T T ^ U T T 1 .

For (a,/?) € fi^(S) x fi^(£) define a © /? e 0 ^ ( 2 © £ ) by a © p = (7r°)*a A (vr1)*^.

232

Cohomology

(Verify that a 0 /? has indeed compact support). For a G Q'cpt(E) and /? G ^ p ( ( i ? ) we have the equalities A P = ^ ( a © /?) G n j + T r ( E )

p'p.a

aAp*p» / S = ^ ( a O / 3 ) G Q - + r r ( £ ) . Hence £>(a, /?) = p*p,a A /? - a A p*p,/? = irKa O /?) - TT°(Q O /?) = /

a©/?.

We now use the fibered Stokes formula to get D(a,(3)=[ JE/E

d£T*(aOP)

+ (-iydE

T is the natural projection £ = Ix(E®E)->E®E. {a, 13)= /

Je/E

T>0/3).

[

T{aQ0).

JS/E

The lemma holds with •

Proof of the Poincare lemma for compact supports such that 0< S<1 f 5(x)dx = 1.

Consider 5 G CQ°(K")

Define the (closed) compactly supported n-form r = 6(x)dx1 A • • • A dxn. We want to use Lemma 7.2.2 in which E is the rank n bundle over a point i.e. E = {pt} x R " A {pt}- The integration along fibers is simply the integration map. p . : J2^(R") -> K

{

0 , degw < n / K » ( j , degw = n

If now o> is a closed, compactly supported form on R n we have o> = (p*ptr) A u). Using Lemma 7.2.2 we deduce U - T A p*pmw = (—l) n dm(T, w). Thus any closed, compactly supported form u on ffin is cohomologous to r Ap*p,w. The latter is always zero if degw < n. When degw = n we deduce that w is cohomologous to (/ US)T. This completes the proof of the Poincare lemma. •

233

Poincare duality

To finish the proof of Theorem 7.2.1 we must construct a Mayer-Vietoris sequence. Let M be a smooth manifold decomposed as an union of two open sets M = UUV. The sequence of inclusions

unv^u,v^M induces a short sequence

o -> wcpt(u n v) -4 tr^u) © %t(v) 4 orcpt{M) -+ o where i(w) = (w,w) j{u,ri) =

fi-w.

The hat denotes the extension by zero outside the support. This sequence is called the Mayer-Vietoris short sequence for compact supports. Lemma 7.2.3 The above Mayer-Vietoris sequence is exact. Proof i is obviously injective. Clearly, Range (i) = ker(j). We have to prove that j is surjective. Let (
VUT, e n*cpt(u)
oyy).

In particular we have

n = 3{-{pvn,'pvn)Hence j is surjective.

•

We get a long exact sequence called the long Mayer-Vietoris sequence for compact supports. • • • - » • Hkcpt(U HV)^

Hkcpt(U) © Hkcpt{V)

- 4 Hkcpt(M)

4 Hkpf

-> • • •

The connecting homomorphism can be explicitly described as follows. If UJ 6 Q,k t{M) is a closed form then d((puL)) = d(—ipvu) on U Pi V. We set Sui = d(ipuu). The reader can check immediately that the cohomology class of <5w is independent of all the choices made. If <j> : N <-» M is a morphism of 9Jt„ then for any MV-cover of {U, V} of M {<^_1([/),<)!i-1(V)} is an MV-cover of N. Moreover, we almost tautologically get a commutative diagram

H^{M)-^H^{UnV) This proves H*pt is a covariant Mayer-Vietoris sequence. •

234

Cohomology

Remark 7.2.4 To be perfectly honest (from a categorial point of view) we should have considered the chain complex

y3 Q = (J) fl^ fc<0 fc<0

and correspondingly the associated homology H

* =

H

^t-

This makes the sequence 0 <- Q.~n i

<- £l~l <- Cl° <- 0

a chain complex and its homology H* is a bona-fide covariant Mayer-Vietoris functor since the connecting morphism 5 goes in the right direction H* —¥ H*~x. However, the simplicity of the original notation is worth the small formal ambiguity so we stick to our upper indices. From the proof of the Mayer-Vietoris principle we deduce the following. Corollary 7.2.5 For any M e 9Jtn and any k < n we have dim H^,t(M) §7.2.2

< oo.

The Poincare duality

Definition 7.2.6 Denote by 9Dt+ the category of n-dimensional, finite type, oriented manifolds. The morphisms are the embeddings of such manifold. The MV functors on 971^" are defined exactly as for 9Jl„. Given M £ 9Jt+ there is a natural pairing ( , )K : Q.k{M) x Q.nc;tk{M) -> R defined by (LJ,rj)K=

/

uAi|.

JM

which is called the Kronecker pairing . We can extend this pairing to any (w, rj) € Q* x n^ as 0 , deg u + deg rj / n \u,T))K | / M W A T ? , degw + degr/ = n This pairing induces maps V = Vk : Qk(M) -> {QP-tk{M)Y (V(u),T])

= <w,r/)K.

235

Poincare duality

( , ) denotes the natural duality between a vector space and its dual, V* x V —> R. We continue to denote by T>(OJ) the restriction to the space of closed, compactly supported (n — /e)-forms, Z^k. If moreover w is closed, this functional on Z^k k vanishes on the subspace B™~t of exact, compactly supported (n — fc)-forms. Indeed, if 7? = dr/, i 6 n ^ * _ 1 ( M ) then (X>(w), TJ> = I uAdn'

St es

=

JM

± f dwAV'

= Q.

JM

Thus, if w is closed V(u>) defines an element of (H^tk{M))*. If moreover u> is exact, a computation as above shows that V(u>) = 0 € {H^~tk{M))*. Hence V descends to a map in cohomology V : Hk(M) -> (H£k{M))* which is the same as saying that the Kronecker pairing descends to a pairing in cohomology. Theorem 7.2.7 (Poincare duality)The Kronecker pairing in cohomology is a duality for all M € 2rt+. Proof

The functor 971+ —>• Graded Vector Spaces

defined by

M->©ff*(M)=®(fl£*(M))' k

k

is a contravariant MV-functor. (The exactness of the Mayer-Vietoris sequence is preserved by transposition. This is where the fact that all the cohomology groups are finite dimensional vector spaces plays a very important role). For purely formal reasons (which will become apparent in a little while) we define the connecting morphism of the functor Hk (Hk(U D V) -4 Hk+l(U U V)) to be (-l) f c 5 ( , where H^tk~l{UnV) A H^k(UUV) denotes the connecting morphism in the DeRham cohomology with compact supports and <$f denotes its transpose. The Poincare lemma for compact supports can be rephrased

The Kronecker pairing induces linear maps VM : Hk{M) ->

Hk{M).

Lemma 7.2.8 Qk Vk is a correspondence of MV functors.

236

Cohomology

Proof We have to check two facts. FACT A. Let M <-t N be a morphism in SDT+. Then the diagram below is commutative. Hk(N) - ^ Hk(M)

Hk{N)

-*--

Hk{M)

F A C T B . If {U, V} is a i W - c o v e r of M e 97t+ then the diagram bellow is commutative Hk(UHV) -^ Hk+1(M) VM

T>unv

ff'([/ny)l-^ffw(M) Proof of FACT A. Let w € Hk(N). For any rj e H^k{M) natural duality between a vector space and its dual) (if °VN{u),rj) =

/

= (^tyVN{u),ri)

w A <j>*r] =

JN

/

((•,•) denotes the

= (VN{u>),tr)}

w \M AT? =

JM<-+N

/

*Lj A 7?

JM

= (VM(*U>),Tl). Hence f o P A f = B M o f Proof of F A C T B . Let ipUy ipv be a partition of unity subordinated to the M^-cover {U, V} of M € 9H+. Consider a closed fc-form w e Qfc(t/ n V). Then the connecting morphism in usual DeRham cohomology acts as „

Choose rj e

fi^fc_1(M)

_ f d(-ifivuj) \ d(tpuu)

on U on V

such t h a t ofr? = 0. We have (Viudu), rj) = / Out Arj JM

=

6ujArj+dujArj

—

9UAIJ

7(7 Jv ./m-iv = — / dfojyo;) A n + / dipu<J) Ar) + dUpvw) A rj. Ju Jv Junv Note t h a t t h e first two integrals vanish. Indeed, over U we have the equality (d(ipvw) A rj = d (tpvv A rj)

237

Poincare duality

and the vanishing now follows from Stokes formula. The second term is dealt with in a similar fashion. As for the last term we have / (dipvuj) A ri = f d(pv A UJ A r] = {-l)desu f Junv Junv Junv = (-l)k

[

Junv

(-l)k(VunVLj,

UJA5V=

w A (d
SV)

{{-l)HtVmVU,Tl).

=

This concludes the proof of Fact B. The Poincare duality now follows from the Mayer-Vietoris principle. •

Remark 7.2.9 Using the Poincare duality we can associate to any smooth map / : M —> N between compact oriented manifolds of dimensions m and respectively n a natural push-forward morphism / . : H*{M) -> H*+q{N)

(q = dim TV - d i m M = n - m)

defined by the composition H*(M)

V

-H {Hm-(M)Y

{f

-7 (Hm-(N)y

V

A

H*+n~m{N)

where (/*)' denotes the transpose of the pullback morphism. Corollary 7.2.10 If M e SDT+ then Hkcpt{M) = Proof

Since Hkpt(M)

{Hn~k(M)Y.

is finite dimensional the transpose ^M •• (HHk(M)Y*

-v

(Hk(M)Y

is an isomorphism. On the other hand, for any finite dimensional vector space there exists a natural isomorphism V" =* v. a Corollary 7.2.11 Let M be a compact oriented n-dimensional manifold. Then the pairing Hk(M)

x Hn-k(M)

->• K (to, r)) .-> J w A 77 JM

is a duality. In particular bk(M) — bn-k(M),

\fk.

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Cohomology

If M is connected H°(M) ^ Hn(M) ^ R so that Hn{M) is generated by any volume form defining the orientation. The symmetry of Betti numbers can be translated in the language of Poincare polynomials as tnPM(\)

= PM(t).

(7.2.1)

Example 7.2.12 Let E 9 denote the connected sum of g tori. We have shown that X (E 9 )

= b0 - h + b2 = 2 - 2g.

Since E s is connected, the Poincare duality implies b2 = b0 = 1. Hence b\ = 2g i.e. PSg{t) = l + 2gt + t2. Consider now a compact oriented smooth manifold such that dim M = 2k. Kronecker pairing induces a non-degenerate bilinear form 3 : Hk(M)

x Hk(M)

-*• M

3(LJ,

The

rf) = / u A t). JM

3 is called the cohomological intersection form of M. When k is even (so that n is divisible by 4) 3 is a symmetric form. Its signature is called the signature of M and is denoted by cr(M). When k is odd 3 is skew-symmetric (i.e. it is a symplectic form). In particular, we deduce the following result. Corollary 7.2.13 For any compact manifold M £ ^Xfltk+2 bik+x(M) is even.

tne

middle Betti number

Exercise 7.2.1 (a) Let P 6 Z[t] be an odd degree polynomial with non-negative integer coefficients such that P(0) = 1. Show that if P satisfies the symmetry condition (7.2.1) there exists a compact, connected, oriented manifold M such that Pu(t) = P(t). (b) Let P e Z[i] be a polynomial of degree 2k with non-negative integer coefficients. Assume P(0) = 1 and P satisfies (7.2.1). If the coefficient of tk is even then there exists a compact connected manifold M € StTtJt s u c n * n a t P~M(t) = P{t)Hint: Describe the Poincare polynomial of a connect sum in terms of the polynomials of its constituents. Combine this fact with the Kiinneth formula. Remark 7.2.14 The result in the above exercise is sharp. Using his intersection theorem F. Hirzebruch showed that there exist no smooth manifolds M of dimension 12 or 20 with Poincare polynomials 1 + t6 + t12 and respectively l+t10+t20. Note that in each of these cases both middle Betti numbers are odd. For details we refer to J. P. Serre, "Travaux de Hirzebruch sur la topologie des varietes", Seminaire Bourbaki 1953/54,n° 88.

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theory

Figure 7.1: A cobordism in R3

7.3 §7.3.1

Intersection t h e o r y Cycles a n d t h e i r duals

Definition 7.3.1 Let M G SDT+. A fc-dimensional cycle in M is a pair (S, : S —• M is a smooth map. We denote by Cjt(M) the set of ^-dimensional cycles in M. Definition 7.3.2 (a)Two cycles (S 0 , #o), (Si, 0i) 6 Cfc(M) are said to be c o b o r d a n t (we write this (So, 4>o) ~ c (Si, 4>i)) if there exists a compact, oriented manifold with boundary E and a smooth map <£ : E —• M such that (al) 3E = (—S0) U Si where —S0 denotes the oriented manifold So with the opposite orientation and U denotes the disjoint union.

(a2)*ta=&, i = 0,l. (b) A cycle (S, ) € Cfc(M) is said to be d e g e n e r a t e if <j> is homotopic to the constant map S -> M. We write (S, 0) ~ c 0. We denote by ©^(M) the set of degenerate cycles. Exercise 7.3.1 Let (S 0 , <j>0) ~ c {Si, 4>\). Prove that ( - S 0 U Si, fa LI <£i) ~ c 0. We denote by 3 * (M) the free abelian group generated by Cjt(M) and by 2$k(M) C 3*(-W) the subgroup generated by cycles cobordant to degenerated ones. We can form the quotient group Sjk{M). For any cycle {S,<j>) G £/t(M) we denote by [S,<j>] its image in Sj^{M).

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Cohomology

Figure 7.2: The dual of a point is Dirac's

distribution

Exercise 7.3.2 (a) Prove that [So, 4>o] + [Si, 4>x] = [So uSufau

4>i] V(5i,
gCk(M)

and -[S,4>] =

[-S,4>}€Sik(M).

(b) Prove that[5, 4>] = 0 in fik(M) if and only if there exists a degenerate cycle (S'fi) such that {S,) ~ c (S', '). Any fc-cycle (S, ) defines a linear map Hk{M) —> R by

Vs

Stokes formula shows that this map is well defined i.e. it is independent of the closed form representing a cohomology class. Indeed, if w is exact, i.e. u> = du' then

f 'dJ = / # V = 0.

Js

Js

In other words, each cycle defines an element in (Hk(M))* which can be identified via the Poincare duality with H™~tk(M). Thus there exists 5S G H™~tk(M) such that [ LJA5S= JM

[ *io Vw £

Hk{M).

JS

5s is called the Poincare dual of (S, ). There exist many closed forms r) e QJl~k{M) representing 5s- When there is no risk of confusion, we continue denote any such representative by 6S.

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theory

Example 7.3.3 let M = K" and S is a point {pt} C Mn. pt is canonically a 0-cycle. Its Poincare dual is a compactly supported n-form w such that for any constant A (i.e. closed 0-form)

f Xu = [ A = A i.e.

1.

Ar'

Thus (5P( can be represented by any compactly supported n-form with integral 1. In particular we can choose representatives with arbitrarily small supports. Their "profiles" look like in Figure 7.2. "At limit" they approach Dirac's delta distribution. Example 7.3.4 Consider an n-dimensional, compact, oriented manifold M. denote by [M] the cycle (M, 1 M ). Then <5[M] = l e H°{M).

We

For any differential form UJ we set (for typographical reasons) |w| = deg a;. Example 7.3.5 Consider the compact manifolds M 6 SDt+ and TV £ 93T+. To any cycle (S,4>) £ CP(M) and (T,tp) € Cq{N) we can associate the cycle (SxT,xip) 6 £ p + g ( M x TV). We denote by 7rM (resp. irN) the natural projection M x TV —y M (resp. M x TV —y TV). We want to prove the equality = {-l){m-p)q5s

W

x <5T

(7.3.1)

where uxvd=f

TT*MU A Tr^n

V(w, n) e 0*(M)

x Q*(TV).

By Kiinneth formula we have H*{M x N) = H*(M) ® #*(TV) and the cohomology of M x TV is generated (as a linear space) by the cross products w x r\. Pick (w, r/) e ft* (M) x ft* (TV) such that deg u> + deg n = (m + n) - (p + 9). Then, using Exercise 7.1.11 /

(«,)Afex{

T

) = (-lf-*l/

JMxN

(ujA5s)x(riA5T). JMxN

The above integral should be understood in the generalized sense of Kronecker pairing. The only time when this pairing does not vanish is when |w| = p and |n| = q. In this case the last term equals

(~iy^(JsuA6s)

(/ r ,A^) = ( - i ) ^ ( / s f U )

= (_!)?(m-p) J JSxT

This establishes equality (7.3.1).

(^

X

^ V o ; A „).

(fTrv

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Cohomology

Example 7.3.6 Consider a compact manifold M G SDTj. Fix a basis (a;*) of H*(M) such that each a;* is homogeneous of degree \tVi\ = dt and denote by u>1 the basis of H*(M) dual to (a;,) with respect to the Kronecker pairing i.e. < W > J ) B = (-1)I U 'I-" U 'I< W J V>« =

4

I n M x M there exists a remarkable cycle, the diagonal A : M —• M x M

IH(I,I).

We claim that the Poincare dual of this cycle is <5A = *M d= E l - 1 ) ' " " 1 ^

X

"i-

(7-3.2)

i

Indeed, for any homogeneous forms a,/? 6 fl*(M) such that |a| + |/?| = n we have (w x r?) A D M = E l - 1 ) ' " ' 1 /

/ JMxM

= E(-l)|lJ,|(-l)l/JHui| /

J MxM

i

=

(a x /?) A (a/ x

Wi )

JMxM

i

(a A u>{) x (/? A «,-)

B-l)W(-l)WVl(/M«A^)(/M/JAWi)

The i-th summand is nontrivial only when |/?| = jcj11 and |or| = |w,-|. Using the equality |w'| + |<J'| 2 = 0(mod 2) we deduce /

(ax0)ABM = E ( 7

JMxM

aAf/)(7

~? \JM Q CJI

= E( ' i

/?Aw 4 )

I \JM

J

? W

>«(/ ' 0K-

From the equalities

we conclude

/ A>x/3)= / aA/J=/ ( E ^ ( ^ , a ) . ) A ( E ( A ^ ) » = / E( W '' a )«(^' a ; j)'' < J < A '^

=

E(W,'a)«(^'a'j)"(W»wJ}''

= ^(-l)KI(»-MI)(a, W i )-(-l) | w , | ( "- | u i l ) <)8,w i >« =

Equality (7.3.2) is proved.

E(Q'W')K(/?'W0K-

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theory

Proposition 7.3.7 Let M 6 tXfl* and (S i ; fa) G €k(M) (a) If (So, <M ~c (Si,
(i = 0,1) - two cycles in M.

Proof (a) Consider a compact manifold E with boundary 9E = So U Si and a smooth map $ : E —> M such that $ \aiz=fa,U fa. For any closed fc-form w € fifc(M) we have 0 = / $*(dw) = / d$*w StO*l 'ofces

/

=

/

I

A*

I

A*

(J = / fau - / 0„W.

(b) is left to the reader, (c) is obvious. To prove (d) consider E = [0,1] x S 0 and $ : [0,1] x S 0 -> M, $(t, x) = 0 o (x) V(t, x) e E. Note that 5E = (-S 0 ) U S 0 so that 5-s 0 + SSo = 5_Sous0 = •

fl£*(M).

This is usually called the homological Poincare duality. §7.3.2

Intersection theory

Consider M 6 fXJl* and S a fc-dimensional compact oriented submanifold of M. We denote by i : S <-¥ M inclusion map so that (S, i) is a fc-cycle. Definition 7.3.8 A smooth map <j> : T -» M from an {n — fc)-dimensional, oriented manifold T is said to be transversal to S (and we write this S ffl) if (a) <j>~1(S) is a finite subset of T; (b) for every x G
(direct sum).

In this case, for each x € 0 - 1 ( S ) we define the local intersection number at x to be (or= orientation) x{

°'

>

f 1 , or {T^x)S) A or{faTxT) = or(T 0 ( x ) M) \ - 1 , o r ( T , ( l ) S ) A o r ( ^ T x T ) = -or(T« x ) Af)

Finally, we define the intersection n u m b e r of S with T to be S-T=

Y,

i*(S,T).

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Cohomology

Figure 7.3: The intersection number of the two cycles on T2 is 1 Our next result offers a different description of the intersection number indicating how one can drop the transversality assumption from the original definition. Proposition 7.3.9 Let M G 971+. Consider S <-> M a compact, oriented, kdimensional submanifold and (T, ) G £„_fc(M) a (n — fc)-dimensional cycle intersecting S transversally, i.e. S (\)(j). Then S • T = f 5SAST

(7.3.3)

JM

where 5. denotes the Poincare dual of •. The proof of the proposition relies on a couple of technical lemmata of independent interest. Lemma 7.3.10 (Localization lemma) Let M € 971+ and (S, ) € £*(M). Then for any neighborhood N of (f>(S) in M there exists 8% 6 Q"~tk(M) such that (a) 6tf represents the Poincare dual Ss € H^~tk(M); (b) supptSf C.AA. Proof Fix a Riemann metric on M. Each point p £ {S) has a geodesically convex neighborhood entirely contained in A/". Cover <j)(S) by finitely many such neighborhoods and denote by N their union. Then TV G 9JT+ and (S,4>) G SDT+(./V). Denote by S G Qk(M) then ui \N is also closed and (S s A , Aw=

{?Aw=

4>*LO.

Intersection

theory

245

Hence, Sg represents the Poincare dual of S in H"ptk(M)

and moreover, supp 5$ C Af.

a Definition 7.3.11 Let M e 9DT+ and S <-» M a compact, A:-dimensional, oriented submanifold of M. A local transversal at p 6 S is an embedding 4>:Bc

Rn~k -» M

(B = open ball centered at 0)

such that 5 rfi0 and 0 _ 1 (S) = {0}. Lemma 7.3.12 Let M e 90?^, S <-> M a compact, fc-dimensional, oriented submanifold of M and (B, 4>) a local transversal at p € S. Then for any sufficiently "thin" closed neighborhood TV of S C M we have S •(£>*) = / / # • Proof Using the transversality S (\\(j>, the implicit function theorem and eventually restricting <> / to a smaller ball, we deduce that (for some sufficiently "thin" neighborhood TV of 5) there exist local coordinates (a; 1 ,..., xn) defined on some neighborhood U of p G M diffeomorphic with the cube {|i*'| < 1, Vt} such that (i)5 n U = {xk+1 = -.- = xn = 0},p=(0,...,0) (ii) The orientation of S n U is defined by dxl A • • • A dxk. (iii) The map : B C K?"* „_M —>• M is expressed in these coordinates as x 1 = 0 , . . . ,xk = O , ^ 1 = y\ ... ,xn =

yn-k.

{rv)Mr\U = {\xi\ < 1/2; j = l , . . . , n } . Let e = ± 1 such that edx1 A . . . A dxn defines the orientation of TM. words

e= 1

S-(B,).

k

For each £ = (re ,..., x ) e S n £/ denote by P ? the (n - fc)-"plane" P€ = { ( e ; i * + 1 , . . . , i " ) ; | ^ ' | k}.

We orient each P^ using the (n - fe}-form dxk+1 A • • • A ds" and set

"(0 = I 4-

In other

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Cohomology

Equivalently

v(t) = IBfBWS where ^ : B —• M is defined by

My\---,yn-k)

(Z;y\---,yn-k)-

=

To any function

n t/) such that s u p p l e {\x{\ < 1/2 ;

i
we associate the fc-form uv = tpdx1 A---Adxk

= cpd£ € Sl%t(S n f/).

Extend the functions a; 1 ,... ,xk G C°°(Ur\/f) to smooth compactly supported functions x{ e C^{M) ->• [0,1] The form ui^ is then the restriction to U n S of the closed compactly supported form u)v = ipix1,..., We have

xk)dxl

A • • • A dxk.

/>A ^=L u * A ^ = h u * = J * ^-

(?-3-4)

The integral over U can be evaluated using the Fubini theorem. Write 6% = fdxk+1

A---Adxn

+Q

where Q is an (n —fc)-formnot containing the monomial dxk+1 A • • • A dxn. Then J uov A 6$ = f f(pdxl A • • • A dxn = e / fip\dxx A • • • A dxn\ F

=nie[

Jsnu

(\dxl A • • • A dxn\ = Lebesgue density)

y»(0 f/"

\JPS

f\dxk^A---Adxn\)m J

= e Js
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theory

247

If in the above equality we use the assumption (iii) we recognize in the left-hand-side the integral JB

The local transversal lemma is proved. • Proof of Proposition 7.3.9

Let (j>-l(S) = { p i , . . . , p m } .

The transversality assumption implies that each pt has an open neighborhood B, diffeomorphic to an open ball such that i = <£|BJ is a local transversal at yt = {xi). Moreover, we can choose the neighborhoods -Bj to be mutually disjoint. Then

S-T = £ S •(£;,&)• i

The compact set K =

does not intersect S so that we can find a "thin", closed neighborhood" N of S '-* M such that i f n A r = 0. Then (f)*5-g is compactly supported in U-Dj

From the local transversal lemma we get j D tfSg = (-1)*<»-*>S • (Bit4>) =

iPi(S,T).

The Proposition is proved. • Equality (7.3.3) has a remarkable feature. Its right-hand-side is an integer but is defined only for cycles S, T such that 5 is embedded and S fflT, while the left-handside makes sense for any cycles of complementary dimensions but a priori it may not be an integer. In any event, we have a remarkable consequence. Corollary 7.3.13 Let ( $ , & ) e £k(M) i = 0,1. If

and ( ( T ^ i ) € C„_fc(M) where M e Ort+,

(a) So ~ c S i , T 0 ~ c T i ,

(b) the cycles Si are embedded and (c) Si !\\TZ

Then So • T0 = S\ • Ti.

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Cohomology

Definition 7.3.14 The homological intersection pairing is the Z-bilinear map 3 = ?M • &k(M) x Sjn.k(M)

-> M

(M e 9Jt+) defined by 0r(5,T)= /

5SA5T.

J M

We have proved that in some special instances 3(S,T) that when M is compact this is always the case.

6 Z. We want to prove

Theorem 7.3.15 Let M e 2K+ be compact manifold. Then for any (S,T) S)k{M) x Sjn-k(M) the intersection number 3{S,T) is always an integer.

e

The theorem will follow from two lemmata. The first one will show that it suffices to consider only the situation when one of the two cycles is embedded. The second one will show that if one of the cycles is embedded then the second cycle can be deformed so that it intersects the former transversally. (This is called a general position result). Lemma 7.3.16 (Diagonal trick)Let M, S and T as in Theorem 7.3.15. Then 3(S,T)

= (-1)""* /

6SxTA5A

JMxM

where A is the diagonal cycle A : M —>• M x M, x i-¥ (x,x). (It is here where the compactness of M is essential, since otherwise A would not be a cycle). Proof

We will use the equality 7.3.1 SsxT = ( - 1 ) " - Ss x ST-

Then (-1)"-* = /

5SxT A SA = /

JMXM

(Ss x ST) A 6A JMXM

= [ A*(5 S x 6T) = / 8s A ST. JM

JM

•

Lemma 7.3.17 (Moving lemma) Let S € C*(M) and T €
Intersection §7.3.3

249

theory

The topological degree

Consider two compact, connected, oriented smooth manifolds M, N having the same dimension n. Any smooth map F : M —• TV canonically defines an n-dimensional cycle i n M x J V IV : M -> M x N x^{x,F(x)). Tp is called the graph of F. Any point y G N defines an n-dimensional cycle M x {y}. Since N is connected all these cycles are cobordant so that the integer Tp • {M x {y}) is independent of y. Definition 7.3.18 The topological degree of the map F is defined by

degF^(Mx{y})-rF. Note that the intersections of TF with M x {y} correspond to the solutions of the equation F(x) = y. Thus the topological degree counts these solutions (with sign). Proposition 7.3.19 Let F : M ->• N as above. Then for any n-form w G Q.n(N)

f F*u) = degF [ u). JM

JN

Remark 7.3.20 The map F induces a morphism U = Hn(N)

^Hn{M)^R

which can be identified with a real number. The above proposition guarantees this number is an integer. Note that if w G Q.n(N) is exact then

Proof of the proposition

/ u> = / F*u = 0.

JN

JM

Thus, to prove the proposition it suffices to check it for any particular form which generates Hn(N). Our candidate will be the Poincare dual 5y of a point y G TV. We have while equality (7.3.1) gives &Mx{y] = SM X Sy = 1 X Sy. We can then compute the degree of F using Theorem 7.3.15 degF = d e g F f 5y = [ JN

(1 x <$„) A &rF

JMxN

"

= I VF(l x 6y) - f F*6y. D JM

JM

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Cohomology

Corollary 7.3.21 ( G a u s s - B o n n e t ) Consider a connected sum of g-tori E = E , embedded in K3 and let © s : E -> S 2 be its Gauss map. Then deg©s=x(S) = 2-2P. This corollary follows immediately from the considerations at the end of Subsection 4.2.4. Exercise 7.3.3 Consider the compact, connected manifolds M0,Mi,N € SDT^ and the smooth maps Ft : Mi -> N, i = 0,1. Show that if F0 is cobordant to Fx then deg F0 = deg Fi. In particular, homotopic maps have the same degree. Exercise 7.3.4 Let A be a nonsingular n x n real matrix. It defines a smooth map

Prove that deg FA = sign det A. H i n t : Use the polar decomposition A = PO (where P is a positive symmetric matrix and O is an orthogonal one) to deform A inside GL(n, tt) to a diagonal matrix. Exercise 7.3.5 Let M —» N be a smooth map (M, TV are smooth, compact oriented of dimension n). Assume y €. N is a. regular value of F i.e. for all x G F~l(y) the derivative DXF : TXM -+ TVN is invertible. For x £ F~1(y) define deg(F,x)

=

1 , DXF preserves orientations — 1 , otherwise

Prove that degF=

Y,

deg(F,x).

F(x)=y

Exercise 7.3.6 Let M denote a compact oriented manifold and consider a smooth map

F:M^M. Regard H*(M) as a superspace with the obvious Z2-grading H*(M) = Heven(M)

© H°dd(M)

and define the Lefschetz number X(F) of F as the supertrace of the pull back F* : H*{M) -+ H*{M). Prove that X(F) = A • r > . and deduce from this the Lefschetz fixed point theorem: X(F) ^ 0 => F has a fixed point.

Intersection §7.3.4

251

theory

Thorn isomorphism theorem

Let p : E —> B be an orientable fiber bundle with standard fiber F and compact, oriented basis B. Let d i m B = m and d i m F = r. In this subsection we will extensively use the techniques of fibered calculus described in Subsection 3.4.5. The integration along fibers

/

=

p,:Wcpt{E)^W{B)

J E/B

satisfies p,dE

{-\)rdBp*

=

so that it induces a map in cohomology P, : H*cpt{E) -+

H*~r(B).

This induced map in cohomology is sometimes called the Gysin map. Exercise 7.3.7 Consider a smooth map / : M -¥ N between compact, oriented manifolds M, N of dimensions m and respectively n. Denote by if the embedding of M in M x N as the graph of / M 3 x i-> (x, f(x)) g M x N. The natural projection M x N -+ N allows us to regard M x N as a trivial fiber bundle over N. Show that the push-forward map / . : H*(M) ->• H*+n~m(N) defined in Remark 7.2.9 can be equivalently defined by / , = 7T» O

(if)t

where 7r, denotes the integration along fibers while (i/). : H*cpt(M) -> H*cpt(N) is the natural morphism defined by the embedding if. Let us return to the fiber bundle p : E —• B. Any smooth section o:B-+

E

defines an embedded cycle in E of dimension m = dim B. Denote by 5a its Poincare dual in Hlpt(E). Using the properties of the integration along fibers we deduce that for any u) e Q,m(B) we have

/ <5CTAp*w= f ( f

JE

JB \JE/B

5a)to. J

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Cohomology

On the other hand by Poincare duality we get

[ 6aAp*Lj = (-l)rm

JE

fp'wAi

JE

= ( - i r / era; = (-1)™ f {P°T" = (-i)rm / <. JB

JB

JB

Hence

'A = I

Sa = ("I)™ € fi°(B).

JE/B

Proposition 7.3.22 Let p : E —> B a bundle as above. If it admits at least one section then the Gysin map p. : H*cpt(E) ->

H*-r(B)

is surjective. Proof

Denote by Ta the map r„ : H'(B)

->

fl^CE)

w H-> (-l) r m 5 C T A p*w = p*w A <J„.

Then ra is a right inverse for p„. Indeed u

= ( - l ) ™ p , { „ A u = (-l) r m p«(5 f f A p ' u ) =p.(T ff w).n

The map p* is not injective in general. If for example (5, <j>) is a fc-cycle in F then it defines a cycle in any fiber 7T_1(&) an consequently in E. Denote by 5s its Poincare dual in H™tr-k{E). Then for any w € fim~*(B) we have / (p.fe) A CJ = / <SS A p*u> = ± f fp'uj VD

= [ (p o 0)*<j = 0 JS

since p o ^ is constant. Hence pt5s = 0. Hence if F carries nontrivial cycles kerp, may not be trivial. The simplest example of standard fiber with only trivial cycles is a vector space. Definition 7.3.23 Let p : E —> B be an orientable vector bundle over the compact oriented manifold B (dimB = m, rank(£') = r). The Thorn class of E, denoted by TE is the Poincare dual of the cycle defined by the zero section Co : B —> E, b^0€Eb. Note that TB € Hrcvt{E).

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theory

Theorem 7.3.24 (Thom isomorphism)Let p : E -» B as in the above definition. Then the map T : H*(B)

-» tf^YX-E) W H T J A p*w

is an isomorphism called the T h o m isomorphism. Its inverse is the Gysin map (-l)""p. : H*cpt(E) -»• Proof

ff-'(S).

We have already established that

P.T =

Mr

To prove the reverse equality rp, = (—l) rm we will use Lemma 7.2.2 of Subsection 7.2.1. For p e Sl*q,t(E) we have (p*P,r B ) A 0 - TE A (p*pt(3) = ( - l ) r d ( m ( r E , P)) where m(rE,P)

G Q^, ( (£). Since p*p,rE = (—l) rm we deduce (-1)""/? = (TE Ap*(p,/?) + exact form

i.e. ( - l H = TEop,(^)

inff;t(£).Q

Exercise 7.3.8 Show that rE = C.l where C* : ff*(M) -> H'c+tAXmM{E) is the pushforward map defined by a section £ : M —> E. §7.3.5

Gauss-Bonnet revisited

We now examine a very special type of vector bundle: the tangent bundle of a compact, oriented, smooth manifold M. Note first the following fact. Exercise 7.3.9 Prove that M is orientable if and only if TM is orientable as a bundle. Definition 7.3.25 Let £ - > M b e a real orientable vector bundle over the compact, oriented n-dimensional smooth manifold. Denote by TE € H*pt(E) the Thom class of E. The Euler class of E is defined by e(E) = QTE G Hn(M) where ( 0 : M —> E denotes the zero section. e(TM) and is denoted by e(M).

is called the Euler class of M

254

Cohomology

Note that the sections of TM are precisely the vector fields on M. Moreover, any such section a : M —> TM tautologically defines an n-dimensional cycle in TM and in fact any two such cycles are homotopic - try an affine homotopy along the fibers of TM. Any two sections 0o,(?i : M —>• TM determine cycles of complementary dimension and thus the intersection number a0 • ax is a well defined integer, independent of the two sections. It is a number reflecting the topological structure of the manifold. Proposition 7.3.26 Let aQ, <7i : M -» TM be two sections of TM. Then / e(M) =<70 -o\JM

In particular, if dim M is odd then / e(M) = 0. JM

Proof The section
= [

JTM

I Sa JTM

A 5a

rM A SCo = f

JM

=

is the

/ TM A TM JTM

QTM

= f e(M). JM

If dim M is odd then

J e(M) = <70 • ox = -<7i • 0O = - J e(M). D JM

JM

Theorem 7.3.27 Let M be a compact oriented n-dimensional manifold and denote by e(M) its Euler class. Then / e(M)=x(M) = ^ ( - l ) ^ ( M ) .

In the proof we will use an equivalent description of x{M). Lemma 7.3.28 Denote by A the diagonal cycle in M x M. Then X(M)

= A • A.

Proof of the lemma Consider a basis (WJ) of H*{M) consisting of homogeneous elements. We denote by (a/) the dual basis i.e. (uji,ojj)K

=

5ij.

Intersection

255

theory

According to (7.3.2) we have

<5A = £ ( - l ) | w V

x

<*•

i

Similarly, if we start first with the basis (wl) then its dual basis is (_l)kl-l^l w . so we also have (taking into account that \u>j\ + |OJJ| • |w,| = n\u>j\ (mod 2))

5A = 'Ei-iyMuj

x J.

i

Using Exercise 7.1.11 we deduce

A A

• =/

(EC-1)1"'1^ X W *)

A

(E(-1)n|w,lwJ x ^ I

( E ( - l ) | w i | ( - l ) n • k K - 1 ) 1 ^ 1 ^ 1 ^ A W j j x (Wi x UJJ)

= /

= E ( - i ) k l ( - i ) " • KK-i^-'-^'k,^). (Wi)^)K. In the last expression we now use the duality equations (^ V)K

= (-l)^H^'

and the congruence \J\ + n\u)i\ + \tJi\2 + \wi\ • H^l = | ^ | + n\ui\ + \ui\2 + \ui\(n - |WJ|) H |U/| (mod 2) to conclude that A.A = £(-1)I^=X(M).

D

(J 1

Proof of theorem 7.3.27 The tangent bundle o f M x M restricts to the diagonal A as a rank 2n vector bundle. If we choose a Riemann metric on M x M then we get an orthogonal splitting T{M x M ) | A = 7 V A e T A . The diagonal map M -+ M x M identifies M with A so that TA = TM. have the following remarkable result. Lemma 7.3.29 NA =* T M .

We now

256 Proof

Cohomology Use the isomorphisms T(M x M) | A ^ TA © NA ^ TM © NA

and T ( M x M) | A = T M © TM.

•

From this lemma we immediately deduce the equality of Thorn classes TNA = TM.

(7.3.5)

At this point we want to invoke a technical result whose proof is left to the reader as an exercise in Riemann geometry. Lemma 7.3.30 Denote by exp the exponential map of a Riemann metric g on M x M. Regard A as a submanifold in N& via the embedding given by the zero section. Then there exists an open neighborhood TV of A C TVA C T(M X M) such that exp |AT: TV -> M x M is an embedding. Let A/" be a neighborhood of A C TVA as in the above lemma and set 9 t = exp(TV). We can view A as a submanifold of both 9 t and TV. Denote by <5/ (resp. <5^f) the Poincare dual of A in 91 (resp. TV). SA is the Thorn class of N& —> A which in view of (7.3.5) means =

<^A

T

NA

= TA =

TM.

We get A •A =

/

sf-

JA

= /

=

/

6% = f C0VA JA

JA

QrM = /

e(M).

IM

Hence / e(M)=X(M).

D

If n is a connected sum of g tori then we can rephrase the Gauss-Bonnet theorem as follows. Corollary 7.3.31 For any Riemann metric ft on a connected sum of
47T

= e(E 9 ) in ff*(£9).

Intersection

257

theory

The remarkable feature of the Gauss-Bonnet theorem is that once we choose a metric we can explicitly describe a representative of the Euler class in terms of the Riemann curvature. The same is true for any compact oriented even dimensional Riemann manifold. In this generality the result is known as Gauss-Bonnet-Chern and we will have more to say about it in the next chapter. We now have a new interpretation of the Euler characteristic of a compact oriented manifold M. Given a smooth vector field X on M, its "graph" in TM

Tx = {(x,X(x))eTxM;

xeM}

is an n-dimensionat submanifold of TM. The Euler characteristic is then the intersection number X (M) = Tx • M where we regard M as a submanifold in TM via the embedding given by the zero section. In other words, the Euler characteristic counts (with sign) the zeroes of the vector fields on M. For example if x{M) ^ 0 this means that any vector field on M must have a zero ! We have thus proved the following result. Corollary 7.3.32 If x(Af) / 0 then the tangent bundle TM is nontrivial. The equality x{S2n) = 2 is particularly relevant in the vector field problem discussed in Subsection 2.1.4. Using the notations of that subsection we can write v{S2n) = 0. We have thus solved "half" the vector field problem. Exercise 7.3.10 Let X be a vector field over the compact oriented manifold M. A point i G M i s said to be a non-degenerate zero of X if X(0) = 0 and

det

(§K^ a

for some local coordinates (xl) near XQ such that the orientation of T*0M is given by dx1 A • • • A dx". Prove that the local intersection number of Tx with M at x0 is given by *x 0 (rx,M) =sign det

dX£ dxi '

h=x

°

(This is sometimes called the local index of X at x0 and is denoted by i(X, x0). From the above exercise we deduce the following celebrated result.

258

Cohomology

Corollary 7.3.33 (Poincare-Hopf)If X is a vector field along a compact, oriented manifold M with only non-degenerated zeros xi,...,Xk then X(M)

1

=

£i(X,xj).

3

Exercise 7.3.11 Let X be a vector field on M" and having a non-degenerate zero at the origin. (a) prove that for all r > 0 sufficiently small X has no zeros on Sr — {\x\ = r}. (b) Consider Fr : ST -> S"" 1 defined by

Prove that i(X, 0) = degF r for all r > 0 sufficiently small. Hint: Deform X to a linear vector field.

7.4

S y m m e t r y and topology

The symmetry properties of a manifold have a great impact on its global (topological) structure. We devote this section to presenting some aspects of this avenue. §7.4.1

Symmetric spaces

Definition 7.4.1 A homogeneous space is a smooth manifold M acted transitively by a Lie group G called the symmetry group. Recall that a left action G x M —>• M

(g, m) >-> g • m

is called transitive if for any m £ M the map

tym:G3gi-^g-meM is surjective. For any point a; of a homogeneous space M we define the isotropy group at x by 3x = {geG

; g-x

= x}.

Lemma 7.4.2 Let M be a homogeneous space with symmetry group G and x,y € M. Then closed subgroup of G; (b) 3X =* 3y.

259

Symmetry and topology Proof that

(a) is immediate. To prove (b), choose g e G such that y = g-x. Then note 3y = g3xg~1-

•

Remark 7.4.3 It is worth mentioning some fundamental results in the theory of Lie groups which will shed a new light on the considerations of this section. Their proofs can be found in the monograph [73]. FACT 1. Any closed subgroup of a Lie group is also a Lie group. In particular, the isotropy groups 3X of a homogeneous space are all Lie groups. They are smooth submanifolds of the symmetry group. FACT 2. Let G be a closed group and H a closed subgroup. Then the space of left cosets G/H = {g • H ; g € G} can be given a smooth structure such that the map

Gx(G/H)^G/H

(gi,g2H)^(gig2)-H

is smooth. G/H becomes a homogeneous space with symmetry group G. All isotropy groups are isomorphic to H. FACT 3. If M is a homogeneous space with symmetry group G and x G M then M is equivariantly diffeomorphic to G/3X i.e. there exists a diffeomorphism 4>: M -> G/3X such that (g-y)=g- <j>{y). We will be mainly interested in a very special class of homogeneous spaces. Definition 7.4.4 A symmetric space is a collection of data (M, h, G, a, i) satisfying the following conditions. (a) (M, h) is a Riemann manifold. (b) G is a connected Lie group acting isometrically and transitively on M G x M 3 (p, m) >->• g-me

M.

(c) a : M x M —> M is a smooth map (mi,m 2 ) !-• omi{mi) such that (cl) Vm 6 M am : M —> M is an isometry and am(m) = m. (c2) amoam = 1 M . (c3) Dam\TtnM= -lTmM(c4) Ggm = gamg~l (d) v : M x G —> G, (m, g) i-> i,„g is a smooth map such that (dl) Vm 6 M, im : G -> G is a homomorphism of G. (d2) im o im = 1G(d3) i 9 m = g\mg~l, V(m, g) e M xG. (e) amga^-{x)

= amgam{x)

= ^(g)

• x, Vm,x € M, g &G.

260

Cohomology

Remark 7.4.5 This may not be the most elegant definition of a symmetric space and certainly it is not the minimal one. Optimizing it will require a substantial amount of work and we refer to [34] for an extensive presentation of this subject. This definition has one advantage. It lists all the properties we need to establish the topological results of this section. The next exercise offers the reader a feeling of what symmetric spaces are all about. In particular, it describes the geometric significance of the family of involutions am. Exercise 7.4.1 Let connection and by R (a) Prove that VR = (b) Fix m 6 M and

(M,h) be a symmetric space. Denote by V the Levi-Civita the Riemann curvature tensor. 0. let j{t) be a geodesic of M such that 7(0) = m. Show that

°"m7(*) = 7 H ) Example 7.4.6 Perhaps the most popular example of symmetric space is the round sphere Sn C M™+1. The symmetry group is SO(n + 1) - the group of orientation preserving "rotations" of K" +1 . For each m e Sn+1 we denote by am the orthogonal reflection through the 1-dimensional space determined by the radius Om. We then set im(T) =CTmTo-"1,V T e S O ( n + l ) . We let the reader check that a and i satisfy all the required axioms. Example 7.4.7 Let G be a connected Lie group and m a bi-invariant Riemann metric on G. The direct product G x G acts on G by {91,92) •h =

g1hg21.

This action is clearly transitive and since m is bi-invariant its is also isometric. Define a : G x G -¥ G agh = gh~lg~x and i:Gx(GxG)->GxG

ig{gug2)

= (sSiS~\552P" 1 )-

We leave the reader check that these data define a symmetric space structure on (G, m). The symmetry group is G x G. Example 7.4.8 Consider the complex grassmannian M = Gk{n,C). S G M can be identified with a rank k, selfadjoint idempotent P : C " ->
Any element

P2.

P = Ps is the orthogonal projection onto the fc-dimensional subspace S C C". Denote by Symn the linear space of selfadjoint n xn complex matrices. The map M 3 S >-> Ps € Symn

261

Symmetry and topology is an embedding of M in Symn.

The linear space Symn has a natural metric

go{A,B)=9\etr{AB*). It defines by restriction a Riemann metric go on M. The unitary group U(n) acts on Symn by A^>T*A

= TAT*

TeU{n)AeSymn.

Note that U(n) * M = M and g0 is J7(n)-invariant. Thus £/(n) acts transitively and isometrically on M. For each subspace S € M define Rs = Ps-

PS± = 2P S - 1.

Rs is the orthogonal reflection through S-1. Note that Rs e U(n) and Rs = 2. The map A>-^ RS*A is an involution of Symn. family of involutions

It descends to an involution of M. We thus get an entire

a:MxM^M,

{PSl,Ps2)

•-> RSl * Ps2-

define i : M x t/(n) ->• [/(n),

isT =

flsTi?S-

We leave the reader check that the above collection of data defines a symmetric space structure on Gk(n,C). Exercise 7.4.2 Fill in the details left out in the above example. §7.4.2

Symmetry and cohomology

Definition 7.4.9 Let M be a homogeneous space with symmetry group G. A differential form u € Q*(M) is said to be (left) invariant if £*gw = ui \/g e G where we denoted by

tg: n*(M) -> n*(M) the pullback defined by the left action by g: rrn-^r g • m. Proposition 7.4.10 Let M be a compact homogeneous space with compact, connected symmetry group G. Then any cohomology class of M can be represented by a (not necessarily unique) invariant form.

262

Cohomology

Proof Denote by dfi9 the normalized bi-invariant volume form on G. For any form u € fi*(M) we define its G-average by uJ =

tgUidfig.

aJ is an invariant form on M. The proposition is a consequence of the following result. Lemma 7.4.11 If a; is a closed form on M then ZJ is closed and cohomologous to u. Proof of the lemma The form u> is obviously closed so we only need to prove it is cohomologous to UJ. Consider a bi-invariant Riemann metric m on G. Since G is connected the exponential map exp : £ G -> G X >-*• exp(iX) is surjective. Choose r > 0 sufficiently small such that exp : Dr = {\X\m

= r ; X G £ G } -> G

is an embedding. Set Br = expD T . We can select finitely many gi,..-,gm that

€ G such

771

G={jBj

(B^gjBr).

Now pick a partition of unity (a.j) C C°°(G) subordinated to the cover (Bj) i.e. 0 < OLJ < 1,

suppQj C Bj,

^2ai

=

!•

Set Oj = /

ajd/j,g.

For any j = 1 , . . . , m define Tj : ft*(M) -»• fi'(M) by Tj-w = /

aj{g)e*gujdfj,g.

JG

Note that a; = J ^ 3}w and dTjUi = Tjdtv. 3

Each Tj is thus a cochain map. It induces a morphism in cohomology which we continue to denote by Tj. The proof of the lemma will be completed in several steps. Step 1. £g = id on H*{M) for all g G G. Let X e £ G such that g = expX. / : / x M -)• M

/ t (m) = exp(tX) • m =

e*exp(tx)m.

Define

263

Symmetry and topology / is a homotopy connecting 1 M with I*. This concludes Step 1. Step 2 Tj = Ojlfl-.(Af).

For t € [0,1] consider 4>j,t • Bj —> G defined as the composition B.

9

4 Br •%' DT f 4 Dtr *% Btr 4 G.

Define Tj>t : Q*(M) -> fi*(M) by 2},(w = / a , - ( f l ) ^

(t) wd/i,

=T^>.

(7.4.1)

We claim that T j 0 is cochain homotopic to T ^ . To verify this claim set t = es, - c o < s < 0 and p s = exp(e s exp" 1 (g)) Vp £ B r . Then t/sw = TjJC,u> = J

a>j(gjg)e*g.gau; = J

a{gjg)tgfg.u}dng.

For each g 6 BT the map ys(m)

= g,(m)

defines a local flow on M. We denote by Xg its infinitesimal generator. Then d r — {Usu) = JB = ]

aj{gjg)(dMiXg

aj(gjg)LXgtgstg.u>dng +

ix9dM)(tgatg.u})diig.

Consequently TjfiU ~ T J M W = U-ooU -U0LO

= - 1 ^ [JB

aj{gjg){diXg

+ iXsd){tgstgju)diJ,g)

ds.

(An argument entirely similar to the one we used in the proof of the Poincare lemma shows that the above improper integral is pointwise convergent). From the above formula we immediately read a cochain homotopy x : fi*(M) —> fl*_1(M) connecting U-oo to U0. More precisely {xH}

\XEM= ~ j ^ yjB aj{gjg) {ixSgfgv}

Now notice that Tj.ou = [JB aj(g)diig\

= a/'.cj

U d(J.g) ds.

264

Cohomology

while Tj,iu> =

TjUj.

Taking into account Step 1 we deduce T, 0 = a,id. Step 2 is completed. The lemma and hence the proposition follow from 1H-(M) = YLa.jlH'(M) = Y^Ti j

= G

~

avera

ge-

D

i

The proposition we have just proved has a greater impact when M is a symmetric space. Proposition 7.4.12 Let (M, h) be an, oriented symmetric space with symmetry group G. Then the following are true. (a) Every invariant form on M is closed. (b) If moreover M is compact then the only invariant form cohomologous to zero is the trivial one. Proof (a) Consider an invariant fc-form u. Fix m £ M and set <2i = O^UJ. We claim <2> is invariant. Indeed, Vp € G Z*gU = < * / > =

(0Vnff)*O>

Since Z)
on M.

In particular diu = {-l)kduj

on M.

The (k + l)-forms dui = a^dw and du) are both invariant and as above we deduce du =
{-l)k+lduj.

The last two inequalities imply du) = 0. (b) Let u) be an invariant form cohomologous to zero i.e. u = da. Denote by * the Hodge *-operator corresponding to the invariant metric h. Since G acts by isometries

Symmetry and topology

265

7] = *u> is also invariant so t h a t drj = 0. We can now integrate ( M is compact) a n d use Stokes theorem t o get / u A *w = / daA = ± / a A dr) = 0.

•/M

JM

VM

This forces w = 0. • Form Proposition 7.4.10 and the above theorem we deduce the following celebrated result of Elie Cartan ([17]) Corollary 7.4.13 (Cartan)Let (M, h) be a compact, oriented symmetric space with compact, connected symmetry group G. Then the cohomology algebra H*{M) of M is isomorphic with the graded algebra Q.*inv{M) of invariant forms on M. In the coming subsections we will apply this result to the symmetric spaces discussed in the previous subsection: the Lie groups and the complex grassmannians. §7.4.3

T h e cohomology of compact Lie groups

Consider a compact, connected Lie group G and denote by £ G its Lie algebra. According to Proposition 7.4.10, in computing its cohomology it suffices to restrict our considerations to the subcomplex consisting of left invariant forms. This can be identified with the exterior algebra A*£ G . We deduce Corollary 7.4.14 H*(G) = H*(£G) £ A •„„£<; where A* nt) £ G denote the algebra of bi-invariant forms on G, while H*(£,Q) denotes the Lie algebra cohomology introduced in Example 7.1.9. Using the Exercise 7.1.2 we deduce the following consequence. Corollary 7.4.15 If G is a compact semisimple Lie group then HX(G) = 0. Proposition 7.4.16 Let G be a compact semisimple Lie group. Then H2(G) = 0. Proof A closed bi-invariant 2-form u on G is uniquely defined by its restriction to £ G and satisfies the following conditions. duj = 0 ^=> u>([X0,Xx},X2)

-CJ([X0,X2},X1)

+ LJ([XUX2],X0)

= 0

and (right-invariance) (LXow)(XuX2)=Q

VX0 G

£G<=*U([X0,X1},X2)-CJ([X0,X2},X1)=0.

Thus u{[X0,X1],X2)

= 0 V*0,XltX2€ £G.

On the other hand since ^(HQ) = 0 we deduce (see Exercise 7.1.2) £ G = [ £ G , £ G ] so that the last equality can be rephrased as u(X,Y)

=0 VX,ye£G. D

266

Cohomology

Definition 7.4.17 A Lie algebra is called simple if it has no nontrivial ideals. A Lie group is called simple if its Lie algebra is simple. Exercise 7.4.3 Prove that SU(n) and SO(m) are simple. Proposition 7.4.18 Let G be a compact, simple Lie group. Then H3(G) Moreover, H3(G) is generated by the Cartan form a(X,Y,Z)

=

= R.

K([X,Y],Z)

where n denotes the Killing pairing. The proof of the proposition is contained in the following sequence of exercises. Exercise 7.4.4 Prove that a simple Lie algebra is necessarily semi-simple. Exercise 7.4.5 Let w be a closed, bi-invariant 3-form on a Lie group G. Then u(X,y,

[Z,T]) = u([X, Y], Z,T)

VX, Y,Z,Te

£G.

Exercise 7.4.6 Let w be a closed, bi-invariant 3-form on a compact, semisimple Lie group. (a) Prove that for any X £ £ G there exists a unique left invariant TJX € O1 (G) such that (ixuj)(Y,Z) = r,x([Y,Z}). Moreover, the correspondence X i-> rjx is linear. HintrUse H\G) = H2{G) = 0. (b) Denote by A the linear operator £G —> £ G defined by K(AX,Y)=T,X(Y).

Prove that A is selfadjoint with respect to the Killing metric. (c) Prove that the eigenspaces of A are ideals of £ G . Use this to prove Proposition 7.4.18. Exercise 7.4.7 Compute / JSU(2)

a and /

a

JSO(3)

where a denotes the Cartan form. (These groups are oriented by their Cartan forms.) Hint: Use the computation in the Exercise 4.1.16 and the double cover SU(2) —> 5 0 ( 3 ) described in the Subsection 6.2.1. Pay very much attention to the various constants.

Symmetry §7.4.4

267

and topology Invariant forms on grassmannians and Weyl's integral formula

We will use the results of Subsection 7.4.2 to compute the Poincare polynomial of the complex grassmannian Gk(n, C). Set £ = n — k. Gk{n,C) is a symmetric space with symmetry group U(n). It is a complex manifold so that it is orientable (cf. Exercise 3.4.2). Alternatively, the orientability of Gk{n, C) is a consequence of the following fact. Exercise 7.4.8 If M is a homogeneous space with connected isotropy groups then M is orientable. We have to describe the [/(n)-invariant forms on Gk(n, C). These forms are completely determined by their values at a particular point in the grassmannian. We choose this point to correspond to the subspace So determined by the canonical inclusion C* ^ Cn. The isotropy of So is the group H = U(k) x U(£). H acts linearly on the tangent space V0 = Ts0Gk{n, C). If UJ is an [/(n)-invariant form then its restriction to V0 is an //-invariant skew-symmetric, multilinear map VQ

x • • • x Vo -»• M.

Conversely, any if-invariant element of A*V0* extends via the transitive action of U(n) to an invariant form on Gk{n, C). Denote by A*nv the space of .ff-invariant elements of A*V0*. We have thus established the following result. Proposition 7.4.19 There exists an isomorphism of graded M-algebras: H*{Gk(n,C))

= AJni).

We want to determine the Poincare polynomial of the complexified graded space, A* TO ®C

PkAt) = Et3

dim

c AL, ® c = PGt(n,C)(0-

3

Denote the action of H on Vo by H B h^Th

e Aut(Vo).

Using the equality (3.4.9) of Subsection 3.4.4 we deduce PhM

= / I d e t ( V 0 + tTh)\2dh

(7.4.2)

where dh denotes the normalized bi-invariant volume form on H. At this point the above formula may look hopelessly complicated. Fortunately, it can be dramatically simplified using a truly remarkable idea of H.Weyl.

268

Cohomology

Note first that the function
H3h^

is a class function i.e.
xTl

where T* = {diag(e I ( \...,e 1 9 *) € [ / ( £ ) } and similarly T ' = {diag (e 1 * 11 ,..., e1*') <E fJW} Each h € U(k) x U(£) is conjugated to diagonal unitary matrix i.e. there exists g e H such that ghg~l g T. We can rephrase this fact in terms of the adjoint action of H on itself Ad: HxH

^H

{g,h)^> Adg{h) = ghg'1.

The class functions are constant along the orbits of the adjoint action and each such orbit intersects the maximal torus T. In other words, a class function is completely determined by its restriction to the maximal torus. Hence, at least theoretically, we should be able to describe the integral in (7.4.2) as an integral over T. This is achieved in a very explicit manner by the next result. Define A n : Tn ->• C by

A„(0i,...A)=

II (eWi-ewO- (i = V=l) l
Proposition 7.4.20 (Weyl's integration formula) Consider a class function tp on the group G = U(ki) x • • • x U(ks). Then

fa ^g)d9

=

fci-^fc! L V^U '''' <s)|A*l(il)|2 ''' lA*-(*«)l2(fti A - ' - A dt°-

Above, dg denotes the normalized bi-invariant volume form on G. For each j = 1 , . . . , s we denoted by t = ti the collection of angular coordinates on Tk = Th> while dt denotes the normalized bi-invariant volume on Tk

Symmetry and topology

269

The remainder of this subsection is devoted to the proof of this proposition. The reader may skip this part at the first lecture and go directly to Subsection 7.4.5 where this formula is used to produce an explicit description of the Poincare polynomial of a complex grassmannian. Proof of Proposition 7.4.20 We will consider only the case s = 1. The general situation is entirely similar. Thus G = U(k) and T = Tk. Denote the angular coordinates on T by (01,..., 9k). Given a class function ip : G —> C form the complex valued form iov =
q:TxG/T^G

Note that if gxT = g{T then g\tgxx = gitg^1 so q is well defined. Pick a real metric m on £ G which is Ad-invariant i.e. Ad;m = m

WgeG.

The natural choice m(X, Y) = - 9 t e tr (XY*)

u{k) = £ G

will do the trick. The Lie algebra £<j splits orthogonally as £G

=

£T®

£G/T-

{£G/T

= £r)

The tangent space to 1 • T € G/T can be identified with £G/TFix x € G/T. Any g eG defines a linear map Lg : TXG/T -* TgxG/T. Moreover, if gx = hx = y then Lg and Lh differ by an element in the stabilizer of a; € G/T. This stabilizer is isomorphic to T and in particular it is connected. Hence, if a; € det TXG/T then Lgtj € det TyG/T and Lhoj € det TyG/T define the same orientation of TyG/T. In other words, an orientation in one of the tangent spaces of G/T "spreads" via the action of G to an orientation of the entire manifold. Thus, we can orient G/T by fixing an orientation on £G/T- We fix an orientation on HQ and orient £? using the form d6l A • • • A dOk. The orientation on HG/T will be determined by the condition (or= orientation) or(£ G ) = o r ( £ r ) A o r ( £ G / r ) The proof Weyl's integration formula will be carried out in two steps. Step 1 / (J = 77 / JG

k\

q*LJ V w .

JTXG/T

Step 2 For any class function tp on G we have

/ JTxG/T

q'uv=

[
270

Cohomology

S t e p 1. We use the equality /

q*u = deg q \ u

JTxG/T

JG

so it suffices to compute the degree of q. Denote by N(T) the normalizer of T in G i.e. N(T) = {geG;

gTg'1 C T}

and then form the Weyl group 2IT =

N(T)/T.

Lemma 7.4.21 2U = (S^-the group of permutations of k symbols. Proof This is a pompous rephrasing of the classical statement in linear algebra that two unitary matrices are similar iff they have the same spectrum (multiplicities included). The adjoint action of N(T) on T = diagonal unitary matrices simply permutes the entries of a diagonal unitary matrix. This action descends to an action on the quotient ?2XS so that 2H C &kConversely, any permutation of the entries of a diagonal matrix can be achieved by a conjugation. Geometrically this corresponds to a reordering of an orthonormal basis. • Lemma 7.4.22 Let a 1 , . . . , ak) £ Mfc such that 1, f^,..., ^ are linearly independent over Q. Set r = (exp(ia 1 ),... ,exp(ia* : )) G Tk. Then the sequence {Tn)n&2, is dense in Tk. (T is said to be a generator of Tk.) For the sake of clarity, we defer the proof of this lemma to the end of this subsection. Lemma 7.4.23 Let r € Tk C G be a generator of Tk. consists of |23J| = A;! points.

Then ^ ( r ) C T x G/T

Proof q{s, gT) = T<==> gsg-1 = r ^=> grg'1 = n

l

In particular, gr g~

n

= s £T,Vn

seT.

n

£ Z. Since (r ) is dense in T we deduce

gTg-lcT^geN{T). Hence 9 _ 1 (r) = {{g-lT9,gT) _1 T

and thus ? ( )

nas tne

eTxG/T;g&

N(T)}

same cardinality as the Weyl group 2B. •

271

Symmetry and topology The metric m on £ G / r extends to a G-invariant metric on G/T. left-invariant volume form on G/T, dv\n- Let v0 = /

JG/T G/T

It defines a

dv m

and set d\i = — dvxnLemma 7.4.24 q'dg = \Ak(t)\2dt A d/j,. In particular, any generator r of Th C G is a regular value of q since Afc(
hs = q(toexp(sX),g0exp(sY)T)

e G.

We want to describe —h^h, ds

6 TXG = £ G .

Using the Taylor expansions exp(sX) = 1 + sX + G(s 2 ) and exp( S y) = 1 + s F + 0(s2 we deduce V % = <7ot0-1(l + ay)*o(l + sX)(l = 1 + s (got^Ytogo1

- sY)g^

+ goXg^1 - goYgo1) +

+ 0(s2) 0(s2).

Hence Ar„ : TX0{T x G/T) * £r © £ G / T -> £ r © £ G / r = £ G can be written as DXQq{X ®Y)=

Ad 90 (Ad ( -. - id)Y + Ad 90 X

or in block form DXoq = Ad9,

0

Ad,-i - 1 P

Ad 9 is an m-orthogonal endomorphism of £ G so that det Ad 9 = ± 1 . On the other hand, since G = U(k) is connected det Ad 9 = det Adi = 1. Hence detDSog = d e t ( A d 1 - i - l i » o / r ) . We can identify £ G / r with { X e w ( f c ) ; Xjj = 0 j =

l,...,k}.

272

Cohomology

Given t = diag (exp(W 1 ),... , exp(i0*)) 6 T ' c U(k) we can explicitly compute the eigenvalues of Ad 4 -i acting on £G/T- They are {exp(-i(0i - 0,)) ; 1 < i± j < k}. Consequently = \/\k(t)\2.

detDXoq = det{Adt-i-l)

n

Lemma 7.4.24 shows that q is an orientation preserving map. Using Lemma 7.4.23 and Exercise 7.3.5 we deduce degg = |2B| = k\. Step 1 is completed. Step 2 follows immediately from Lemma 7.4.24. Weyl's integration formula is proved. • Proof of Lemma 2 We follow Weyl's original approach ([75],[76]) in a modern presentation. Let X = C(T, C) denote the Banach space of continuous complex valued functions on T. We will prove that

If U C T is an open subset and / is a continuous, non-negative function supported in U (/ ^ 0) then for very large n

This means that f(r') / 0 i.e. T> £ U for some j . To prove the equality (7.4.3) consider the continuous linear functionals Ln, L : X ->C

M/) = ^ £ / ( ^ ) • 3=0

and

L(f) = f fdt. JT

We have to prove that lim Ln(f)

= L(f)

/€X.

(7.4.4)

It suffices to establish (7.4.4) for any / G <S where 5 is a subset of X spanning a dense subspace. We let S be the subset consisting of the trigonometric monomials e c ( 0 \ . . . , 6k) = exp(iCi^1) • • • exp(iC^ fc )

C = (Ci, • • •, Ct) 6 %"•

273

Symmetry and topology

The Weierstrass approximation theorem guarantees that this <S spans a dense subspace. We compute easily L

^) = ^TlS^ (a) = ^ T l e c ( a ) - l •

Since 1, ^ r Q i , . . . , ^ak all C € Zk. Hence "

are linearly independent over Q we deduce that e^(a) / 1 for lim LJeA

= 0 = / e<-d£ = L(eA-

Lemma 7.4.22 is proved. • §7.4.5

The Poincare polynomial of a complex grassmannian

After this rather long detour we can continue our search for the Poincare polynomial ofGt(n,C). Let So denote the canonical subspace C* <-» C". The tangent space of Gk{n, C) at So can be identified with the linear space <£ of complex linear maps Cfc —> Ce, t — n — k. The isotropy group at So is H = U(k) x U{tj. Exercise 7.4.9 Prove the isotropy group H acts on <£ = {L : Ck -* Ce} by {T,S)-L

= SLT*

VL€<E,TeU{h),S

&U(l).

Consider the maximal torus Tk x Te C H formed by the diagonal unitary matrices. We will denote the elements of Tk by g = ( e i , . . . , ej.), eQ = exp(ir a ) and the elements of Tl by e = ( e i , . . . , ei), ej = exp(i^). The normalized measure on Tk is denoted by dr while the normalized measure on Te is denoted by d6. The element (e, e) 6 Tk x Te viewed as a linear operator on (£ has eigenvalues {eaej ; 1 < a < k

l<j<£}.

Using the Weyl integration formula we deduce that the Poincare polynomial of Gk {n, C) is PkM

=

= ~

^JTkxTtmi+t^j\2\AMr))\2\^(e(0))\2drAde fTkxTtn\ea+tej\2\AMT))\2\Me((>))\2dT

Ad6.

We definitely need to analyze the integrand in the above formula. Set hA*) = I ] > a + <e,)Afc(e)A£(e)

274

Cohomology

so that P

w W = TiH fu

JkAVhMdr

A d$.

(7.4.5)

We will study in great detail the formal expression Jk,t{t\ x'> V) = I[(x<* + *%•)• The Weyl group 2U = ©* x &e acts on the variables (x; y) by separately permuting the T-components and the ^/-components. If (p, ip) € 2U then JkAt;a(x)'>ip(y))

=

JkA^^y)-

Thus we can write Jj.^ as a sum

^MW = Eid<3d(a:)^(l/) where Qd(x) and i?d(y) are symmetric polynomials in x and respectively y. To understand the nature of these polynomials we need to introduce a very useful class of symmetric polynomials, namely the Schur polynomials. This will require a short trip in the beautiful subject of symmetric polynomials. An extensive presentation of this topic is contained in the monograph [50]. A partition is a compactly supported, decreasing function A:{1,2,...}^{0,1,2,...}. We will describe a partition by an ordered finite collection (Ai, A2,..., An) where Ai > • • • > An > A n+1 = 0. The length of a partition A is the number L(A) = max{n ; A„ ^ 0}. The weight of a partition A is the number |A| =

£ V n>\

Traditionally, one can visualize a partition using Young diagrams. A Young diagram is an array of boxes arranged in left justified rows (see Figure 7.4). Given a partition (Ai > • • • > An) its Young diagram will have Ai boxes on the first row A2 boxes on the second row etc. Any partition A has a conjugate A defined by An = # 0 ' > 0 ; A,- > n } . The Young diagram of A is the transpose of the Young diagram of A (see Figure 7.4). A strict partition is a partition which is strictly decreasing on its support. Denote by Vn the set of partitions of length < n and by "P* the set of strict partitions A of length n - 1 < L(X) < n. Clearly V*CVn. Denote by 6 = 6n G V*n the partition (n-l,n-2,...,l,0,...).

275

Symmetry and topology

Figure 7.4: The conjugate of (6,5,5,3,1) is (5,4,4,3,3,1) Remark 7.4.25 The correspondence VnB\>^\

+ 6n£V*

is a bijection. To any A e V* we can associate a skew-symmetric polynomial a A (:ri,...,x n ) = d e t ( a ^ ) =

Y,

e

(°)xa(i)-

a 6 e„ Note that asn is the Vandermonde determinant as(xu ...,xn)=

det(x" _ 1 _ I ) = Y[(xt - x,) =

An(x).

For each A G Vn we have A + 5 € "P* so that a^+s is well defined and nontrivial. Note that a\+s vanishes when xt = Xj so that the polynomial a\+s(x) is divisible by each of the differences (x( — Xj) and consequently is divisible by as- Hence s ( x )

=

^

)

is a well defined polynomial. It is a symmetric polynomial since each of the quantities a\+s and as is skew-symmetric in its arguments. S\(x) is called the Schur polynomial corresponding to the partition A. Note that each Schur polynomial 5A is homogeneous of degree |A|. We have the following remarkable result. Lemma 7.4.26 \i(t)=

£

trMS-x(x)S-x(y)

where Vk,i = {A ; Ai < fc L(A) < I}. For each A € T>k,i we denoted by A the complementary partition A = (k - Ac, k - X1-1, . . . , k - Ai).

Cohomology

276

6

1 0

7

j

4 4 2

1

0

*"

5 3

3 1

|||

Figure 7.5: The compiementarj of (6, 4,4, 2,1,0) £ P7,6 is (7,6, 5, 3,3,1) G P 6 , 7 Geometrically, the partitions in Vk,i are precisely those partitions whose Young diagrams fit inside &£xk rectangle. If A is such a partition then the Young diagram of the complementary of A is (up to a 180° rotation) the complementary of the diagram of A in the I x k rectangle (see Figure 7.5). For a proof of Lemma 7.4.26 we refer to [50] Section 1.4, Example 5. The true essence of the Schur polynomials is however representation theoretic and a reader with a little more representation theoretic background may want to consult [49], Chapter VI (Sec 6.4, Thm .V) for a very exciting presentation of the Schur polynomials and the various identities they satisfy, including the one in Lemma 7.4.26. Using (7.4.5), Lemma 7.4.26 and the definition of the Schur polynomial we can describe the Poincare polynomial of Gk(n, C) as

™=mL

TkxT<

£ *M,

^+ik

(e)aJ+S/(e)

dr A (

(7.4.6)

A€7>M

The integrand in (7.4.6) is a linear combination of trigonometric monomials e\l • • • e\ • e i ' " ' ' ei' where the r's and s' are nonnegative integers. Note that if A, /j, € Vk,i are distinct partitions then the terms a-x+6(e) and an+s{e) have no monomials in common. Hence jTkaX+M)aa+6(£)

dr = 0.

Similarly J iaj+5(e)a1J+s{e)de

= 0 if A / /z.

On the other hand a simple computation shows that

fTk\a-x+M2dr and JT<

*\+6\

= k\

277

Symmetry and topology

+1

•C 1

R l+\

,k+l

R

Figure 7.6: The rectangles Rt+i and Rk+1 are "framed" inside Rke+l In other words, the terms /

l

Nl/2

°A+«(£)aA+i(e)

KM)

form an orthonormal system in the space of trigonometric (Fourier) polynomials endowed with the L2 inner product. We deduce immediately from (7.4.6) that (7.4.7) ASPM

The map PH3AH>A6

V,k

is a bijection so that

PkA*)=

£

*2|A| =-fytW-

(7-4-8)

AeP,,t

Computing the Betti numbers i.e. the number of partitions in Vk/ with a given weight is a very complicated combinatorial problem and currently there are no exact general formulas. We will achieve the next best thing and rewrite the Poincare polynomial as a "fake" rational function. Denote by b^^w) the number of partitions A 6 V^k with weight |A| = w. Hence

Alternatively, bk,e(w) i s the number of Young diagrams of weight w which fit inside a k x £ rectangle. Lemma 7.4.27 h+i,i+i(w) = bk,t+1(w) + h+l>e(w - £ - 1)

278

Cohomology rectangle Rk+1 inside the (k + 1) x (1 + l)-rectangle

Proof Look at the (k + l)x£ Rke+l (see Figure 7.6). Then

bk+i,e+i{w) = # {diagrams of weight w which fit inside Rk+l} = #{diagrams which fit inside Rk+1} +#{diagrams which do not fit inside

Rk+1}.

bk+i,e = #{diagrams which fit inside

Rk+1}.

On the other hand

If a diagram does not fit inside Rk+1 this means that its first line consists of £ + 1 boxes. When we drop this line we get a diagram of weight w — £ — 1 which fits inside the k x (£ + 1) rectangle Rt+i of Figure 7.6. Thus, the second contribution to bk+u+i(w) is h,e+l(w - £ - 1). • The result in the above lemma can be reformulated as pk+u+1(t)

= Pk+U(t)

+ t2(m)PM+iW-

Because the roles of k and £ are symmetric (cf. (7.4.8)) we also have Pk+u+l(t)

= Pk4+1(t)

+

t2(k+Vpk+l,e(t)-

These two equality together yield P w + 1 ( l - ^ + 1 ) = P f c + 1 ,,(l-* 2 ( * + 1 ) ). Let m = k + £ + 1 and set Qd,m{t) = Pd,m-d{t) = PQ im C) W- The ^ a s t equality can be rephrased as Qk+l,m(t)

= Qk,m • i _ f2(k+l)

so that 1 _ f2(rn-k)

Qk+l,m{t) =

1

_

t2(fc+1)

j — fUm-k+l)

j " 3 ^

J _ ^2(n-l)

1 _*4

•

Now we can check easily that bl<m-i(w) = 1 i.e. Qi,mW = l + *2 + t 4 + --- + t 2 ( m - 1 ) Hence

(1 _ t2(m-k+l)j = <

9*,">(f) =

^(m.CjW

l-t2m 1-i2 . . . ( ! _ f2m)

Q _ t2\ . . . n _ f2*\

( 1 - t2) • • • ( 1 - t 2 m )

_ 2

~ (1 - 1 ) • • • (1 - t 2fc )(i - 1 2 ) • • • (i -1 2 (™-*))'

279

Cech cohomology

Remark 7.4.28 (a) The invariant theoretic approach in computing the cohomology of Gk{n, C) was used successfully for the first time by C. Ehresmann[24]. His method was then extended to arbitrary compact, oriented symmetric spaces by H. Iwamoto [38]. However, we followed a different avenue which did not require Cartan's maximal weight theory. (b) We borrowed the idea of using the Weyl's integration formula from Weyl's classical monograph [74]. In turn, Weyl attributes this line of attack to R.Brauer. Our strategy is however quite different from Weyl's. Weyl uses an equality similar to (7.4.5) to produce an upper estimate for the Betti numbers (of U(n) in his case) and then produces by hand sufficiently many invariant forms. The upper estimate is then used to established that these are the only ones. We refer also to [70] for an explicit description of the invariant forms on grassmannians. Exercise 7.4.10 Show that the cohomology algebra of QP" is isomorphic to the truncated ring of polynomials R[x}/{xn+1) where x is a formal variable of degree 2 while (xn+1) denotes the ideal generated by xn+l. Hint: Describe A| nv CP n explicitly.

7.5

Cech cohomology

In this last section we return to the problem formulated in the beginning of this chapter: what is the relationship between the Cech and the DeRham approach. We will see that these are essentially two equivalent facets of the same phenomenon. Understanding this equivalence requires the introduction of a new and very versatile concept, namely that of a sheaf. This is done in the first part of the section. The second part is a fast paced introduction to Cech cohomology. A concise yet very clear presentation of these topics can be found in [36]. For a very detailed presentation of this subject we refer to [29]. §7.5.1

Sheaves and presheaves

Consider a topological space X. The collection Ox of its open sets can be organized as a category. The morphisms are the inclusions U °-» V. A presheaf of abelian groups on X is a contravariant functor S : Ox —> A b . In other words, S associates to each open set an abelian group Sa and to each inclusion U °-> V a group morphism ry : Sy —>• Su such that if U •-• V M- W then r w — rv ° rw- If £ G <Sv then for any [ / ^ F w e set x\uaH 4(x)

G 5,

280

Cohomology

If / € Su then we define dom / = U. The presheaves of rings, modules, vector spaces are defined in an obvious fashion. E x a m p l e 7.5.1 Let X be a topological space. For each open set U C X denote by C(U) the space of continuous functions U —» R. The assignment U !-• C(U) defines a presheaf of R-algebras on X. The maps ry are determined by the restrictions \u- C(V) -> C(U). If X is a smooth manifold we get another presheaf U (->• C°°(U), of smooth function. More generally the differential forms of degree k can be organized in a presheaf fi*(»). If E is a smooth vector bundle then the E-valued differential forms of degree k can be organized as a presheaf of vector spaces

U^QkE{U) = nk(E\u). If G is an abelian group equipped with the discrete topology then the G-valued continuous functions C(U, G) determine a presheaf called the constant G-presheaf which is denoted by G_x. Definition 7.5.2 A presheaf S on a topological space X is said to be a sheaf if the following hold. (a) If (Ua) is an open cover of the open set U and f,geSu satisfy / \ua= 9 \ua Va then f = g. (b) If (Ua) is an open cover of the open set U and / „ e Sua satisfy fa \uanu0= fp krw f l then there exists / 6 Su such that f\ua=

VC/a n Up / 0

fa, Va.

E x a m p l e 7.5.3 All the presheaves discussed in Example 7.5.1 are sheaves. E x a m p l e 7.5.4 Consider the presheaf S over R defined by S(U) = continuous, bounded functions / : U —> R. We let the reader verify this is not a sheaf since the condition (b) is violated. The reason behind this violation is that in the definition of this presheaf we included a global condition namely the boundedness assumption. Definition 7.5.5 Let X be a topological space and R a commutative ring with 1. (a) A space of g e r m s over X ( "espace etale " in the french literature) is a topological space £ together with a continuous map ir : £ —>• X such that (al) 7r is a local homeomorphism (that is each point e G £ has a neighborhood U such that ir\u is a homeomorphism onto the open subset TT{U) C X). (a2)The stalk £x = 7r_1(a;) is an i?-module Vi 6 X. (a3) The module operations (u, v) H* (ru + sv) (r,s 6 R, u,v € -n~l{x)) depend continuously upon u, v. (b) A section of a space of germs 7r : £ -> X defined over U C X is a continuous function s : U —• £ such that s(x) G £x Mx e X. The spaces of sections defined over U will be denoted by £{U).

281

Cech cohomology

Example 7.5.6 (The space of germs associated to a presheaf) Let 5 be a presheaf of abelian groups over a topological space X. For each x G X define an equivalence relation ~ x on U3x

by / ~x 9 <==*• 3 open U B x such t h a t / \v= g \u . The equivalence class of / G \Ju3x $u is denoted by [f}x and is called the germ of / at x. Set Sx = {[/]*; d o m / 3 z } . and

s=\_\sx. xEX

There exists a natural projection n : S —> X which maps [f]x to x. The "fibers" of this map are TT~1(X) = <SX - the germs at i € X . Any f € Su defines a subset f(U) = {[/]„ ; u G [/} C 5 We can define a topology in S by indicating a basis of neighborhoods. A basis of open neighborhoods of [f]x G <S is given by the collection {g{U) ;UBx,g£Su

[g]x = [/],}.

(We leave the reader check that this collection of sets satisfies the axioms of a basis of neighborhoods as discussed e.g. in [43].) With this choice of topology each / G Sy defines a continuous section of 7r over U [/] : U B u M> [/]„ G Su. Note that each fiber Sx has a well defined structure of abelian group [f]x + [g]x = [(/ + g) \v]x U 3 x is open and U C d o m / n domg. (Check this addition is independent of the various choices.). Since n : f{U) —> U is a homeomorphism it follows that IT : S —> X is a space of germs. It is called the space of germs associated to the presheaf 5 . If the space of germs associated to a sheaf <S is a covering space we say that S is a sheaf of locally constant functions (valued in some discrete abelian group). When the covering is trivial i.e. it is isomorphic to a product X x { discrete set} then the sheaf is really the constant sheaf associated to a discrete abelian group.

282

Cohomology

Example 7.5.7 (The sheaf associated to a space of germs) Consider a space of germs £ A X over the topological space X. For each open subset U C X denote by £{U) the space of continuous sections U —> £. The correspondence U H-> £(U) clearly a sheaf. £ is called the sheaf associated to the space of germs. Proposition 7.5.8 (a) Let £ A X he a space of germs. Then £. = £. (b) Let S be a presheaf over the topological space X. S is a sheaf if and only if S = S. Definition 7.5.9 If S is a presheaf over the topological space X then <S = <S is called the sheaf associated to S. Exercise 7.5.1 Prove the above proposition. Definition 7.5.10 (a) Let A morphism between the (pre)sheaves of abelian groups (modules etc.) S and S over the topological space X is a collection of morphisms of abelian groups (modules etc.) hv : Su —> Su, one for each open set U C X, such that when V C U hv ° r^ — fy o hy. The Ty denote the restriction morphisms of S while the fy denote the restriction morphisms of S. A morphism h is said to be injective if each hy is injective. (b) Let S be a presheaf over the topological space X. A sub-presheaf of S is a pair (T, i) where T is a presheaf over X and i : T —> S is an injective morphism. i is called the canonical inclusion of the sub-presheaf. Let h : S —>• T be a morphism of presheaves. The correspondence U i-> ker hy C Su defines a presheaf called the kernel of the morphism h. It is a sub-presheaf of <S. Proposition 7.5.11 Let h : S -4 T be a morphism of presheaves. If both <S and T are sheaves then so is the kernel of h. The proof of this proposition is left to the reader as an exercise. Definition 7.5.12 (a)Let £* ^V X (i = 0,1) be two spaces of germs over the same topological space X. A morphism of spaces of germs is a continuous map h : £Q —» £\ such that (al) iti o h = 7To i.e. /i(7rj1(a;)) C 7rf 1(a;) Vz e X. (a2) For any x e X the induced map hx : 7r^"1(a;) -> 7rf1(x) is a morphism of abelian groups (modules etc.). The morphism is called injective if each hx is injective. (b) Let £ A X be a space of germs. A subspace of germs is a pair (T, j) where T is a space of germs over X and j : T —> £ is an injective morphism. Proposition 7.5.13 (a) Let h : £Q —> £\ he a morphism between two spaces of germs over X. Then h(£o) -^ X is a space of germs over X called the image of h and denoted by Im h. It is a subspace of £ x .

283

Cech cohomology Exercise 7.5.2 Prove the above proposition.

Lemma 7.5.14 Consider two sheaves S and T and let h : S —> T be a morphism. Then h induces a morphism between the associated spaces of germs h : S —> T• The definition of h should be obvious. If / € Sv and x 6 U then h([f]x) = [h(f)]x where h(f) is now an element of T(U). We leave the reader check that h is independent of the various choices and it is a continuous map S —> T with respect to the topologies described in Example 7.5.6. The sheaf associated to the space of germs Im/i is a subsheaf of T called the image of h and is denoted by Im h. Exercise 7.5.3 Consider a morphism of sheaves (over X) h : S —• T. Let U C X be an open set. Show that a section g ETu belongs to (Im h)u if an only if for every x € X there exists an open neighborhood Vx C U such that g \vx— h(f) for some

/eSv„. Definition 7.5.15 (a) A sequence of sheaves and morphisms of sheaves • • • —> Sn -4 Sn+i

-4 >S„+2 —• • • •

is said to be exact if Im hn = ker hn+i, Vn. (b) Consider a sheaf S over the space X. A resolution of S is a long exact sequence 0 -> S A S0 4 5 i 4 • • • -> Sn % Sn+1 -> • • •. Exercise 7.5.4 Consider a short exact sequence of sheaves 0 -> 5_i -> S0 ->• Si -> 0. For each open set U define S(U) = S0(U)/S-i(U). (a) Prove that U *-* S(U) is a presheaf. (b) Prove that Si = S = the sheaf associated to the presheaf S. Example 7.5.16 (The D e R h a m resolution)Let M be a smooth ra-dimensional manifold. Using the Poincare lemma and the Exercise 7.5.3 we deduce immediately that the sequence

o^R

M

^ n°M A nlM A • • • A o,nM ^ o

is a resolution of the constant sheaf RM. Q.kM denotes the sheaf of fc-forms on M while d denotes the exterior differentiation.

284 §7.5.2

Cohomology Cech cohomology

Let U \-> S(U) be pre-sheaf of abelian groups over a topological space X. Consider an open cover i l = (t/ a )as^t of X. A g-simplex (q € Z+) is an ordered (q + l)-uple a = (a 0 , Q i , . . . , aq) € Aq+1 such that

Ua=ff]Uai^0. o The set of all g-simplices is denoted by ii( ? ). Their union

U-"fo> is denoted by Af(il) and is called the nerve of the cover. Define

C<(S,U) = II

S

° (S* = S(Ua)).

The elements of C(S,ii) are called Cech g-cochains (subordinated to the cover il). In other words, a g-cochain c associates to each g-simplex a an element (c, a) 6 S„For each g-simplex a = (a0,... ,aq) we define its j - t h boundary as the (q — 1)simplex &o = &qo = (a0,..., otj,..., aq) e il(,-i) where as usual a hat indicates a missing entry. Exercise 7.5.5 Prove that 9J_ X ^ = d^Zld'q for j > i. We can now define an operator

5:Co-1{S,il)->C'l(S,iX) which assigns to each (q — l)-cochain c a g-cochain 6c whose value on a g-simplex a is given by

(5c,a) =

f:(-iy(c,cVa)\u,.

Using the Exercise 7.5.5 above one deduces immediately the following result. L e m m a 7.5.17 52 = 0 so that

o <-> c°(s,u) 4 c 1 ^,!!) -4 • • • 4 cq(s,ii) 4 • • • is a cochain complex. The cohomology of this cochain is called the Cech cohomology of the cover il with coefficients in the pre-sheaf S.

285

Cech cohomology

Example 7.5.18 Let i l and S as above. A 0-cochain is is a correspondence which associates to each open set Ua € i l an element cQ € S(Ua)- It is a cocycle if for any 1-simplex (a, /?) of the nerve we have C/3 -

CQ =

0.

A 1-cochain associates to each 1-simplex (a, (5) an element Cal3 G

S{Uap).

This correspondence is a cocycle if for any 2-simplex (a, /?, 7) we have C/?7 — C a 7 + Ca/} = 0.

For example if X is a smooth manifold and i l is a good cover then we can associate to each closed 1-form w e Q1(M) a Cech 1-cocycle valued in Rx as follows. First, select for each Ua a solution fa G C°°(Ua) of d/„ = oi. Since d(/ Q — /^) = 0 on f a| a we deduce there exist constants cap such that fa — Jp = cap. Obviously this is a cocycle and it is easy to see that its cohomology class is independent of the initial selection of local solutions fa. Moreover, if UJ is exact this cocycle is a coboundary. In other words we have a natural map

H1(X)^H1(Af(iX)Mx)We will see later this is an isomorphism. Definition 7.5.19 Consider two open covers i l = (Ua)aeA and 2J = ( V ^ g s of the same topological space X. We say 2J is finer than i l and we write this i l -< 2J if there exists a map Q : B —> A such that

v0cum

V(3eB.

The map Q is said to be a refinement map. Proposition 7.5.20 Consider two open covers i l = {Ua)aeA and 93 = (VJj)/jgB of the same topological space X such that i l -< 93. Fix a sheaf of abelian groups S. Then the following are true. (a) Any refining map Q induces a cochain map

0.:©c%u,s)->©c(aj,s). (b) If r : A —• B is another refinement map then Q, is cochain homotopic to r„. In particular, any relation i l X 23 defines a unique morphism in cohomology t^l : H*(il,S)

-> H*(V3,S).

(c) If i l X 23 -< 223 then 213 ! 2D 23 i l = 2 3 °*U-

J

286

Cohomology

Proof

(a) We define Q. : C(Xl) -s- C?(2J) by S(Va) B
VC e C ( i l ) a €

VM)

q

where by definition g(a) € A is the g-simplex {Q{PQ), . . . , Q{Pq))- The fact that £>, is a cochain map follows immediately from the obvious equality Q o &q = 8% o p.

(b) We define ft, : 2J(,-i) ->• it( g ) by hj-(A), • • • ,/?,-i) = (e(A>),..., e{Pj),r{Pj), • •-.»•(/?,_!)). The reader should check that fcj( = £ ( - i y ( C , ^ ( a ) ) |v„

Vc e C(ii)

Va e 2J(,-i).

We will show that S o X ,(c) + X,+i ° *(*) = g.(c) - r.(c)

Vc e C ' ( i l ) .

Let a = (/?0, • • •, Pq) € 9J(?) and set e(ff) = (Ao, • • •, A,), r(a) = ( p 0 , . . . , nq) £ il ( ,) so that ftj(cr) = (A0, ...,\j,fij,...,

nq).

Then q

9+1

= £(-i) J '(£(-inc,9* + 1 ft>)>k) ^E(- 1 )ME(- 1 ) fc ( c '( A o,-.-,A,-,... > A j)ft ,...,/i,))iv
\fc=0

+ ZX-l) m (c, (A°' • • •' K Hi • • •. A*. • • •. M«)> k :

i(-l)ME(-l)* , W ,.--,^)>l j=0

\fc=0

Cech cohomology

287 i

+ Y,

(- 1 ) m ( c >( A o,---,A.,-,A i j,---,Wb---,M«)>l

t=j+i i

+ Z X - ! ) J {(c> (•*<>> • • •. ^ j - i . Mj, • • • > M«)> k +(c, (A0, • •., Aj, / x j + 1 ) . . . , //,)) | v „} • The last term is a telescopic sum which is equal to (c, (A 0 ,..., A,) \v„ - ( c , (/x0, • • •, M,)) k = (e.c,CT>- (r.c, a). If we change the order of summation in the first two term we recover the term

(Sxc,a). (c) is left to the reader as an exercise. • We now have a collection of groups {H*{ii, S) ; i l - open cover of X} and maps {»g:.ff%U,S); i l ^ S T } such that ,i i l = i d

ti

and

lU = J 0 J o l i l whenever il -< 2J -< 2U. We can thus define the inductive limit = / lim/f*(il,«S).

H'(X,S)

if* (X, 5) is called the Cech cohomology of the space X with coefficients in the pre-sheaf S. Let us briefly recall the definition of the direct limit. One defines an equivalence relation on il by i T ( i l ) B f ~ g G # • ( » ) <=> 30B V U, Of : i f f / = z g V We denote the equivalence class of / by / . Then lim/T(it)= *

(Uff*CU,$))/. \il

At limit we get maps *iI:/T(.U,lS)->ir(X,<S).

288

Cohomology

Example 7.5.21 Let S be a sheaf over the space X. For any open cover i t = (UQ) a 0-cycle subordinated to i t is a collection of sections fa 6 S(Ua) such that every time Ua n Up ^ 0 fa\uafi=

fp\ua0

•

According to the properties of a sheaf, such a collection defines a unique global section / € S(X). Hence H°(X,S) = S(X). Proposition 7.5.22 Any morphism of pre-sheaves h : So —> <Si over X induces a morphism in cohomology

h,:H*(X,S0)^H%X,S1). Sketch of proof

Let i t be an open cover of X. Define

ht:C'(iX,S0)^C(il,S1) by (h,c, a) = hu((c, a}) Vc 6 C(il, S0) a e il ( g). The reader can check easily that ht is a cochain map so it induces a map in cohomology

h^:H*(ii,S0)^H*(ii,S1) which commutes with the refinements i^/. The proposition follows by passing to direct limits. • Theorem 7.5.23 Let 0 -> 5_i 4 S 0 ^ <5i -> 0 be an exact sequence of sheaves over a paracompact space X. Then there exists a natural long exact sequence • H"{X, S-i) 4 H"(X, So) ^ H"(X, Si) 4 Hq+1(X, 5_i) -> • • •. Sketch of proof For each open set U C X define S(U) = S0{U)/S-i(U). Then the correspondence U H-> S(U) defines a pre-sheaf on X. Its associated sheaf is isomorphic with Sx (see Exercise 7.5.4). Thus for each open cover i t we have a short exact sequence 0 -> C»CU,5_i) A C 9 (it,«S 0 ) 4 C(.U, 5) -*• 0. We thus get a long exact sequence in cohomology • / / ' ( i t , 5_i) -»• / / ' ( i t , So) -> / / ' ( i t , S) -> / / « + 1 ( i i , A") -> • • •. Passing to direct limits we get a long exact sequence • H"{X, S-i) -> H"{X, So) -» / / ' ( X , 5) -> / / ' + 1 ( X , 5_i) - • • • • . To conclude the proof of the proposition we invoke the following technical result. Its proof can be found in [68].

Cecil cohomology

289

Lemma 7.5.24 If two pre-sheaves <S, S' over a paracompact topological space X have isomorphic associated sheaves then H*(X,S)^H*{X,S'). Definition 7.5.25 A sheaf S is said to be fine if for any locally finite open cover iX = (Ua)a£A there exist morphisms ha : S —¥ S with the following properties. (a) For any a & A there exists a closed set Ca such that Ca C Ua and ha(Sx) = 0 for x £ Ca where Sx denotes the stalk of S at x G X. The set Ca is called a support of ha and and we write this supp ha C Ca( b ) £ a ha = ls- This sum is well defined since the cover i l is locally finite. Example 7.5.26 Let X be a smooth manifold. Using partitions of unity we deduce that the sheaf £lkx of smooth fc-forms is fine. More generally, if E is a smooth vector bundle over X then the space Q.kE of E-valued fc-forms is fine. Proposition 7.5.27 Let 5 be a fine sheaf over a paracompact space X. H"{X,S) £ 0 f o r g > 1.

Then

Proof Because X is paracompact any open cover admits a locally finite refinement. Thus it suffices to show that for each locally finite open cover il — {Ua)aeA the cohomology groups Hq(il, S) are trivial for q > 1. We will achieve this by showing that the identity map C(il, S) —¥ C(il,S) is cochain homotopic with the trivial map. We thus need to produce a map Xq:C{ii,S)-^C''-1{iX,S) such that X«+1<5» + 5q-\*

= id.

(7.5.1)

Consider the morphisms ha : S ~> S associated to the cover i l postulated in the definition of a fine sheaf. For each a € A, a € iX(,-i) and / € C'(il, S) we construct (ta{f),o) € S(Ua) as follows. Consider the open cover of IJ„ {V = UanU„,W

= UtT\Ca}

(supp/iQcCa).

Note that haf \vnw= 0 and according to the axioms of a sheaf ha(f \v) can be extended by zero to a section (ta(f), a) € S(Ua). Now, for every / e C(ii, S) define q l by X feC<~ (il,S) q

(x V),°) = Y,(W),v)a

The above sum is well defined since the cover i l is locally finite. We leave the reader check that x9 satisfies (7.5.1). •

290

Cohomology

Definition 7.5.28 Let S be a sheaf over a space X. A fine resolution is a resolution

o -> s ^ s0 4 Si -4 • • • such that each of the sheaves <S, is fine. T h e o r e m 7.5.29 ( A b s t r a c t D e R h a m t h e o r e m ) Let 0 ^ S ->• S0 $ Sx $ • • • be a fine resolution of the sheaf <S over the p a r a c o m p a c t space X. Then 0 -> SQ{X) 4 Si{X) 4 • • • is a cochain complex and there exists a natural isomorphism Hq{X) =*

H"(Sq{X)).

Proof The first statement in the theorem can be safely left to the reader. For q > 1 denote by Zq the kernel of the sheaf morphism dq. We set for uniformity ZQ = S. We get a short exact sequence of sheaves 0 ->• Zq -> Sq -> Zq+1 - » 0 q>0.

(7.5.2)

We use the associated long exact sequence in which Hk(X, Sq) = 0 for k > 1 since Sq is a fine sheaf. This yields the isomorphisms Hk-\X,

Zq+1) £ Hk(X, Zq)

k>2.

We deduce inductively that Hm{X,Z0)^H\X,Zm_l)

m>\.

Using again the long sequence associated to (7.5.2) we get an exact sequence H°(X, V i ) ^ ' H°(X, Zm) -». H\X,

Zm_x) -> 0.

We apply the computation in Example 7.5.21 we get H\X,

2 m _ 0 3 Zm(X)K~x

(«S m _iP0).

This is precisely the content of the theorem. D Corollary 7.5.30 Let M be a smooth manifold. Then H*(M,RM)^H*(M).

(7.5.3)

291

Cecil cohomology Proof

M is paracompact. We conclude using the fine resolution 0 -+ 1

M

M - ^ M - ^ ^ M ->••••

D

Exercise 7.5.6 Describe explicitly the isomorphisms H1{M)^H1{M,RM) and H2(M)^H2{M,RM). Remark 7.5.31 The above corollary has a surprising implication. Obviously the Cech cohomology is a topological invariant and thus, so must be the DeRham cohomology which is denned in terms of a smooth structure. Hence if two smooth manifolds are homeomorphic they must have isomorphic DeRham groups! Such exotic situations do exist. In a celebrated paper [54], John Milnor has constructed a family of nondiffeomorphic manifolds all homeomorphic to the sphere S7. More recently, the work of Simon Donaldson in gauge theory was used by Michael Freedman to construct a smooth manifold homeomorphic to M4 but not diffeomorphic with M4 equipped with the natural smooth structure. (This is possible only for 4-dimensional vector spaces!) These three mathematicians, J. Milnor, S. Donaldson and M. Freedman were awarded Fields medals for their contributions. Theorem 7.5.32 Let M be a smooth manifold and iX = (Ua)aeA M i.e. Ua ^ l d i m M . Then

a

good cover of

Ht(ii,mM)^H*{M).

Proof Let Zk denote the sheaf of closed fc-forms on M. Using the Poincare lemma we deduce Zk{U„) = dQ."-1^). We thus have a short exact sequence

o -> cq{n, zk) -> c(ii, n*-1) 4 c(u, zk) -> o. Using the associated long exact sequence and the fact that Q,k~1 is a fine sheaf we deduce as in the proof of the abstract DeRham theorem that H^il,RM)^H"-1(U,Z1)^--s* H\il,

Z"-1) s H°(U, Z")/d,H°{iX, W1)

£ Z"{M)ldQ.q-x{M)

£ H*{M).

•

292

Cohomology

Remark 7.5.33 The above result is a special case of a theorem of Leray: if S is a sheaf on a paracompact space X and i l is an open cover such that H"(Ua,S)

=0

\/q>laeU{k)

then H*(il,S) = H*(X,S). For a proof we refer to [29]. When G_M is a constant sheaf (where G is an arbitrary abelian group) we have a Poincare lemma (see [25], Chapter IX, Thm. 5.1) H"(Rn,G)

=0

q>\.

Hence for any good cover iX H'(M,GM)=H*(U,GM).

Example 7.5.34 Let M be a smooth manifold and i l = (Ua)a^A a good cover of M. A 1-cocycle of R M is a collection of real numbers /a/g - one for each pair (a,/3) € A2 such that Ua0 ^ 0 satisfying fa0 + hi + fja = 0 whenever Uap7 ^ 0. The collection is a coboundary if there exist the constants fa such that fap = fp — fa. This is precisely the situation encountered in Subsection 7.1.2. The abstract DeRham theorem explains why the Cech approach is equivalent with the DeRham approach. Remark 7.5.35 Often in concrete applications one finds it is convenient to work with skew-symmetric Cech cochains. A cochain c G C(S, il) is skew-symmetric if for any q simplex a = (ao, • • •, o-q) and for all ip € &q+i we have (c, (a0,...,

a,)) = e{tp)(c, ( a ^ o ) , . . . , Q^,)))-

One can then define a "skew-symmetric" Cech cohomology following the same strategy. (The various intervening formulae will be more elaborate.) The resulting cohomology coincides with the cohomology described in this subsection. For a proof of this fact we refer to [68].

Chapter 8 Characteristic classes We now have sufficient background to approach a problem formulated in Chapter 2: find a way to measure the "extent of nontriviality" of a given vector bundle. This is essentially a topological issue but, as we will see, in the context of smooth manifolds there are powerful differential geometric methods which will solve a large part of this problem. Ultimately, only topological techniques yield the best results.

8.1 §8.1.1

Chern-Weil theory Connections in principal G-bundles

In this subsection we will describe how to take into account the possible symmetries of a vector bundle when describing a connection. All the Lie groups we will consider will be assumed to be matrix Lie groups i.e. Lie subgroups of a general linear group GL(n, K). This restriction is neither severe, nor necessary. It is not severe since, according to a nontrivial result (Peter-Weyl theorem), any compact Lie group is isomorphic with a matrix Lie group and these groups are sufficient for the applications in geometry. It is not necessary since all the results of this subsection are true for any Lie group. We stick with this assumption since most proofs are easier to "swallow" in this context. For a matrix Lie group G, its Lie algebra JJ is a Lie algebra of matrices in which the bracket is the usual commutator. Let M be a smooth manifold. Recall that a principal G-bundle P over M can be defined by an open cover (Ua) of M and a gluing cocycle 9ap • Uap —> G.

The Lie group G operates on its Lie algebra g via the adjoint action Ad:G->

GL{g),

g .-*• Ad(g) e GL{g)

where Ad{g)X = gXg-1 293

VX e g.

294

Characteristic classes

We denote by Ad(P) the vector bundle with standard fiber g associated to P via the adjoint representation. In other words, Ad(P) is the vector bundle defined by the open cover (Ua) and gluing cocycle Ad(gaP) : UaP -> GL{g). The bracket operation in the fibers of Ad(P) induces a bilinear map [•,•] : Qk(Ad{P))

x tf{Ad(P)) -»

Qk+e{Ad(P))

defined by [w* ® X, rf ® Y] = (w* A T)1) ® [X, Y] k

(8.1.1)

£

for all tj 6 fi*(M), ?7 g 0 £ (M) and X, F e n°(ylrf(P)). Exercise 8.1.1 Prove that for any u>, rj, € fi*(Ad(F)) the following hold. K ^ ] = -(-l)H-M[,,w].

(8.1.2)

[[u,,7?],^] = [[w, fl,77] + (-1)M-I«[ W , fa, *}].

(8.1.3)

In other words, fT(yW(F)), [, ]) is a super Lie algebra. Definition 8.1.1 (a) A connection on the principal bundle P is a collection Aa € ^(Uc) <8> 0 satisfying the transition rule A

f>ix) = 9«p{x)dgap{x) + g-p(x)AQ{x)ga/3(x)

Vz € Uaf).

(b) The curvature of the above connection is defined by the collection Fa € fl2([/Q)® g where Fa = dAa +

-[Aa,Aa].

Proposition 8.1.2 (a) The set 21(F) of connections on F is an affine space modeled by Q^AdiP)). (b) The collection (Fa) defines a global j4d(F)-valued 2-form. (c) (The Bianchi identity) dFa + [Aa,Fa] = Q VQ. Proof (a) If (Aa, {Ba) £ 21(F) then their difference Ca = Aagluing rules Cp = g/3aCagj3a so that it defines an element of Q1(Ad(P)). (b) We need to check that the forms Fa satisfy the gluing rules F0 =

g-lFag

(8.1.4) Ba satisfies the

295

Chern-Weil theory where g = ga0 = g^.

We have

= d(g-ldg + g-xAag)

+ -\g~ldg + g~lAag, g~ldg +

g~lAag}.

Set m = g~*dg. Using 8.1.2 we get Fp = +d(g~1Aag)

dm-\--[m,m]

+ [vj,g-lAag\

+ -{g-lAag,g-lAag\.

(8.1.5)

We will check two things. A. The Maurer-Cartan structural equations. dm + -[m,m] = 0. B. d{g~lAag)

+ [m, g~lAag] =

g-x{dAa)g.

Proof of A. Let us first introduce a new operation. Let gl{n, K) denote the associative algebra of K-valued n x n matrices. There exists a natural operation A : Slk{Ua) ® gl(n, K) x Qe(UQ) ® gl(n, K) -> Slk+e{Ua) ® p/(n, K) uniquely defined by (<J* ®A)A

{rf ®B) = (u>k A rf) ® (A • B)

(8.1.6)

where uk € ft*(£4), ?/ € ft£(£/a) and A, B G p/(n,K) (see also Example 3.3.11). The space gl(n7 K) is naturally a Lie algebra with respect to the commutator of two matrices. This structure induces a bracket [, ] : Slk(Ua) ® gl(n, K) x tf(Ua) ® gj.(n, K) -> ttk+e{Ua) ® gi(n, K) defined as in (8.1.1). A very simple computation yields the following identity. uA7]=-[ui,r]}

^u),r]&Q1(Ua)®gi{n,K).

(8.1.7)

Assume the Lie group lies inside GL(n, K) so that its Lie algebra g lies inside gl(n, K). We can think of the map gap as a matrix valued map so that we have dm = d{g~ldg) = (dg'1) Adg=

-(g'1

• dg • g~l)dg = -{g~ldg)

A {g~xdg)

296

Characteristic classes = -G7As7

=

— -[m, UJ\.

Proof of B. We compute d{g-lAQg) = -g'1

= {dg-lAa)

• g + g~\dAa)g

• dg • g-1 A Aa • g + g-l{dAa)g = -w A g~lAag + g~lAa

+

+ ( s " M a S ) A g~xdg g-1{dAa)g

Azu +

-\w, g-xAag] + -[w, gxAag] +

2l

(8.1.2)

[nj,g ^cgj

g^A^g

g'^dA^g

x

+g

{dAa)g.

Part (b) of the proposition now follows from A, B and (8.1.5). (c) First, we let the reader check the following identity d[w)T/] = [«L;>^] + (-l)M[w)d»7]

(8-1-8)

where w, rj 6 Q,*(Ua) Q. Using the above equality we get d(Fa) = l-{[dAa,Aa] - [Aa,dAa]}

= [Fa,Aa] - \[\Aa,Aa], The proposition is proved.

Aa]

(8

(8

= 2 ) [dAa,Aa]

= 3 ) [Fa,AQ].

•

Exercise 8.1.2 Let ua G iik(Ua) ® g satisfy the gluing rules w/3 = gpawag0a

on

Ua0.

In other words, the collection uia defines a global /c-form w e £lk(Ad(P)). the collection dtoa + [Aa,ua\

Prove that

defines a global .<4d(P)-valued (k + l)-form on M which we denote by d^uj. Thus, the Bianchi identity can be rewritten as dAF{A) = 0 for any A € 2l(P).

Chern-Weil theory §8.1.2

297

G-vector bundles

Definition 8.1.3 Let G be a Lie group and E —¥ M a vector bundle with standard fiber a vector space V. A G structure on E is defined by the following collection of data. (a) A representation p : G —> GL(V). (b) A principal G-bundle P over M such that E is associated to P via p. In other words, there exists an open cover (Ua) of M and a gluing cocycle 9a/3 '• Uap —> G

such that the vector bundle E can be defined by the cocycle P(gap) •• Ua0 ->

GL(V).

We denote a G-structure by the pair (P,p). Two G-structures (P*, p,) on E (i = 1,2) are said to be isomorphic if pi = p2 and the principal G-bundles Pi are isomorphic. Example 8.1.4 Let E —>• M be a rank r real vector bundle over a smooth manifold M. A metric on E allows us to talk about orthonormal moving frames. They are easily produced from arbitrary ones via the Gramm orthonormalization technique. In particular, two different orthonormal local trivializations are related by a transition map valued in the orthogonal group 0(r) so that a metric on a bundle allows one to replace an arbitrary collection of gluing data by an equivalent (cohomologous) one with transitions in 0(r). In other words, a metric on a bundle induces an 0(r) structure. The representation p is in this case the natural injection 0(r) «-» GL(r, M). Conversely, an 0(r) structure on a rank r real vector bundle is tantamount to choosing a metric on that bundle. Similarly, a Hermitian metric on a rank k complex vector bundle defines an U(k)structure on that bundle. Let E — (P, p, V) be a G-vector bundle. Assume P is defined by an open cover (Ua) and gluing maps 9ag • Uap —> G. 1

If the collection {Aa 6 0 ([/ a ) ® g} defines a connection on the principal bundle P then the collection p,{Aa) defines a connection on E. Above, p, : g —> End (V) denotes the derivative of p at 1 e G. A connection of E obtained in this manner is said to be compatible with the G-structure. Note that if F(Aa) is the curvature of the connection on P then the collection p„(F(Aa)) coincides with the curvature F(p„(Aa)) of the connection p„(Aa). For example, a connection compatible with some metric on a vector bundle is compatible with the orthogonal/unitary structure of that bundle. The curvature of such a connection is skew-symmetric which shows the infinitesimal holonomy is an infinitesimal orthogonal/unitary transformation of a given fiber.

298 §8.1.3

Characteristic classes Invariant polynomials

Let V be a vector space over K = R, C. Consider the symmetric power Sk(V)

C (V*)®*

which consists of symmetric, multilinear maps
->K.

Note that any

=
••-,»).

This follows immediately via the polarization formula ip{vi

'---Vk)

=

hdt1---dtkPv{tiVi+---+tkVk)-

If dim V = n, then fixing a basis of V we can identify Sh(V*) with the space of degree k homogeneous polynomials in n variables. Assume now that A is a K-algebra with 1. Starting with ip € S*(V*) we can produce a K-multilinear map ip = ipA : (A®V)

x • • • x (A®V)

-* A

uniquely determined by v(«i ®vi,...,ak®vk)

= tp{vu ..., vk)aia2 • • • a^ £ A.

If moreover the algebra A is c o m m u t a t i v e then (pA is uniquely determined by the polynomial Pv{x)

=
x e A ® V.

This is no longer the case if A is not commutative. Example 8.1.5 Let V = gl(n,C). For each matrix T € V we denote by ck(T) the coefficient of A* in the characteristic polynomial

cx(T) = det ( l - ^ T ) = Y, cdT)\k,

(i = >/=!).

ck{T) is a degree k homogeneous polynomial in the entries of T. For example c l ( T) = - ^ t r r ,

c n (T) =

(^)

n

detT.

Via polarization, ck(T) defines an element of Sk(gl(n, C)*).

299

Chern-Wei! theory

If A is a commutative C-algebra with 1 then A ® gUp., C) can be identified with the space gl{n, A) of n x n matrices with entries in A- For each T € gl{n, A)

det 1

( -Si T ) e - A W

and C).(T) continues to be the coefficient of \k in the above polynomial. Consider now a matrix Lie group G. The adjoint action of G on its Lie algebra g induces an action on Sk(g*) still denoted by Ad. We denote by Ik(G) the >ld-invariant elements of Sk(g*). It consists of those ip € Sk(g*) such that 9Xk9"1) = p(-Xi, • • •, -X"*) for all A"i,...,X f c e fl. Set

/*(G) = 0/*(G) *>o and

r*(G) = n J*(G). fc>0

The elements of I*(G) are usually called invariant polynomials. 7**(G) can be identified (as vector space) with the space of Ad-invariant formal power series with variables from g*. E x a m p l e 8.1.6 Let G = GL(n,C)

so that g = g/(n,C). The map

p/(n, C) 3 Jf i-> tr exp(X) defines an element of I**(GL(n, C)). To see this we use the "Taylor expansion"

exp(X) = £ - L r * /t>o K -

which yields trexppO = £ ± t r X \ fc>0 Kk

For each fc, t r X * G I (GL(n,C))

since

tr( S X f f - 1 )*=trflXV 1 =trX*. Proposition 8.1.7 Let y> G Ik(G).
...,Xk)

Then for any

X,X1,...,Xk€g

+ --- +
The proposition follows immediately from the equality ft\t=ov(etxXle-t\...,etxXke-tx)

= 0. Q

(8.1.9)

300 §8.1.4

Characteristic classes The Chern-Weil theory

Let G be a matrix Lie group with Lie algebra g and P —> M a principal G-bundle over the smooth manifold M. Assume P is defined by an open cover (Ua) and a gluing cocycle gap '• Uag -> G.

Pick A G 21(F) defined by the collection Aa e ^(Ua) defined by the collection Fa = dAa +-[Aa,

® g. Its curvature is then

Aa].

Given <j> e /*(G) we can define as in the previous section (with A =

£leven(Ua),

V = o) p
(Fa,...,Fa)en2k(ua).

Because <j> is yld-invariant and Fp = gpaFag^ P^(F Q ) = P^Fp)

we deduce on Uap

so that the locally defined forms P^{Fa) patch-up to a global 2/c-form on M which we denote by (F(A)). Theorem 8.1.8 (Chern-Weil) (a) The form <j>{F{A)) is closed VA e 21(F). (b) If A0, A1 £ 21(F) then the forms {F{A1)) are cohomologous. In other words, (f>(F(A)) defines a cohomology class in H2k(M) which is independent of the connection A £ 21(F). Proof

We use the Bianchi identity dFa = — [Aa,Fa\. The Leibniz' rule yields d{Fa, ...,Fa)

= <j>{dFa, Fa,...,Fa)

= -<j>{[Aa, Fa], Fa,...,Fa)

+ --- + {Fai . . . , F Q , dFa)

(Fa, ...,Fa,

[Aa, Fa})

(8

= 9 ) 0.

(b) Let A1 e 21(F) (i = 0,1) be defined by the collections

A^en^U^^g. Set Ca = A1a- A°a and for 0 < t < 1 we define Ala e H 1 ^ ) ® g by Ala = A°a + tCa. The collection (A*a) defines a connection A* e 21(F) and t H ^ ' e 21(F) is an (affine) path connecting A0 to A1. Note that C = (CQ) = A*. We denote by F* = (F£) the curvature of A f . A simple computation yields Fla = F° + i(dC Q + [A°a, Ca]) + y [C a , C J .

(8.1.10)

301

Chern-Weil theory Hence Fla = dCa + [A°a, Ca] + t[Ca, Ca] = dCa + K , Ca]. Consequently dt 0 ( O - 4>(K) = Si {*&>

n,-,**a)

+ -

+
= Si {ttdCa, F<,..., F<) + • • • + 4>{Fl ..., F<, dCa)}dt + jf * {<MK, c„], F%,-,K)

+ --- + *(Fl • • • > * t K . ca])} dt.

Because the algebra Ueven(Ua) is commutative we deduce for all a e &k and any wi,..., wfc € Q.even(Ua) ® fl. Hence ^ i ) - (dCa + [Al Ca], Fl,..., Ffa)dt. Jo

We claim that (dCa + K , CJ, J*,..., Fa) = # ( C Q , J*,..., F^). Using the Bianchi identity we get d<j>{Ca,Fl...,Fl) = <j>(dCa, Fa,---, F*a) + $(Ca, dF*a,---,F*a) + --- + (Ca, Ft,,---, dFl) = HdCa, Fl...,

F<) - (Ca, K , *£], F*a,..., F<) <£(CQ, j £ , . . . , 1*, K , J*]) = 0(dCQ + K , C a ] , F ^ , . . . , F ^ ) -^(K,CQ],F^...,F^)-^Q)KIF^],F(JI...,F^) (Ca,Fa,...,Fta,[Ata,Fta}) {B )

^ d>(dCa +

[Ata,Ca},Fta,...,Fta)

Hence ^(Fi) - 4>(F°) = d /"' fc0(Co, J*,..., F^)di ./o = d Z"1 fc^.Pi,... ,Fl)dt HI dT^AlAl). (8.1.11) ./o Since C/j = gpaCag^ and F/j = g/3aFagaa o n ^a/9 w e conclude from the Ad-invariance of <j> that the collection T(Ala, A°) defines a global (2k — l)-form on M which we denote by T(Al,A°) and we name it the (/^transgression from A° to A1. We have thus established the transgression formula HF(A1)) - 4>(F(A0)) = dT+(A\A0). (8.1.12) The Chern-Weil theorem is proved.

•

302

Characteristic classes

Example 8.1.9 Consider a matrix Lie group G with Lie algebra g and denote by Po the trivial principal G-bundle over G, P0 = G x G. Denote by w the tautological 1-form w = g~xdg g Q1(G) g. Note that for every left invariant vector field X g g we have w(X) = X. Denote by d the trivial connection on P0. Clearly d is a flat connection. Moreover, the Maurer-Cartan equation implies that d + m is also a flat connection. Thus for any g Ik{G) cf>(F(d)) = (F(d + w)) = 0. The transgression formula implies that the form T> = Tj,(d + w,d) = k f (tv,F{d + tw), •••,F{d + tw))dt g Q 2 * _1 (G) Jo is closed. Clearly r^ is closed. A simple computation using the Maurer-Cartan equations shows that r

* = ¥^ ( / 0 1 ( * 2 " 1 )*" l r f i ) • ,

lVfc_,

^w'I07'07]''"'[vM)

fc 22fc-1A:!(fc - 1)' ofc

We thus have a natural map r : 7*(G) -> Hodd(G) called transgression. The elements in the range of r are called transgressive. When G is compact and connected then a nontrivial result due to the combined efforts of H. Hopf, C. Chevalley, H. Cartan, A. Weil and L. Koszul states that the cohomology of G is generated as an K-algebra by the transgressive elements. We refer to [18] for a beautiful survey of this subject. Exercise 8.1.3 Let G = SU(2). The Killing form K is a degree 2 Ad-invariant polynomial on su(2). Describe rK g fi3(G) and then compute

L

rK

G

Compare this result with the similar computations in Subsection 7.4.3. Let us now analyze the essentials of the Chern-Weil construction. Input: (a) A principal G-bundle P over a smooth manifold M (defined by an open cover (Ua) and gluing cocycle gap : Uap —>• G). (b) A connection A € 2l(P) defined by the collection Aa

gfi1(f/a)®0

303

Chern-Weil theory satisfying the transition rules

4s = 9a0dgaj3 + ga(jAagai3 on Uap. (c) <£ € Ik(G). Output: A closed form (f>(F(A)) e Q,2k(M) whose cohomology class is independent of the connection A. We denote this cohomology class by <j>(P). Thus, the principal bundle P defines a map, called the C h e r n - W e i l correspondence crop : I*{G) -> H*(M) 4>^<j>{P). One can check easily the map crop is a morphism of K-algebras. Definition 8.1.10 Let M and N be two smooth manifolds and F : M —> N a smooth map. If P is a principal G-bundle over N defined by an open cover (Ua) and gluing cocycle gap : Uap —• G then the pullback of P by F is the principal bundle F*(P) over M defined by the open cover F _ 1 ([/ t t ) and gluing cocycle F-l(Ua())

4 Ua0 9^ G.

The pullback of a connection on P is defined similarly. The following result should be obvious. Proposition 8.1.11 (a) If P is a trivial G-bundle over the smooth manifold M then <j>{P) = 0 € H*{M) for any £ I*(G). (b) Let M —> N be a smooth map between the smooth manifolds M and N. Then for every principal G-bundle over N and any (f> e P{G) we have # f ( P ) ) = F*(«(P)). Equivalently, this means the diagram below is commutative. /*(G) C-Hf iT(AT)

\

IF-

cro F . ( P ) >v • H*(M) Denote by ^P G the collection of smooth principal G-bundles (over smooth manifolds). For each P G V($G we denote by Bp the base of P. Finally, we denote by T a contravariant functor from the category of smooth manifolds (and smooth maps) to the category of abelian groups.

304

Characteristic classes

Definition 8.1.12 An ^-valued G-characteristic class is a correspondence «p G 3 P n . c(P) e

HBp)

such that (a) c(P) = 0 if P is trivial and (b) J r (F)(c(P)) = c(F*(P)) for any smooth map F : M -¥ AT and any principal G-bundle P -> AT. Hence, the Chern-Weil construction is just a method of producing G-characteristic classes valued in the DeRham cohomology. Remark 8.1.13 (a) We see that each characteristic class provides a way of measuring the nontriviality of a principal G-bundle. (b) A very legitimate question arises. Do there exist characteristic classes (in the DeRham cohomology) not obtainable via the Chern-Weil construction? The answer is negative but the proof requires an elaborate topological technology which is beyond the reach of this course. The interested reader can find the details in the monograph [58] which is the ultimate reference on the subject of characteristic classes. (b) There exist characteristic classes valued in contravariant functors other then the DeRham cohomology. E.g., for each abelian group A the Cech cohomology with coefficients in the constant sheaf A defines a contravariant functor H*(—,A) and using topological techniques one can produce H*{—, j4)-valued characteristic classes. For details we refer to [58] or the classical [71].

8.2

Important examples

We devote this section to the description of some of the most important examples of characteristic classes. In the process we will describe the invariants of some commonly encountered Lie groups. §8.2.1

The invariants of the torus T"

The n-dimensional torus T" = (7(1) x • • • x U(l) is an abelian group so that the adjoint action on its Lie algebra t n is trivial. Hence F(Tn)

= S*((t n )*).

In practice one uses a more explicit description. This is obtained as follows. Pick angular coordinates 0 < 8' < 2n (1 < i < n) and set ~^-.d0]. 27ri The x'jS form a basis of (t™)* and now we can identify Xj1 =

/•(rjanin,..,^.

Important §8.2.2

305

examples C h e r n classes

Let E be a rank r complex vector bundle over the smooth manifold M. We have seen that a Hermitian metric on E induces an f/(r)-structure (P, p) where p is the tautological representation

p:U{r)^GL{r,C)Exercise 8.2.1 Prove that different Hermitian metrics on E define isomorphic {/(restructures. Thus, we can identify such a bundle with a principal U(r) bundle in a tautological way. A connection on the principal bundle is then equivalent with a linear connection V on E compatible with a Hermitian metric ( , ) i.e. V x (Aw, v) = A{(V x u, v) + (u,

Vxv)}

VA e C, u, v € C°°(E), X € Vect (M). The characteristic classes of E are by definition the characteristic classes of the tautological principal [/(r)-bundle. To describe these characteristic classes we need to elucidate the structure of the ring of invariants I*(U(r)). I*(U(r)) consists of symmetric, r-linear maps cf>: u(r) x • • • x u(r) -¥ K invariant under the adjoint action

u(r) 9 1 4 TXT'1

6 u(r), T € U(r).

It is convenient to identify such a map with its polynomial form P*{X) =

(X,...,X).

The Lie algebra u(r) consists of r x r complex skew-adjoint matrices. Classical results of linear algebra show that for any X € u(r) there exists T € U(r) such that TXT~l is diagonal TXT*1 = idiag(Xi,..., A r ). The set of diagonal matrices in u(r) is called the C a r t a n a l g e b r a of u(r) and is denoted by C„(r). It is a (maximal) abelian Lie subalgebra of u(r). Consider the stabilizer 9a(r) = {T€ U(r) ; TXT'1 = X VX € C„ (r) } and the normalizer 9lp ( p ) = {T € U(r) ; TC^T'1

C C„ (r) }.

306

Characteristic classes

5t/( r ) is a normal subgroup of ^Sluir) so we can form the quotient

W{U{r))dM mu{T)/$u(r) called the Weyl group of U(r). As in §7.4.4 we see that it is isomorphic with the symmetric group <5r because two diagonal skew-Hermitian matrices are unitarily equivalent if and only if they have the same eigenvalues (including multiplicities).We see that P^ is j4d-invariant if and only if its restriction to the Cartan algebra is invariant under the action of the Weyl group. The Cartan algebra is the Lie algebra of the (maximal) torus Tn (consisting of diagonal unitary matrices) and as in the previous subsection we introduce the variables X; = 3

-6Bi2?ri

3

The restriction of P$ to Ca(r) is a polynomial in the variables x\,...,xT. The Weyl group <5r permutes these variables so that P^ is j4d-invariant if and only if P$(xi,..., xr) is a symmetric polynomial in its variables. According to the fundamental theorem of symmetric polynomials the ring of these polynomials is generated (as an M-algebra) by the elementary ones Ci

=

E j Xj

Thus r(U(r))=R[c1,c2,...,cT}. In terms of matrices X G u{r) we have J2ck(X)tk

= det ( l - ^X)

G

r{U{r))[t).

The above polynomial is known as the universal Chern polynomial and its coefficients are called the universal Chern classes. Returning to our rank r vector bundle E we get the Chern classes ck(E) = ck(F(V))

G H2k(M)

and the Chern polynomial ct{E)

= det ( l - ^ T F ( V ) ) .

V denotes a connection compatible with a Hermitian metric ( , ) on E while F ( V ) denotes its curvature.

307

Important examples

Remark 8.2.1 The Chern classes produced via the Chern-Weil method capture only a part of what topologists usually refer to characteristic classes of complex bundles. To give the reader a feeling of what the Chern-Weil construction is unable to capture we will sketch a different definition of the 1st Chern class of a complex line bundle. The following facts are essentially due to Kodaira and Spencer [45]; see also [32] for a nice presentation. Let L —> M be a smooth complex Hermitian line bundle over the smooth manifold M. Upon choosing a good open cover (Ua) of M we can describe L by a collection of smooth maps zap : Uap —> f/(l) = S 1 satisfying the cocycle condition zaf}Zf}-,z-,a = 1 Va,/?,7.

(8.2.1)

If we denote by C°°(-,5 1 ) the sheaf of multiplicative groups of smooth S^-valued functions we see that the family of complex line bundles on M can be identified with the Cech group HX{M, C°°(-, S1)). This group is called the smooth Picard group of M. The group multiplication is precisely the tensor product of two line bundles. We will denote it by Pic°° (M). If we write zQ/3 = exp(27ri#Q/g) (8pa = — 9ap £ C°°(Ua0,R)) we deduce from (8.2.1) we deduce that V[/Q/37 ^ 0 #a/3 + #/37 + $ 7 a

=

napj

€ Z.

It is not difficult to see that Vf/Q/37j ^ 0 riffjs — nQ7<5 + naps — n a ^ 7 = 0. In other words nQ/g7 defines a Cech 2-cocycle of the constant sheaf Z. On a more formal level we can capture the above cocycle starting from the exact sequence of sheaves 0 - • Z -> C°°(-, R) exp^''") C°°(-, S 1 ) -> 0. The middle sheaf is a fine sheaf so its cohomology vanishes in positive dimensions. The long exact sequence in cohomology then gives 0 ->• Pic 00 (M) A H2(M, Z) ->• 0. The cocycle (na/37) represents precisely the class S(L). The class S(L), L e Pic 00 (M) is called the topological 1st Chern class and is denoted by cf(L). This terminology is motivated by the following result of Kodaira and Spencer, [45]: "The image of cf(L) in the DeRham cohomology via the natural morphism H*{M,Z)

-> H*{M,R) <* H*DR(M)

308

Characteristic classes

coincides with the 1st Chern class obtained via the Chern-Weil procedure.". The Chern-Weil construction misses precisely the torsion elements in H2(M,Z). For example if a line bundle admits a flat connection then its Chern class is trivial. This may not be the case with the topological one. The line bundle may not be topologically trivial. §8.2.3

Pontryagin classes

Let E be a rank r real vector bundle over the smooth manifold M. An Euclidean metric on E induces an 0(r) structure (P, p). The representation p is the tautological one p : 0(r) -> GL(r, M). Exercise 8.2.2 Prove that two metrics on E induce isomorphic 0(r)-structures. Hence, as in the complex case, we can naturally identify the rank r real vector bundles with principal <9(r)-bundles. A connection on the principal bundle can be viewed as a metric compatible connection in the associated vector bundle. To describe the various characteristic classes we need to understand the ring of invariants /•(O(r)). As usual we will identify the elements of lk(0(r)) with the degree k, Ad-invariant polynomials on the Lie algebra o(r) consisting of skewsymmetric r xr real matrices. Fix P e lk(0(r)). Set m = [r/2] and denote by J the 2 x 2 matrix J =

0 1

-1 0

Consider the Cartan algebra CO(T)

{A X J®--

© A m J 6 o(r) ; Aj 6 K} Am J © 0 S o(r) ; A^ 6 » }

r = 2m r = 2m + 1

Ce(r) is the Lie algebra of the (maximal) torus T

Re, © • • • ® Rem € 0(r) Re, © • • • © Rem © 1]K € °(r)

:2m > r = 2m + l

where for each 6 € [0,2ir] we denoted by Re the 2 x 2 rotation cos 0 — sin 9 sin 9 cos 9

Re =

As in Subsection 8.2.1 we introduce the variables XA = 3

——dBi.

2n

}

309

Important examples

Using standard results concerning the normal Jordan form of a skew-symmetric matrix we deduce that for every X € o(r) there exists T £ 0(r) such that TXT~l € C2(r)Consequently, any Ad-invariant polynomial on o(r) is uniquely defined by its restriction to the Cartan algebra. Following the approach in the complex case, we consider &>(r) = {T e 0(r) ; TXT'1 % )

= I V I e C„(r)}

= {T € O(r) ; T C ^ T - 1 C C B(r) }.

ffo(r) is a normal subgroup in 9 t ( 0 ( r ) ) so we can form the Weyl group 2»(0(r)) = Stow/SoMExercise 8.2.3 Prove that 2U(0(r)) is the subgroup of GL(m,R) involutions <Jij : \X\, . . • , X{, . . . , Xj, . . . , Xm)

generated by the

I V {Xij . . . , Xjj . . . , Xj, . . . , £ m J H-^ ^Xi, . . . , — Xj, • • • , ^ m j -

Sj : ( X i , . . . , £ j , . . . , XmJ fc

The restriction of P G 7 (0(r)) to C2(r) is a degree k homogeneous polynomial in the variables x\,...,xm invariant under the action of the Weyl group. Using the above exercise we deduce that P must be a symmetric polynomial P = P(xi,..., xm) separately even in each variable. Invoking once again the fundamental theorem of symmetric polynomials we conclude that P must be a polynomial in the elementary symmetric ones Pi = Y,j A ?2

=

Pm

—

Z/i<j xi

x

j

2 2 l' ' ' xm

x

Hence /*(0(r)) = K[p 1 ,... ) p [ r / 2 ]]. In terms of X e o(r) we have PtiX) = Y.PA*)t2i

= det ( l - ±X)

€

I'lO(r))[t\.

The above polynomial is called the universal Pontryagin polynomial while its coefficients Pj(X) are called the universal Pontryagin classes. The Pontryagin classes pi(E),.. .,pm(E) of our real vector bundle E are then defined by the equality

ME) = I > W

= det (l - ^ F ( V ) ) 6 H'{M)[t]

where V denotes a connection compatible with some (real) metric on E, while -F(V) denotes its curvature. Note that Pj(E) 6 Hij(M).

310 §8.2.4

Characteristic classes The Euler class

Let E be a rank r, real oriented vector bundle. A metric on E induces an 0(r)structure but the existence of an orientation implies the existence of a finer structure, namely an SO(r)-symmetry. The groups 0(r) and SO(r) share the same Lie algebra so(r) = o(r). The inclusion i : SO(r) ^

0{r)

induces a morphism of K-algebras i* : l'{0{r))

-+

I*{SO{r)).

Because so{r) = o(r) one deduces immediately that i* is injective. Lemma 8.2.2 When r is odd then i* : l*(0(r))

-» I*(SO(r))

is an isomorphism.

Exercise 8.2.4 Prove the above lemma. The situation is different when r is even, r = 2m. To describe the ring of invariants I*(SO(2m)) we need to study in greater detail the Cartan algebra

c2(2m) = {Ai J e • • • e \mJ e o(2m)} and the corresponding Weyl group action. The Weyl group 2F(50(2m)), defined as usual as the quotient W(SO(2m)

= 9ft so(2m) /3r S o(2m)

is the subgroup of GL(Co(2m)) generated by the involutions Oij : ( A i , . . . , A{,..., A j , . . . , Xm) — i > ( A i , . . . , \j,...,

A;,..., Amj

and where e"i,..., £m = ± 1 and E\ • • • em = 1. (Check this!) Set as usual Xi = — A;/27r. The Pontryagin invariants Pj(xu...,xm)=

(xii---xij)2

J2 l
continue to be 2U(50(2m)) invariants. There is however a new invariant A ( z i , . . . , z m ) = Y[XJ. i

In terms of

x = A1Je---eAmJecS0(2m)

Important

311

examples

we can write

where Pf denotes the pfaffian of the skewsymmetric matrix X viewed as a linear map M2m —> tt2m, when R 2m is endowed with the canonical orientation. Note that Pm = A 2 . Proposition 8.2.3 7*(50(2m)) 3 R[Zlt Z 2 , . . . , Zm; Y}/(Y2 - Zm)

(Zj = Pj,

Y = A)

where (Y2 — Zm) denotes the ideal generated by the polynomial Y2 — Zm. Sketch of proof We follow an approach used by H. Weyl in describing the invariants of the alternate group ([74], Sec. II.2). The isomorphism will be established in two steps. Step 1 I*(SO(2m)) is generated (as an M-algebra) by p i , . . . , p m , A. Step 2 The kernel of the morphism R[Zl,...,Zm,;Y}

A/*(50(2m))

denned by Zj >->• p3•„ Y i-> A is the ideal (Y2 - Zm). Proof of Step 1 Note that W{SO(2m)) has index 2 as a subgroup in 9B(0(2m)). Thus 2U(SO(2m)) is a normal subgroup and <8 = 2U(0(2m))/2n(SO(2m)) S Z 2 . & = {1, c} acts on I*{SO{2m))

by

(eF)(a; 1 ,a;2,...,x m ) = F{-xx,x2,

• • • ,xm) = ••• - F(xux2,

and moreover r ( 0 ( 2 m ) ) = ker(l - e). For each F € J*(SO(2m)) we have F+ d=f (1 + e)F € ker(l - e) so that F+ =

P(Pl,...,Pm).

On the other hand, F~ "^ (1 -

t)F

is separately odd in each of its variables. Indeed, F~{-x1,x2,---,xm)

= eF-(x1,...,zm)

•. •,

-xm)

312

Characteristic classes = e(l - e)F(xu

...,xm)

= -(l-

e)F(xu

...,xm)

= -F

{xu ...,

xm).

Hence, F~ vanishes when any of its variables vanishes so that F~ is divisible by their product A = xi • • • xm F~ = A-G. Since eF~ = -F~ and eA = - A we deduce eG = G i.e. G € 7*(0(2m)). Consequently, G can be written as G =

Q(p1,...,pm)

so that F~ =

A-Q(Pl,...,Pm).

Step 1 follows from F = ±(F+ + F-) = l-{P{Pl,... ,Pm) + A • Q(pu ..

.,Pm)).

Proof of Step 2 From the equality det X = Pf(X)2

VX 6 so{2m)

we deduce

(r 2 -Z m )Cker so that we only need to establish the opposite inclusion. Let P = P{ZU Z2, • • •, Zm\ Y) € kerip. Consider P as a polynomial in Y with coefficients in R[Zi,..., Zm\. Divide P by the quadratic polynomial (in Y) Y2 — Zm. The remainder is linear R = A(Z1,...,Zm)Y

+

B(Z1,...,Zm).

Since Y2 — Zm, P 6 kerV' we deduce R € ker^. Thus A(pi, • • • ,p m )A + B(pu .. .,pm) = 0. Applying the morphism e we get -A{pu

... ,p m )A + B(Pl,...

Hence A = B = 0 so that P is divisible by Y2 - Zm.

,pm) = 0. •

Let E be a rank 2m, real, oriented vector bundle over the smooth manifold M. As in the previous subsection we deduce that we can use a metric to naturally identify E with a principal SO(2m)-bundle and in fact, this principal bundle is independent of the metric. Finally, choose a connection V compatible with some metric on E.

Important

313

examples

Definition 8.2.4 (a) The universal Euler class is defined by e = e(X) = _ L - P / ( _ A : )

e

7m(50(2m)).

(b) The Euler class of E, denoted by e(E) G H2m(M) represented by the 2m-form

is the cohomology class

e(V) = ( ^ V ( - n V ) ) (According to Chern-Weil theorem this cohomology class is independent of the metric and the connection.) Example 8.2.5 Let (S, g) be a compact, oriented, Riemann surface and denote by V s the Levi-Civita connection. The the Euler form £

(d) = -jzs{g)dvg 4ir~

9

coincides with the Euler form e(V ) obtained via the Chern-Weil construction. Remark 8.2.6 Let £ be a rank 2m, real, oriented vector bundle over the smooth, compact, oriented manifold M. We now have two apparently conflicting notions of Euler classes. A topological Euler class etop(E) 6 H2m(M) defined as the pullback of the Thorn class via an arbitrary section of E. A geometric Euler class egeom{E) e H2m(M) defined via the Chern-Weil construction. The most general version of the Gauss-Bonnet theorem, which will be established later in this chapter, will show that these two notions coincide! §8.2.5

Universal classes

In each of the situations discussed so far we discussed characteristic classes for vector bundles with a given rank. In this section we show how one can coherently present these facts all at once, irrespective of rank. The algebraic machinery which will achieve this end is called inverse limit. We begin by first describing a special example of inverse limit. A projective sequence of rings is a a sequence of rings {Rn}n>o together with a sequence of ring morphisms Rn t^ Rn+i. The inverse limit of a projective system (Rn, 4>n) is the subring

limRn C n # n n>0

consisting of the sequences (x\,x2, • • •) such that 4>„(xn+i) = xn, Vn > 0.

314

Characteristic classes

E x a m p l e 8.2.7 Let Rn = ^ [ [ X ^ . . .,Xn]] be the ring of formal power series in n variables with coefficients in the commutative ring with unit 9t. (RQ = 9t.) Denote by n '• Rn+i —> Rn the natural morphism defined by setting Xn+i = 0. The inverse limit of this projective system is denoted by 9t[[Xi, X2, • • •]}• Given a sequence Fn e 9l[[X]] (n > 1) such that F„(0) = 1 we can form the sequence of products (1, F1(X1), F1(X1)F2(X2),...,

FX(XX) • • •

Fn{Xn),...)

which defines an element in Dtfpfi, X2, •..]] denoted by Fi(Xi)F2(X2) •• •. When F\ = F2 = • • • Fn = • • • = F the corresponding elements is denoted by (F)°°. Exercise 8.2.5 (a) Let SK^a;]]1' denote the set of formal power series F 6 5H[[a;]] such that F(0) = 1. Prove that (fH[[x]]k, •) is an abelian group, (b) Prove that VF, G € ^{[xtf {F • G)°° = (F)°° • (G)°°

Similarly, given Gn E ^[[^jji {ji ^ 1) such that Fn(0} — 0 we can from the sequence of sums (0, Gi(Xi), GiiXJ

+ G2(X2),...,

G1(X1) + • • • e

Gn(Xn),...)

which defines an element in •SH[[X1,X2,.. •]] denoted by Gi(Xi) + G2(X2) -i . When G\ = G2 = • • • = Fn — • • • = F we denote the corresponding element (G)^. In dealing with characteristic classes of vector bundles one naturally encounters the increasing sequences £7(1) - » V{2) M- • • • (8.2.2) (in the complex case) and (in the real case) 0(1) ^ 0(2) - > • • • .

(8.2.3)

We will discuss these two situations separately. The complex case The sequence in (8.2.2) induces a projective sequence of rings R <- I"(U(l))

<- r*(U(2))

i

.

(8.2.4)

We know that I"(U(n)) = © n [fo]]= the ring of symmetric formal power series in n variables with coefficients in R. Set ©<»[[*,•]]= l i m © " .

315

Important examples

Given a rank n vector bundle E over a smooth manifold M and € I**(U(n)) the characteristic class cj)(E) is well defined since u d l m M + 1 = 0 for any u 6 fi*(M). Thus we can work with the ring /**([/(n)) rather than P(U(n)) as we have done so far. An element

= (0i,^, • • •) e r * ( t / M ) =' iimr*(t/(n)) = e~[fo]] is called universal characteristic class. If S is a complex vector bundle we set 4>{E) = (j>r{E) where r = v&nkE. More precisely, to define (j>T{E) we need to pick a connection V compatible with some Hermitian metric on E and then set 4>(E) = &(F(V)). Example 8.2.8 We denote by q. n variables (n)

(n > A;) the elementary symmetric polynomial in V"^ l
Then the sequence

(0,...,o,cik\c{r >,...)

defines an element in <5°°[[:rj]] denoted by c^ which we call the universal A;-th Chern class. Formally, we can write l
We can present the above arguments in a more concise form as follows. Consider the function F(x) = (1 + tx) € 9t[[a;]] where 9 t = R[i\. Then (F)°° defines an element in
{F)°°{xux2,...)

We can perform the above trick with any F £ M.[[x]] such that -F(O) = 1. We get a semigroup morphism

m[x}]\-)3F^(Fre(eea[[xj}],-). (see Exercise 8.2.5) One very important example is T

1

1

i

oo

r>

The coefficients B^ are known as the Bernoulli numbers. The product

316

Characteristic classes

defines an element in <5°°[[:r.,]] called the universal Todd class and denoted by Td. Using the fundamental theorem of symmetric polynomials we can write Td = 1 + Tdi + Td 2 + •• • where Td n G ©°°[[XJ]] is an universal symmetric, homogeneous "polynomial" of degree n, hence expressible as a combination of the elementary symmetric "polynomials" Ci,c2, — By an universal "polynomial" we understand element in the inverse limit M[x\,X2, • • •] = limR[a;i,a:2,...]. An universal "polynomial" P is said to be homogeneous of degree d if it can be represented as a sequence

P=(PUP2,...) where Pm is a homogeneous polynomial of degree d in m variables and Pm+l(xli

x

2, • • • > xmi 0 ) = Pm{X\,

...

Xm).

For example Tdi = -Cl,

Td 2 = —{c\ + c 2 ), Td 3 = — c l C 2 etc.

Analogously, any function G 6 K[[x]] such that G(0) = 0 defines an element &co[[xj]}

(GU = G(Xl) + G(x2) + •••€

We have two examples in mind. First, consider G(x) = xk. We get the symmetric function sk = (a;*)oo = (G)oo = x\ + x\ + • • • called the universal k-th power sum. We will denote these by s^. Next, consider G(x) = e 1 - 1 = V

=-.

m>l

We have

(C- - 1)„ = E i(^m)oc = E i»™ 6 e°°[fe']] m>l m-

m=l

m

-

Given a complex vector bundle E we define ch(jE;) = r a n k £ + (ex - !)«,(£). ch(E) is called the Chern character of the bundle E. If V is a connection on E compatible with some Hermitian metric then we can express the Chern character of E as c h ( £ ) = tr (eF)

Important examples

317 00

1

= rank(£) + £ - i t r ( . F ( V r * ) k=i

K

-

where A is the bilinear map ft* (End (£)) x ft* (End (£)) -> Q i+J '(End {E)) defined in Example 3.3.11. Proposition 8.2.9 Consider two complex vector bundles E\,E2 over the same manifold M. Then ch{E1 © E2) = chiEi) + ch{E2) and ch(Ex ® E2) = ch(E1) • ch(E2) € H*{M).

Proof Consider a connection V 1 on Ei compatible with some Hermitian metric ht, i = 1,2. Then V 1 ©V 2 is a connection on E\®E2 compatible with the metric h\®h2 and moreover F(V 1 ffiV 2 ) = F ( V 1 ) © F ( V 2 ) . Hence exp(F(V 1 © V 2 )) = exp(F(V 1 )) © exp(F(V 2 )) from which we deduce the first equality. As for the second equality consider the connection V on E\ ® E2 uniquely defined by the Leibniz rule V(si ® s2) = (V x si) ® s2 + si ® (V 2 s 2 ),

Si

6 C00^)

where the operation

®: n*(Fi) x n £ (F 2 ) -» n*+/(Je1 ® F 2 ) is defined by (w* ® si) ® (7/'s2) = (w* A rf) ® (si ® s 2 ) Si S C°°(Fi), w* € fi*(M) and r/ e fi'(M). We compute the curvature of V using the equality F(V) = (d v ) 2 . If s, € then F ( V ) ( S l ® s2) = d v {(V 1 si) ® s 2 + S l ® (V 2 s 2 )} = {{F{Vl)Sl)

(8.2.5) Cx{Ei)

® s 2 - ( V ^ i ) ® (V 2 s 2 ) + ( V ^ ) ® (V 2 s 2 ) + s x ® (F(V 2 ))} = F(V1)®lfi2 + lBl®F(V2).

The second equality in the proposition is a consequence the following technical lemma.

318

Characteristic classes

Lemma 8.2.10 Let A (resp. B) be a skew-adjoint, n x n (resp. m x m) complex matrix. Then tr (exp(A l c ™ + l c » ® B)) = tr (exp(^)) • tr (exp(B)). Proof of the lemma Pick an orthonormal basis (e;) of C n (resp. an orthonormal basis (fj) of C m ) such that with respect to this basis A = diag ( A j , . . . , An) (resp. B = diag ( ^ i , . . . , fJ.m))- Then with respect to the basis (e* fj) of C" <8> C m we have (A l c ™ + l c » ® B) = dio5(Aj + ^ - ) . Hence exp(^ ® 1 C ". + 1,£» ® B) =

diag^eK)

so that tr ( 4 ® l C m + l c » ® B) = YleXieN The proposition is proved.

=

tr

( e/1 ) • t r ( e B )-

n

D

Exercise 8.2.6 ( N e w t o n ' s formulae) Consider the symmetric polynomials ck=

]T

xh---xik€R[xi,...,xn}

l
and (r > 0) 1)

•

•

• 1 • " •71J-

3

Set

n

- £.,'£).

/(*) = .7=1

(a) Show that 1 /'(*) = - £ •s.r- . /(*) r

(b) Prove the Newton formulae •iySr.

jC^- = 0

VI < r < n.

(c) Deduce from the above formulae the following identities between universal symmetric polynomials. Si=ci,

s 2 = c 1 - 2 c 2 , s 3 = Ci - 3cic 2 + 3c 3 .

Important

319

examples

The real case

The sequence (8.2.3) induces a projective system J**(0(l))<-I**(0(2))<

.

We have proved that J**(0(n)) = ©' n/2 '[[a^]]= the ring of even, symmetric power series in [n/2] variables. The inverse limit of this system is 7* * (O(oo) = l i m / " ( 0 ( r ) ) = ©<»[[:#] ^ lim © " [ [ : # ] . As in the complex case, any element of this ring is called a universal characteristic class. In fact, for any real vector bundle E and any 4> £ ©""M3^]] there is a well defined characteristic class <j>{E) which can be expressed exactly as in the complex case, using metric compatible connections. If F e lR[[z]]l> then (F(a;2))°° defines an element of ©°°[[a$]. In topology, the most commonly encountered situations are the following.

The universal characteristic class (F(x))°° is denoted by A and is called the A — genus. We can write A = 1 + Aj + A 2 + • • • where A^ are universal, symmetric, even, homogeneous "polynomials" and as such they can be described using universal Pontryagin classes

E

2 X

jl

2 X

' ' ' jm

•

1<J1 < - * »

The first couple of terms are Ai = ~ ,

A2 =

27,32.5(-4p2

+ 7p?) etc.

B . Consider

F(x) = v

^ _ = i + lx

tanh^/z

3

+

JL3? + ... = i +

45

^

'

X2(-i)^l^Bkxk. (2k)\

The universal class (F(x2))°° is denoted by L and is called the L-genus. As before we can write L = 1 + L! + L 2 + • • • where the L^'s are universal, symmetric, even, homogeneous "polynomials". They can be expressed in terms of the universal Pontryagin classes. The first few terms are Li = - P i , L 2 = —(7p 2 - p j ) etc. o 45

320

8.3

Characteristic classes

Computing characteristic classes

The theory of characteristic classes is as useful as one's ability to compute them. In this section we will describe some methods of doing this. Most concrete applications require the aplication of a combination of techniques from topology, differential and algebraic geometry and Lie group theory that go beyond the scope of this book. We will discuss in some detail a few invariant theoretic methods and we will present one topological result more precisely the Gauss-BonnetChern theorem. §8.3.1

Reductions

In applications, the symmetries of a vector bundle are implicitly described through topological properties. For example, if a rank r complex vector bundle E splits as a Whitney sum E = Ei ®E2 with rankjBj = Ti then E, which has a natural U(r)- symmetry, can be given a finer structure of U(ri) x C/(r2) vector bundle. More generally assume that a given rank r complex vector bundle E admits a Gstructure (P, p), where P is a principal G bundle and p : G —> U(r) is a representation of G. Then we can perform two types of Chern-Weil constructions: using the U(r) structure and using the G structure and in particular we obtain two collections of characteristic classes associated to E. One natural question is whether there is any relationship between them. In terms of the Whitney splitting E = Ei © E2 above, the problem takes a more concrete form: compute the Chern classes of E in terms of the Chern classes of E\ and E2- Our next definition formalizes the above situations. Definition 8.3.1 Let ip : H —> G be a smooth morphism of (matrix) Lie groups. (a) If P is a principal //-bundle over the smooth manifold M defined by the open cover ({/„) and gluing cocycle hap '• Uap —» H

then the principal G-bundle defined by the gluing cocycle ga0 = G

is said to be the (^-associate of P and is denoted by (p(P). (b) A principal G-bundle Q over M is said to be (^-reducible if there exists a principal //-bundle P -» M such that Q = G in the above definition induces a K-algebra morphism tp* •. r{G) The elements of ker ip* are called universal The following result is immediate.

->

r(H).

identities.

Computing characteristic classes

321

Proposition 8.3.2 Let P be a principal G-bundle which can be reduced to a principal ^-bundle Q. Then for every rj G ker
Proof

Denote by ip„ the differential of tp at 1 e H £ G .

Pick a connection [Aa) on Q and denote by (Fa) its curvature. Then the collection
v(
ck(E®F)=

Yl

(8.3.1) Equivalently, this means

ct(E)-Cj(F).

i+j=k

To check this, pick a Hermitian metric g on E and a Hermitian metric h on F. g®h is a Hermitian metric on E © F. Hence, E © F has an U(r + s) structure reducible to an U(r) x U(s) structure. The Lie algebra of U(r) x U(s) is the direct sum u(r) © u(s). Any element X in this algebra has a block decomposition XE X =

Xr 0

0 X.,

= xr®xs

where XT (respectively Xs) is an r x r (resp. sxs) complex, skew-adjoint matrix. Let i denote the natural inclusion u(r) ©«(s) <-> u(r + s) and denote by 4 " ' e I*(U{v))[t] the Chern polynomial.

322

Characteristic classes

We have

i*(c[r+s))(xr e x.) = det (i r + , - ^Xr e x.) =

det(l

r

-^X

r

).det(l

s

-^

s

)

The equality (8.3.1) now follows using the argument in the proof of Proposition 8.3.2. Exercise 8.3.1 Let E and F be two complex vector bundles over the same manifold M. Show that Td{E ®F) = Td(E) • Td(F). Exercise 8.3.2 Let E and F be two real vector bundles over the same manifold M. Prove that pt{E®F)=Pt{E)-Pt{F) (8.3.2) where pt denotes the Pontryagin polynomials. Exercise 8.3.3 Let E and F be two real vector bundles over the same smooth manifold M. Show that L{E®F) = L(E)-L(F) A{E®F)

=

A{E)-A{F).

Example 8.3.4 The natural inclusion R" •-+ C induces an embedding i : 0(n) °-> U(n). (An orthogonal map T : R n —»• R n extends by complexification to an unitary map T(£ : C n —> C n ). This is mirrored at the Lie algebra level by an inclusion o(n) ^-¥ u(n) and we obtain a morphism : P(U(n))

i

-+

P{0(n))

We claim that j*(c2fc+i) = 0

and

t'Coa) = (-I)*P*. Indeed, for X G o(n) we have ,

1

x 2/fc+l «+l

•(c2,+1)(X)= (- — ) \

Z7T1/

£ l<:,<-
^PO-.-'WA-)

Computing characteristic classes

323

where Xj(X) are the eigenvalues of X over C. Since X is in effect a real skewsymmetric matrix we have

XiiX) = \^X)

=-\j(X).

Consequently i*(c2k+1)(X)

= i*(c2k+l(X))=

(-—) \

= (-1)2fc+1 (-i) 2 f c + 1 \

Z7T1/

£

Z7T1/

s

\h(X)-..\i2k+l(X)

l
* . w • • • w*)=-»•(<*+!)(*)•

l
The equality i*(c2k) = {—l)hpk is proved similarly. From the above example we deduce immediately the following consequence. Proposition 8.3.5 If E -> M is a real vector bundle and E®C is its complexification then = (-l)kc2k(E®C), A = 1,2,.... (8.3.3) Pk(E) In a more concentrated form this means pt{E)=p_t(E)

=

cit{E®C).

Exercise 8.3.4 Let E —• M be a complex vector bundle of rank r. (a) Show that ck(E*) = (-l)kck{E)

i. e. ct(E') =

c.t(E).

(b) One can also regard E as a real, oriented vector bundle E^. Prove that

E(-i)^ 2 V(%) = ct(£;)-c_((s) k

i.e. p i t ( £ t t ) = ct{E) • c_ ( (£). and moreover Cr(E)=e(Em).

Exercise 8.3.5 (a) The natural morphisms I**(U(r)) -> l"(0(r)) induce a morphism $oo : I"(U(oo)) -+ / " ( O ( o o ) ) .

described above

324

Characteristic classes

As we already know, for any F e K[[a;]]b, (-F)00 is an element of P*{U(oo)) and if moreover F is even then (F)°° can be regarded as an element of I**(0(oo)). Show that for every F € ffi[[i]]1' $0O((F)co)

= (F • F-)°° G 7"(0(oo))

where F~(x) = F{-x). (b) Let £ be a real vector bundle. Deduce from part (a) that Td(E®C) =

A(E)2.

E x a m p l e 8.3.6 Consider the inclusion i: SO{2k) x SO{2l)) ^ SO{2k + 21). This induces a ring morphism i* : r{SO(2k

+ 21) -> r(SO{2k)

x SO(2£)).

Note that r{SO{2k)

x 50(2^)) ^ /*(S0{2k)) ® R /*(S0(2£).

Denote by e'"' the Euler class in /* (SO (2;/)). We want to prove that

Let X = Xk © Xt € so(2fc) © so(2£). Modulo a conjugation by (S,T) 6 50(2fc) x SO(2£) we may assume that Xk = A i J e - - - © A f c J and A^ = jUiJffi •••® mJ where as usual J denotes the 2 x 2 matrix

JJ = [[°1 - 01 " We have S(e^)(X)

= (-ij

Ax • • • A* • W • • • W = eW(Xk)

= eW®eM(X f c ®^).

•

e^(Xe)

325

Computing characteristic classes The above example has an interesting consequence.

Proposition 8.3.7 Let E and F be two real, oriented vector bundle over the same manifold M and having even ranks. Then e(E 0 F) = e(E) • e(F) where as usual the • denotes the A-multiplication in

Heven(M).

Example 8.3.8 Let £ be a rank 2k real, oriented vector bundle over the smooth manifold M. We claim that if E admits a nowhere vanishing section f then e(E) = 0. To see this, fix an Euclidean metric on E so that E is now endowed with an S'0(2A;)-structure. Denote by L the real line subbundle of E generated by the section £. Clearly, L is a trivial line bundle and E splits as an orthogonal sum E = L © L^. The orientation on E and the orientation on L defined by. £ induce an orientation on L1- so that L1- has an SO(2k — restructure. What we have just said shows that the SO(2k) structure of E can be reduced to an 5 0 ( 1 ) x SO(2k - 1) =* SO(2k - l)-structure. Denote by i* the inclusion induced morphism

r{so{2k)) -> r(so(2k -1)). Since i*(e(fc)) = 0 we deduce from the Proposition 8.3.2 that e(E) = 0. The result proved in the above example can be reformulated more suggestively as follows. Corollary 8.3.9 Let E be a real oriented vector bundle of even rank over the smooth manifold M. If e(E) ^ 0 then any section of E must vanish somewhere on M! §8.3.2

The Gauss-Bonnet-Chern theorem

If E —> M is a real oriented vector bundle over a smooth, compact, oriented manifold M then there are two apparently conflicting notions of Euler class naturally associated toE. The topological Euler class e«op(-E) = QTE where TE is the Thorn class of E and £0 : M —>• E is the zero section. The geometric Euler class „ (p^-f egeam(*)-{

(£yPf(-FW) Q

if rank (E) is even if rank (S) is odd

where V is a connection on E compatible with some metric and 2r = r a n k ( £ ) . The next result, which generalizes the Gauss-Bonnet theorem, will show that these two notions of Euler class coincide.

326

Characteristic classes

Theorem 8.3.10 (Gauss-Bonnet-Chern)Let E A M be a real, oriented vector bundle over the compact oriented manifold M. Then etop(E) =

egeom{E).

Proof We will distinguish two cases. A rank (E) is odd. Consider the automorphism of E \:E-+E

«4-«Vue£.

Since the fibers of E are odd dimensional we deduce that i reverses the orientation in the fibers. In particular, this implies KXTE

= -7T,T£ =

7T,(-TE)

where IT, denotes the integration along fibers. Since 7r» is an isomorphism (Thom isomorphism theorem) we deduce i*TE = -TEHence etop(E) = -C0*iVB.

(8.3.4)

On the other hand notice that Coi* = CoIndeed Coi* = (Ko)* = (-Co)* = ( 0 * (Co = -Co)The equality etop = e 9eom now follows from (8.3.4). B . rank (E) = 2k. We will use a variation of the original argument due to Chern ([19]). Let V denote a connection on E compatible with a metric g. The strategy of proof is very simple. We will explicitly construct a closed form u> 6 Q!^t(E) such that (i) 7r„w = 1 e n°(M). (ii) Qu, = e(V) = (2w)-kPf(-F(W)). The Thom isomorphism theorem coupled with (i) implies that u> represents the Thom class in H^t(E). (ii) simply states the sought for equality e top = egeom. Denote by S(E) the unit sphere bundle of E S(E) = {ueE;

\u\g = 1}.

S(E) is a compact manifold and dim S{E) = d i m M + 2 f c - l .

327

Computing characteristic classes

Denote by ir0 the natural projection S(E) —¥ M and by TT^E) —> S(E) the pullback of E to S(.E) via the map •KQ. TTQ{E) has an 50(2A:)-structure and moreover it admits a tautological, nowhere vanishing section T : S(EX) Be^eeEx

= K ( £ ) x ) e (x G M).

Thus according to Example 8.3.8 we must have ege0m«(E))

= 0G

H2k(S(E))

where e9eom(7TQ^) denotes the differential form e9eom(7roV) = J L p / ( - F ( j r ; v ) ) . Hence there must exist V 6 fl2A_1(S(.E)) such that dip =

egeam{ixlE).

The decisive step in the proof of Gauss-Bonnet-Chern theorem is contained in the following lemma. Lemma 8.3.11 There exists * = # ( V ) G

fi2*"1^^))

such that

d*(V) = egeom(Tr*0(E))

(8.3.5)

and /

W(V) = - 1 G

fi°(M).

(8.3.6)

JS(E)/M

The form \1/(V) is sometimes referred to as the global angular form of the pair (E, V). For the clarity of the exposition we will conclude the proof of the GaussBonnet-Chern theorem assuming Lemma 8.3.11 which will be proved later on. Denote by r : E —> M+ the norm function E Be

H->

|e|j

If we set E° = E \ {zero section} then we can identify £°^(0,oo)xS(£)

e^(\e\±e).

Consider the smooth cutoff function p = p(r) : [0, oo) -J- R

328

Characteristic classes

such that p(r) = - 1 for r € [0,1/4] and p(r) = 0 for r > 3/4. Finally define u, =

W (V)

= -p'(r)dr

A tf (V) - p(r)7r*(e(V)).

w is well defined since p'(r) = 0 near the zero section. Obviously w has compact support on E and satisfies the condition (ii) since CJw = - p ( 0 X o V e ( V ) = e(V). From the equality f

JE/M

p(r)7r*e(V) = 0

we deduce f

JE/M

ui = - [

JE/M

p'Mdr A * ( V )

=- r p'(r)dr-1 Jo

*( v ) JS(E)/M

*(V)(8^6)1.

= -(p(l)-p(0))/ JS(E)/M

To complete the program outlined at the beginning of the proof we need to show that u) is closed. duj = p'{r)dr A d*(V) - p'(r) A 7r*e(V) (8

= 5 ) p'(r)dr A {7rJe(V) - 7r*e(V)}.

The above form is identically zero since 7Tge(V) = 7r*e(V) on the support of p'. Thus ui is closed and the theorem is proved. • P r o o f of L e m m a 8.3.11 We denote by V the pullback of V to ir^E. The tautological section T : S(E) —>• ir^E can be used to produce an orthogonal splitting •KQE =

L®L±

where L is the line bundle spanned by T while LL is its orthogonal complement in •KQE with respect to the pullback metric g. Denote by P : 7r0*£ -> ir*0E the orthogonal projection onto LL. Using P we can produce a new metric compatible connection V on TTQE by

V = (trivial connection on L) © PVP. We have an equality of differential forms ^ e ( V ) = e(V) =

1 (2TT)

:P/(-F(V)).

Computing characteristic classes

329

Since the curvature of V splits as a direct sum F(V) = 0 © F'(V) where F'(V) denotes the curvature of V \LJ. we deduce P/(F(V)) = 0. We denote by V the connection V + t(V - V) so that V° = V and V1 = V. If F1 is the curvature of V we deduce from the transgression formula (8.1.11) that 7 r 0 *e(V)=e(V)-e(V)

=d{{^)kkLlpfv^>Ft>-->Ft)dt}We claim that the form

*(V) = ( ^ ) kkfolPf(V-V,F\...,

F*)dt

satisfies all the conditions in Lemma 8.3.11. By construction d-Jr(V) = 7r*e(V) so all that we need to prove is [

JS(E)IM

It suffices to show that for each fiber Ex of E we have / ^ ( V ) = -l. Along this fiber TTQE is naturally isomorphic with a trivial bundle TT;E\EX^(EXXEX^EX).

Moreover V restricts as the trivial connection. By choosing an orthonormal basis of Ex we can identify w^E \EI with the trivial bundle R2* over R2fc. The unit sphere S(EX) is identified with the unit sphere S2k~x C R2*. The splitting L © LL over S(E) restricts over S{EX) as the splitting R2* = v © TS210-1 where v denotes the normal bundle of S2*""1 •-* R2*. The connection V is then the direct sum between the trivial connection on v and the Levi-Civita connection on

TS2k-\

330

Characteristic classes

Fix a point p € 5 2 f c _ 1 and denote by ( x 1 , . . . ,x 2 * - 1 ) a collection of normal coordinates near p such that the basis ( ^ | p ) is positively oriented. Set d{ = ^ 7 for % = 1 , . . . , 2k — 1. Denote the unit outer normal vector field by d0. For a = 0 , 1 , . . . , 2fc — 1 set fa = da\p. The vectors fa form a positively oriented orthonormal basis of K2*. We will use Latin letters to denote indices running from 1 to 2k — 1 and Greek letters indices running from 0 to 2k — 1. Vida = ( V A ) " + (V { cU T where the superscript v indicates the normal component while the superscript r indicates the tangential component. Since at p 0 = Via,- =

(VidjY

we deduce Vidj = (Vidj)"

at p.

Hence

Vja,- = (Via„a0)ao = -(a,-, Vi&jaaRecall that V is the trivial connection in M2* and we have Vido\P= (-gfdo I | p = f, = di\p. Consequently Vidj = ~5jid0 2k x

If we denote by 9' the local frame of TS ~ equality as

at p.

to (3,) then we can rephrase the above

vdj = -(e1 + --- + e2k-1)®d0. On the other hand Vid0 = dt i.e.

va 0 = 91 ® ai + • • • + 0 2 * -1 ® a2t_!.

Since (x 1 , •'• •, x2k x) are normal coordinates with respect to the Levi-Civita connection V we deduce that Vda = 0, Va so that

A=(V-V)|„=

0 01 92 6>2*-

924-1

0 0

Computing characteristic classes

331

Denote by F° the curvature of V° = V at p. Then

F° = 0 e R where R denotes the Riemann curvature of V at p. The computations in Example 4.2.13 show that the second fundamental form of the embedding S2k-i

^

R

»

coincides with the induced Riemann metric (which is the first fundamental form). Using Teorema Egregium we get (R{di, dj)dk, dt) = SuSjk - 6ik5jeIn matrix format we have F° = 0 0 ( % ) l

where fi^ = 6 A (P. The curvature Fl at p of V ' = V + tA can be computed using the equation (8.1.10) of subsection 8.1.4 and we get Fl = F° + t2A A A = 0 © (1 We can now proceed to evaluate *

(V).

*(V) lp= O" kll = {^)k

t2)F°.

Pf{A (1

' " t2)F°'''''(1 " t2)F°)dt

k ^J\l - t2)*"1*) Pf(A, F°, F°, • • •, F°).

(8.3.7)

We need to evaluate the pfaffian in the right-hand-side of the above formula. Set for simplicity F = F°. Using the polarization formula and Exercise 2.2.19 in Subsection 2.2.4 we get "J (A, r, r, • • •, r J =

2__, £((y)Aaoai-t'o2cr3 ' ' ' ^
.

"£&2k

For i = 0,1 define & = {a e &a

; at = 0}.

We deduce 2kk\Pf(A,F,---,F) +{-l)k

= {-l)k

Y,

^e1

£ rt6°

^ ^ ( - ^ ' J A ^ 2 A ^ 3 A---A6C 7 "- 2 A ^ " " 1

f W ^ A f

A0" 3 A - - - A I T " - 2 A P - 1 .

332

Characteristic classes

For each a € G° we get a permutation 4>: (0-1,172, •••,CT2k-l)

€ ©2*-l

1

such that e(o-) = e(<£). Similarly, for a € (5 we get a permutation = (<7o, ^ 2 , • • • , 02/fc-l) € © 2 * - l

such that e(cr) = e(<j>). Hence 2kk\Pf{A,F,---,F)

= 2{-l)k+1

*T, e ( ^ A

0*""1

= 2(-l)* +1 (2A;-l)!0 1 A---A0 2 * : - 1 2(-l)k+1dvolff,k-i.

=

Using the last equality in (8.3.7) we get

*(V) |„= ( ^ ) * * ( j f ( l - t 2 )* - 1 *) • 2(-l)*+1(2fc -

l)\dvols^

2irk

Using the Exercise 4.1.15 we get that f and consequently

/^.•w-ifc-.^GCd-fl"*)

-^-D.Gf'1-*1^)The above integral can be evaluated inductively using the substitution t = cos tp Ik = (/ o X (l - t 8 )*- 1 *) = [

\**
= (cos^) 2fc - 2 sin^|J /2 + (2* - 2) I"1*(cas
= (2k - 1)4_! - (2* - 2)4 so that Ik

_2fc-3

-2k=lIk-1-

One now sees immediately that / JS2k-l

Lemma 8.3.11 is proved.

•.

*(V) = - 1 .

333

Computing characteristic classes

Corollary 8.3.12 (Chern [Ch])Let {M,g) be a compact, oriented Riemann Manifold of dimension 2n. If R denotes the Riemann curvature then

The next exercises provide another description of the Euler class of a real oriented vector bundle E —>• M over the compact oriented manifold M in terms of the homological Poincare duality. Set r = rank E and let TE be a compactly supported form representing the Thom class. Exercise 8.3.6 Let $ : N —> M be a smooth map, where N is compact and oriented. We denote by $ * the bundle map $*E -> E induced by the puUback operation. Show that <3>*TE = ( $ # ) * T £ € Qr($*E) is compactly supported and represents the Thom class of $*£. In the next exercise we will also assume M is endowed with a Riemann structure. Exercise 8.3.7 Consider a nondegenerate smooth section s of E i.e. (i) Z = s _ 1 (0) is a codimension r smooth submanifold of M. (ii) There exists a connection V° on E such that V°xs / 0 along Z for any vector field X normal to Z (along Z). (a) (Adjunction formula) Let V be an arbitrary connection on E and denote by Nz the normal bundle of Z H M. Show that the adjunction map

a v : Nz

-+E\z

defined by X M- Vxs

X e C°°(7VZ)

is a bundle isomorphism. Conclude that Z is endowed with a natural orientation. (b) Show that there exists an open neighborhood 9 t of Z «-• Nz <-> TM such that exp : 91 -¥ exp(9t) is a diffeomorphism. (Compare with Lemma 7.3.30.) Deduce that the Poincare dual of Z in M can be identified (via the above diffeomorphism) with the Thom class of Nz. (c) Prove that the Euler class of E coincides with the Poincare dual of Z .

334

Characteristic classes

The part (c) of the above exercise generalizes the Poincare-Hopf theorem (see Corollary 7.3.33). In that case the section was a nondegenerate vector field and its zero set was a finite collection of points. The local index of each zero measures the difference between two orientations of the normal bundle of this finite collection of points: one is the orientation obtained if one tautologically identifies this normal bundle with the restriction of TM to this finite collection of points while and the other one is obtained via the adjunction map. In many instances one can explicitly describe a nondegenerate section and its zero set and thus one gets a description of the Euler class which is satisfactory for most topological applications. Example 8.3.13 Let r n denote the tautological line bundle over the complex projective space OP". According to Exercise 8.3.4 ci(r„) = e ( r „ ) when we view r n as a rank 2 oriented real vector bundle. Denote by [H] the (2n — 2)cycle defined by the natural inclusion i : OP"" 1 ^ OP",; [z0 : . . . *„_!] H-> [z0 : . . . : zn^

: 0] € OP".

We claim that the (homological) Poincare dual of Ci(r n ) is —\H]. We will achieve this by showing that Ci(r*) = —Ci(Tn) is the Poincare dual of [H]. Let P be a degree 1 homogeneous polynomial P G C[zo,.. .,zn]. For each complex line L <-> C" + 1 the polynomial P defines a complex linear map L —¥ C and thus an element of L* which we denote by P\i. We thus have a well defined map OP"9Z,H->P|LeL* = < | L and the reader can check easily that this is a smooth section of T£ which we denote by [P]. Consider the special case P 0 = zn. The zero set of the section [P0] is precisely the image of [H]. We let the reader keep track of all the orientation conventions in Exercise 8.3.7 and conclude that CX(T*) is indeed the Poincare dual of [H]. Exercise 8.3.8 Let Us = S2\ {north pole} and UN = S2\ {south pole} ^ C. The overlap UN n Us is diffeomorphic with the punctured plane C* = C \ {0}. For each map g : C* -> U(l) = S 1 denote by Lg the line bundle over S 2 defined by the gluing map

gNS-UNnUs^U(l),

gNS{z) = g{z).

Show that where degg denotes the degree of the smooth map
Chapter 9 Elliptic equations on manifolds Almost all the objects in differential geometry are defined by expressions involving partial derivatives. The curvature of a connection is the most eloquent example. One is often led to studying such objects with specific properties. For example, we inquired whether on a given vector bundle there exist flat connections. This situation can be dealt with topologically, using the Chern-Weil theory of characteristic classes. Very often, topological considerations alone are not sufficient and one has look into the microstructure of the problem. This is where analysis comes in and more specifically, one is led to the study of partial differential equations. Among them, the elliptic ones play a crucial role in modern geometry. This chapter is an introduction to this vast and dynamic subject which has numerous penetrating applications in geometry and topology.

9.1 §9.1.1

Partial differential operators: algebraic aspects Basic notions

We first need to introduce the concept of partial differential operator (p.d.o. for brevity) on a smooth manifold M. To understand what we are looking for we begin with the simplest of the situations, M = MN. Perhaps the best known partial differential operator is the Laplacian A : C°°{MN) -> C^QR")

Au = - £ . 9 i 2 u i

where as usual 9; = -g^-. This is a scalar operator in the sense it acts on scalar valued functions. Note that our definition of the Laplacian differs from the usual one by a sign. The Laplacian defined as above is sometimes called the geometric Laplacian. Next in line is the exterior derivative d:ttk(RN)^-Slk+l(RN). 335

336

Elliptic equations on manifolds

This is a vectorial operator in the sense it acts on vector valued functions. A degree k form u> on R " can be viewed as a collection of (\) smooth functions or equivalently, as a smooth function UJ : M.N —> M\k'. Thus d can be viewed as an operator

d : C°°{RNM"h

->

C O O (R J V ,K(*+0).

It is convenient to think of C°°(RN, R") as the space of smooth sections of the trivial bundle R" over RN For any smooth K=R, C-vector bundles E, F over a smooth manifold M we denote by Op(E, F) the space of K-linear operators C°°(£) ->

C^iF).

Op(E,E) is an associative K-algebra. The spaces C°°{E), C°°{F) are more than just K-vector spaces. They are modules over the ring of smooth functions C°°(M). The partial differential operators are elements of Op which interact in a special way with the above C°°(M)-module structures. First define PDO°{E,F) = Horn {E,F). Given T € P D O ° , u € C°°(E) and / e C°°{M) we have T(fu)

- f(Tu)

=0

or, in terms of commutators [T,f}u = T{fu)-f(Tu)

= 0.

(9.1.1)

Each / € C°°(M) defines a map ad(f):Op(E,F)^Op(E,F) by ad{f)T = Tof-foT=[T,f]

\/TeOp(E,F).

Above, / denotes the C°°(M)-module multiplication by / . equality (9.1.1) as P D O ° ( £ , F ) = {TeOp

{E,F) ; ad(f)T

We can rephrase the

= 0 V/ e C™{M)} = ; kerad.

Define PDO<m>(£;, F) = kero
This point of view is especially useful in induction proofs. Example 9.1.2 Denote by R the trivial line bundle over RN. The sections of R are precisely the real functions on RN. We want to analyze P D O ( 1 ) = P D O ( 1 ) ( l , R ) . Let L e P D O ^ , u, f € C°°(R Ar ). Then

[L,f]u = where cr(/) G C°°(RN).

o(f)-u

On the other hand, for any / , g G C°°{RN)

o{fg)u = [L, fg}u = [L, f](gu) + f([L, g]u) = a{f)g • u + fa{g) • u. Hence o{jg) = o{f)g + fcr(g). In other words the map / i-> a(f) is a derivation of C°°(RN) and consequently (see Exercise 3.1.2) there exists a smooth vector field X on RN such that a(f)=X-f, V/eC°°(Rw). Let n = L(l) G C°°{RN).

Then for all u G C 0o (R /v ) we have

Lu = L(u • 1) = [L, u] • 1 + u • L(l) = X -u + fx-u.

Lemma 9.1.3 Any L G P D O ( m ) ( £ , F) is a local operator i.e. Vu G C°°{E) supp Lu C supp u.

Proof We argue by induction over m. For m — 0 the result is obvious. Let L G P D O ( m + 1 ) and u G C°°(£). For every / G C°°(M) we have L{fu) = [L,f]u + fLu. Since [L, / ] G P D O ( r a ) we deduce by induction s u p p l ( / w ) C supp u U supp / V / G C°°(M). For any open set O such that 0 3 suppu we can find / = 1 on suppu and / = 0 outside O so that fu = u). This concludes the proof of the lemma. • The above lemma shows that in order to analyze the action of a p.d.o. one can work in local coordinates. Thus understanding the structure of an arbitrary p.d.o. boils down to understanding the action of a p.d.o. in P D O ( m ) ( K p , Kq) where Kp and Kg are trivial K-vector bundles over RN. This is done in the exercises at the end of this subsection.

338

Elliptic equations on manifolds

Proposition 9.1.4 Let E, F, G be smooth K-vector bundles over the same manifold M.lIPe P D O ( m ) ( F , G ) and Q e P D O ( n ) ( £ , F ) then PoQ e P D O ( m + n ) ( £ , G ) . Proof We argue by induction over m + n. For m + n = 0 the result is obvious. In general, if / € C°°(M) [PoQ,f\

= [P,f]oQ

+

Po[Q,f\.

By induction, the operators in the right-hand-side have orders < m + n — 1. The proposition is proved. • Corollary 9.1.5 The operator L = Y,

°a(*)d Q : C°°(RN) -> C°°{RN)

{da = d?1 • • • d%N)

\a\<m

(a = (QI, . . . , a if) 6 Z J ,

\a\ = Y, OH) is a P-d.o. of order < m.

Proof According to the computation in Example 9.1.2 each partial derivative <% is a 1st order p.d.o. According to the above proposition multiple compositions of such operators are again p.d.o.'s. D Lemma 9.1.6 Let E, F —> M be two smooth vector bundles over the smooth manifold M. Then for any P 6 P D O (E, F) and any / , g € C°°(M) ad(f) • (ad(g)P) = ad(g) • (ad(f)P). Proof ad(f) • (ad(g)P) = [[P, g], / ] = [[P, / ] , g] + [P, [/, g] = [[P,f},g] = ad(g)-(ad(f)P).

a

From the above lemma we deduce that if P G P D O ( m ) then for any C {M) the bundle morphism ca

ad(h)ad{f2)---ad{fm)P does not change if we permute the / ' s . Proposition 9.1.7 Let P 6 PBO{m){E,F), that at a point x0 G M dfi{x0) = dgi(x0) £ T*oM

/ i ) f t e C°°{M) (i = l , . . . , m ) such

Vi = 1 , . . . , m.

Then {ad{fi)ad{f2)

• • • ad{fm)P)

| I 0 = {ad{gx)ad{g2) • • • ad{gm)P} \Xo .

Algebraic aspects

339

In the proof we will use the following technical result which we leave to the reader as an exercise. Lemma 9.1.8 For each XQ € M consider the ideals of C°°(M) m I 0 = {/ e C°°(M) • f(x0) = 0} 3 I 0 = {/ e C°°{M) ; f(x0) = 0, df(x0) = 0} Then 3xo =

m

lo

i.e. any function / which vanishes at x0 together with its derivatives can be written as / = X)9jhj 9j, hj £ m I 0 . i Exercise 9.1.1 Prove the above lemma. Proof

Let P 6 P D O ( m ) and ft, g{ € C°°(M) such that dfi(x0) = dgi{x0)

Vi = 1 , . . . , m.

Since ad(const.) = 0 we may assume (eventually altering the / ' s and the g's by additive constants) that fi(x0) = 9i(xQ)

Vi

We will show that {ad(f1)ad(f2)

• • • ad(fm)P}

| x o = {ad(9l)ad{f2)

• • • ad{fm)P}

\xo .

Iterating we get the desired conclusion. Let = fi-gi

and set Q = ad(f2) • • • ad(fm)P

G P D O ( 1 ) . We have to show that

{ad{ e 3 l 0 so according to the above lemma we can write 4> = YlaiPj

a

3,0j€mxo-

3

We have {adf»Q}U,=

l^adiajP^Q

(9.1.2)

340

Elliptic equations on manifolds

= £{[Q, «,-]&} L + £ K [ Q , £•]} \xo= 0. J i This proves the equality (9.1.2) and hence the proposition.

•

The proposition we have just proved has an interesting consequence. Given P G PDO ( m ) (£> F ) , z 0 6 M and £ e C°°(M) (i = 1 , . . . , m) the linear map i^ad{fi)

• • • ad(fm)P}

\xo: Exo - • FXo

depends only on the quantities £ = dfi(x0) £ T*oM. Hence for any f* G T*oM (i = 1 , . . . , m) the above expression unambiguously induces a linear map a(P)(Zu...,U)(P)--EX0^FX0 which moreover is symmetric in the variables &. Using the polarization trick of Chapter 8 we see this map is uniquely determined by the polynomial

a(P)(fl = am(P)(0 =

aiP)^^). m

If we denote by n : T*M -» M the natural projection then for each P G P D O ( m ) ( F , F) we have a well defined map am(P){-)eRom{n*E,TT*F) where n*E and n*F denote the pullbacks via n. Along the fibers of T*M am{P){^) looks like a degree m homogeneous "polynomial" with coefficients in Horn {Exo, Fxo). Proposition 9.1.9 Let P G P D O ( m ) ( £ , F ) and Q G PDO ( n ) (F,G). Then 0-m+n(Q ° P) = O-n(Q) °CT m (P).

Exercise 9.1.2 Prove the above proposition. Definition 9.1.10 A p.d.o. P G P D O ( m ) is said to have order m if am{P) ^ 0. In this case am{P) is called the (principal) symbol of P . The set of p.d.o.'s of order m will be denoted by P D O m . Definition 9.1.11 Let P G P D O m ( £ , F). P is said to be elliptic if for any x G M and any ZeT*M\ {0} om{P){0 : Ex-> Fx is a linear isomorphism.

Algebraic aspects

341

The following exercises provide a complete explicit description of p.d.o.'s on RN. Exercise 9.1.3 Consider the scalar p.d.o. on RN described in Corollary 9.1.5 a

a[x)da-

£= E \a\<m

Show that

om{pm= E M*)r= E o-W 1 •••&"• |a|<m

|a[<m

N

N

Exercise 9.1.4 Let C : C°°{M ) ->• C°°{®L ) be a p.d.o. of order m. Its principal symbol has the form ffm(i)(0= E "«{*)€• \a\<m

Show that a

L~ E

a{xW

|a]=m

is an p.d.o. of order < m — 1. Conclude that the only scalar p.d.o. on RN are those indicated in Corollary 9.1.5. Exercise 9.1.5 Let L € P D O ( m ) ( £ , F ) and u € C°°{E). Show the operator C°°{M) Bf^

[L,f}u € C°°(F)

belongs to P D O ( m ) ( l M , F). Exercise 9.1.6 Let Kp and Kq denote the trivial K-vector bundles over RN of rank p and respectively q. Show that any L e P D O ( m ) ( K p , l « ) has the form

L= E

Mx)d°

\a\<m

where for any a Aa € C°°(K N ,Hom (K P ,K ? )). Hint: Use the previous exercise to reduce the problem to the case p = 1. §9.1.2

Examples

At a first glance, the notions introduced so far may look too difficult to "swallow". To help the reader get a friendlier feeling towards them we included in this subsection a couple of classical examples which hopefully will ease this process. More specifically, we will compute the principal symbols of some p.d.o.'s we have been extensively using in this book. In the sequel • will denote the multiplication by a smooth scalar function.

342

Elliptic equations on manifolds

Example 9.1.12 (The euclidian Laplacian)This is the second order p.d.o. A : C°°(RN) -> C ^ l " )

A = - E

ft2

i

Let / G C ^ R " ) . Then (ft = £ - ) ad(/)(A) = - £ a d ( / ) (ft) 2 = - E W ( / ) ( $ ) ° ^ + ft o ad(/)(ft)}

= - E ( / * ' • $ + $(/»<•)} i

= ~ E ( / x i • #• + /*<*' • +/x- • di) i

= (A/)--2E/«'-ft. Hence ad(/) 2 (A) = ad(/)(A/-) - 2 E /.< • «*(/) (ft) i

2

= -2E(/ I 0 - = -2M/| 2 -. t

If we set £ = df in the above equality we deduce

* 2 (A)(o = -iei 2 •. In particular this shows A is an elliptic operator. Example 9.1.13 (Covariant derivatives)Consider a vector bundle E —¥ M over the smooth manifold M and V a connection on E. We can view V as a p.d.o. V : C°°{E) -J- C°°(T*M ® E). Its symbol can be read from a d ( / ) V , / € C°°(M). For any « G C°°(£) (ad(/)V)w = V(/w) - / ( V u ) =

df®u.

By setting £ = df we deduce ffi(V)(0=£® i.e. the symbol is the tensor multiplication by £. Example 9.1.14 (The exterior derivative)Let M be a smooth manifold. The exterior derivative d : n*{M) -*• Q*(M) is a first order p.d.o. To compute its symbol consider u G fl*(M) and / G C°°(M). Then (ad(f)d)u) = d(fuj) — fdu> = df Aw. If we set f = d/ we deduce <7i(d) = e(£) = the left exterior multiplication by £.

343

Algebraic aspects

Example 9.1.15 Consider an oriented, n-dimensional Riemann manifold (M, ft*(M) where * is the Hodge *-operator *:ft*(M)->-nn-*(M). 6 is a first order p.d.o. and moreover CJI(6) = *a1(d)* = *e(£) * .

This description can be further simplified. Fix £ G T*M and denote by £* € TjM its metric dual. For simplicity we assume |£| = |£*| = 1. Extend C1 = £ to an oriented orthonormal basis ( £ \ . . . ,£ n ) of TX*M and denote by £* the dual basis of TXM. Consider u £ AkT*M a monomial d£7 where / = ( » i , . . . , ijt) denotes as usual an ordered multi-index. Note that if 1 ^ / then 1 Cr1(£ )w

= 0.

(9.1.3)

If 1 € / e . g . 7 = ( 1 , . . . , A ; ) then *c(0(*w) = *(£J A £* +1 A • • • A O = ( - l ) ^ - * ) ^ - 1 ) ^ A • • • A f* = -(-l)"-»i(Dw

(9.1.4)

where i/n]fc = nk + n + 1 is the exponent introduced in Subsection 4.1.5 while i(£*) denotes the interior derivative along £*. Putting together (9.1.3) and (9.1.4) we deduce

(9-1.5)

Example 9.1.16 (The Hodge-DeRham operator) Let (M,g) be as in the above example. The Hodge-DeRham operator is d + d" :Q*{M)

-+ST{M)

k

where d* = (—l)""' 5. Hence d + d* is a first order p.d.o. and moreover

a(d +
(e(o - t(D)2 = -«mo+iiCHt))

= -lei2 •.

Notice that d + d* is elliptic (since the square of its symbol is invertible).

344

Elliptic equations on manifolds

Definition 9.1.17 Let E —)• M be a smooth vector bundle over the Riemann manifold (M,g). A second order p.d.o.

L:Cco(E)^C'x{E) is called a generalized Laplacian if o2(L){£) = — If ljNotice that all generalized Laplacians are elliptic operators. §9.1.3

Formal adjoints

For simplicity, all vector bundles in this subsection will be assumed complex, unless otherwise indicated. Let EX,E2 —» M be vector bundles over a smooth oriented manifold. Fix a Riemann metric g on M and Hermitian metrics (-, •)* on E{, 2 = 1,2. We denote by dvg = *1 the volume form on M defined by the metric g. Finally, C^(Ei) denotes the space of smooth, compactly supported sections of E{. Definition 9.1.18 Let P € P D O (EUE2). The operator Q e P D O {E2, Ex) is said to be a formal adjoint of P if Vu € C^iEJ and Vw e C^{E2) I (Pu, v)2dvg = / (u, Qv)xdvg. JM

JM

Lemma 9.1.19 Any P € P D O (E\,E2) Proof

admits at most one formal adjoint.

Let Qu Q2 be two formal adjoints of P. Then Wv e C$°(E2) I (u, (Qi - Q2)v)idvg = 0 VM e C o 0 0 ^ ) . JM

This implies (Qx - Q2)v = 0 V u £ C^{E2). If now v G C°°(E2) is not necessarily compactly supported then , choosing (a) C C™(M) a partition of unity, we conclude using the locality of Q = Qi — Q2 that Qv = J2 aQ(av) = 0. D The formal adjoint of a p.d.o. P g PT>0(Ei,E2) (whose existence is not yet guaranteed) is denoted by P*. It is worth emphasizing that P* depends on the choices of g and (•, •),. Proposition 9.1.20 (a) Let L0 6 P D O ( £ 0 , £ i ) and Lx G P D O ^ , ^ ) admit formal adjoints L\ € P D O (Ei+i,Ei) (i = 0,1) ( with respect to a metric g on the base and metric (•, •)., on Ej, j = 0,1, 2. Then L^LQ admits a formal adjoint and (LILQ)*

(b) If L e P D O ( m ) ( £ , 0 , £ i ) then L* e

= L*0L\.

VDO{m)(EuE0).

345

Algebraic aspects Proof

(a) For any u; € C^(Ei)

we have

/ (LiL0uo, u2}2dvg = / {L0u0,Llu0)ldvg=

/ (M 0 , L*0Llu2)0dvg.

(b) Let / e C°°{M). Then (ad(f)LY

= ( I o / - / o L ) ' = -[I',/] =

-ad(f)L\

Thus ad(f0)adUi)

• --adUmW

= ( - l ) m + 1 ( a d ( / o M / i ) • • -ad(/ m )L)* = 0. a

The above computation yields the following result. Corollary 9.1.21 If P e P D O ( m ) admits a formal adjoint then om{p*) =

{~\)mom{py

where the * in the right-hand-side denotes the conjugate transpose of a linear map. Let E be a Hermitian vector bundle over the oriented Riemann manifold Definition 9.1.22 A p.d.o. L e P D O (E,E) L = L*.

(M,g).

is said to be formally selfadjoint if

The above notion depends clearly on the various metrics Example 9.1.23 Using the integration by parts formula of Subsection 4.1.5 we deduce that the Hodge-DeRham operator d + d* :Q*(M)->Q*(M) on an oriented Riemann manifold (M, g) is formally selfadjoint with respect with the metrics induced by g in the various intervening bundles. In fact d* is the formal adjoint of d. Proposition 9.1.24 Let (£ i ,(-,-) i ) (i = 1,2) be two arbitrary Hermitian vector bundle over the oriented Riemann manifold (M,g). Then any L 6 P D O ( £ i , £ 2 ) admits at least (and hence exactly) one formal adjoint L*. Sketch of proof We prove this result only in the case when E\ and E2 are trivial vector bundles over K^. However we do not assume the Riemann metric over RN is the euclidian one. The general case can be reduced to this via partitions of unity and we leave the reader check it for him/her-self.

346

Elliptic equations on manifolds

Let Ei = C p and E2 = Cq. By choosing orthonormal moving frames we can assume the metrics on Et are the euclidian ones. According to the exercises at the end of Subsection 9.1.1 any L e P D O ( m ) ( £ i , £ 2 ) has the form

L= £

Aa(x)da

\a\<m

where Aa e Cco(RN, Mqxp(C)). transpose

Clearly, the formal adjoint of Aa is the conjugate

To prove the proposition it suffices to show that each -^

ePBO1

(EuEr)

admits a formal adjoint. It is convenient to consider the slightly more general situation. Lemma 9.1.25 Let X = X{£:

e Vect {RN) and denote by Vx the first order p.d.o.

Vxu = xi4-u u e c n c ) . OXi

Then Vx = -Vx-div9(X) where divg(X)

denotes the divergence of X with respect to the metric g.

Proof of the lemma Let u,v 6 C^(UN,CP)Choose R > 0 such that the euclidian ball BR of radius R centered at the origin contains the supports of both u and v. From the equality X • (u, v) = {Vxu, v) + {u, Vxv) we deduce JRN ( V X W , v)dvg = J = [

(Vxu,

X-(u,v)dvg-f

(u,Vxv)dvg. J

JBR

v)dvg

Set / = {u,v) e C£°(RN,C) and denote by a € Q,1^) respect to the Riemann metric g i.e. (a,P)g=(3{X)

(9.1.6)

BR

V/?€n'(M").

Equivalently a = gijX'dx3'.

the 1-form dual to X with

347

Algebraic aspects The equality (9.1.6) can be rewritten /

{Vxu,v)dvg=

JBR

df(X)dvg-

{u,Vxv)dvg *'J&

JBR

= J

(df, ct)gdvg - J^N (u,

Vxv)dvg.

The integration by parts formula of Subsection 4.1.5 yields /

(df, a)gdvg =

J BR

(df A *ga) \9BR + / JdBft

fd*advg.

JBR

Since / = 0 on a neighborhood of of 8BR we get / JBR

Since d*a = — divg(X) /

JBR

X-(u,v)dvg

(df,a)gdvg=

JBR

(u,v)d*advg.

(see Subsection 4.1.5) we deduce = —

JBR

(u,v)diVg(X)dvg=

/

JBR

(u, —

divg(X)v)dvg.

Putting all the above together we get J

(Vxu,v)dvg

= JmN(v; ( - V * -

dWg(X))v)dvg

i.e. V x = - V x - div 9 (X) = - V x - 4 = E di(^/\g\X').

(9.1.7)

\9\ i

The lemma and consequently the proposition is proved.

•

E x a m p l e 9.1.26 Let E be a rank r smooth vector bundle over the oriented Riemann manifold (M, g), d i m M = m. Let (•, •) denote a Hermitian metric on E and consider a connection V on E compatible with this metric. V defines a first order p.d.o. V : Cco(E) -+ C°°(r*M ® E). The metrics g and (•, •) induce a metric on T*M®E. We want to describe the formal adjoint of V with respect to these choices of metrics. As we have mentioned in the proof of the previous proposition this is an entirely local issue. So we fix Xo G M and denote by (x1,... ,xm) a collection of g-normal coordinates on a neighborhood U of x. Next, we pick a local synchronous frame of E near x0 i.e. a local orthonormal frame (eQ) such that a t XQ VjeQ = 0 Vi = 1 , . . . , m.

348

Elliptic equations on manifolds

The adjoint of the operator dxk® : C°°{E \v) -> C°°{TU

E \v)

is the interior derivative (contraction) along the vector field g-dual to the 1-form dxk i.e. {dxk®Y = Ck d=f g^-iidj). Since V is a metric connection we deduce as in the proof of Lemma 9.1.25 that Vk* =

-Vk-dWg{dk).

Hence

V* = Y.v** ° c" = E ( ~ v * - div,(3t)) o Ck k

k

= -Y(Vk

+ dk(log(^\))oCk.

(9.1.8)

k

In particular, since dkg = 0 at x0 we get

v*u=-£vfc°c*. k

The covariant Laplacian is the second order p.d.o. A = A v : C°°(E) -> C°°{E),

A = V*V.

To justify the attribute Laplacian we will show that A is indeed a generalized Laplacian. Using (9.1.8) we deduce that over U (chosen as above) we have A

= - ( E ( V * + d"logy/\9\) ° Ck\ o hzdxJ = -

fe(V,

®VJ}

+ dk log yfc\) J o (g"i • Vj)

= - E{5 k J 'V, + dkgki + gkjdk log J\g~\} ° V,k,j

kj

\/\g\

The symbol of A can be read easily from the last equality. More precisely

= -\Z\29.

Algebraic aspects

349

In the following exercise we use the notations in the previous example. Exercise 9.1.7 (a) Show that

gkerke =

i cM^*) \J\g\

where T\t denote the Christoffel symbols of the Levi-Civita connection associated to the metric g. (b) Show that Ay = -Tr 9 (V T * M ® B V B ) where S7T~M®E is the connection on T*M ® E obtained by tensoring the Levi-Civita connection on T*M and the connection VE on E while Tr 3 : C^iT'M®2

® E) -> C°°{E)

denotes the double contraction by g TtgiSij ® u) = gijSijU. Let E and M as above. Proposition 9.1.27 Let L € P D 0 2 ( £ ) be a generalized Laplacian. Then there exists a unique metric connection V on E and SH = £H(£) € End (E) such that

The endomorphism 9t is known as the Weitzenbock remainder of the Laplacian L. Exercise 9.1.8 Prove the above proposition. Hint Try V denned by V / g r a d W M = t{(Agh)u

- (ad{h)L)u}

f,he

C°°{M), u € C*>(E).

Exercise 9.1.9 (General Green formula) Consider a compact Riemannian manifold (M, g) with boundary dM. Denote by n the unit outer normal along dM. Let E, F -* M be Hermitian vector bundles over M and suppose L e PDO* (E, F). Set go = g |dM> E0 = E \SM and F0 = F \QM- The Green formula states that there exists a sesquilinear map BL : C'X{E) x C°°{F) ->• C^idM) such that / (Lu,v)dv{g) JM

= [ JdM

BL(u,v)dv{g0)

+ f JM

(u,L*v)dv(g).

350

Elliptic equations on manifolds

Prove the following. (a) If L is a zeroth order operator (i.e. L is a bundle morphism) then Bi = 0. (b) If Lx € P D O (F, G) and L2 6 P D O {E, F) then BLIL2

(u, v) = BLl (L2u,

v) + BLi

(u,

L\v).

(c) BL.(v,u)

=

-BL(u,v).

(d) Suppose V is a Hermitian connection on E and X G Vect(M). Set L = Vx '• C°°(E) -> C°°(E). Then BL{u,v) = (u,v)g(X,n). (e) Let L = V : C°°(E) -+ C°°(T*M ® E). Then BL(u,v)

=

(u,inv)E.

where in denotes the contraction by n. (f) Denote by V the section of T*M \QM ff-dual to n. Suppose L is a first order p.d.o. and set J := OL{P)-

Then

BL{u,v)

=

{Ju,v)F.

(g) Using (a)-(f) show that for all u e C°°(E), v G C°°(F) and any z 0 e dM the quantity BL(u, v)(xo) depends only on the jets of u, v at xo of order at most k — 1.

9.2

Functional framework

The partial differential operators are linear operators in infinite dimensional spaces and this feature requires special care in dealing with them. Linear algebra alone is not sufficient. This is where functional analysis comes in. In this section we introduce a whole range of functional spaces which represent the suitable enviroment for p.d.o.'s to live in. The presentation assumes the reader is familiar with some basic principles of functional analysis. As a reference for these facts we recommend the excellent monograph [15] or the very comprehensive [79]. §9.2.1

Sobolev spaces in M^

Let D denote an open subset of RN. We denote by Ljoc(D) the space of locally integrable real functions on D i.e. Lebesgue measurable functions / : RN —> K such that Va G C?{RN) of G L 1 ^ ) Definition 9.2.1 Let / G L}0C(D). A function g G L}0C(D) is said to be the weak fc-th partial derivative of / if f g
JD

V^ G C?{D).

351

Functional framework Lemma 9.2.2 Any / G L}0C(D) admits at most one weak partial derivative. The proof of this lemma is left to the reader. Warning Not all locally integrable functions admit weak derivatives. Exercise 9.2.1 Let / G C°°{D). also its weak A:-th derivative.

Prove that the classical partial derivative dkf is

Exercise 9.2.2 Let H G L}oc{&) denote the Heaviside function, H(t) = 1 for t > 0, H(i) = 0 for t < 0. Prove that H is not weakly differentiate. Exercise 9.2.3 Let fuf2£ g\ + 52 weakly.

L}0C{D). If dkft = ft G L}oc weakly then dk{h + f2) =

The definition of weak derivative can be generalized to higher order derivatives as follows. Consider L : C°°{D) -> C°°(£>) a scalar p.d.o. and f,g G L\0C{D). Then we say that Lf = g weakly if / gcpdx = / fUipdx JD

V G C0°°(Z3).

JD

Above, L* denotes the formal adjoint of L with respect to the euclidian metric on

mN.

Exercise 9.2.4 Let f,g G C°°{D). Prove that Lf — g classically <=$• Lf = g weakly. Definition 9.2.3 Let k G Z+ and p G [l,oo]. The Sobolev space Lk'p(D) consists of all the functions / G LP(D) such that for any multi-index a satisfying \a\ < k the mixed partial derivative daf exists weakly and moreover daf G ^(D). For every / G Lk*(D) we set i/p

a

Pdx

||/lk P =||/lk P , 0 =(E / \d f\ \\a\
if p < oo while if p = oo ||/lk»=

£esssup|c><7|.

When k = 0 we write ||/|| p instead of ||/||o, p . Definition 9.2.4 Let k € Z+ and p G [1, oo]. Set Lk£(D) = {/ € L L ( ^ ) ; ¥>/ 6 L fc ' p (^) VV 6 C0°°(D)}.

352

Elliptic equations on manifolds

Exercise 9.2.5 Let f(t) = \t\a, t 6 I , a > 0. Show that for every p > 1 such that a > (P - l ) / p Theorem 9.2.5 Let k e Z+ and 1 (RN) is dense in Lk*{RN). (c) If 1 < p < oo the Sobolev space Lk'p(RN) is reflexive. The proof of this theorem relies on a collection of basic techniques frequently used in the study of partial differential equations. This is why we choose to cover the proof of this theorem in some detail. We will consider only the case p < oo leaving the p = oo situation to the reader. Proof Using the exercise 9.2.3 we deduce that Lk>p is a vector space. From the classical Minkowski inequality ( v

/ v

\ 1/P

(El^+^J

/ v

\ l/P

P

p

\ 1/J>

+(X>l ) •

<(£N )

we deduce that || • ||fc.p is a norm. To prove that Lk,p is complete we will use the well established fact that V is complete. Convention To simplify the notations, throughout this chapter all the extracted subsequences will be denoted by the same symbols as the sequences they originate from. Let (/„) C Lk'p(RN) be a Cauchy sequence i.e. lim ||/ m - / „ | k p = 0. m,n—•oo

'r

In particular, for each multi-index |a| < k the sequence (9 a /„) is Cauchy in L ^ R " ) and thus 3 % 3 ft, V\a\
/ M N a Q / n ^ = (-i)H/ MN /„av Since d Q / n —• ga and / n —» / in 77 and ip £ Lq(RN) where l/q = 1 — 1/p) we conclude J^N ga •
= lim(-l)H |

/„ • d°
f • r
353

FunctionaJ framework

Part (a) is proved. To prove that Cp°(]Rw) is dense we will use mollifiers. Their definition uses the operation of convolution. Given / , g e L1 (RN) define (/ * 9){x) = JRN f{x -

y)g{y)dy.

We leave the reader check that / * g is well defined (i.e. y H-> f(x — y)g{y) G L1 for almost all x). Exercise 9.2.6 If / e L1^) / * g e LP(mN) and

and g g 1 / ( 1 " ) then / * g is well defined. Moreover II/*SIIP
(9-2.1)

To define the mollifiers one usually starts with a function p 6 CQ°(K W ) such that p > 0,

suppp C {|:r| < 1}

and / „ pdx = 1. Next, for each 5 > 0 we define =

8-Np{x/6).

JRNPsdx

= l.

Pi{x)

Note that suppp,; C {\x\ < 6} and

The sequence {ps) is called a mollifying sequence. The next result describes the main use of this construction. Lemma 9.2.6 (a) For any / € L}oc(MN) the convolution ps * f is a smooth func£ion/(This explains the term mollifier. ps smoothes out the asperities), (b) If / € V>(RN) (1 / as<5—>0.

Proof of the lemma Part (a) is left to the reader as an exercise in the differentiability of integrals with parameters. To establish part (b) we will use the fact that C^(RN) is dense in LP(RN). Fix s > 0 and choose g € C^(SLN) such that ll/-flllp<e/3.

354

Elliptic equations on manifolds

We have

\\pt * f -

<

/UP

UP*

* (/ -

+ \\P6 * g - slip + lis - /||p.

Using the inequality (9.2.1) we deduce \\P6 * f ~ /||p < 2 | | / - g\\p + Up, * g - 5 || p .

(9.2.2)

We need to estimate \\ps * g — g\\p. Note that Ps*g (x) = JRN p(z)g(x - 8z)dz and

9

^ = ImN

p z x dz

(M) -

Hence \ps*g{x)

-g[x)\

< J^N p(z)\g{x - Sz) - g(x)\dz < 5sup \dg\.

Since s u p p p s * g C {x + z ; x G s u p p g ; z e suppp s } there exists a compact set K C RN such that supp (ps * g - g) C K

V5e(0,l).

We conclude that IIP* * 9 ~ g\\P < (JK 5 p (sup \dg\Ydx^

" = vol (Ky/'Ssup

\dg\.

If now we pick 6 such that vol(K) 1 / p Jsup|dp| < e / 3 we conclude from (9.2.2) that \\P6*f-f\\P<eThe lemma is proved.

D

The next auxiliary result describes another useful feature of the mollification technique especially versatile in as far as the study of partial differential equations is concerned. Lemma 9.2.7 Let / , g € L}0C(RN) such that 9*/ = g weakly. Then dk(ps*f) = Ps*gMore generally, if

Q

L = Y, «^ |a|<m

is a p.d.o with constant coefficients aQ 6 1 and Lf = g weakly then L(P6 * / ) = Ps * g classically.

355

Functional framework

Remark 9.2.8 The above lemma is a commutativity result. It shows that if L is a p.d.o. with constant coefficients then [L,

Ps*]f

= L{ps *f)-pt*

(Lf) = 0.

This fact has a fundamental importance in establishing regularity results for elliptic operators. Proof of the lemma It suffices to prove only the second part. We will write Lx to emphasize that L acts via derivatives with respect to the variables x = (a; 1 ,... ,xN). Note that L{P6 * f) = JRAL*Ps(x - V))f(y)dy. (9.2.3) Since

iript{x-y)

and

=

-wpt{x~y)

dC = -di we deduce from (9.2.3) that

L{ps *f) = JRN{L*yps{x - y))f{y)dy = J^N ps{x - y)g{y)dy = ps * g since Lf — g weakly and y >-» ps{x — y) € C%?(RN) Vs. The lemma is proved.

•

After this rather long detour we return to the proof of Theorem 9.2.5. Let / g Lk*(MN). We will construct /„ G C?(RN) such that /„ -» / in Lkj" using two basic techniques: truncation and mollification. Step 1: Truncation The essentials of this technique are contained in the following result. Lemma 9.2.9 Let / € Lk'p(UN). Consider for each fl>0a smooth function rjR € C^(RN) such that r)(x) = 1 for |x| < R, r]R(x) = 0 for |x| > R + 1 and \dqR[x)\ < 2 Vx. Then T]R • f G Lkj,(RN) VR > 0 and moreover VR

" / -* f as # —> oo.

Proof We consider only the case k = 1. The general situation can be proved by induction. We first prove that di(r]Rf) exists weakly and as expected dtivR • f) = {diT]R) -f + rin- dif. Let ip e C?(RN).

Since T]R

we have

JRN(diVR)
356

Elliptic equations on manifolds

This confirms our claim. Clearly (<%%)/ + r)Rdif e LP(RN) so that nRf € L l p ( R w ) . Note that d{r)R = 0 for | s | < R and \x\ > R + 1. In particular we deduce ($»te)/ -*• 0 a.e. Clearly |(^7?ij)/| < 2\f(x)\ so that by the dominated convergence theorem we conclude (dm) • f % 0 as R -> oo. Similarly ?7fl9i/ ->• 9 i / a.e. and |7?H9,/| < |<9j/| which implies rjudif —> 9 j / in ZA The lemma is proved.

•

According to the above lemma the space of compactly supported Lfc'p-functions is dense in Lk'p(RN). Hence it suffices to show that any such function can be arbitrarily well approximated in the Z^'P-norm by smooth, compactly supported functions. Let / € Lk'p(RN) and assume e s s s u p p / C {|a;| < R}. Step 2: Mollification ps * f L-^ f as 5 -> 0. Note that each p$ * / is a smooth, function supported in {\x\ < R + 5}. According to Lemma 9.2.7 da(Ps *f)=Ps* (d°f) V|a| < k. The desired conclusion now follows using Lemma 9.2.6. To conclude the proof of Theorem 9.2.5 we need to show that Lk'p is reflexive if 1 n

V(RN)

f H- ( d Q / ) H < *

|a|
which is continuous 1-1 and and has closed range. Indeed, if 9 a /n % fa then arguing as in the proof of completeness we deduce that fa = 9af0 weakly where f0 = l i m / n . Hence (/ Q ) = Tf0. We now conclude that Lk'p(MN) is reflexive as a closed subspace of a reflexive space. Theorem 9.2.5 is proved. •

357

Functional frameworJc Remark 9.2.10 (a) For p = 2 the spaces Lk,2(RN) inner product is given by (">v)'< =

are in fact Hilbert spaces. The

fBis['Eaf'u-ff>vdx\dx.

(b) If D C WLN is open then Lk'p(D) is a Banach space, reflexive if 1 < p < oo. However C^{D) is no longer dense in Lk'p{D). The closure of C^{D) in Lk'p{D) is denoted by LQ'P(D). Intuitively, Lkl'p{D) consists of the functions u G Lk'p(D) such that dju — = 0 ondD Vi = 0 , l , . . . , A - l where d/dv denotes the normal derivative along the boundary. The above statement should be taken with a grain of salt since at this point it is not clear how one can define u \gr> when u is defined only almost everywhere. We refer to [Adm] for a way around this issue. The larger space C'X{D) n Lk>p{D) is dense in Lk,p(D) provided the boundary of D is sufficiently regular. We refer again to [3] for details. Exercise 9.2.7 Prove that the following statements are equivalent. (a) u e Ll'"(RN). (b) There exists a constant C > 0 such that for all

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