Conformal Differential Geometry and Its Generalizations

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Conformal Differential Geometry and Its Generalizations

I

MAKS A. AKIVIS VLADISLAV V. GOLDBERG

CONFORMAL DIFFERENTIAL GEOMETRY AND ITS GENERALIZATIONS

PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts

Founded by RICHARD COURANT Editor Emeritus: PETER HILTON Editors: MYRON B. ALLEN Ill, DAVID A. COX, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume.

CONFORMAL DIFFERENTIAL GEOMETRY AND ITS GENERALIZATIONS

MAKS A. AKIVIS Ben-Gurion University of the Negev Beer-Shera, Israel VLADISLAV V. GOLDBERG New Jersey Institute of Technology Newark; New Jersey

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC. Brisbane Chichester New York

Toronto

Singapore

This text is printed on acid-free paper. Copyright © 1996 by John Wiley & Sons, Inc. Published simultaneously in Canada.

All rights reserved. This book is protected by copyright. No part of it. except brief excerpts for review. may be reproduced. stored in a retrieval system. or transmitted in any form or by any means, electronic, mechanical, photocopying. recording. or otherwise, without permission from the publisher. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons. Inc.. 605 Third Avenue, New York, NY 10158-0012. Ubrary of Congress Catafogfag-in-Pubfleadon Data

Akivis, M. A (Maks Aizikovich) Conformal differential geometry and its generalizations / Maks A. Akivis and Vladislav V. Goldberg. p. cm. - (Pure and applied mathematics) "A Wiley-Interscience publication." Includes bibliographical references (p. - ) and indexes. ISBN 0-471-14958-6 (cloth alk. paper) 1. Geometry, Differential. 1. Gol 'dberg, V. V. (Vladislav Viktorovich) II. Title. Ill. Series: Pure and applied mathematics (John Wiley & Sons : Unnumbered) QA641.A587 1996 :

516.3'63-dc20

10987654321

96-31348

Contents Introduction CHAPTER 1 1.1

1.2 1.3 1.4

2.2 2.3 2.4 2.5

3.2 3.3 3.4

HYPERSURFACES IN CONFORMAL SPACES

Fundamental objects and tensors of a hypersurface Invariant normalization of hypersurfaces The rigidity theorem and the fundamental theorem Curvature lines of a hypersurface Geometric problems connected with the tensor c;j Notes

CHAPTER 3 3.1

CONFORMAL AND PSEUDOCONFORMAL SPACES

Conformal transformations and conformal spaces Moving frames in a conformal space Pseudoconformal spaces Examples of pseudoconformal spaces Notes

CHAPTER 2 2.1

ix

SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Geometry of a submanifold in a conformal space Submanifolds carrying a net of curvature lines Submanifolds in a pseudoconformal space Line submanifolds of a three-dimensional projective space Notes

v

1

8 14 19

28

31 31

40 45 52 61 70

73 73

89 100 108 115

A

CONTENTS

CHAPTER 4 4.1 4.2

4.3

CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

A manifold with a conformal structure Weyl connections and Riemannian metrics compatible with a conformal structure A conformal structure on submanifolds of a conformal space

4.4

5.1

5.3 5.4 5.5 5.6

5.7

6.2 6.3 6.4 6.5 6.6

132 141

space

150

Notes

160

THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

163

Structure equations of the CO(2, 2)-structure The CO(1, 3)-structure and the CO(4, 0)-structure The Hodge operator Completely isotropic submanifolds of four-dimensional conformal structures Four-dimensional webs and CO(2, 2)-structures Conformal structures of some metrics in general relativity Conformal structures on a four-dimensional hypersurface

163 169 176

Notes

217

CHAPTER 6 6.1

119

A conformal structure on a hypersurface of a projective

CHAPTER 5

5.2

119

GEOMETRY OF THE GRASSMANN MANIFOLD

183 193

202 208

221

Analytic geometry of the Grassmannian and the Grassmann mapping

222

Geometry of the Grassmannian G(1, 4) Differential geometry of the Grassmannian Submanifolds of the Grassmannian G(m, n) Normalization of the Grassmann manifold Homogeneous normalization of the Grassmann manifold

232

Notes

265

236 244 252 260

vii

CONTENTS

CHAPTER 7

MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES 267 267

7.3

Almost Grassmann structures on a differentiable manifold Structure equations and torsion tensor of an almost Grassmann manifold The complete structure object of an almost Grassmann manifold Manifolds endowed with semiintegrable almost Grassmann

281

7.4

structures Multidimensional (p + 1)-webs and almost Grassmann

292

7.5

7.1 7.2

274

structures associated with them

301

7.6

Grassmann (p + 1)-webs

305

7.7 7.8

Transversally geodesic and isoclinic (p + 1)-webs Grassmannizable d-webs Notes

309 314 319

Bibliography

323

Symbols Frequently Used

355

Author Index

359

Subject Index

363

Introduction This book presents the foundations and applications of local conformal differential geometry and the theory of conformal, Grassmann, and almost Grassmann structures.

Conformal differential geometry was developed within the framework of classical differential geometry at the end of the last and the beginning of this century. It included concepts such as surfaces with isothermic or spherical curvature lines, canal surfaces, congruence of circles, triply orthogonal systems of surfaces, and conformal differential invariants and conformally invariant differential quadratic forms of a surface (G. Darboux, G. Fubini, A. Ribaucour, A. Voss, and others). L. Berwald's paper of 1927 contains a survey of works in this area. However, in the 1920s affine and projective differential geometries became independent branches of differential geometry (E. tech, G. Fubini, E. J. Wilczynski, and others), while conformal differential geometry lagged behind in its development. This phenomenon can be explained by the fact that in works on affine and projective differential geometry, the coordinate systems natural for these geometries, namely the affine and projective systems, had been used, whereas in the works on the conformal differential geometry the investigations were conducted in the rectangular Cartesian coordinate system. Only G. Thomsen in 1923-1925 and E. Vessiot in 1926-1927 started to use the pentaspherical coordinates (introduced long before by G. Darboux) and tensor analysis in their studies in conformal differential geometry. A part of W. Blaschke's book of 1929 and T. Takasu's papers of 1928-1938, and his book of 1938 were devoted to the differential geometry of the conformal space C", the Laguerre space, and the space whose fundamental group is the group of spherical transformations of S. Lie. In all these works the differential geometry of submanifolds of spheres was considered. The 1918 paper of H. Weyl was very important for the development of conformal differential geometry. In this paper H. Weyl studied conformal invariants of Riemannian metrics and their relation to general relativity, which was intensively developing at that time. Following Weyl's ideas, in the 1920s and 1930s E. Cartan, V. Hlavaty, S. Sasaki, J. A. Schouten, I. M. Thomas, T. Y. Thomas, K. Yano, and others intensively developed the theory of multidimensional conformally connected spaces. However, in most of these works ix

INTRODUCTION

X

the conformal differential geometry of submanifolds was constructed by means

of Riemannian geometry. The authors of these works did not go beyond obtaining the Frenet equations and finding their integrability conditions. The complete bibliography of these works can be found in the book Ricci Calculus by J. A. Schouten (1924). Along with proper conformal geometry, pseudoconformal geometry is also of great importance. One of the reasons is that pseudo-Riemannian metrics are used in general relativity, and these metrics lead to the study of pseudoconformal structures. After World War II the geometry of submanifolds of the conformal space C" was intensively developed. As apparatus, tensor methods and the method of exterior forms and moving frames were applied (M. A. Akivis, A. P. Norden, V. I. Vedernikov, L. L. Verbitsky, and others). In addition the conformal theory of manifolds of spheres of different dimensions was investigated (R. M. Geidelman, B. A. Rosenfeld, V. I. Vedernikov, and others). Although multidimensional conformal differential geometry is important for other parts of differential geometry and in other branches of mathematics, and

there are numerous papers on the subject, there is as yet no book in which multidimensional conformal differential geometry has been presented systematically.

The last book devoted to the theory of conformal structures was the book by S. Sasaki published in 1948 in which conformal connections on submanifolds were the subject of study. However, there is the need for a book on conformal

and almost Grassmann structures, since these structures find applications in a number of branches of mathematics and physics. The present book will fill the indicated gap in the literature on differential geometry. There exists a connection between conformal geometry and the geometry of Grassmann and almost Grassmann structures. It was F. Klein who noted that geometry of the manifold of straight lines of a three-dimensional space is equivalent to the geometry of a four-dimensional pseudoconformal space. Grassmann

and almost Grassmann structures on a manifold are close to conformal structures, since both kinds of structures are determined on a manifold by a field of cones. The difference is that for conformal structures these cones are cones of second order, while for Grassmann and almost Grassmann structures they are more complicated algebraic cones called Segre cones. This is the reason for studying the conformal, Grassmann, and almost Grassmann structures in the framework of a unified theory. We tried to combine all these and a series of other topics in this book. Before studying conformal and almost Grassmann structures on a differentiable manifold, we consider differential geometry of conformal and pseudoconformal spaces, and Grassmann manifolds and their submanifolds. This allows us to present a clear geometric treatment of theory of conformal, pseudoconformal, and almost Grassmann structures and to construct their realizations on submanifolds of conformal and projective spaces. Using multidimensional webs, we obtain other realizations of these structures.

INTRODUCTION

xi

In this book we conduct all our considerations over the field R of real numbers. As a result, in studying conformal spaces and conformal structures, we distinguish proper conformal spaces and structures, and pseudoconformal spaces and structures of different signatures. We emphasize the general properties of these spaces and structures and the differences existing between them. We give special attention to four-dimensional pseudoconformal structures, since this kind of structure plays an important role in general relativity. Spacetime in general relativity is a four-dimensional Riemannian manifold of signature (1, 3). Since many features of general relativity are of a conformal invariant nature, it is interesting to study pseudoconformal structures of signature (1, 3). Along with these kinds of conformal structures, on a real four-dimensional conformal structure, one also can consider conformal structures of signatures (4, 0) and (2, 2). Unlike the previous investigations, we consider conformal structures on a real manifold M. Moreover we apply complexification not of the manifold M itself but only of its tangent spaces T=(M), and consider in these spaces coordinate transformations preserving the real part of these tangent spaces and the symmetry with respect to it. The study of Grassmann and almost Grassmann structures is important, since these structures find a wide variety of applications in the theory of hypergeometric functions, integral geometry, representation theory, field theory, theory of multidimensional webs, etc. In our theoretical considerations we include many examples and realizations. Many results presented here appeared earlier in journal articles. However, in our book these results are considered from a unified point of view and by a unified method, which is often different from the original presentation. The book also contains some results that have not been unpublished.

The Contents of the Book. The book consists of seven chapters. In Chapter 1 we give those facts from the theory of conformal and pseudoconformal spaces C." of any signature q that are necessary for further exposition. In particular, we introduce polyspherical coordinates and consider the method of moving frames, some examples of pseudoconformal spaces, and the Grassmannian of straight lines of the three-dimensional projective space. The geometry

of this Grassmannian is equivalent to the geometry of the four-dimensional pseudoconformal space CZ of signature two. As another example, we study the geometry of Lie hyperspheres in an n-dimensional Euclidean space that leads to an (n + 1)-dimensional pseudoconformal space of signature one. In Chapters 2 and 3 we develop the theory of submanifolds of conformal spaces, first for a hypersurface and further for a submanifold of any codimension. For these submanifolds we construct an invariant normalization and find its geometric characterization, prove the rigidity theorem and the fundamental theorem on determination of a submanifold by a system of tensors, and study some special types of submanifolds in conformal and pseudoconformal spaces. In particular, we study canal hypersurfaces of different kinds and hypersurfaces in a conformal space of Lorentzian signature. Moreover we consider the line

xii

INTRODUCTION

submanifolds of a three-dimensional projective space as submanifolds of the pseudoconformal space C. We do not consider the theory of curves, since it has been thoroughly studied in many books and papers. We also do not consider the geometry of manifolds of spheres and circles of conformal space, since this theory has also been studied in detail. In Chapter 4 we investigate conformal and pseudoconformal structures on a manifold of arbitrary dimension, derive the structure equations, and introduce and study the tensor of conformal curvature. In addition to the general theory of conformal structures, we consider conformal structures induced on submanifolds of a conformal space and on hypersurfaces of a projective space. Here we find rather wide classes of submanifolds carrying conformally flat conformal structures and connect the study of these classes of hypersurfaces with the theory of canal hypersurfaces considered in Chapter 2. In Chapter 5 we consider the real theory of four-dimensional conformal structures of all possible signatures, study their isotropic fiber bundles and completely isotropic submanifolds on these structures, compute their Hodge tensor and tensor of conformal curvature, and connect the splitting of the tensor of conformal curvature with the geometry of these bundles. We also consider here different realizations of conformal and pseudoconformal structures, and study conformal structures of some metrics in general relativity. For studying of four-dimensional conformal structures, we use their connection with fourdimensional webs.

In Chapters 6 and 7 we consider Grassmann and almost Grassmann structures. First of all, we study the analytic and differential geometry of the Grassmannian, some interesting classes of submanifolds of the Grassmannian G(m, n), its normalizations (in particular, the harmonic and homogeneous nor-

malization), and stereographic projection onto an (m + 1)(n - m)-dimensional flat space. Next, we define almost Grassmann structures as a generalization of the Grassmann structure, and for them we find the structure equations and the structure tensors whose vanishing gives a locally flat almost Grassmann structure. As was the case for a four-dimensional conformal structure, the

structure tensors split into pairs of subtensors, and the vanishing of any of these subtensors leads to a manifold endowed with a semiintegrable almost Grassmann structure. Further we consider the connection between the theory of almost Grassmann structures and the theory of webs, and use this connection for finding examples of semiintegrable and locally flat almost Grassmann structures. Sections in the book are numbered within each chapter, and formulas and figures are numbered within each section. Each chapter is accompanied by a set of notes containing remarks of historical and bibliographical nature and some supplementary results pertinent to the main content of the book. A fairly complete bibliography, a list of notations, and an index are given at the end of the book. Bibliographic references give the author's last name followed by the first

two letters of the author's last name and the last two digits of the year in

INTRODUCTION

xiii

square brackets, for example, Kobayashi [Ko 72]. Note that in the bibliography, in addition to the original article being cited, reviews of the article in major mathematical reviews journals (Jahrbuch fur Fortschritte der Mathematik, Zentralblatt fur Mathematik, Mathematical Reviews) are referenced.

General Remarks for the Reader. The book is intended for graduate students whose field is differential geometry, as well as for mathematicians and

teachers conducting research in this subject. This book can also be used in special graduate courses in mathematics. In our presentation we use the tensorial methods in combination with the methods of exterior differential forms and moving frames of the Cartan. The reader is assumed to be familiar with these methods, as well as with the basics of modern differential geometry. Many concepts of differential geometry are explained briefly in the text, and some are given without any explanation. As references, the books Kobayashi and Nomizu [KN 63], Michor and Slovak [KMS 91], Sternberg [St 64], and Bryant et al. [BCGGG 91] are recommended. We also recommend our book Akivis and Goldberg [AG 93], in the first chapter

of which the methods used in that book and in the current book are briefly explained. We will often refer to our book Akivis and Goldberg [AG 93], especially when we consider projective realizations of conformal and almost Grassmann structures. All functions, vector and tensor fields, and differential forms are assumed to be differentiable sufficiently many times. As a rule we use the index notations in our presentation. We believe this allows us to obtain a deeper understanding of the essence of problems in local differential geometry.

Note also that if we impose a restriction on a submanifold, then, as a rule, we assume that this condition holds at all points of this submanifold. More precisely, we consider only that domain of the submanifold where this restriction holds.

Acknowledgements. The completion of this book would not have been possible without the support provided to the authors by the Mathematisches Forschungsinstitut Oberwolfach (MFO), Germany. A large portion of the book was written at MFO during the summer of 1994 and the fall of 1995. We express our deep gratitude to Professor Dr. M. Kreck, the director of MFO, for the opportunity to use excellent facilities at MFO. In the fall of 1995 our work at MFO was partially supported by the Volkswagen-Stiftung (RiP-program at MFO). We are also grateful to the Mathematics Departments of Ben-Gurion University of the Negev, Israel, and of New Jersey Institute of Technology, Newark,

New Jersey, for the assistance provided during our writing of the book. The work of the first author was also partially supported by the Israel Ministry of Absorption and the Israel Public Council for Soviet Jewry. We express our sincere gratitude to B. A. Rosenfeld for numerous discussions, J. Vilms for reading most of the chapters and making many useful suggestions, M. Lomonosov for his valuable remarks, Z. Waksman for providing organizational support, and to L. V. Goldstein for her invaluable assistance in

xiv

INTRODUCTION

preparing the manuscript for publication. We are also very grateful to the people at John Wiley & Sons, Inc. for their patience and kind cooperation.

Psagot, Israel Livingston, New Jersey

Maks A. Akivis Vladislav V. Goldberg

Chapter 1

Conformal and Pseudoconformal Spaces 1.1

Conformal Transformations and Conformal Spaces

1. We will define conformal space by means of Euclidean space. Different definitions of Euclidean space can be found in many books on geometry (e.g., see Rosenfeld [Ro 96), §0.5.3 and 3.1.1; Dieudonn6 [D 64), §5.1 and App. II, no. 9). The basic elements of a Euclidean space are points and subspaces of different dimensions. If there are n+1 points in this space that do not belong to a proper subspace, and any k, where k < n, points belong to its proper subspace, then the Euclidean space is n-dimensional and is denoted by R". Proper subspaces

of R" can be of dimension m, where m = 1,.. . , n - 1. Such subspaces are denoted by R. One-dimensional subspaces of the space R" are called straight lines, and (n-1)-dimensional subspaces of the space R' are called hyperplanes. An ordered pair of points x and y of the space R" determines a vector x-b for which x is the initial point, and y is the terminal point. The equality of vectors with different initial points is defined in a regular manner. All vectors with the same initial point x form a vector space, which is called the tangent space to R" at x and is denoted by Tz(R") or T.. In this tangent space the scalar product is defined also in a regular manner, and by means of this product the length of a vector, the angle between two vectors, and the distance between two points

can be determined. The scalar product of the vectors e = it and 77 = Ft is denoted by (l:, q). The scalar product is a nondegenerate symmetric bilinear form the corresponding quadratic form (1;, ) of which is positive definite. A motion of the Euclidean space is a transformation preserving the distance between pairs of points. It is obvious that motions also preserve scalar products of vectors and angles between vectors. 1

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

2

It is easy to see that motions of the space R' form a group G. Let us consider the subgroup H. of the group G which consists of transformations leaving a point x fixed:

Hz= {gEG:gx=x}. This subgroup is called the isotropy group of the point x. The subgroup H. transforms the space T= into itself and is isomorphic to the n-dimensional orthogonal group 0(n) : Hs °_' 0(n). Another important subgroup of the group G is the subgroup T(n) of parallel translations which leave the vectors of the space R" invariant. The isotropy groups H., and Hs, of points x and y are connected by the relation:

Hy = t(H:tc',

j.

where t( is the parallel translation determined by the vector An orthogonal transformation V E 0(n) and a parallel translation t( are connected by the following relation: 'ot('p-1

= tW(O'

where gyp({) is the vector which is obtained from the vector l; by an orthogonal transformation W.

The group G acts transitively on R" and is isomorphic to the semidirect product of the subgroups T(n) and 0(n): G °_I 0(n) x T(n), and the group T(n) is an invariant subgroup of the group G. Thus the Euclidean space R" can be defined as a pair (S, G) consisting of the set S of points of the space R" and the group G of motions of this space. In the general case, a homogeneous space is defined in the same manner (e.g., see Alekseevskii, Vinberg, Solodovnikov [AVS 88], Ch. 1, §1). 2. A conformal transformation of a domain D C R" is a mapping cp: D -* D, where D C R", which preserves angles between curves. As follows from Subsection 1.1.1, motions of the Euclidean space R" preserve angles between curves, and hence they are conformal transformations. However, there exist other conformal transformations that are not motions.

If n = 2, we can introduce complex coordinates on the Euclidean plane R2, and then any analytic function w = f (z), z E D, determines a conformal transformation f : D -+ D, provided that f'(z) iA 0 for points z E D. The situation is different for n > 2. In this case the following theorem holds (see Liouville [Lio 50]):

Theorem 1.1.1 (Liouville) If n > 3, then a conformal transformation is the composition of a motion, a homothety and an inversion.

We recall that an inversion in a hypersphere S with center at a point a and radius r is a transformation which sends a point x E R" into a point y E R"

1.1

Conformal Transformations and Conformal Spaces

3

such that y belongs to the straight line ax and Iaxl Iayl = r2 where Iaxl and l ayl are the distances between corresponding points. Moreover there is no point corresponding to the center a of inversion. To include the center in the domain of the mapping determined by the inversion, we enlarge the Euclidean space R" by the point at infinity, oo, and let it correspond to the center of inversion. Then hyperplanes of the space R" correspond to hyperspheres passing through the point at infinity. Thus it is natural to call hyperplanes improper hyperspheres and consider the usual hyperspheres as proper.

It is clear now that motions and homotheties map proper hyperspheres into proper ones and improper hyperspheres into improper ones. As to inversions, they map improper hyperspheres into proper ones passing through the center of inversion, and vice versa. Taking this into account, we can formulate the Liouville theorem in the following manner: if n > 3, then conformal transformations map hyperspheres into hyperspheres. Here a hypersphere is understood as proper or improper. We consider now a Euclidean space R" enlarged by the point at infinity oo. We denote this new space by C": C" = R" U too). After the space R" has been enlarged by the point at infinity, this space becomes a compact differentiable manifold which is homeomorphic to an n-dimen-

sional sphere S". This is the reason that the operation of adding of the point at infinity to the space R" is called a compactification of the Euclidean space. In the space C" we further consider all conformal transformations that map C" onto itself. Such transformations form a group G which is called the group

of conformal transformations. The pair (C", G) is said to be the conformal space. It is also called the Mobius space. For simplicity we will denote the conformal space by the symbol C". Note that if n = 2 (i.e., in the plane), we can consider a class of transformations defined by linear-fractional functions f (z) _ cz+d of a complex variable z,

where det (: d) Q. Transformations of this class transfer circles into circles. They form a group, and the geometry defined by this group in the plane is called the Mobius geometry. The plane itself is the conformal space C2.

Note also that in addition to the compactification of Euclidean space considered above, which is the enlargement of R" by the point at infinity and which leads to the conformal space C", there is another compactification of R", which is the enlargement of R" by the hyperplane at infinity and which leads to the projective space P". However, in this book we will be interested mostly in the first compactification, although the notion of a projective space will also be used.

4. To study the conformal space in more detail, we introduce polyspherical coordinates. Consider a rectangular Cartesian coordinate system (x',. .. , x") in the space R". In these coordinates the equation of a hypersphere S has the

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

4

form

n

n

8° > (x$)2 + 2 E s'x' + 2an+1 = 0.

(1.1.1)

i=1

i=1

We will call the set of numbers so, Si and an+1 the polyspherical (or (n + 2)spherical) coordinates of the hypersphere S C Rn. If n = 3, the numbers so, s', s2, s3, and s4 are called the pentaspherical coordinates of the sphere S in the space R3. It is obvious that these coordinates are defined up to an arbitrary factor.

If s° # 0, then the hypersphere S is proper, and if so = 0, then the hypersphere S is improper. Hyperplanes of the space Rn can be considered as particular cases of hyperspheres corresponding to the value so = 0. We also identify the points of the space Rn with hyperspheres of radius 0. The standard equation of a hypersphere S of radius r whose center a has coordinates a', i = 1,. .. , n, is n

E(x'

- a')2 = r2.

i=1

By completing squares in equation (1.1.1), we find that the coordinates a' and the radius r of the hypersphere S are given by 1

s'

a' _ -so; r2 =

(80)2

n

(a')2 - 2s0s' 1).

(1.1.2)

Thus, for polyspherical coordinates of a point X E R" (a hypersphere of radius 0), we have the condition n

(X, X) :_ 1:(ai)2 - 2309n+1 = 0. i=1

For a real hypersphere S, the left-hand side of equation (1.1.3) is positive. If this expression is negative, then S is an imaginary hypersphere (or a hypersphere of imaginary radius). Suppose that we have two hyperspheres P and Q (with centers at points a ,pn+1 and b) whose polyspherical coordinates are the sets of numbers, po,p', q1 , qn+1, respectively. We define the angle between two intersecting and q0,

hyperspheres P and Q as the angle between two circles (the intersections of P and Q with any two-dimensional plane passing through the centers of P and Q). If one of these hyperspheres, say Q, is a hyperplane (i.e., if q° = 0), then the angle between P and Q is defined as the angle between the circle and the straight line (the intersections of P and Q with any two-dimensional plane passing through the center of P and perpendicular to Q). If both hyperspheres P and Q are hyperplanes (i.e., if p° = q° = 0), the angle between P and Q is the angle between these hyperplanes.

Conformal Transformations and Conformal Spaces

1.1

5

Figure 1.1.1

It is easy to prove that this angle does not depend on the choice of the 2-plane by means of which it was defined.

In the first case this angle is that angle of the triangle with vertices at the centers a and b of P and Q and one of the common points of P and Q, which is the angle opposite to the side ab (or the adjacent angle if the angle described above is obtuse) (see Figure 1.1.1). By the law of cosines, the first of these angles is connected with centers at a and b and the radii r1 and r2 of P and Q by the following formula: Ia - b12 = ri + r2 - 2r1r2 cos

where Ia - bI is the distance between the points a and b in the Euclidean space Rn.

Since the angle between P and Q is cp or 7r-gyp and since cos(7r-W)

it follows that cos2p =

cos gyp,

(r?+,_Ia_bI2) 2 2r1 r2

(1.1.4)

If we calculate Ja - bl,r1 and r2 by means of (1.1.2) and substitute their values into equation (1.1.4), we can easily find that

_ cos2 P where

(P, Q)2

(P,P)(Q,Q)'

n

(P

Q) = > pigi _ pOgn+1 - pn+l qo i=1

is a bilinear form which is called polar to the quadratic form (1.1.3). The form (P, Q) is called the scalar product of the hyperspheres P and Q. This expression

6

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

makes sense even if one or both hyperspheres P and Q are points. All points X lying on the hypersphere P satisfy the equation (P, X) = 0. If one or both of the hyperspheres P and Q is improper, then it is easy to show that formula (1.1.5) is valid in this case as well. If the hyperspheres P and Q are both real and have no common points, then the right-hand side of equation (1.1.5) is greater than 1, and the angle W is pure imaginary. The linear transformations of polyspherical coordinates preserving equation (1.1.3) map points of the space C" into points and hyperspheres into hyperspheres. Moreover these transformations preserve the expression for cos2 gyp, that is, they preserve angles between hyperspheres. This is the reason that such transformations are called conformal transformations of the space C". It is easy to see that these transformations form a group-the group G of conformal transformations of the space C". 5. The polyspherical coordinates of a hypersphere were defined up to an arbitrary real factor. Thus we can consider them as homogeneous coordinates of a point of a projective space Pn+1 of dimension n + I (regarding projective space, see Akivis and Goldberg [AG 93], §1.3). Points X of the space C" are hyperspheres of radius 0, and their coordinates satisfy equation (1.1.3) which defines a hypersurface Q" of second order (a hyperquadric) in the space P"+1 Thus to each point X E C" there corresponds a point of the hyperquadric Q" in the space P"+1. This one-to-one point correspondence is called the Darboux mapping, and Q" is called the Darboux hyperquadric. Since equation (1.1.3) of the hyperquadric Q" can be reduced to a canonical form containing n + 1 positive squares and one negative square, this equation determines an oval hyperquadric not carrying real rectilinear generators. This hyperquadric divides the space Pnt1 into two parts, exterior and interior. The tangent hyperplane to the oval hyperquadric Q" at its arbitrary point x does not have real points common with Q" except the point x. However, from the complex point of view, the tangent hyperplane Tz(Q") intersects Q" along an imaginary cone C. of second order with real vertex x. This cone is called the isotropic cone of the hyperquadric Q" at the point x. To conformal transformations of the space C", there correspond projective transformations of the space P"+' that map the hyperquadric Q" into itself. Linear transformations preserving equation (1.1.3) are determined up to a factor c. Since the matrix A of such a transformation is of order n + 2, its determinant is defined up to the factor c"+2. For n odd, this determinant can be always reduced to the value 1 by taking an appropriate value of c. Thus, for n odd, the group G of conformal transformations is isomorphic to the group SO(n + 2, 1) of pseudoorthogonal transformations with determinant equal to 1:

G

SO(n + 2,1).

For n even, the sign of the determinant will not be changed if we multiply the matrix A by a negative number c. Since the matrices A and -A define the

1.1

Conformal Transformations and Conformal Spaces

7

same conformal transformation, we have

G 2t O(n + 2, 1)/Z2, where O(n + 2, 1) is the pseudoorthogonal group of signature (n + 2, 1) and Z2 is the cyclic group of second order. There is a common notation for the groups SO(n+2, 1) and O(n+2, 1)/Z2:

PO(n + 2, 1)

SO(n + 2, 1)

l O(n + 2, 1)/Z2

if n is odd, if n is even

(see Rosenfeld [Ro 96], §0.8.8). Thus the fundamental group' of the conformal

space C" is the group PO(n + 2,1). The interior part of the hyperquadric Q" provides the Klein interpretation of an (n + 1)-dimensional hyperbolic (Lobachevsky) space H"+'. The group PO(n + 2, 1) is isomorphic to the group of motions of the space H"+' Since a hypersphere P of the space C" has the equation (P, X) = 0, in the space P"+', there corresponds to P a hyperplane that is determined by the same linear equation as P. The pole of this hyperplane with respect to the hyperquadric Q" lies outside of Q". Conversely, to each point of the space P"+' lying outside of the hyperquadric Q", there corresponds a real hypersphere in

the space C". To the points X of the space P"+' lying inside of the hyperquadric Q", there corresponds a hypersphere of imaginary radius in the space C", since for them (X, X) < 0. 6. In the same manner as one constructs the stereographic projection of a sphere onto a plane, one can construct the stereographic projection of a con-

formal space C" onto an Euclidean space R". To this end, we consider a realization of the conformal space C" on a hyperquadric Q" of the projective space Pn+', which we have constructed above. We fix a point z E Q" (see Figure 1.1.2) and call it the pole. Next we project the hyperquadric Q" from the pole z onto the hyperplane E" C P"+', not passing through z in such a way that to the point z E Q" there corresponds the point y of intersection of the straight line zx with the hyperplane E" : y = zz fl E". To the pole z, there corresponds more than one point on E", namely, the whole (n - 1)-plane E"-' of intersection of E" with the tangent hyperplane Tz(Q") to Q" at the point z: Ts (Q") fl E" = E"-'. We will call this hyperplane the plane at infinity of the hyperplane E". The isotropic cone C. of the hyperquadric Q" intersects the plane E"-' along the imaginary quadric U"-2 of dimension n - 2. This quadric U"-2 can be taken as the absolute on E". As a result the structure of the Euclidean space R" is induced on E". For n = 3, such an interpretation of a Euclidean space goes back to F. Klein (see Klein [KI 28], Chs. 4 and 7). I We use the term "fundamental group" for the group of transformations of a homogeneous

space. Since we will not use this term for the Poincare group iri(X) (the first homotopy group), there will be no ambiguity.

8

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

Figure 1.1.2

In the same manner as for n = 2, one can prove that the mapping of the hyperquadric Q' minus the point z, Q" \ {z}, onto the Euclidean space R" is conformal; that is, this mapping preserves angles between curves and maps hyperspheres into hyperspheres.

1.2

Moving Frames in a Conformal Space

1. Consider an n-dimensional conformal space C" referred to polyspherical

coordinates. A system of reference or a conformal moving frame in the space C" is a system consisting of n + 2 linearly independent hyperspheres. In particular, some of elements of a conformal moving frame can be points.

In this section and later in this chapter, we will use the following index ranges:

It will be convenient for us to take as a moving frame a system consisting of two points Ao and An+1 and n linearly independent hyperspheres Ar, passing through these points (for n = 2, see Figure 1.2.1). The frame elements of such frames satisfy the following analytical conditions:

(Ao,Ao) = (An+I,An+1) = (Ao,Ar) = (An+1,Ar) = 0, where, as in Section 1.1, ( , ) denotes the scalar product.

(1.2.1)

1.2

Moving Frames in a Conformal Space

9

A2

Figure 1.2.1

In addition we normalize the points AO and

by the condition

(Ao,An+1) = -1,

(1.2.2)

(A,., A.) = 9rs-

(1.2.3)

and set

Since the frame elements are linearly independent, we have det(gr,) 34 0. We denote the family of frames in question by R(C"). Any hypersphere P C Cn can be represented as a linear combination of the elements Aoi A,, and An+1 of the conformal moving frame. If P = x°Ao + xrAr + xn+1 An+1, Q = y°Ao + yrAr + yn+1 An+1

are the decompositions of hyperspheres P and Q with respect to the moving frame, then their scalar product (P, Q) has the following expression:

(P Q) = gr.xrys - x0yn+1 - xn+IYO

(1.2.4)

If X = x°Ao + xrAr + xn+'An+1 is a point of the space Cn, then its coordinates satisfy the equation (X, X) := 9raxrx' - 2x°xn+1 = 0.

(1.2.5)

Since this quadratic form is different from the quadratic form (1.1.3) only in notation, its first term, the quadratic form gr,x'x', is positive definite, and the entire quadratic form (X, X) is of signature (n + 1, 1). As in the preceding formulas, in all formulas we will adhere the well-known Einstein summation convention.

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

10

Figure 1.2.2

We can consider equation (1.2.5) as the equation of a hyperquadric Q" in the projective space P1+1, which is referred to a moving frame consisting of n + 2 linearly independent points A0, Ar and An+1, where the hyperquadric Q" is the image of the space C" under the Darboux mapping. It follows from equations (1.2.1) that the vertices of the moving frame are located with respect to the hyperquadric Q" in the following manner: the

points A0 and Ant1 lie on Q", and the points Ar constitute a basis of the (n - 1)-dimensional plane of intersection of two n-dimensional planes that are tangent to Q" at the points Ao and An+1 (for n = 2, see Figure 1.2.2). 2. In the space C" we consider the set of all frames of the type indicated above. The equations of infinitesimal displacement of these frames can be written in the form: dA{ = wE A,1,

(1.2.6)

where the 1-forms wE are differential forms of the parameters of the group PO(n + 2, 1). Here and in what follows, we consider only the connected component of the identity of the Lie group in question. The forms wf can be considered also as coordinates of a vector in the Lie algebra of the group PO(n+2,1). The number of linearly independent forms among the forms w, is equal to the number of independent parameters of this group. If we differentiate equations (1.2.1)-(1.2.3), we obtain all linear conditions that the forms wE must satisfy: n+l WO

n+l _ 0 0 - n+l - 0+ w0 + Wn+1 ' 0,

j wr+l - 9rsW = 0, l` d9rs = 9r1Ws + 9lsWr

Wr0 - 9rsWn+1 = 0,

(1.2.7)

(1.2.8)

1.2

Moving Frames in a Conformal Space

11

If all forms w{ in equations (1.2.6) are linearly independent, then these equations are equations of infinitesimal displacement of a moving frame in the projective space P"+1 Equations (1.2.7) and (1.2.8) single out the subgroup leaving the hyperquadric (1.2.5) invariant, and this subgroup is isomorphic to the group PO(n + 2,1) of conformal transformations of the space C". We will now write in more detail the equations (1.2.6), taking into account equations (1.2.7) and (1.2.8): dAo

dAr

= woAo + WrAr, = w°Ao + wrA8

dAn+1 =

9IL

+ grsw'An+1,

(1.2.9)

A, - wpAn+1

Here (grs) is the matrix inverse of the matrix (g,.9), that is, g,.sg" = d;., where b1 is the Kronecker symbol. In addition to equations (1.2.7) and (1.2.8), the forms wn satisfy the structure equations of the conformal space C": dw("

= wE A wS,

(1.2.10)

which are necessary and sufficient conditions for complete integrability of equations (1.2.6).

We can now write in more detail the structure equations (1.2.10), taking into account equations (1.2.7) and (1.2.8): dwoo = wo n w°,

dwo=woAwo+woAw;, dw,'.

= w A wo + w* A wi + grlgsuw0 A wo,

(1.2.11)

dw°=w°Awp+wr' Aw°. The family R(C") of frames in question can be considered as a frame bundle whose base is the space C" and whose fiber is a set of frames with a fixed point x = A0. This frame bundle is determined by the projection

7r:R(C")-1C", where 7r is defined by the relation

a{Ao,A1,..., A,,,A"+1} = Ao. Since from equations (1.2.9) it follows that JA

o = wo°Ao +woAr,

the forms war, which we will denote by wr. wr = wo, are base forms of the frame bundle 1(C"). They are also called horizontal forms of the frame bundle R(C").

12

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

For w' = 0, equations (1.2.9) take the form 6Ao = noAo,

eAr = a°Ao + ir,'.A,, bAn+l = 7rn+l A, - 7roAn+1,

(1.2.12)

where b is the restriction of the operator d to a fiber of the frame bundle in question (i.e., for wr = 0), and Tr1 = w"(6) = will.=o.

Equations (1.2.12) are the equations of infinitesimal displacement of the station

ary subgroup of the point x = A0, and the forms occurring in these equations are invariant forms of this subgroup. These forms ao, 7r°, Ar and 1rn+1 are fiber forms of the frame bundle R(C"). They are also called vertical forms of the frame bundle R(C"). Note that the stationary subgroup of the point x is also called the isotropy group of the point x. We will denote this group by H. Since by (1.2.11) we have dwo=Woo AWoo +wonWoo

then, by the Frobenius theorem, the system

wo =0 is completely integrable, and its first integrals can be taken as nonhomogeneous coordinates of the point x = Ao. However, on the family R(C") of frames, the group PO(n+2,1) of conformal transformations acts intransitively, since the quantities gr, are invariants of these frames, and frames with different invariants cannot be mapped to each other by a transformation of the group PO(n + 2,1). Nevertheless, we can select a transitive subfamily in the family R(C") of frames. For example, for such a transitive subfamily, we can take the subfamily of orthogonal frames for which the hyperspheres A,. are orthonormal, that is, for which g,., = b,.,, where

b is the Kronecker symbol. We denote this subfamily by Ro(C"). For this subfamily, equations (1.2.8) take the form wr+1 = wo' wo = wn+t, w* + w; = 0,

(1.2.13)

and the third structure equation from (1.2.11) becomes dw* = wo A wo + w; A wi + wo n w,.

(1.2.14)

The number of parameters on which the group PO(n + 2, 1) of conformal transformations of the space C" depends is equal to the number of linearly independent 1-forms w{ occurring in equations (1.2.6). By (1.2.7) and (1.2.13)

1.2

Moving Frames in a Conformal Space

13

the independent 1-forms are wo, w$, w,'. for r < s, and w°. The number of these forms is equal ton + 1 + 2n(n - 1) + n = 2 (n + 1)(n + 2). 3. If we take wr = in the equation dAo = w°Ao +wrAr,

(1.2.15)

the point AO will describe a curve (it). Consider the hyperspheres P = x°Ao +xrAr +xn+lAn+i which are tangent to this curve at the point A0. For these hyperspheres we have (Ao, P) = 0, (dtAo, P) = 0,

where dt is the operator of differentiation with respect to the parameter t. It follows from the latter equations that

xn+i = 0, \rx'* = 0, where Ar = gr,a'. This shows that the hypersphere d1A0, passing through the point Ao and orthogonal to all hyperspheres P, will be orthogonal also to the curve (lt). Suppose that we are given two curves, (lt) and (m,), passing through the point Ao and defined by the equations

wr = Ardt, wr = prdr. Then the angle V between these two curves is defined as the angle between the hyperspheres dtA0 and d,A0 which are orthogonal to the curves (lt) and (m,),

respectively. Here dt and d, are the operators of differentiation with respect to the parameters t and r. By formula (1.1.4) we find that

_

cos

(dtAo, d,Ao)2

(dtAo,deAo) - (d,Ao,d,Ao)

Since by (1.2.15), (1.2.1), and (1.2.3) we have 9

(dAo, dAo) = 9r.wrw',

(1.2.16)

a conformal structure is defined in a neighborhood of the point A° by the quadratic form gr,wrw' which is positive definite in the space C". Thus the formula for cos 2 V can be rewritten as follows: Cost W _

(9r. \r,*), gr,J\r,\' 9pgltpµa

The equation

9rswrw' = 0

(1.2.17)

defines an isotropic cone C. Since the quadratic form (1.2.16) is positive definite, this cone is pure imaginary. The Darboux mapping maps this isotropic

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

14

cone into the cone of rectilinear generators of the hyperquadric Q" C P"+1 which passes through the point corresponding to x = A°. In the case under consideration, these rectilinear generators are pure imaginary. 4. In the sequel we will use the following theorem on determination of a family of frames by components of their infinitesimal displacement.

Theorem 1.2.1 Let D be a p-dimensional domain of parameters ul, ... , UP, p < (n + 1)2. Suppose that in D are given functions gr, such that det(g,.s) 0 0 and 1-forms 11C, satisfying Pfa flan equations (1.2.7) and (1.2.8) and structure equations (1.2.10). Then, in the conformal space C", up to a conformal transformation, a unique p-parameter family 1Z° of frames is determined for which the functions grs define a relatively invariant form g, and the 1-forms wE are components of infinitesimal displacement of frames of the family W.

This theorem follows from Cartan's theorem on equivalence of two systems of 1-forms (see Cartan [Ca 08]; Gardner [Gar 89]). The maximal number of parameters (n + 1)2 corresponds to the case when

we have an arbitrary oblique family of frames {AE} in the space C". For such a family the forms wo, w', w,', and w° are linearly independent, and the remaining 1-forms are expressed in terms of these independent forms by means of the tensor g,,. If the family of frames in question consists of orthonormal frames, then the maximal number of parameters is equal to the number of parameters on which the group of conformal transformations of the space C" depends; that is, this number is equal to (n + 1)(n + 2). 2

1.3

Pseudoconformal Spaces

1. In the same way as a conformal space was constructed in Section 1.1 by means of a Euclidean space, a pseudoconformal space can be constructed on the basis of a pseudo-Euclidean space. We will perform this construction, but in abbreviated form, emphasizing the main features. Let Rq be an n-dimensional pseudo-Euclidean space of index q (see Wolf [Wol 77], p. 64). The fundamental form of R9 can be reduced to the form (x1)2 + ... + (XP)2 - (xP+1)2 - ... - (x")2,

where n = p + q. If q = 0, then we have a proper Euclidean space that was considered in Section 1.1. Thus we will suppose in this section that q > 0. At any point x of the space RQ , there is an isotropic cone defined by the equation 0. For q > 0 this cone is a real cone of second order. As we (lid earlier, we will denote this cone by C. A hypersphere S in the space Rq is defined by the equation s°[(x1)2 + ... + (9P)2 - (X p+1)2 + ... - (xn)2) nn P P- P+1 P+1 +2(s 1x +...+$x s x 1

- ...s- x)+2s n+1 -0. =

(1.3.1)

1.3

Pseudoconformal Spaces

15

As before, the numbers so, s', r = 1, ... , n, and s"+1 are called the polyspherical

coordinates of the hypersphere S C R". If so = 0, then the hypersphere S becomes a hyperplane of the space RQ . Assuming that so $ 0 and completing squares in equation (1.3.1), we reduce this equation to the form (x1 - a1)2 +

. .

. + (x9 - a")2 - (x"

- an+1)2 - ... - (x" - a")2 = r2, (1.3.2)

where

at = s T2 =

(802 1 (81)2 +

... + (Sp)2 - (3D+1)2 - ... - (8")2 - 288"t1 1. (1.3.33)

Equation (1.3.2) shows that the point a(al , ... , a") is the center of the hypersphere S. If r2 varies, then equation (1.3.2) determines a family of concentric hyperspheres with center at the point a. Moreover the number r2 can take not only positive but also negative and zero values. For r2 > 0, the hypersphere S has

a real radius; for r2 < 0, it has an imaginary radius; and for r2 = 0, it has zero radius. But in the space Rq all these hyperspheres are real hypersurfaces of second order (see Figure 1.3.1). A hypersphere of zero radius is a cone of second order with vertex at the point a, and this cone is called isotropic and is denoted by Ca. Note that the pseudo-Euclidean space Ri is very important in special relativity. Usually the metric form of this space is written as follows:

\C S) =

-(xo)2

+ (x1)2 + (x2)2 + (x3)2+

where xo = ct is the time coordinate, and (x' , x2, x3) are the space coordinates. The space R1 is called the Minkowski space. Isotropic cones of this space are the light cones (see Einstein [Ein 05)).

In the pseudo-Euclidean space R,', the inversion in a hypersphere S with center at a point a is defined exactly in the same manner as it was defined in the Euclidean space R" (see Subsection 1.1.2). However, in contrast to the space R", under an inversion in the space R" not only does the center a of the hypersphere S not have an image but also all points of the isotropic cone Ca with vertex at the point a do not have images. To include these points in the domain of the mapping defined by the inversion in Ra , we enlarge the

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

16

(x1)2

+ (x2)2 - (x3)2 = r2

n=3, q=1 r2 > 0: r2 < 0: r2 = 0:

a sphere of real radius

-a hyperboloid of one sheet a sphere of imaginary radius -a hyperboloid of two sheets a sphere of zero radius -an isotropic cone

Figure 1.3.1

space Ra not only by the point at infinity, oo, corresponding to the point a but also by the isotropic cone Co,, with vertex at this point. The manifold obtained as the result of this enlargement is denoted by Co :

C, =RqU{C.} and is called a pseudoconforvnal space of index q. As a result, the noncompact pseudo-Euclidean space Ro becomes a compact

pseudoconformal space C. n. This is the reason that the operation described above is called the compactification of the Rq. Note that compactification of the Minkowski space Ri produces a pseudoconformal space C, which is widely used in special relativity. Just like conformal space C", the pseudoconformal space Ca is homogeneous.

2. As we saw earlier, homogeneous coordinates so, s', and sn}1 of a hypersphere S C Ca can be considered as coordinates of a point of a projective space P"+1. It follows from relation (1.3.3) that under this mapping, to hyperspheres of zero radius there correspond points of the space P"+1 lying on the hyperquadric Qo determined by the equation (s1)2 + ... + (#P)2 - (sP+1)2 - ... - (s")2 - 2sos"+1 = 0,

(1.3.4)

where n = p + q. As in the case of a proper conformal space (see Section 1.1), this hyperquadric is called the Darboux hyperquadric. However, for a pseudoconformal space Cq this hyperquadric is not oval anymore, since the

1.3

Pseudoconformal Spaces

17

signature of the quadratic form on the left-hand side of equation (1.3.4) is equal to (p + 1, q + 1). This hyperquadric carries rectilinear generators and might also carry plane generators. If q < p, then the hyperquadric Q11 carries q-dimensional plane generators. The points of the hyperquadric QQ" can be considered as images of the vertices x of cones C,. which are spheres of zero radius. Thus there is a one-to-one correspondence between points of the space Ca and points of the hyperquadric

QQ C P"+1. Just as for a proper conformal space, the mapping we have constructed is called the Darboux mapping. Under the Darboux mapping, to isotropic cones of the space CQ there correspond the asymptotic cones of the hyperquadric Q11 that are intersections of the hyperquadric Qn with its tangent subspace. Since the left-hand side of the equation of the hyperquadric Qo has signature (p + 1, q + 1), the group of transformations of the space Pn}1 sending Qq to itself is isomorphic to the group SO(n + 2, q + 1) for n odd and to the group O(n + 2, q + 1)/Z2 for n even. Because of this we denote by PO(n + 2, q + 1) the fundamental group of transformations of the space Cq :

PO(n + 2,q + 1) :=

r SO(n + 2,q + 1)

if n is odd,

tl O(n + 2, q + 1)/Z2 if n is even.

We will prove that the space CQ" is homeomorphic to the manifold (SP X SQ)/Z2. In fact the equation (1.3.4) of the hyperquadric Qq, into which the Darboux mapping sends bijectively the space Cn, in some frame {A(} can be reduced to the form: (xe)2 +

... + (xp)2 - (xp+1)2 - ... - (xn+1)2 = 0.

It follows from this equation that the subspaces LP = AO A ... A Ap and that L9 = Ap+1 A ... A A"+, do not have common points with the hyperquadric Qn . Let X E La and Y E LP be arbitrary points of these subspaces. Then the subspaces a = LP A X and 0 = LQ A Y intersect the hyperquadric Qa along quadrics QP and QQ of dimensions p and q, which are homeomorphic to the spheres SP and SQ. Moreover, through any point x of the hyperquadric Q", there passes a quadric QP and a quadric QQ, but the latter two quadrics, in addition to the point x, have the second common point y (see Figure 1.3.2). Thus Q

Cq

Q' - (S' X S9)/Z2

In particular, a pseudoconformal plane C1 is homeomorphic to a quotient of a two-dimensional torus by the group Z2. This factorization leads again to a two-dimensional torus: C; Sl X S1. In the same manner as was done for the space Cn, we can define the stereographic projection of the space CQ onto a space Ro . To this end, we consider the Darboux mapping of the space CQ onto a hyperquadric Qq C P"+1 , fix

a point z E QQ, and project points x E Q" onto a hyperplane E" C P"+1 Q

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

18

Figure 1.3.2

not passing through the point z. Under the Darboux mapping, to the point z there corresponds the (n - 1)-plane E' in which the tangent hyperplane TZ (Qq) intersects E": E" 1 = Ts(Qq)f1E". The intersection of the asymptotic cone C, of the hyperquadric Qq with the plane E"-' is a real quadric U' 2 of dimension n - 2 whose equation is (xl )2 +

... + (x") - (xD+1)2 - ... - (x")2 = 0,

where n = p + q. The quadric U"-2 can be taken as an absolute of the hyperplane E". Since the quadratic form in the left-hand side of the above equation is of signature (p, q), this absolute induces the structure of the pseudo- Euclidean space Rq on E". Thus the stereographic projection maps a pseudoconformal space Cq with a fixed point z onto a pseudo-Euclidean space Rq. 3. A moving frame in the space C." is introduced in the same manner as in the space C". Each frame of the family 1Z(C") consists of frames composed by two points Ao = x and A"+1, and n independent hyperspheres A, passing through these points. Moreover all equations (1.2.1)-(1.2.3) still hold, but the quadratic form gr,x''x' determined by the tensor g,., has signature (p, q), where p + q = n. In this frame the equations of the hyperquadric Qq preserves the form (1.2.5), the equations of infinitesimal displacement and the structure equations have the form (1.2.6) and (1.2.10) as before, and the forms" satisfy equations (1.2.7) and (1.2.8). The quadratic form g defined by equation (1.2.16) is the second fundamental form of the hyperquadric Q. This form is not positive definite but of signature (p,q). Equation (1.2.17) determines the family of real isotropic cones of the space C,' to which, under the Darboux mapping, there corresponds the family of asymptotic cones C., of the hyperquadric Q, with its vertices x E Qq .

1.4

Examples of Pseudoconformal Spaces

19

The fundamental group PO(n + 2, q + 1) acts intransitively on the family 1(C") of conformal frames. This group will act transitively, for example, on the subfamily R°(C") of pseudoorthogonal frames, which is defined by the conditions:

(Ar, As) = 0, r 0 s;

(Ar, Ar) = -1, r > p; (1.3.5)

(Ar, Ar) = 1, r < p;

that is, 1

grs =

0

ifros,

1

ifr = s < p,

(1.3.6)

-1 ifr=s>p.

However, the subfamily R°(C") is not always convenient for us, and we will use other transitive subfamilies of the family R(C") of frames. We will perform all further considerations in a proper conformal space C". However, all following constructions can be easily made in a pseudoconformal spaces Ce" of any index q.

1.4

Examples of Pseudoconformal Spaces

1. We now consider two classical examples leading to pseudoconformal geometry. The first of these examples is given by the line geometry of a real three-dimensional projective space. We consider a real three-dimensional projective space P3 (e.g., see Dieudonne (D 64]) and denote by x1 , x2, x3, x4 homogeneous coordinates of a point x of this space with respect to a frame {M1, M2, M3, M4}. Thus we have

x=x'M;,

i = 1,2,3,4.

Consider another point y = y'M;, and denote by p the straight line passing through the points x and y so that p = x A y, where, as everywhere earlier, the symbol A denotes the exterior product of the points x and y, or of the vectors of the same name in the four-dimensional vector space V4 from which the space p3 is obtained by dividing out by the set of real numbers R: P3 = (V4 - {0})/R. The set of straight lines of the projective space p3 is a differentiable manifold which is denoted by G(1, 3). It is called the Plucker manifold, since J. Phicker was the first who studied such manifolds (see Plucker [P1 68]; Klein [Kl 26b], vol. 2, pp. 5-10). The straight line p is determined not only by the points x and y but also by any other pair of points

u=ax+8y, v=-yx+by, a6 -p-y34 0, lying on this straight line. Consider the two matrices: 2

C y,

y2

3

y3

4

yA

J

and

U1

U2

U3

U4)

VZ

V3

v4

20

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

composed of the coordinates of these points. One can see that the determinants

p'J =

xi

xj y,

and q'' = I

.

I

Y,

differ only by the common factor p =

u Vi

vj u'

I01

qt = Pp 'j, and thus they can be taken as homogeneous coordinates of the straight line p and can be also considered as coordinates of a point p E P5. However, the coordinates p'3 of the straight line p are not arbitrary numbers. They are connected by the quadratic relation p12p34 + p23p14 + p31p24 = 0,

(1.4.1)

arising from the identity z1

x2

yl

y2

zl yl

x2 y2

x3 y3

z4 y4

x3 y3

x4 y4

=0.

Equation (1.4.1) determines a hypersurface of second order in a five-dimensional

projective space P5. To any straight line p C P3, there corresponds a unique point with projective coordinates p'3, belonging to the hyperquadric (1.4.1). The mapping of the set of straight lines of the space p3 onto the four-dimensional quadric (1.4.1) is called the Plucker mapping, and the hyperquadric (1.4.1) itself is said to be the Plucker hyperquadric. Let us denote it by fl(1,3). The mapping G(1,3) -4 f2(1,3) was first considered in Klein [Kl 72b]. Since this mapping is one-to-one and differentiable and the hyperquadric 11(1, 3) is a differentiable manifold, the set G(1,3) is itself a differentiable manifold. By means of a linear transformation of coordinates of the type p12 = a + t, p34 = s - t, we can reduce the left-hand side of equation (1.4.1) to the form containing three positive and three negative squares, and thus having signature (3, 3).

We will denote by the same letter a straight line in p3 and the point on the hyperquadric f2(1,3) corresponding to this straight line. Let us introduce new notations for the Plucker coordinates of the line p by setting p31

p12 = p° p23 = pl = p2 p34 = p5, p14 = p4, p24 = p3.

Then equation (1.4.1) of the hyperquadric fl(1,3) becomes p°ps + plp4 + p2p3 = 0.

(1.4.2)

This equation can be written as follows: 2p°ps = gijp'pt,

(1.4.3)

1.4

Examples of Pseudoconformal Spaces

21

where i, j = 1, 2, 3, 4, and the quantities gig are the entries of the matrix

( 90 )

0

-1

0 0 0

0 0

-1

-1

0

0

-1

0

0

0

0

(1 . 4 . 4)

Then the quadratic form on the right-hand side of equation (1.4.3) is of signature (2, 2). But, as we saw in Section 1.3, a hyperquadric of this kind is endowed with the structure of the pseudoconformal space C24. Hence, the geometry of the Grassmannian G(1, 3) is equivalent to the geometry of the pseudoconformal space C21-

2. We denote by (p, p) the left-hand side of equation (1.4.2) multiplied by 2. Then (1.4.5)

(p, p) = 2P°p5 + 2p'p4 + 2p2p3.

We also denote by (p, q) the bilinear form that is polar to the quadratic form (P, p):

(p, q) = P°q5 + p1g4 + P2g3 + P3g2 + P4q' +

p5g0.

(1.4.6)

Then the condition (p, p) = 0 means that the point p E P5 lies on the hyperquadric 1I(1,3) and represents a straight line of the space p3. If p and q are two straight lines in p3 determined by pairs of points x, y and u, v and having Pliicker coordinates x` pI = yi

xj

I

and

_

yj

ui

uj

vi

vj

,

then equation (1.4.6) takes the form xI

x2 y2

x3

yl ul

u2

y3 u3

vl

v2

v3

x4 y4 u4 v4

It follows that the straight lines p and q intersect one another if and only if (p, q) = 0. Such straight lines p and q determine a pencil of straight lines in P3, and a rectilinear generator on the hyperquadric 11(1, 3) corresponds to this pencil. Expressions (1.4.5) and (1.4.6) make sense not only for points of the hy-

perquadric tl(1,3) but also for any points of the space P5 in which this hyperquadric is located. If s is a fixed point of the space P5, then the equation (s, p) = 0 determines a hyperplane a that is the polar hyperplane of the point s with respect to the hyperquadric 11(1,3). If s V 11(1,3), then the intersection a fl 11(1, 3) is a nondegenerate three-dimensional quadric that corresponds to a three-parameter family of straight lines in the space P3. The latter family is called a linear complex. We will denote this complex by the same letter s. If s E 11(1, 3), then its polar hyperplane a is tangent to the hyperquadric 11(1, 3)

22

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

at the point s, and the intersection a fl 12(1, 3) is a real cone of second order. We denote this cone by C,. In the space C2 this cone is an isotropic cone with its vertex at the point s. This cone is the image of a special linear complex. of the space P3 that consists of all straight lines of P3 intersecting the straight line s. Let r and s be two linear complexes in the space P3, whose images are two points in P5 which we denote by the same letters. A linear congruence in p3 is a collection of straight lines belonging simultaneously to both complexes; that is, it is the set r fl s. To find the geometric meaning of a linear congruence, we

consider the straight line r A s determined in P5 by the points r and s. The parametric equation of this line is t = Ar + µs.

The location of this line with respect to the hyperquadric 11(1,3) depends on the quadratic trinomial (t, t) = A2 (r, r) + 2aµ(r, s) + µ2(s, s),

(1.4.7)

whose discriminant 0 is equal to A = (r, s)2 - (r, r)

(s, s).

If 0 < 0, then the straight line r A s has no common points with the hyperquadric 12(1, 3). The linear congruence r fl s, corresponding to such a line, is called elliptic. If 0 > 0, then the straight line r A s has two common points

p and q with the hyperquadric Q(1, 3). The linear congruence r fl s, corresponding to such a line, is called hyperbolic. Such a congruence consists of all

straight lines of the space p3 intersecting two straight lines p and q, which are called the directrices of the linear congruence r fl s. If A = 0, but not all coefficients of the quadratic trinomial (1.4.7) vanish, then the points p and q coincide, the straight line r A s in P5 is tangent to the hyperquadric 11(1, 3), and the congruence r fl s is called parabolic. Finally, if all coefficients of the quadratic trinomial (1.4.7) vanish, then the straight line rAs lies on the hyperquadric 12(1, 3); that is, this line is a rectilinear generator of 12(1, 3). This straight line r A s is the image of a pencil of straight lines of p3 determined by the intersecting lines r and s. The linear congruence r A s degenerates in this case into a two-parameter family of straight lines lying in a 2-plane r of the pencil r A s. Such a degenerate linear congruence is called a plane field of straight lines. The image of such a plane field of straight lines is a two-dimensional plane generator of the hyperquadric 12(1, 3). Since the set

of 2-planes r in the space P3 depends on three parameters, the hyperquadric 0(1,3) carries a three-parameter family of two-dimensional plane generators corresponding to 2-planes of the space p3. Moreover the hyperquadric 12(1,3) carries also a second three-parameter family of two-dimensional plane generators corresponding to the bundles of straight lines of the space P3, since the bundles of straight lines, just as plane fields, are linear images in the space P3.

Examples of Pseudoconformal Spaces

1.4

23

Thus, the hyperquadric 11(1, 3) carries two families of two-dimensional plane generators each of which depends on three parameters. Considering the preimages of these generators in P3, one can easily prove that any two generators of one family have a common point on the hyperquadric 11(1,3) and that generators of different families either have no common points or have a common straight line. Lets be a fixed point of the hyperquadric 11(1, 3), and let C, be an isotropic

cone with vertex at s and at the same time the special linear complex in P3 corresponding to this cone. The line s in P3 possesses a one-parameter family of points, and a one-parameter family of 2-planes passes through the line s. This implies that the cone C, carries two one-parameter families of two-dimensional plane generators, and the projectivization PC3 with center at s of this cone is a ruled surface of second order in a three-dimensional projective space which, in turn, is the projectivization of a four-dimensional tangent subspace T,(1)(1, 3)). (For more details on projectivization see Akivis and Goldberg [AG 93], pp. 2324.)

3. Suppose that {M1i M2, M3, M4) is a moving frame in the space p3. The equations of infinitesimal displacement of this frame have the form dMi = 8 M;,

i,j = 1,2,3,4.

(1.4.8)

The forms O , occurring in these equations, satisfy the structure equations of the space p3: dB; =BkAO(, i,j,k=1,2,3,4. (1.4.9) In addition we assume that the frame {M1} in p3 is normalized by the condition (Ml, M2, M3, M4)

which implies that the forms 6 vi

(1.4.10)

connected by the equation

9+8+9+9=O.

(1.4.11)

The lines

f ao=MiAM2, a1=M2AM3, a2=M3AM1, 1 as=M3AM4, a4=M1AM4, a3=M2AM4

(1.4.12)

form a frame on the Grassmannian G(1,3), and the corresponding points in the space P5 form a frame in this space. By (1.4.10) and (1.4.12), the vertices of this frame satisfy the conditions (ao,a5) = (al,a4) = (a2,a3)

(at,a,,)=0,

+p34 5-

(1.4.13)

Equations of infinitesimal displacement of this frame can be written as dat _ -E a,,,

0, 1, ,77=0,1,2,3,4,5,

(1.4.14)

24

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

where the forms wE are connected with the forms 9, by the following relations: 91 +02

(wE)

-93 -e3

-91 +03 3 -91

922

-03

-94

oZ

0

-922

93

0

-02

0] +03

o

-03

91

02 + 0;

92

e2

9;

91 + 94

0;

03

03

93 + 04

-B;

9;

024

0

0 -034

0

-94

of

(1.4.15)

These forms satisfy equations (1.2.8) where now the matrix (gi,) has the form (1.4.4), and hence they are the components of infinitesimal displacement of a frame in the pseudoconformal space C24. Thus, the fundamental groups of the spaces C2 and p3 are isomorphic to one another. We now consider the quadratic form g = (dao, dao) defining a conformal

structure in the space C. By (1.4.14) and (1.4.15), this form can be written as follows:

g=

(03021

- 0 9i).

(1.4.16)

The equation g = 0 determines the isotropic cone Co at the point ao E f1(1, 3)

(or, which is the same, ao E Cz) that has the same structure as the cone C. described above. All curves in CZ satisfying this equation are isotropic. In the space p3 the developable surfaces correspond to these isotropic curves. In the space C2 we consider now two nonisotropic curves ll and 12 passing through the point a°. As we saw in Section 1.2, the condition of orthogonality of these curves can be written in the form g(d1,d2) = (dlao,d2ao) = 0,

(1.4.17)

where the operators d1 and d2 denote differentiation along the curves ll and 12 at the point ao. In the space P3 the condition (1.4.17) is the condition of harmonic intersection of ruled surfaces. This notion in line geometry was introduced in Cartan [Ca 31] and was studied in the papers Akivis (A 48, 50] and Vasilyev [Va 48].

4. As the second example of pseudoconformal geometry, we consider Lie sphere geometry. This geometry can be introduced as follows: We consider the Darboux mapping of the conformal space C" (see Subsection 1.1.5). Under this mapping the image of a point X E C" is a point of the hyperquadric Q" of the space Up to notations the equation of Q" has the form (cf. (1.1.3)): Pn+1.

n

E(xi)2 - 2x°x"+1 = 0.

(1.4.18)

i=1

By a real linear transformation of coordinates, equation (1.4.18) can be reduced to the form n E(xi)t + (xn+1)2

(X, X) = -(x°)2 + i=1

= 0.

(1.4.19)

1.4

Examples of Pseudoconformal Spaces

25

Let X be a hypersphere of the space C". Under the Darboux mapping the image of the hypersphere X is a point of the space P"+1 lying outside of the hyperquadric Q", and for coordinates of points of X we have n

(X X) _ _(x°)2 + E(xi)2 + (xn+1)2 > 0.

(1.4.20)

i=1

To introduce Lie sphere geometry, we normalize coordinates of points of X by the condition n

(X,X) _ -(x°)2 +

(xi)2 + (xn+1)2 = 1

(1.4.21)

i=1

and define homogeneous coordinates yo, y', yn+1 and y"+2 in such a way that

x° =

yo

yn+2'

xi = y'

yn+2'

x"+1 =

yn+1

yn+2'

(1.4.22)

Then equation (1.4.21) takes the form n

(Y,Y) _ -(y°)2 + E(y`)2 + (yn+1)2 -

(yn+2)2 = 0.

(1.4.23)

i=1

The left-hand side of this equation is a quadratic form of signature (n + 1, 2). The numbers y°, y'(i = 1, ... , n), y"+1 and yn+2, can be taken as homogeneous coordinates of a point Y in a projective space pn+2 of dimension n + 2. Then equation (1.4.23) determines a real hyperquadric in the space Pn+2. We denote this hyperquadric by L and call it the Lie hyperquadric. If in the space C" a hypersphere X with coordinates xo, xi, xn+1 is given, then from equations (1.4.22) and (1.4.23) the coordinates yo, y', y"+1 and y"+2 are determined not only up to a common factor but also up to a sign of y"+2 Thus to a hypersphere X E C" there correspond two points on the Lie hyperquadric L. To make this correspondence one-to-one, we need to furnish the hypersphere X with an orientation. Then there will be a one-to-one correspondence between the oriented hyperspheres of the space C" and the points of the Lie hyperquadric L. This correspondence is called the Lie mapping. This construction is represented in Figure 1.4.1. In it the hyperboloid represents the Lie hyperquadric L in the space Pn+2, and the plane x represents P"+', defined by the equation y"+2 = 0, which is the image of the subspace the conformal space C" under the Darboux mapping. To the Darboux hyperquadric Q" itself there corresponds the intersection of the hyperquadric C and

the plane n. The image of a hypersphere of the space C" is a point X E it

26

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

Figure 1.4.1

lying outside of the hyperquadric Q". Denote by 0 the pole of the hyperplane

r with respect to the hyperquadric C. The straight line XO meets C in two points Yl and Y2 which are the images of two oriented hyperspheres defined by the hypersphere X of the space C". These oriented hyperspheres are the basic elements of Lie sphere geometry. Their coordinates y°, y', y"+1, and y"+2 are called (n + 3) -spherical coordinates. A parabolic pencil of hyperspheres in C" is a one-parameter family of hy-

perspheres tangent to one another at a point (see Figure 1.4.2). Under the Darboux mapping, to such a pencil there corresponds a straight line ZX of the hyperplane r that is tangent to the Darboux hyperquadric. Under the Lie mapping to this straight line there correspond two rectilinear generators ZY1 and ZY2 on the Lie hyperquadric (see Figure 1.4.1) which represent two pencils of oriented hyperspheres defined by hyperspheres of the original parabolic pencil.

A parabolic pencil of hyperspheres determines a hyperplanar element of the space C" (see Figure 1.4.2). Under the Lie mapping, to this element there corresponds a pair of oriented hyperplanar elements each of which is represented

by a rectilinear generator of the Lie hyperquadric. It is easy to compute that the manifold of rectilinear generators of this hyperquadric C, and consequently of the manifold of oriented hyperplanar elements, is of dimension 2n - 1. In fact this dimension coincides with the dimension of the manifold of tangents to the Darboux hyperquadric Q". Since dim Q" = n, and the tangent lines to Q"

Figure 1.4.2

at a fixed point x depend on n - 1 parameters, the dimension of the manifold of rectilinear generators of the Lie hyperquadric G equals n + (n - 1) = 2n - 1.

Note that the manifold of oriented hyperplanar elements of an arbitrary n-dimensional differentiable manifold has the same dimension 2n - 1. The fundamental group of Lie sphere geometry is isomorphic to the group of projective transformations of the space P"+2 sending the Lie hyperquadric L into itself. Hence this group is isomorphic to the pseudoorthogonal group SO(n + 2, 2). Therefore Lie sphere geometry is the geometry of a pseudoconformal space Ci +i . Under transformations of this group, the points of the hyperplane y"+z = 0 can be transferred into points not belonging to this hy-

perplane. This means that in Lie sphere geometry, a point and an oriented hypersphere of the space C" are indistinguishable. But parabolic pencils of oriented hyperspheres, and hence oriented hyperplanar elements, are invariant under transformations of this group. Thus the fundamental group of Lie sphere geometry is a subgroup of the pseudogroup of contact transformations of an n-dimensional differentiable manifold (see Lie and Scheffers [LS 96]).

28

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

NOTES2 1.1. Conformal transformations in a Euclidean plane were first considered by L. Euler (Eu 69]. L. Euler [Eu 77a, b) also applied conformal transformations to the cartography. Conformal transformations of a three-dimensional Euclidean space were first considered by J. Liouville [Lio 50) in an appendix to the book Application de !'analyse d la gdom&rie by G. Monge. Liouville's theorem was proved by Liouville in the same appendix. Later different proofs of Liouville's theorem for n > 3 have been published in many textbooks on differential geometry (e.g., see Blaschke [BI 21), §49, p. 101) or Dubrovin, Fomenko, and Novikov (DFN 92], §15, p. 138). It was F. Klein [KI 72a] who in his historical Erlanger program gave the first accurate and precise definition of conformal space. In the same publication F. Klein also presented the idea of projective interpretation of a conformal space and defined the group of conformal transformations. 1.2. If n = 3, polyspherical coordinates become pentaspherical coordinates of two-dimensional spheres. These coordinates were first introduced and applied in Darboux (Dar 731 (see also the book Darboux [Dar 17]). Polyspherical coordinates were defined by F. Klein (e.g., see Klein (KI 28), §50). Cartan (Ca 23] defined moving frames in the space C". However, he used the diagonal form of the tensor g,, while K. Yano [Y 39c] considered a more general moving frame, which is essentially the frame we constructed in this section. The pentaspherical moving frames were first used in Demoulin [Demo 05, 21, 26b (see also Backes (Ba 50, 51a, b]). 1.3. E. Cartan [Ca 20c] introduced the notion of pseudoconformal space. Haantjes [Haa 37] studied the geometry of pseudoconformal spaces. For more on pseudoconformal spaces, see the book Rosenfeld [Ro 96], Ch. 3, §8). Liouville's theorem in a pseudoconformal space was proved in Haantjes [Has 37) (see also §19 of the book Schouten and Struik (SS 38)). The Minkowski space was introduced by H. Minkowski [Min 09) and named after

him. This space led to the idea of the space-time of special relativity. The paper Poincard (Po 061 had been published a few years earlier in a specialized mathematical

journal, and for a long time scientists had no knowledge of it. This is the reason that the four-dimensional pseudo- Euclidean space which models the space-time of special relativity is referred to as Minkowski space rather than, more appropriately, as Poincar6 space.

1.4. The coordinates of a straight line were introduced by J. PIucker (P1 46) in 1846. Two years earlier H. Grassmann ([Gra 44, 62]) defined the coordinates of a linear subspace L'"1 embedded in a projective space P` 1. However, H. Grassmann just gave very general considerations and did not give applications to the space P3. Independently of Grassmann and Plucker, in 1859 A. Cayley (Cay 591 found these coordinates for r = 2 and n = 4, that is, for a straight line in P3. The Plucker mapping was suggested by J. Plucker (P1 68). F. Klein (Kl 72b) indicated the reduction of the equation of Plucker's hyperquadric.0(1, 3) C P5 to the sum of squares and the correspondence between the set of straight lines in P3 and the points of f2(1, 3) C P5. Later this correspondence was studied in detail in Segre (Seg 85).

The terms "congruence" and "complex" (and "linear complex") were introduced 2The numbers 1.1, 1.2, etc., refer to section numbers.

Notes

29

in Plucker [P1 461 for two- and three-parameter families of straight lines in P3. The term "congruence" is explained by the fact that Plucker considered a congruence as a set of coinciding lines of two complexes. For more detail, on Lie sphere geometry, see the books Klein [KI 26a] (§§25-27, 64, 70, 73) and Blaschke [BI 29] (Chs. 5, 6, and 9) and the recent book Cecil [Ce 92] which is devoted entirely to this subject.

Chapter 2

Hypersurfaces in Conformal Spaces 2.1

Fundamental Objects and Tensors of a Hypersurface

V"-1 1. We will start from consideration of the theory of real hypersurfaces in the real proper conformal space C". This theory is close to the theory of hypersurfaces in the pseudoconformal space C.". However, there are certain differences between these two theories. These differences will be considered in more detail in Section 3.3. Let V"-1 be a smooth, connected, and simply connected hypersurface in the space C", that is, a differentiable submanifold of dimension n - 1 in C". With any point x E V"-1, we will associate the family of conformal frames whose vertex A0 coincides with the point x, whose hypersphere An is tangent to the hypersurface Vn-1 at the point x, and whose hyperspheres Ai, i = 1,.. . , n - 1, are orthogonal to V"-1 at the point x. We denote by A"}1 the second intersection point of the hyperspheres Ai and An (see Figure 2.1.1). This family of frames is the bundle 7Z1(Vn-1) of frames of first order associated with the hypersurface V"-1. The base of this frame bundle is the hypersurface V"-1, and its fiber is the collection of frames with a fixed point

x=A°. For the frames of the bundle R1(V"-1), we have conditions (1.2.1). As in Section 1.2, we will suppose that condition (1.2.2) holds. In addition we normalize the hypersphere An by the condition (A", An) = 1. This can always be done since the quadratic form, determined by the tensor (g,., _ (9'i ° ) is positive definite, and from this it follows that g"" > 0. The quantities gig are determined by a first-order neighborhood of the hypersurface V"-1 and form a nonsingular (0, 2)-tensor. The quadratic form 31

2. HYPERSURFACES IN CONFORMAL SPACES

32

Figure 2.1.1

9 = 9iglt' defined by this tensor is also positive definite. This form can be also written as

9 = 9ijuj, where g'.' is the inverse tensor of the tensor gi, and ti = gigk. In this chapter we will use the following index ranges:

1 < i,j,k,I,... < n - 1; 0 < ,q,(,... < n + 1. We will now write all conditions that the frame elements satisfy: (Ao, Ao) = (An+1, An+1) = (Ao, An) = (An+1, An) = 0, (Ao, Ai) = (An, A,) = (An+1, Ai) = 0, (A,, Aj) = gij,

(2.1.1)

(Ao, An+1) = -1, (An, An) = 1,

where, as in Section 1.1, the symbol (, ) denotes the scalar product of the corresponding frame elements. By (2.1.1), equations (1.2.7) and (1.2.8) take the form n+1 w0

n+l

wi

0

= wn+1 = 0,

- 9ijwoj = 0,

w° - 9+jwn+1 = 0, 9ijwn + win = 0,

n+l - + 0 + Wn+l Wo n+1 wn

- w0n = 0, wn - n+1 = 0,

(2.1.2)

n = 0, wn

dgij = 9ikw4 + 9kjwi .

Since the hypersphere An is tangent to the hypersurface Vn'1, we have

2.1

Fundamental Objects and Tensors of a Hypersurface

33

(dAo, An) = 0. By the first equation of (1.2.6), this implies that

wn=0

(2.1.3)

dAo = w0Ao +w'A;,

(2.1.4)

and that where w' = wo. From (2.1.4) it follows that the forms w' are linearly independent. They are base forms of the frame bundle 1V (V n-1) From (2.1.2) and (2.1.3) we also find that wn+1 = 0.

(2.1.5)

We now take exterior derivatives of equation (2.1.3) by applying the structure equations (1.2.10) of the conformal space Cn. As a result we obtain w° h w` = 0.

(2.1.6)

Applying Cartan's lemma (e.g., see Bryant et al. [BCGGG 91], p. 320 or Akivis and Goldberg [AG 93], p. 9), we obtain

w," _ \,iw', Aii = Ap.

(2.1.7)

If we fix a point x on the hypersurface Vn-' (i.e., set w' = 0), then the matrix (wi) of the forms w," can be written as follows: 0 7r0 0

(ir) _

0 7ri1

0 0

'ro

0

0

0

g' it

n

0 0 0

(2.1.8)

in -ir0

where trf = w (b) = wE L, =o (see Subsection 1.2.2). The entries of this matrix are fiber forms of the frame bundle R1(V n-1). They are invariant forms of the stationary subgroup H = (V n-1) of the point x E V n-1. This subgroup leaves invariant the tangent element (x,TT(Vn-1)) of the hypersurface Vn-1 consist-

ing of a point x E Vn-1 and the tangent subspace TZ(Vn-1). Geometrically this tangent element is defined by the point Ao and the pencil of hyperspheres An + sAo which is tangent to the hypersurface V11-1 at the point Ao. Let us consider some subgroups of the group HI(Vn-1). To this end we write the values of the exterior differentials of invariant forms of the stationary subgroup Hi (V n-1) on this subgroup: dlro = 0,

(2.1.9)

drr _irkAi,

(2.1.10)

dan = irn A 4,

(2.1.11)

d7r° = 7r° A 7ro + ir; A

a°.

(2.1.12)

2. HYPERSURFACES IN CONFORMAL SPACES

34

Equation (2.1.9) shows that iro is an invariant form of the one-parameter subgroup of the group Hi (V"-1) which is isomorphic to the group H of homotheties. From equations (2.1.10) it follows that ir; are invariant forms of the subgroup of the group H, (V tt-1) which is isomorphic to the general linear group GL(n - 1). Similarly, as equations (2.1.9) and (2.1.11) show, 7rn and ao are invariant forms of the subgroup of transformations that transfer the pencil

of tangent hyperspheres A + sAo into itself. Finally, for iro = ir = 0, from equations (2.1.11) and (2.1.12) it follows that rro and rr° are invariant forms of the group T(n) of parallel translations of the space R". The latter group is an invariant subgroup of the stationary subgroup Hs (V"-1) Taking the exterior derivatives of equation (2.1.7), we arrive at the exterior quadratic equations: WO

(dAij - Akjw{ - Aik(J +

+ 9ijwn) A w' = O.

(2.1.13)

Applying Cartan's lemma to equations (2.1.13), we find that

dAij - Akjwk - \,,, " + A;jwo0 + gijw° = A;jkwk,

(2.1.14)

where Aijk are symmetric with respect to all indices.

The procedure that we used to derive equations (2.1.7) from (2.1.3) and equations (2.1.14) from (2.1.7) is called the prolongation of a system of equations. This procedure can be applied p times provided that the hypersurface V"-1 is p times differentiable. We will assume that p > 4. Applying this procedure one more time, we obtain dA;jk

-Aljkwi - \iikw' - A,,, 4 + 2A;jkwp

(2.1.15)

+3(A(ij9k), - 9(1jAk)I)wn+1 - 3A(;jAk),wn = I\ijklwl

where the quantities A;jkl are symmetric with respect to all indices, and the parentheses in the indices denote the cycling; for example, 1

9(ijAk)1 = 3(9ijAkl +9jkAil +9kiAjl)

Note that the quantities A;j in (2.1.7) are determined in a differential neighborhood of second order, the quantities A;jk in (2.1.14) are determined in a differential neighborhood of third order, and the quantities A;jkl in (2.1.15) are determined in a differential neighborhood of fourth order of a point of the hypersurface V"-1 2. Let us find the law of transformation of the basis forms wi under transformations of the stationary subgroup HH(V"'1). Since, by (1.2.10), we have dw` = w2 A (wj' - bj'wo),

the bilinear equation associated with this equation, has the form dwi(b) - bwi(d) = w'(d)(wj'(b)

-

w}(a)(w;(d) - 6 wo(d))

2.1

Fundamental Objects and Tensors of a Hypersurface

35

(see H. Cartan [CaH 671, §§2.2 and 2.3). Suppose that d is the symbol of differentiation with respect to the base variables, and d is the symbol of differentiation with respect to the fiber variables. Then w'(b) = 0,ww(b) = 7r', and we will write simply w' instead of w'(d). As a result we obtain bw' =

burro).

We set

V6w' = bw' +w'(7r; - burro).

(2.1.16)

Then the previous equation can be written as V6w' = 0.

(2.1.17)

Denote a'., = 7r,' - biro. These forms satisfy the same structure equations as the forms

ari

= n n 7rk,

and they are invariant forms of the general linear group GL(n - 1). The hypersurface Vn-1 is a differentiable manifold, and the forms w' consti-

tute a co-basis in the tangent subspace T2(Vn-1). Thus any vector t E T.(Vn-1) can be expressed as = wi(l)e;, where ei is a basis in the subspaceT=(Vn-1). The formulas (2.1.17) express in differential form the law of transformation of coordinates of an arbitrary tangent vector under a change of frames in the tangent subspace TZ (V n-1) Consider the differential form 1 of degree p: (2.1.18)

The condition for this form to be invariant is 64 = 0. Differentiating equation (2.1.18), applying equations (2.1.17) and the invariance condition, we find that 5Ti,i,...1

-Tki,...i,(7rk - brro)-...-T1112

It is convenient to denote the left-hand side of this equation by O6Ti,i2...i Then this equation can be written as (2.1.19)

O6Ti,i,...i, = 0.

The coefficients Ti2 12...i, of the invariant form D of degree p defined by equation (2.1.18) constitute a symmetric (0, p) -tensor, and the equations (2.1.19) express in differential form the law of transformation of coordinates of this tensor under

transformations of the stationary subgroup If the form $ is relatively invariant, then it satisfies the condition b$ = In this case the coordinates Ti, i,...i, of this form constitute a relative symmetric (0, p)-tensor and satisfy the equations: H=(Vn-1).

V6Ti,i,...i, = dTi,i,...i,.

(2.1.20)

2. HYPERSURFACES IN CONFORMAL SPACES

36

If in addition, 19 = kwo, then the relative tensor T;,i,...1, is called a tensor of weight k.

For further constructions, we will need also absolute and relative (q, p)-tensors. These tensors are defined, respectively, by the following equations:

=

06T;1i2 ...

0

. J9 =

067;1'

(2.1.21)

i.::

For such tensors the operator V6 has the following structure: v6T . 12...jq 1 it i2...iy

=

bT?'...jv

t, i2...io

-

13 q...jq (Irk

ki2...ip

&l

23

7r°.. 0) -

-Tj'j'"'j°(Irk 0) +Tk* iP -6k7r° ,i2...ip iti2...k +0

1'j9...k( 1q

k

.

-6"7r° k 0)

( 2.1.22 )

1v 0

For example, if w' = 0, the last equations of (2.1.2) can be written in the form

V6gij = 2gijiro. These equations show that the quantities g;j form a relative symmetric (0, 2)tensor of weight 2. The entries g'j of the matrix (g'1), which is inverse for the matrix (gij), satisfy the equation

V6g'j = -2g''7ro;

(2.1.23)

that is, they form a relative symmetric (2,0)-tensor of weight -2. In addition to tensors we will use quantities with a more complicated structure called geometric objects. The differential equations that these quantities satisfy, besides the tensorial part (2.1.21), contain some linear combinations of the fiber forms 7rn and 7r°, and the coefficients of these combinations are components of the same object. If we denote by 7rA all invariant forms of the stationary subgroup H=

(Vn-1),

then in the general case the differential equations satisfied by the coordinates of a geometric object have the form

6f"+FA(e)7rA=0, where F({°) are some algebraic functions. These equations must be completely integrable (see Laptev [Lap 53], p. 295). In particular, the components of tensors satisfy equations (2.1.21), where, for absolute tensors, the forms 7rA are the invariant forms Vi , i, j = 1,...,n-1, of the general linear group GL(n - 1). For relative tensors, the forms 7rA are the invariant forms 7r; and woo of the group GL(n - 1) x H.

3. In Subsection 2.1.1, on a hypersurface V"-1, we introduced the quantities g;j, A;j, Aijk, ..., connected with differential neighborhoods of V"-1 of orders 1, 2, 3, .... If w' = 0, the last equations of (2.1.2), and equations (2.1.14) and (2.1.15) take the form 6gij - gkj7rk - gik7rj = 0, bAij - Akj,r{ - A1k7rj + Aijiro + gij7rn = 0, bA;jk - >,rjk7r; - Airk7r; - a;j;7rk + 2a;jkao + 3(Alijgkir - 9(ijAk)t)7rh+, = 0.

2.1

Fundamental Objects and Tensors of a Hypersurface

37

By means of the operator V5, the latter equations can be written in the following form:

Vjgij = 29ijio,

(2.1.24)

Obaij +9ijnn = Aijio,

(2.1.25)

Vd\ijk + 3(A(ijgk)t - 9(ij)k)l)1rn+1 = \ijklr

,

(2.1.26)

As was indicated above, the quantities gij form a relative (0, 2)-tensor of weight 2. The quadratic form 9 = 9ijuj 'w

(2.1.27)

defined by this tensor is also a relatively invariant form of weight 2. As was indicated earlier, this form determines a conformal structure on the hypersurface V"-1. Note that we will denote by the same letter g the relatively invariant

quadratic form determining a conformal structure in the space C" or on its submanifold V'". This should not be confusing, since a domain of this form will be clear from the context. From equations (2.1.25) we see that the quantities \ij do not form either a tensor or a geometric object, since equations (2.1.25) contain the quantities gij. However, the quantities \ij form a geometric object jointly with the tensor g;j, and this geometric object is a basis of the pencil of tensors a;j - sg;j. This pencil of tensors determines an invariant pencil of second fundamental forms: 4i(2)(8) = ()1;j - sg;j)W'w3.

(2.1.28)

Similarly the quantities a;jk do not form a geometric object, since equations (2.1.26) contain the quantities gij and A,,. However, the quantities 1ijk form a geometric object jointly with the quantities g;j and \ij. Thus the geometric objects {g;j}, {aij,g;j}, A,,,g;j} arise. They are called the fundamental geometric objects of orders 1, 2, 3, respectively. In the same manner the quantities aijkl appearing in equations (2.1.15) form a fundamental geometric object of fourth order jointly with the quantities gij, a;j, and aijk. In a similar way, by means of consecutive prolongations of differential equations of the hypersurface V"-1, we can construct fundamental geometric objects of higher orders of V"-1 4. As we have already noted, the quantities Aij do not form a geometric object, since equations (2.1.25) contain the additional term g;j7r°. However, by means of the quantities A,, and the tensor gij one can construct a relative (0, 2)-tensor. To this end, we consider the quantity A=

1

n-

19,E

(2.1.29)

By (2.1.23) and (2.1.25), this quantity satisfies the differential equation:

as = -airs -

(2.1.30)

38

2. HYPERSURI ACES IN CONFORMAL SPACES

This equation shows that the quantity A forms a one-component geometric object determined in a differential neighborhood of second order of V"-1. Equations (2.1.25) and (2.1.30) allow us to eliminate the term containing the form 1rn. In fact, if we set (2.1.31) aif = Aif - Ag+;,

we find that

Vaaii = a,j1r ;

(2.1.32)

that is, the quantities a,3 form a relative (0, 2)-tensor of weight 1 which is connected with a differential neighborhood of second order of V". This tensor is called the second fundamental tensor of the hypersurface Vn-t The quadratic form 4i(2) = a,3w'w'

(2.1.33)

belongs to the pencil (2.1.28) of second fundamental forms determined in a differential neighborhood of second order of l'"-1. Relations (2.1.32) show that the form 4?(2) is a relatively invariant form of weight 1. Since equations (2.1.29) and (2.1.31) imply that

aii9'f = 0,

(2.1.34)

the tensor aid is apolar to the tensor g, Geometrically this means that it is possible to inscribe an orthogonal (n-I)-hedron formed by the tangent vectors el,... , en_1 into the cone determined by the equation .

ai1wiw3 = 0

(see Akivis and Goldberg [AG 931, pp. 214-216). For n = 3 (i.e., on a twodimensional surface V2), the last equation determines two directions at a point x E V2 that are orthogonal by means of (2.1.31). On the whole surface V2, this equation determines an orthogonal net, which is called the conformally asymptotic net. Note that by condition (2.1.34), rank (a;,) # 1: this rank is either 0 or it is

greater or equal to 2. In fact, if rank (aid) = 1, then there exists a coordinate system in which all components of this tensor vanish except all. But then condition (2.1.34) reduces to the form al1gll = 0. Since g" # 0, it follows that all = 0, and rank (ail) = 0. Consider now equations (2.1.26) satisfied by the quantities Aijk. It follows from these equations that these quantities do not form a tensor. However, it is possible to construct a tensor from them if we suppose that the tensor aif is nondegenerate, that is, if det(ail) i4 0. To this end, we will first write equations (2.1.30) assuming that the point x is not fixed on the hypersurface V"-1:

dA=-Awo-w,,,+Akwk.

(2.1.35)

2.1

Fundamental Objects and Tensors of a Hypersurface

39

By (2.1.23) and (2.1.14), the quantities Ak are expressed in terms of the quantities Aijk as follows:

119'jAijk. (2.1.36) n Differentiating equations (2.1.36) with respect to the fiber variables and applying (2.1.26), we obtain Ak =

V6Ai + Aiao - aii1"+1 = 0.

(2.1.37)

Equations (2.1.37) show that the quantities Ai do not form a geometric object, since this equation contains the components of the tensor aij in addition to the quantities A1. However, the quantities Ai allow us to construct two geometric

objects connected with a differential neighborhood of third order of V1-1. These two objects have the form

µ' = a"Aj, µi = 9ijµ3,

(2.1.38)

where a'j are the components of the relative (2, 0)-tensor which is the inverse tensor of the tensor aij. In fact it follows from equations (2.1.37), (2.1.24) and (2.1.32) that these quantities satisfy the following systems of differential equations: D6µ' + 21j'7ro - a;,+1 = 0, V6pi - 7r9 = 0. (2.1.39) The object µ' allows us to construct a relative (0, 3)-tensor connected with a differential neighborhood of third order of V"-1. Comparing equations (2.1.26) and (2.1.39), it is easy to prove that the quantities

aijk = Aijk + 3(a(ij9k)( - 9(ijak),)µl

(2.1.40)

satisfy the following system of differential equations: O6aijk = aijk7rp;

(2.1.41)

that is, they form a symmetric relative (0, 3)-tensor of weight 1. It is easy to see that this tensor is apolar to the tensor gij: aijkg`3 = 0.

(2.1.42)

5. Let us construct one more (0,2)-tensor connected with a differential neighborhood of fourth order of Vn-1. To this end we write the second group of equations (2.1.39) not assuming anymore that the point x E Vn-1 is fixed:

Viii - w? = µ40) ,

(2.1.43)

where Vµi = dpi - µk(w,E - 6, wo). In general, the quantities µij are not symmetric, and they can be algebraically expressed in terms of the components

of the geometric object of fourth order. These quantities are connected with the tensors gij and aij by Z(n - 1)(n - 2) algebraic conditions.

40

2. HYPERSURFACES IN CONFORMAL SPACES

Prolonging equations (2.1.43), we find that for a fixed point x E V"-1, the quantities pij satisfy the equations V514i = Aij7rn - Aiaj9 - Aj7r° +gijpkrrk.

(2.1.44)

By means of equations (2.1.30) and (2.1.39), we can eliminate the forms it and 1r9 from equations (2.1.44). As a result we find that the quantities

cij = p,, + ltwj + AAij -

1(l,kµk + AZ)gij

(2.1.45)

satisfy the differential equations

Oscij = 0,

(2.1.46)

and thus they form an absolute (0, 2)-tensor. In Section 2.2 we will prove that the tensor cij satisfies the conditions 0,

(2.1.47)

where the brackets denote alternation with respect to the indices i and j.

2.2

Invariant Normalization of Hypersurfaces

Rl(V"-1) which are 1. We now construct subbundles of the frame bundle associated with neighborhoods of second and third orders of a hypersurface Vn-1

Consider a pencil of hyperspheres X = An + sAo tangent to the hypersurface V"-1, and choose a hypersphere in this pencil which is intrinsically associated with V"-1. To this end, applying (2.1.8), we find the differential of the hypersphere X with respect to the fiber variables: bX = (6s + situ + ir°)Ao.

(2.2.1)

Since the condition for the hypersphere X to be invariant is 6X = WX, then from (2.2.1) it follows that

bs = -sera - an.

(2.2.2)

Equation (2.2.2) shows that the quantity a forms a one-component geometric object, which is called the normalizing object of first kind of the hypersurface Vn-1 Comparing equations (2.2.2) and (2.1.35), we see that s = A is a solution of equation (2.2.2). Thus the hypersphere C" = An + AAo

(2.2.3)

is connected intrinsically with the point x = Ao E V"-1. It is called the central tangent hypersphere of the hypersurface Vn-1 at the point x.

2.2

Invariant Normalization of Hypersurfaces

41

Since the geometric object A is determined by a second-order differential neighborhood of V"'', the central tangent hypersphere is determined by the same differential neighborhood. Let us find a common point X of the hyperspheres Cn and A. We represent this common point in the form

X =x°Ao+x'Ai+x"An+An+1. The coordinates of this point must satisfy the conditions: (X, X) = 0, (X, Cn) = 0, and (X, A,) = 0. From these conditions it follows that

X'=O, x" = A, and x° = 2 A2. Thus the common point of hyperspheres C" and A, is Bn+1 = An+i + AAn + 1 A2Ao.

The points CO = Ao, Bn+l, and the hyperspheres C" and A, form a family of frames determined by a second-order differential neighborhood of the point

A° E V"-'. The stationary subgroup H= (V"-') of this family of frames is a subgroup of the group Hy (V"-' ), which was considered in Section 2.1. It leaves invariant not only the tangent element (x,TT(V"-' )) but also the central

hypersphere Cn attached to a point x E V". This subgroup is isomorphic to the group (GL(n -1) x H) x T(n -1), where T(n - 1) is an (n -1)-dimensional group of translations and x denotes the semidirect product. The forms V; , 7r0, and 7r9 are invariant forms of the subgroup H= (V"-' ). The constructed family

of frames is a fiber of the frame bundle R2(V"-') of second order which is obtained from the frame bundle R' (V n-') by the reduction defined by the equation 7ro = 0.

Let us compute the scalar product (d2 A0,Cn). By means of equations (2.1.1), we find that

(d2Ao, Cn) = (A,, - Agi1)wV = aijwV.

(2.2.4)

It is then clear that the quadratic form (see (2.1.33)) 4,(2) = a,1w'w'

(2.2.5)

determines the deviation of the hypersurface Vn-' from the central tangent hypersphere Cn in a neighborhood of the point x E V"-' . The equation aijwiwi = 0

determines the cone of directions on V"-', along which the central tangent hypersphere Cn has a second-order tangency with the hypersurface With respect to a second-order frame 1Z.,, (V"-' ), equation (2.1.7) takes the form Vn-1.

w" = a,1w3.

(2.2.6)

2. HYPERSURFACES IN CONFORMAL SPACES

42

If at a point x E

V"-1 the tensor aij vanishes, then the hypersphere C.

has a tangency of second order with the hypersurface V' 1 at this point. Such points are called umbilical points of the hypersurface V"-1

Theorem 2.2.1 If n > 2 and the tensor aij is identically equal to 0 on the hypersur/ace V"-1 (i.e., all points of V"'1 are umbilical), then this hypersurface is a hypersphere or an open subset of a hypersphere. Proof. Differentiating equation (2.2.3), we obtain

dCn = (AkAo - 9'l ajkAi)wk

Since by hypothesis aij = 0, then dC,, = AkWkAo.

Let us prove now that the condition aij = 0 also implies that Ai = 0. To prove this, we note that if aij = 0, then Aij = Agij. Differentiating this and applying (2.1.2) and (2.1.14), we find that

Aijkwk -gij(dA+wn+AWp) = 0. By (2.1.35) we obtain the equations k (Aijk - 9ijAk)w = 0,

from which, by linear independence of 1-forms wk, we arrive at the equations Aijk = 9ijAk.

(2.2.7)

By alternating equation (2.2.7) with respect to the indices j and k and using the symmetry of Aijk, we find that 9ijAk - 9ikAj = 0.

Contracting the latter equation with the tensor gij, we obtain (n - 1)Ak - Ak = 0, or

(n - 2)Ak = 0, from which for n > 2, it follows that Ak = 0;

that is, dCn = 0. This means that for n > 2, the hypersphere Cn is fixed, and the hypersurface V"-1 coincides with C,, or is its open subset. If n = 2, then the space C" is two-dimensional, the hypersurface V"-1 is a curve, and the condition aij = 0 is identically satisfied and means that the

2.2

Invariant Normalization of Hypersurfaces

43

circle C" osculates this curve. But it is well-known that the osculating circle can vary from point to point. 2. The hyperspheres Ai, which are orthogonal to the hypersurface Vn-1, have a common circle, which is orthogonal to V"' at the point x. However,

this circle is not invariantly connected with the point x E V"-1. We will construct a circle that is orthogonal to V"-' and is intrinsically connected with the point x E V"''. To this end we consider the hyperspheres Xi = Ai + xiA0

(2.2.8)

and write the condition for the bundle determined by these hyperspheres to be invariant: oXi = o; Xi. Differentiating equations (2.2.8) with respect to the fiber parameters, we obtain

bXi = (Vaxi + i°)Ao + 7rj A;.

It follows that the condition for the bundle of normal hyperspheres to be invariant is Vdxi + 7r9 = 0.

(2.2.9)

This equation shows that the quantities xi form a geometric object which is called the normalizing object of second kind of the hypersurface V"-' Comparing equation (2.2.9) with the second equation of (2.1.39), we see that one of the solutions of (2.2.9) is

xi = -{`i,

(2.2.10)

and thus an invariant bundle of normal hyperspheres is generated by the hyperspheres

Ci = Ai - µiAo

(2.2.11)

The intersection of the hyperspheres Ci determines the normal circle to V"-' at the point A° which is intrinsically associated with V"-'. This circle is determined by a third-order differential neighborhood of the point A° E V". Note that the construction of the hyperspheres Ci is possible only under assumption of nondegeneracy of the tensor aid, since only under this assumption can the object pi be constructed (see Subsection 2.1.4). Let us find a common point X of the hyperspheres C" and Ci. As on p. 41, We represent this common point in the form

X = x°A° +x'Ai +x" A" + A"+1 Since X is a point, we have (X, X) = 0, and the conditions for X to belong to the hyperspheres C,, and Ci can be written as (X, C,,) = 0 and (X,Ci) = 0. From the first condition we obtain -2x° + gijx'x3 + (x")2 = 0.

'. HYPERSURFACES IN CONFORMAL SPACES

44

By (2.2.3) and (2.2.11), the remaining two conditions imply that

x"=A and x'_-9''µj. Thus 1

xe = 2 (µ+µ' + A2), where

µ` =

9`.1

µj-

Therefore the point of intersection of the hyperspheres C and C; has the form Cn+1 = An+1 - µ'A; + AAn +

1

(µ,µ''\2)Ao

2

(2.2.12)

The point Cn+1, as well as the hyperspheres C are determined by a third-order Vn-1 differential neighborhood of the point Ao E The points Co = Ao and Cn+1 and the hyperspheres C; and Cn determine a family of frames, which is intrinsically connected with the point x E Vn-1.

The stationary subgroup H'(Vn-1) mapping this family of frames into itself is a subgroup of the stationary subgroup HZ(Vn-1), which we considered in Subsection 2.2.1. The subgroup H'(Vn-1) leaves invariant the tangent element (x,TZ(V"-1), the central hypersphere Cn, and the point Cn+l attached to a point x E Vn-1. This subgroup is isomorphic to the direct product GL(n - 1) x H. Its invariant forms are the forms it and iro. The family of frames we have constructed is a fiber of the frame bundle 13(V"-') with the base Vn-1. This bundle was obtained from the frame bundle 1Z2(Vn-1) by the reduction defined by the equation 7r° = 0. The stationary subgroups H (V n-1), H' (V n-1) and H= (V n-1) are connected by the following inclusions:

Hl(Vn-1)

H2(Vn-1)

H3(Vn-1).

We emphasize one more time that our construction of the bundle 1t 3(V"-1) of frames of third order is possible only under assumption of nondegeneracy o the tensor a, . 3. We will now write the equations of infinitesimal displacement of frames of third order, which we constructed in Subsection 2.2.2, in the form 0,17

C,1, ,n=0,I__n+1.

(2.2.13)

Since the elements CE of the frame {CF} satisfy the same conditions (2.1.1) that the elements AE of the original frame {At} satisfy, the forms a satisfy

equations (2.1.2). But since the group H. ,',(V-') is smaller than the group N' (V"-1), the forms an and aP can be expressed in terms of the basis forms. If we substitute for the points Cf in (2.2.13) their expressions in terms of At,

The Rigidity Theorem and the Fundamental Theorem

2.3

45

then a simple calculation will lead to the following expressions for the forms a" in terms of the forms f :

ok =Wk a" = 0 ao = Wo + pkWk, ai = w; - gik(pkWj - pJWk),

(2.2.14)

aj = aijW , an = 0, a? = -cijwj, a!'+' = gijwj. Now it is easy to prove conditions (2.1.47) for the tensor cij. Exterior differentiation of equation an = 0 gives

4 A a° = 0 or g'jajn A o° = 0.

If we substitute for aj and a° their values taken from (2.2.14), we obtain the desired relation (2.1.47). Consider now the submanifold described by the point Cn+1. The tangent subspace to this submanifold is determined by the differential of this point:

dCn+i = g'iaPCi = -giiCjkWkCi,

(2.2.15)

where the tensor cij is defined by formulas (2.1.45). Thus the dimension of the submanifold described by the point Cn+1 is equal to the rank of the tensor cij. In the general case, if the tensor cij has the maximal rank n - 1, then this

point describes a hypersurface V. Since by (2.2.15) we have (dCn+1, Cn) = 0,

this hypersurface is the second sheet of the envelope of the family of central tangent hyperspheres Cn of the hypersurface Vn-1. The first sheet of this envelope is the hypersurface V"-1 itself. The following theorem summarizes constructions we have made:

Theorem 2.2.2 If the ranks of the tensors aij and cij of the hypersurface V n-' have their maximal value n -1, then the fundamental geometric object of third order of V"-1 allows us to construct an invariant normalization of Vn-' by means of the family of central tangent hyperspheres Cn, and the family Vn-' and intersect the central of invariant circles which are orthogonal to hypersphereof V"-1 at the points of their tangency with the two hypersurfaces V"-1 afd Vn-1; these hypersurfaces are the two sheets of the envelope of the family of central tangent hyperspheres C".

2.3

The Rigidity Theorem and the Fundamental Theorem

1. Consider two smooth, oriented, connected, and simply connected hypersurfaces V"-' and V"-1 of the conformal space C". Suppose that there is a one-toone correspondence f : V"-1 -> V"-1 under which f (x) = i, where x E Vn-1

46

2. HYPERSURFACES IN CONFORMAL SPACES

and 7 E V"-1. The correspondence f induces a mapping f. of the tangent -1): bundle T(Vn-1) onto the tangent bundle T(V f.: T(V--') -a T(V _1) such that f.JV.,-, = f and f.ITT(V"-1) is a linear nondegenerate mapping (e.g., see Kobayashi and Nomizu [KN 63), vol. 1, p. 8). Under this mapping any geometric object fl C Ti(V"-1) will be mapped onto the corresponding object f.f2 of the subspace Tz(V -1). We will now prove the theorem on conformal rigidity of hypersurfaces.

Theorem 2.3.1 Let n > 4 and

V"-1 and Vn-1 be two hypersurfaces without

umbilical points in a real conformal space C". Suppose that there is a one-toone correspondence f : V"-1 -+ V-1 between points of these hypersurfaces, and that at corresponding points of V"-1 and V"-1 the following conditions hold:

g = r2 f.g and '(2) =

r 76 0,

(2.3.1)

where g and g are the first fundamental forms of the hypersurfaces Vn-1 and V-1, and 4'(2) and T(2) are their second fundamental forms. Then the hypersurfaces V"-1 and V -1 are conformally equivalent. Note that the form of relations (2.3.1) is explained by the fact the quadratic forms g and Z(2) are relatively invariant forms of weights 2 and 1, respectively (see Subsections 2.1.3 and 2.1.4).

Proof. Let x E Vn-1 and 7 E V-1 be two corresponding points of the hypersurfaces V"-1 and V n-1, and let W. be a conformal transformation map-

ping x = Ao into Y = Ao and the central tangent hypersphere C" into the central tangent hypersphere On. Then the equations of V"-1 and V"-1 have the form

wo = 0, wo = 0.

(2.3.2)

Moreover the basis forms of V"-1 and V"-1 are equal:

wo=4i

(2.3.3)

Since the first fundamental forms of the hypersurfaces V"-1 and V n-1 are

9 = g,jwV, g = g,,w'w', and their second fundamental forms have the form

4i(2) = aijw'ci, ''(2) ='aijw'u)i, where g'1ai3 = 0 and g jai,j = 0 (see (2.1.27), (2.1.33), and (2.1.34)), it follows that relations (2.3.1) are equivalent to the relations gi,j = r2gi3

(2.3.4)

The Rigidity Theorem and the Fundamental Theorem

2.3

47

and

aij = raij.

(2.3.5)

Here and in what follows we write simply gij, aij, etc. instead of f.gij, f.aij, etc.

Since r $ 0, then by renormalizing the point A0, this factor can be reduced to 1. In fact, setting

we find that dAo

=d(1 ) Ao+1(woAo+w'A,) r `r

and that (dTo, dTo) = T,9ijw'w' = 9ijw'w3

Thus we obtain

9ij = gij. Of course, in order to preserve condition (Ao,

(2.3.6)

-1, we must also nor-

malize the point A,,+1 as follows: A,,+1 = rA,,+1.

It is easy to check that under the transformation Ao = Ao, we have -4 -4 4P(2) = (d2Ao, C,) = T (d2Ao, Cn) = T t(2) = f.

(2),

where C,, and C,, are the central tangent hyperspheres of the hypersurfaces V"-1 and V-1, respectively. It follows that

aij = aij.

(2.3.7)

Note that in (2.3.6) and (2.3.7) we wrote gij and aij instead of g;j and aij. Taking the exterior derivatives of equations (2.3.3), we obtain

[0;-w- a;(o -wo)]nwj=0. Applying Cartan's lemma to these equations, we find that 0 j , - wf = b (wo - wo) + Tjkwk,

(2.3.8)

where T,'k = Tkj. It is easy to prove that the quantities TTk form a (1, 2)-tensor, which is called the deformation tensor of the tangent bundle. Differentiating equations (2.3.6), we obtain

9ik(w,-wj)+9kj(W -wk)=0. Substituting for 0, - w, the values taken from (2.3.8), we find that 2gij(wo - woo) + (9ikT,i + 9kjT,i)w' = 0.

(2.3.9)

2. HYPERSURFACES IN CONFORMAL SPACES

48

It follows that the 1-form wo - w(O) is expressed in terms of the basis forms w': 000 - Wo = sjw'.

(2.3.10)

Next we make the transformation A; = A; + x;Ao in the pencil of normal hyperspheres. Since Ao = A0, we have dAo = woAo + w'Af = woAo + w'(Ai - xiAo).

It follows that '-G W')

By (2.3.10), from this we find that 1000 = woo + (s; - x;)W

We can see now that by setting x1 = s,, we reduce relation (2.3.10) to the form 000 = W0O*

(2.3.11)

By (2.3.11), equations (2.3.9) take the form g

9jlTI!k = 0.

By cycling these equations with respect to the indices i, j, and k and subtracting the first equation from the sum of the last two equations, we obtain the conditions

T = 0,

by means of which equations (2.3.8) become w'i = Wf.

(2.3.12)

Taking the exterior derivatives of equations (2.3.11), we obtain the exterior quadratic equation (w - w°) A w' = 0,

from which, by Cartan's lemma, it follows that

w -W° = t;iwi, t$3 = ti;.

(2.3.13)

Taking the exterior derivatives of (2.3.12), we obtain

WiA(E- wP)+cii A 0, -w;,Aw +9`k9il(wk-wk)Aw'=0. By (2.3.7), the second and third terms on the left-hand side cancel out. Substituting for w - w° in the remaining terms the values taken from (2.3.13), and using the fact that the forms w' are linearly independent, we find that

-tikdi + tilUk + g""(tmk9il - tmI9ik) = O.

2.3

The Rigidity Theorem and the Fundamental Theorem

49

Contracting this relation with respect to the indices i and k, we arrive at the equation

(n - 3)tj1 = -tgil,

(2.3.14)

where t = g'mt;,,,. Since n > 4, by contracting the latter equation with the tensor get, we find that (2n - 4)t = 0. It follows that t = 0, and consequently tji = 0. Equation (2.3.13) now takes the form

w =w°.

(2.3.15)

= w; .

(2.3.16)

By (2.2.6) and (2.3.7) we have

Taking the exterior derivatives of (2.3.16), we obtain

w; +I A (W - wn) = 0.

(2.3.17)

By (2.1.2), even for n > 3, the forms w"+1 are linearly independent. Hence equations (2.3.17) imply that w = W0n'

(2.3.18)

Since exterior differentiation of (2.3.16) and (2.3.18) leads to identities the system of equations (2.3.2), (2.3.3), (2.3.6), (2.3.11), (2.3.12), (2.3.15), (2.3.16), and (2.3.18) is completely integrable.

Moreover equations (2.3.2), (2.3.3), (2.3.6), (2.3.11), (2.3.12), (2.3.15), (2.3.16), and (2.3.18) show that all components of an infinitesimal displacement of moving frames of second order associated with the hypersurfaces V"-1 V n-' can be and V"-1 coincide. Thus, by Theorem 1.2.1, the hypersurface obtained from the hypersurface V"-I by means of a conformal transformation. -I Therefore, the hypersurfaces V"-1 and V are conformally equivalent. E As we can see from the equation (2.3.14), the proof of Theorem 2.3.1 fails if n = 3. For this case it is necessary to add certain additional conditions to (2.3.1) that are connected with a third-order differential neighborhood (see Schiemankgk and Sulanke [SSu 80)). 2. Next we will prove the fundamental theorem on determination of a

hypersurface Vn-1 C C" by a system of tensors. By Theorem 1.2.1, a hypersurface V" -I C C" will be determined up to a conformal transformation of the space C" if all forms w, are expressed in terms of the differentials of a certain system of parameters. If we take the frame bundle R.3 (V"-' ), which we have constructed in Section

2.2, then formulas (2.2.14) show that the forms a", o; +1, o; , on, and o° are already expressed in terms of the basis forms a' = w'. We need only to express the forms ao and a; in terms of the forms a'.

2. HYPERSURFACES IN CONFORMAL SPACES

50

In order to find an expression for the form 0100, we will assume, as we did in

Section 2.2, that det(a;3) 0 0, and define the relative invariant

a=

{::}

(2.3.19)

Applying the rule of differentiation of determinants, we find the differential equation which the invariant a satisfies, dloga = -a00 + bkak,

(2.3.20)

where bk =

I aijagk, n-1

(2.3.21)

is a (0,1)-tensor expressed in terms of the covariant derivatives of the tensor a;j and the tensor a;j itself. Now the form a00 is expressed as follows:

00 = -d log a + bkak. To avoid unnecessary complications in computing the forms curvilinear coordinates u1, ... , u"'1 and assume that 01' = du'.

(2.3.22)

we introduce (2.3.23)

Exterior differentiation of these equations gives the following exterior quadratic equations: (aj' A dui = 0,

from which, by Cartan's lemma, it follows that ai = bja00

+ f,kduk, r k = Ii,.

(2.3.24)

The 1-forms aj' satisfy the equations similar to the last equations of (2.1.2). Substituting their values (2.3.24) into these equations, we find that d9;, = 9+k(b a00 + qdu') + 9kJ(6kaoe + r du1) or

d91,, - (9+11'-k +g1i1;k)duk = 2.911a00.

Equations (2.3.22) can be written in the form 0 k ao = akdu,

where al. =bk - 2,1,o

a

(2.3.25)

2.3

The Rigidity Theorem and the Fundamental Theorem

51

Substituting these values for ao into the previous equations, and taking into account that du' are linearly independent, we find that

9ij,k - 9ilr;k - 9ljrik = 29ijak,

(2.3.26)

where gij,k = -. euCycling these equations with respect to the indices i, j, and k and subtracting the first equation from the sum of two others, we obtain

r = L9k1(9jl.i + 9r;.j - 9;j.:) - 6 ai - dkaj + 9ijak,

(2.3.27)

where ak = 9k'al. By formula (2.3.24), the forms ai are expressed now in terms of the tensors gij, aij and their derivatives. Using the structure equations, we now find the expression for dad - a Aak:

dad - a A ak = (cjk6l + gjk9`hChl - g$hajkahl)duk A du'.

(2.3.28)

As usual, we will use the following notation for the left-hand side of (2.3.28):

dog - a k A ak = 2

A du',

(2.3.29)

where R'kl is the Riemannian tensor. Since the forms a,' are known, then the left-hand side of (2.3.28) is expressed in terms of the tensors gij, aij, and their derivatives of order not higher than 2. Thus the quantities Rj'kl in formulas (2.3.29) are also expressed in terms of the tensors gij, aij, and their derivatives of order not higher than 2. Note that the exterior 2-form dad - a A ak is the curvature form of the Weyl connection, which is induced on the hypersurface V` by the bundle of invariant frames {CE} (see more details on this in Section 4.3). The tensor Rj'kt is the curvature tensor of this connection. From equations (2.3.28) and (2.3.29) it follows that Rjkl =

5 c3, + gih(9jkCh/ - 9j1Chk) - gih(ajkahl - ajlahk).

(2.3.30)

These equations allow us to express the tensor cij in terms of the tensors gij, aij,

and their derivatives of order not higher than 2. To obtain these expressions, we contract equations (2.3.30) with respect to the indices i and I and use the apolarity of the tensors gij and aij. As a result we obtain Rjk = (n - 3)Cjk + Cgjk + gihajiahk,

(2.3.31)

where Rjk = Rjiki and c = g'jcij. The tensor Rjk is the Ricci tensor of the Weyl connection indicated above. If we contract equation (2.3.31) with the tensor gjk, we find that

gjk Rjk = (2n - 4)c + I,

2. HYPERSURFACES OF CONFORMAL SPACES

52

where I = gjkg'^aj;ahk. From this, for n > 2, it follows that c

I

2(n --2)

(R - I),

(2.3.32)

where R = gjkRjk is the scalar curvature of the Weyl connection. Now, for n > 3, from (2.3.31) we find the quantities cjk: cjk = n 13 [Rjk

- 2(nI 2)gjk(R - I) - g"ajiahk].

(2.3.33)

Thus the tensor cjk is also expressed in terms of the tensors g;j, a;j, and their derivatives of order not higher than 2. It follows from the structure equations of the space C" that the tensors gij and a;j satisfy the equation (2.3.30), which is similar to the Gauss equation in the theory of surfaces of the Euclidean space, and the equation

(da;j - akjo; -

A duj = 0,

(2.3.34)

which is similar to the Mainardi-Codazzi equations. Thus we have proved the following fundamental theorem of the theory of hypersurfaces of the space C":

Theorem 2.3.2 Suppose that in a connected and simply connected domain D

of the space of variables u1,...,u", the tensors gij and aij are given, and they satisfy the equation g'ja;j = 0 and equations (2.3.30) and (2.3.34), where the tensor c;j is expressed by formula (2.3.33). Then, for n > 3, in the space C", up to a conformal transformation of C", there exists a unique hypersurface V' for which the tensors g;j and a;j are the first and second fundamental tensors, respectively.

The case n = 3 is considered in detail in Schiemankgk and Sulanke (SSu 801.

2.4

Curvature Lines of a Hypersurface

1. We can define the curvature lines on a hypersurface V"-' in the conformal

space C" using the same definition as that used in a Euclidean space or a Riemannian space (e.g., see Schouten and Struik [SS 38], Pt. II, §8). In our notation, this definition will have the following form. Consider a symmetric affinor:

aj = g'ikakj,

(2.4.1)

which is constructed by means of the tensors gij and aij and is called the affinor of Burali-Forti (see Burali-Forti [Bur 121). To any direction on the hypersurface

Curvature Lines of a Hypersurface

2.4

53

V"-1, emanating from the point AO and defined by values l;' of the basis forms co', this affinor sets in correspondence the direction defined by the values

ri' =

(2.4.2)

The directions that emanate from the point AO and are invariant with re-

spect to the affinor a., (the eigendirections of the affinor ai) are called the principal directions of the hypersurface V"-1. Since the tensors ail and gij are nondegenerate at the point AO and the tensor g,j is positive definite, at this point there exist n - 1 mutually orthogonal principal directions. If at a point AO of a hypersurface V"-1 the affinor a; has n - 1 distinct eigenvalues, then exactly n - 1 principal directions emanating from this point. If p eigenvalues of the affinor all coincide, then they define a p-dimensional domain LP of principal directions, and each direction of LP is a principal direction. The curves of V"'1, enveloping the principal directions, are called the curvature lines of

V"-1. If at any point of the hypersurface the affinor a has n - I mutually distinct eigenvalues, then through any point of V"-1 there pass exactly n - 1 mutually orthogonal curvature lines (see Figure 2.4.1 for the case n = 2). If this is the case, we say that the hypersurface V"-1 carries a net of curvature lines.

2. There is another way to define the principal directions and the curvature lines. Consider a hypersphere C"+sCo that is tangent to the hypersurface

V"'1 at its point A0. Let us find those directions on V"-1 along which this hypersphere has a second-order tangency with V"-1. Such directions are determined by the equation (d'Co,C" + sCo) = (a;j - sg;j)o'o' = 0. Thus the desired directions constitute a cone of second order with its vertex

at the point Ao. In general, the rank of the quadratic form (a;j - sg;j)a'ai is equal to n - 1, since the tensor g;q is nondegenerate. However, for some values of s, this rank can be reduced. These values of s are determined by the

Figure 2.4.1

54

2. HYPERSURFACES OF CONFORMAL SPACES

equation (2.4.3) det(a,,, - sg,,) = 0. Suppose that a1, ...,a.-, are roots of equation (2.4.3). Then the hypersphere Bh = C" + ahC0 has a second-order tangency with the hypersurface ['"-1 along the directions, defined by the equation

(a,, - ah9ji)oY = 0.

(2.4.4)

Since the rank of the quadratic form in the left-hand side of equation (2.4.4) is less than n - 1, equation (2.4.4) determines a degenerate cone. If ah is a simple root of equation (2.4.3), then the rank of the quadratic form in the left-hand side of equation (2.4.4) is equal to n - 2, and the vertex of the cone (2.4.4) is a one-dimensional direction defined by the system of equations

(a*, - ahg,j)a = 0.

(2.4.5)

If ah is a root of multiplicity p of equation (2.4.3), then the rank of the quadratic form in the left-hand side of equation (2.4.4) is equal to n - p - 1, and the vertex of the cone (2.4.4) is a p-dimensional direction, also defined by the system (2.4.5). The direction at Ao E V"-, determined as indicated above, is a principal direction of the hypersurface V"-1. In fact equation (2.4.4) coincides with the characteristic equation of the affinor the roots of this equation coincide with the eigenvalues of this affinor, and system (2.4.5) determines the principal directions of 1,,"-1 Since the principal direction corresponding to an eigenvalue ah belongs to the cone (2.4.4), the hypersurface V"'1 has a second-order tangency with the hypersphere Bh along this direction. The hyperspheres Bh are called the contact hyperspheres of the hypersurface V` (cf. Klein (Kl 26a), §26).

3. Suppose that the affinor a of the hypersurface V"'1 has distinct eigenvalues a1i ... , a"_1. Then, at any point Ao E V"-1, there exists a unique system of n - 1 mutually orthogonal principal directions. Let us choose the hyperspheres C; of our moving frame in such a way that they are orthogonal to the corresponding principal directions of the hypersurface V"'. Then they will be mutually orthogonal. In addition we normalize the hyperspheres C, by the condition (C,, C,) = 1. As a result we obtain

9ii = bi

(2.4.6)

where b;j is the Kronecker symbol. Now the principal direction orthogonal to the hypersphere Ch is defined by the system of equations

o` = 0 for i 0h. Since this direction is an eigendirection of the affinor a'., we have

a; = bija

(2.4.7)

Curvature Lines of a Hypersurface

2.4

55

where ai are eigenvalues of the affinor a, or

aij = b;jai. The system (2.4.7) defines the hth family of curvature lines on the hypersurface Vn-1.

By (2.4.6) and (2.1.2), equations (2.2.14) connecting the forms ai imply

that In particular, these equations imply that ai = 0. In this equation there is no summation with respect to the index i. In what follows in this section, summation is assumed over the indices i, j, k only if there is the summation

sign E. Next, differentiating equations aij = dijai and applying (2.1.14), (2.1.35) and (2.1.2), we find that

(a1 - aj)ai = E aijkak+ k

dai + aiao =

aijkak k

where aijk are determined by equations (2.1.40). Since ai 54 aj, the latter equations imply the following expressions for the forms o; : aijkak a; =- Eka; - aj

i 54 j.

(2.4.8)

4. The hypersphere Ch, passing through the point Co and orthogonal to the hth principal direction, along with the hypersphere C. and the point Co, define an (n - 2)-dimensional element Ln-Z. The set of all such elements on the hypersurface Vn-1 is determined by the Pfaffian equation ah = 0.

(2.4.9)

If equation (2.4.9) is completely integrable, then on the hypersurface V"-1, this equation defines a one-parameter family of (n - 2)-dimensional submanifolds that are tangent to the elements L"-2. This family of submanifolds is orthogonal to the hth family of curvature lines and carries the remaining n - 2 families of curvature lines. The net of curvature lines on the hypersurface Vn-1 is said to be holonomic if equation (2.4.9) is completely integrable for any value of the index h. Now it is not difficult to prove the following theorem:

Theorem 2.4.1 Suppose that the tensor aij of a hypersurface Vn-1 has distinct eigenvalues and V"'1 is referred to the net of curvature lines. Then the hypersurface V"-1 carries a holonomic net of curvature lines if and only if all components of its tensor aijk with mutually different indices i, j, and k vanish.

2. HYPERSURFACES OF CONFORMAL SPACES

56

Proof. The condition of complete integrability of equation (2.4.9) can be written in the form dah A ah = 0,

from which it follows that

a'Aa'Aah=0. Substituting here for the forms a;' their values taken from (2.4.8), we find that E.i.k aihk

a' A ok A oh = 0.

ai - ah

(2.4.10)

If i., k, and h are distinct, it follows from (2.4.10) that aihk

akhi

ai - ah

ak - ah

Since the tensor aiik is symmetric, it follows from the last equation that

(ak - ai)aikh = 0, and since ak # a1, we find that (2.4.11)

aikh = 0

for mutually distinct values of the indices i, k, and h. U If a hypersurface V"-1 carries a holonomic net of curvature lines, then condition (2.4.11) must be satisfied for all values of h. Thus, for such a hypersurface, only those components of the tensor aiik that have at least two identical indices are nonvanishing. The forms a of such a hypersurface can be reduced to the following form

a; =

ai

a. (aiiio'

+ aiiioi).

(2.4.12)

We do not consider here the problem of existence of hypersurfaces carrying a holonomic net of curvature lines, since in Section 3.2 we will be able to solve this problem in a more general setting.

5. Let us find how the contact hypersphere Bi = C,, + aiC0 varies as the point Co moves over the hypersurface V11-1. We have

dBi = > aiikakCo - E(ak - ai)akCk. k

k#i

This implies that, in general, the hypersphere Bi depends on n - 1 parameters. As the point Co moves along the ith curvature line ak = 0, k 0 i, we have

dBi = aiiiQ'Co

2.4

Curvature Lines of a Hypersurface

57

This means that the hypersphere B, + dBi belongs to the parabolic pencil of hyperspheres whose basis consists of the point Co and the central hypersphere C. .

Now suppose that aiii = 0. Then dBi = >[aiikCo - (ak - ai)Ck)Ok. k#i

This implies that now the hypersphere B, depends only on n - 2 parameters. As the point Co moves along the ith curvature line all = 0, k # i, the hypersphere Bi remains fixed. Thus the hypersurface V' ' under consideration is the envelope of the (n - 2)-parameter family of hyperspheres Bi. The characteristics of this family of hyperspheres are circles along which the hyperspheres B, intersect the hyperspheres Ck, k i4 i. A parametric equation of these circles can be written in the following form: P = Co + tCi -

t2

2 (Cn+i

1

- aiC" - 2a; Co)

Namely these circles are the ith family of curvature lines of the hypersurface V"-1. Thus we have proved the following result:

Theorem 2.4.2 Suppose that the tensor aij of a hypersurface Vn-' has n -1 distinct eigenvalues and V"-' is referred to the net of curvature lines. Then the hypersurface Vn-I is the envelope of an (n - 2)-parameter family of hyperspheres if and only if at least one diagonal component aiii of the tensor ai2k vanishes.

The hypersurfaces described in Theorem 2.4.2 are called the canal hypersurfaces (see Figure 2.4.2 for n = 3).

Figure 2.4.2

2. HYPERSURFACES OF CONFORMAL SPACES

58

Figure 2.4.3

If on a hypersurface V"-' the conditions a;,, = 0 hold for all values of i, then this hypersurface is the envelope of n -1 families of hyperspheres, each of which depends on n - 2 parameters. Such a hypersurface is a multidimensional analog of the well-known Dupin cyclide (see Figure 2.4.3 for n = 3).

Since the tensor ail, is apolar to the tensor gij, in our moving frame we find that if n = 3, then the condition aiii = 0 implies that 0; that is, the conditions aiJk = 0 hold for any values of the indices i, j, and k. Since aijk is a tensor, it vanishes not only in the special frame under consideration but also in any first-order frame. Thus we have proved the following theorem: Theorem 2.4.3 A two-dimensional surface V2 of a three-dimensional conformal space C3 is a Dupin cyclide if and only if its tensor aijk, determined by a differential neighborhood of third order, vanishes.

This differential geometric characterization of the Dupin cyclide is of an invariant nature; that is, it does not depend on either the choice of a coordinate system on the surface V2 or the choice of a conformal frame associated with the surface V2. Since the tensor aijk can also be calculated for a surface V2 of a three-dimensional Euclidean space, the invariant characterization of the Dupin cyclides we have obtained is also valid for the Dupin cyclides in the Euclidean space R3.

6. We now consider a hypersurface V"-' whose affinor a; has a multiple eigenvalue at each point Ao E V"''. Suppose, for example, that an-m = an-m+1 = ... = an-I = a,

m > 2,

is an eigenvalue of multiplicity m of this affinor. Then the rank of the tensor aid - agij is r = n - m - 1, and the system of equations (2.4.5) determines an m-dimensional subspace L'" of principal directions at the point Ao.

Curvature Lines of a Hypersurface

2.4

59

We choose a first order frame of V"-1 in such a way that its hyperspheres A1,..., An-,n-1 are tangent to the direction L'a, and the hyperspheres

An-m, ... , An-1 are orthogonal to L". We will use the following ranges of indices:

a, b, c, ... = 1,...,n - m - 1; p, q, s = n - m, ... , n - 1; and

i, j, k, ... = 1, ... , n - 1. Then, in the chosen frame, we have aap = 0,

gap = 0,

(2.4.13)

apq - agpq = 0,

and the system of quantities (2.4.14)

bab = aab - agab

forms a nonsingular square matrix of order r. The formulas for differentiation of the tensors g;3 and aid become dgab = gacab + 9cbaa, (2.4.15)

0 = gabap + 9pgan+

dgpq = gp,aq + 9,gay

and daab = aacab + acbaa - aabap + aabkak,

0 = aabap + apgaa

+ aapkak,

dapq = apsac + a,gap' - apgag +

2.4.16)

apgkak.

If we differentiate the last of equations (2.4.13) and apply equations (2.4.15)(2.4.16), then we obtain the equations 9pq(da + aap) = apgkak,

from which we find that da + aao = bkak and

apgk = gpgbk

But the tensor apgk is symmetric. So, by taking k = s in the last equation, we find that apq, = gpgb, = gspbq = gqsbp.

Contracting this equation with the tensor gpq, we obtain

mb, = b,. Since we assume that m > 2, it follows that

b,=0,

s=n-m,...,n-1,

2. HYPERSURFACES OF CONFORMAL SPACES

60

and as a result we arrive at the following equation:

da + aao = bca`,

c = 1,...,n - m - 1.

(2.4.17)

Next, differentiating equations (2.4.14), we find that dbab = bncab + bcbO - ba6ao + (babc - g°Dbc)a` + aabpap.

This shows that the system of quantities bnb forms a symmetric relative tensor. Finally, from the middle equations of (2.4.15) and (2.4.16), we obtain b°6a6p = -a°pkak.

(2.4.18)

It follows that the forms op are principal forms, and by the middle equations of (2.4.15), the forms aQ are also principal forms. Now let us consider the hypersphere B = C,, + aC°. We have dB = (da + aao)Co - g°6b6,a°C°. By (2.4.17), this implies that

dB = a`Bc, where B = b,,Co - g°6b6cC°

Thus the hypersphere B depends only on r parameters. By equations (2.4.18) and symmetry of a°,q, we have da° = a6 A (a6° + b°`acepap),

where b°` is the inverse tensor of the tensor bah. This implies that the system of equations o° = 0 is completely integrable on the hypersurfaceV"-1. On V"-1 these equations define m-dimensional submanifolds along which the hypersphere B is fixed. But these m-dimensional submanifolds are the characteristic submanifolds of the r-parameter family of hyperspheres B. Therefore each of these submanifolds is the intersection of the hypersphere B and the independent hyperspheres B,, and each is an m-dimensional sphere. Thus, if m eigenvalues of the tensor a,3 of the hypersurface V"-1 coincide, then this hypersurface is the envelope of an r-parameter family of hyperspheres (r + m = n - 1) and carries an r-parameter family of m-dimensional spherical generators along which these hyperspheres are fixed. Such a hypersurface V"-1 is called an m-canal hypersurface. It is easy to prove the converse: if a hypersurface Vn-1 is an m-canal hypersurface where m > 2, then its tensor a,3 has m equal eigenvalues. In fact an m-canal hypersurface is the envelope of an r-parameter family of hyperspheres

2.5

Geometric Problems Connected with the Tensor cij

61

B, where r = n-m-1. Since B is a tangent hypersphere, it can be represented as a linear combination of the central tangent hypersphere C and the point Co; that is, the hypersphere B has the following decomposition:

B=C,+aCo. Then

dB = (da + aoo)Co + (an + ao`)Ci, where

o" + act = -g ik(akj - agkj)o j. i

i

Since the hypersphere B depends on r parameters, the rank of the matrix (akj - agkj) is equal to r. But this can be the case only if a is an eigenvalue of multiplicity m of the tensor akj. Thus we have proved the following result:

Theorem 2.4.4 For m > 2, a hypersurface

is an m-canal hypersurface if and only if its tensor aij has an eigenvalue of multiplicity m. Vn-1

Let us consider separately the case r = 0. In this case all eigenvalues of the tensor aij coincide. Since this tensor is apolar to the tensor gij, all these eigenvalues are equal to 0, and aij = 0. By Theorem 2.2.1, a hypersurface V` I is a hypersphere if and only if its tensor aij vanishes. Note that according to the terminology of this subsection, the canal hypersurfaces described in Theorem 2.4.2 are 1-canal hypersurfaces. However, while

the characterization of m-canal hypersurfaces for m > 2 is given in terms of the tensor aij itself, the characterization of 1-canal hypersurfaces is given in terms of the first derivatives of the tensor aij.

2.5

Geometric Problems Connected with the Tensor ci9

1. Let us find a geometric meaning of the tensor cij of a hypersurface V"-1 To this end we consider the second semienvelope V"-1 of the envelope of the family of central hjperspheres C,, of V"-1. As we have proved in Section 2.2, the hypersurface V11-1 is described by the vertex Cn+1 of the invariant frame of the original hypersurface V"-1. We recall one more time that to be able to construct the point Cn+1 in Section 2.2, we imposed the condition det (aij) 0 0. We assume that this condition is satisfied throughout Section 2.5. The infinitesimal displacement of a point of the hypersurface by the equation dCn+1 = -oOCn+l + an+1C

where

i 0 -- -gikckjo. j n+1 = gik ok

,

V"-1 is defined

62

2. HYPERSURFACES OF CONFORMAL SPACES

If we define the affinor c' by setting Cj = g'kckj,

(2.5.1)

then we obtain

dCn+1 = -ooCn+i - cca Ci. Thus the affinor c, determines an infinitesimal displacement of a point on the hypersurface Vn-' Consider the quadratic form (dC"+1, dC"+1) which defines the conformal

structure on the hypersurface V"'. We have (dCn+l,dCC+1) = gklckCjltQ.t = 9 CkiCIjQlQ]

Thus the tensor gk'ckictj defines the conformal structure on the second semienvelope V"-1 of the envelope of the family of central hyperspheres C" of the hypersurface V"-1 2. Next we consider the hypersurfaces V"-' C C", on which the tensor cij is symmetric. We will prove that such hypersurfaces are characterized by the fact that the curvature lines on both semienvelopes of the envelope of the family of central hyperspheres of Vn-1 correspond to one another. This result comes from the following lemma:

Lemma 2.5.1 If the tensor a; is nonsingular, then the symmetry of the tensor cij of a hypersurface V'- is equivalent to the fact that the affinors ar and c 1

commute.

Proof. The conditions gklclkiajll = 0,

which the tensor cij of an arbitrary hypersurface Vn-1 satisfies, can be written in the form aictj = ciaij.

Contracting these equations with the tensor gjk, we find that aicljgjk = ciai .

(2.5.2)

If the tensor cij is symmetric, then

ctjgjk = cjlgik = ci . This equation and equation (2.5.2) imply that a1 Ci = c!`a;;

in words, the affinors ni and cil commute. Conversely, since the tensor a; is nonsingular, the last condition and equations (2.5.2) imply that the tensor cij is symmetric.

2.5

Geometric Problems Connected with the Tensor cij

63

It is well-known (e.g., see Gantmacher [Ga 53], Ch. 9, §15) that if the affinors a and c commute, then they can be simultaneously reduced to diagonal form. Since the affinor r; is symmetric, then at any point to E Vn-1, there exists an orthogonal basis in which both affinors are simultaneously reduced to diagonal forms. This basis is the basis of principal directions of both hypersurfaces, Vn-' and V"-', since in this basis the quadratic forms (dCo, dCo), (dCn, dCn), and (dCn+,, dCn+i) are simultaneously reduced to sums of squares. This immediately implies that the curvature lines of the hypersur-

faces V"-' and Vn-' correspond to each other. Note that in the three-dimensional case the correspondence of the curvature lines of the surfaces V2 and [V2 is a criterion for the surface V2 to be isothermic

(e.g., see Blaschke [BI 29], §72). Thus the hypersurfaces with a symmetric tensor cij may be called isothermic. We note a few more properties of hypersurfaces Vn-' C C" with a symmetric tensor C. First of all, it is easy to see that on such hypersurfaces the differential 1-form co is a total differential, since, if cij = chi, then

do°°=o'Aol =cijo'no2 = 0. Thus oo = dcp. In view of this, the tensor gig, which determines on V"' a conformal structure, can be normalized to be covariantly constant on Vn-' in the Weyl connection mentioned in Section 2.3. In fact it follows from formulas (2.1.2) that Vgij = 2gijdcp. Consider now the tensor g'ij = agi,. For this tensor we have

Vgit = gijdo + aVgi, = gii(da + 2adcc).

If we set da + 2adcc = 0, then we find that a = e-2,0.

Therefore gig = e-2,pgj1, and Vt,, = 0. Thus the tensor gi,j defines a Riemannian metric on the hypersurface V"-'. It follows from equation (2.3.31) that the Ricci tensor of this metric is symmetric. The converse is obvious: if the family of invariant frames constructed in Section 2.2 induces on the hypersurface V"-' a Riemannian metric, then the

form oo = 0, and it is easy to prove that the tensor cij is symmetric on the

hypersurface V`. Finally, we consider the following geometric property of hypersurfaces V"-' C C" with a symmetric tensor ci3: the congruence of circles

P=Co-tCn+2t2Cn+,, which is associated with such a hypersurface, is normal, that is, it admits a oneparameter family of orthogonal hypersurfaces. (Such congruences are called the Ribaucour congruences.) In fact it is easy to show by calculations that

2. HYPERSURFACES OF CONFORMAL SPACES

64

the congruence of circles indicated above is normal if and only if the equation

dlog t+ao = 0 is completely integrable. On the other hand, this equation is completely integrable if and only if the tensor cij is symmetric. The following theorem combines all results obtained in this subsection:

Theorem 2.5.2 If det (aij) $ 0, then for a hypersurface

Vn-1 of the space

C", the following statements are equivalent:

i. The tensor c;j is symmetric. H. The curvature lines of the hypersurfaces other.

Vn-1 and Vn-1 correspond to each

iii. The Weyl connection, induced by the invariant frame bundle on the hyper-

surface V"-', is Riemannian. iv. The congruence of circles, which are orthogonal to the hypersurfaces Vn-1 is normal. and

3. Let us consider the hypersurfaces Vn-1 C Cn, on which the tensor G j is proportional to the tensor g;j:

c,, = ag;j.

(2.5.3)

Since the tensor c;j is symmetric, the curvature lines of the hypersurfaces V"-1 Vn-1 correspond to each other. Moreover, since in this case we have and x

(dCn+l+dCn+1) = a 9ijW

V+

the hypersurfaces Vn-1 and Vn-1 are in conformal correspondence. It is easy to prove the converse: if the hypersurfaces V' ' and Vn-1 are in conformal correspondence and their curvature lines corresponds to each other, then equation (2.5.3) holds. By (2.2.14), it follows from equation (2.5.3) that

ao = -aa; +1 Exterior differentiation of this equation implies that (da + 2aao) A a7+1 = 0.

For n > 2, by means of liner independence of the forms a; +1, it follows from the last equation that

da + 2aao = 0. It is easy to check that this equation is completely integrable.

2.5

Geometric Problems Connected with the Tensor c;,

65

By means of condition (2.5.3), the differentials of the points Co and Cn+1 have the following form: dCo = ooCo + o'C1, dCn+l = -aoCn+l + aa'C,.

(2.5.4)

Since

d(Cn+1 - aCo) = -oo(Cn+1 - aCo),

it follows that the hypersphere Cn+1 - aCo of the pencil of hyperspheres Cn+l +sCo is fixed when the frame moves along the hypersurface Vn-1. Moreover, since (Cn, CO) = (Cn, Cn+1) = 0,

all central hyperspheres of Vn-1 are orthogonal to this fixed hypersphere

D=Cn+1 - aCo.

(2.5.5)

This means that the hypersurface V"-1 can be obtained from the hypersurface V"-1 by the inversion mapping in the fixed hypersphere D. Conversely, if for the hypersurface V"-', the second semienvelope Vn-1 of the envelope of the family of its central hyperspheres can be obtained from V"-1 by the inversion mapping in some fixed hypersphere D, then condition (2.5.3) holds. In fact in this case, since (Cn+1, Cn+t) = 0, we have Cn+1 = D + aCo,

(2.5.6)

where a = - 2 D,Co Let us find the differential of the hypersphere D:

dD = -ooD - (da + 2aoo)Co + (a;,+1 - aao)C;. Since the hypersphere D is fixed, it follows that da + 2aao = 0 and a;,+1 - aao = 0.

The last equation implies condition (2.5.3). Thus we have proved the following result:

Theorem 2.5.3 A conformal correspondence between hypersurfaces V"-' and V"-1, which preserves the curvature lines of these hypersurfaces, occurs if and only if these hypersurfaces are connected by an inversion mapping in some fixed hypersphere.

Note that by (2.1.1), it follows from formula (2.5.5) that

(D, D) = 2a. This means that for a > 0 the hypersphere D is real, for a < 0 it is imaginary, and for a = 0 the hypersphere D degenerates into a point (see Subsection

2. HYPERSURFACES OF CONFORMAL SPACES

66

1.1.4). For all three cases formula (2.5.5) defines an inversion in the hypersphere D, but for a = 0 this inversion is degenerate. The hypersurfaces V"-' whose tensor c;j is a multiple of the tensor g;j can be divided into three classes depending on whether the factor of proportionality a is positive, negative, or 0.

Let us first study the case a = 0. In this case a = 0 on the hypersurface V"-', the hypersphere D coincides with the point C"+1, and this point is fixed. A conformal space with a fixed point is equivalent to a Euclidean space Vn-' for which the fixed point is the point at infinity. Thus the hypersurface can be considered as a hypersurface of a Euclidean space. Having done this, we

see that the central hypersphere C" of V` is its tangent hyperplane, and its normal hyperspheres C; are its normal hyperplanes. The differential 1-form ao will become a total differential, and by normalizing the point A0, we can reduce this form to 0. The tensor a;j is the second fundamental tensor of the Euclidean

geometry of V` 1, and the contraction nl Ig'ja;j is the middle curvature H of the hypersurface Vn-1 in the Euclidean space R". But, since the tensors g;j and a;j are apolar, the hypersurface V"-1 is minimal. Thus, if a = 0, the hypersurface V11-1 is conformally equivalent to a minimal hypersurface of an Euclidean space. Conversely, if a hypersurface Vn-1 C R" is minimal, then its middle curva-

ture H = "1 g'ja;j = 0, and consequently \q = a;j. The tangent hyperplanes 1

of the hypersurface Vn-1 coincide with its central hyperspheres, and under compactification of the Euclidean space, they will become hyperspheres passing through the point C"+1, which is fixed. Thus we have c;j = 0. Consider now the case a 54 0. As was indicated above, for a > 0, this V"-' hypersphere is real, and for a < 0 it is imaginary. The hypersurfaces Vn-1 are connected by the transformation of inversion in the hypersphere and D. The hypersphere D can be taken as the absolute in the space C". If this hypersphere D is real, then the space C" with the absolute D is a conformal model of the hyperbolic (Lobachevsky) space H". If this hypersphere D is imaginary, then the space C" with the absolute D is a conformal model of the elliptic space S". In both cases, the central hyperspheres of V` are orthogonal to the absolute D. Thus they represent the tangent hyperplanes to the hypersurface V` in the corresponding geometry. The condition g'ja;j = 0 means that the hypersurface V"-1 is a minimal hypersurface of the hyperbolic space H" or the elliptic space S". 4. We conclude this section by giving a projective interpretation of the conformal theory of hypersurfaces. Under the Darboux mapping, a hypersurface V"-' of the space C" is transformed into a submanifold U"-1 of codimension 2 of a projective space Pnt1 belonging to a hyperquadric Q" C P"+1 _ Moreover, under the Darboux mapping, the family of first-order conformal frames of the hypersurface V"-1 are transformed into the family of first-order projective frames of the submanifold Un-1, and all geometric notions and objects of the hypersurface V"-1 C C" can be interpreted in terms of the projective differ-

2.5 Geometric Problems Connected with the Tensor cij

67

ential geometry of the submanifold Un-' C Pn+' We consider some of these interpretations that are specific to the theory of hypersurfaces in the space C".

The correspondence between first-order frames of V"-' C C" and U"-1

Ao E

C pn+' under the Darboux mapping implies that the image of the point V"-1 C C" is the point Ao E U"-' C P"+', the image of the normal

hyperspheres Ai of V"-' C C" are the points Ai E TA0(U"-') C P"+', and the image of the tangent hypersphere An E C" and the point An+i C C" are the points An E P"+1 and An+l E P"+' The points AO and Ai compose a basis of TA0(U"-' ), and the points A0, An and An+l in P"+' determine the normal E2 of the first kind of the submanifold U"-' C P"+1. Moreover, since

(A.,Ai) = (An+1,Ai) = 0,

the normal E2 is polar conjugate to the tangent subspace TA0(U"-') with respect to the hyperquadric Q". Let us consider a second-order differential neighborhood of the point U"-1 C P"+' . We have Ao E d2Ao = wiw; An + wiw; +' An+l

(mod Ao, Ai).

Thus the quadratic forms

WV' = Aijwi w' and w'w +l = gijwiwj

are the second fundamental forms of the submanifold U"-' C pn+' at the point Ao. Equations (2.1.25) and (2.1.24), which the coefficients \ij and gij of these forms satisfy, show that these forms compose an invariant pencil. If we transfer to the invariant frame {CO} C P"+', we obtain d2C0 =

The quadratic forms 41" (2)

C', + Oiai +l Cn+1

(mod CO, Ci).

'j(2)

and n+1 = aia"+l = gija+ = 0'O" i = a O'a

also compose a basis of the pencil of second fundamental forms of the submanifold U"-' C P"+1 at the point Co, but now each of these forms is itself invariant.

Since the principal directions of V"-' C C" at the point AO are mutually conjugate with respect to these second fundamental forms, the directions of Un-1 C pn+l corresponding to the principal directions of V"-' C C" are conjugate. Thus the net of curvature lines of V"-' C C" corresponds to the net of conjugate lines of U"-' C P"+1 (on conjugate lines and conjugate nets, see Akivis and Goldberg [AG 93), §§3.1-3.4).

If the projective frame {C{} C P"+' of U"-' corresponds to the invariant conformal frame {CE} C C" of V"-', then we have dCo = aooCo + a'Ci, dCn+l = -aooCn+1 + an+1Ci

68

2. HYPERSURFACES OF CONFORMAL SPACES

The points Co and Cn+1 of the frame {Cf} C P1+1 define the straight line Co ACn+1 in the space P"+1. If the frame moves along U"-1, the straight line Co ACn+1 describes an (n -1)-parameter family of straight lines, which can be considered as a hypersurface U" in the space Pn+1 The preceding equations show that a first-order neighborhood of a straight line of this family belongs entirely to the n-dimensional plane CoAC1A...ACn_1ACn+1. Thus the tangent hyperplane to the hypersurface U" is not changed when its point moves along the straight line Co A Cn+1, and depends only on n - I parameters. Therefore U" is a tangentially degenerate hypersurface of rank n - 1. Following the book Akivis and Goldberg [AG 93] (p. 113), we will denote such a hypersurface by n n-1' Each rectilinear generator Co A Cn+1 of the hypersurface Un_ 1 carries n - I foci, and there are n - 1 developable surfaces passing through Co A Cnt1 and belonging to Un_,. In fact a focus F = xCo +Cn+1 of the generator Co A Cn+1 is defined by the condition dF =_- 0

(mod Co, Cn+1)

Since

dF = (dx + xoo)Co - aoCn+1 + (0`n+1 + xo'')Cii,

the foci and developable surfaces are determined by the following system of equations: 0;,+1 + xoi = 0. Since

aniti = -gi Ckja"+ this system can be reduced to the form

(Cij - xgij)oj = 0.

(2.5.7)

Nontrivial solutions of of this system define developable surfaces on Un_,. This

system has nontrivial solutions aj if and only if det (cij - xgij) = 0. If we solve this equation, then we find n-1 values of x, each of which determines a focus on the straight line C0 ACn+1, and for any such x, from equations (2.5.7)

we find a system of values of the forms of determining a developable surface on Un_,. The surface U"_, is also called a focal family of straight lines. 5. Let us investigate the structure of the surface Un_, if the tensor cij of the hypersurface Vn-1 is symmetric. As we have proved earlier, in this case the tensors gij, aij, and ci j can be simultaneously reduced to diagonal forms. This implies that the conjugate lines of the submanifold U"'1 and the submanifold U", described by the vertex Cn+1 of our frame, correspond to one another. Moreover it follows from system (2.5.7) that the conjugate lines correspond to the developable surfaces on the submanifold Un_1. In other words, the

2.5 Geometric Problems Connected with the Tensor c;j

69

developable surfaces of the focal family U,",_1 intersect the submanifolds U"-' and U"-' along the lines of the conjugate nets. According to the conventional

terminology, the submanifolds U"-' and on-' and the focal family Un_1 are conjugate to one another. Let us further prove that the focal family U.n_1 is normal, that is, it admits a one-parameter family of orthogonal trajectories if and only if the tensor c;j is symmetric. In fact a displacement of the point X = xCo + Cn+1 is orthogonal to the straight line Co A C"+1 if and only if (dX, Y) = 0,

where Y = -xCo + C"+1 is the point that is conjugate to the point X with respect to the hyperquadric Q". But this condition reduces to the following Pfaffian equation: d log x + 2ao = 0.

It is easy to see that this Pfaffian equation is completely integrable because of the symmetry of the tensor c;i.

Figure 2.5.2

Figure 2.5.1

Figure 2.5.3

70

2. HYPERSURFACES IN CONFORMAL SPACES

If the tensor c,j is connected with the tensor gii by the condition cij = a9i,j, then, as follows from Subsection 2.5.3, the focal family Un_1 is a cone whose vertex is the fixed point D = Cn+1 - aCo. Moreover, if a = 0, then the vertex

D of this cone lies on the hyperquadric Q"; if a > 0, then the vertex D lies outside of the hyperquadric Q'; and if a < 0, then the vertex D lies inside the hyperquadric Q" (for n = 2 see Figures 2.5.1-2.5.3). NOTES 2.1-2.2. The conformal differential geometry of a surface V2 was studied initially within classical differential geometry. For example, in the nineteenth century G. Darboux, A. Ribaucour, and others investigated in three-dimensional Euclidean space E3 surfaces with isothermic or spherical curvature lines, canal surfaces, systems of circles, triply-orthogonal systems of surfaces, etc. In the beginning of the twentieth century, some papers appeared in which the authors found the law of transformation of the most important differential invariants and invariant quadratic forms of the classical theory of surfaces under conformal transformations of the space E3. Based on these considerations, conformal differential invariants and invariant quadratic differential forms of V2 C E3 were constructed in the papers of Demoulin [Demo 26), Fubini [Fu 09), Tresse [Tr 94], Voss [Vo 80], and other geometers. A survey of these papers is in Berwald [Berw 27). Chen [Ch 73b, 74), and Hsiung and Mugridge [HM 79) considered conformal invariants of submanifolds in a multidimensional Euclidean space E". Chern (C 86] described a conformal invariant of a three-dimensional submanifold on an n-dimensional Riemannian manifold. Conformal properties of surfaces V2 C E3 were also considered by Blaschke [BI 29], Delens [Del 27), Eisenhart [Ei 23), and Takasu [Ta 38). In 1923 in the paper Thomsen (Tho 23], for the first time pentaspherical coordinates and tensor calculus were used to study the conformal differential geometry of surfaces. In this paper G. Thomsen constructed three conformally invariant quadratic differential forms of V2 C C3 and used them to develop the foundations of conformal differential geometry of surfaces. E. Vessiot (Ves 26a, b; 271 also studied conformal differential geometry of V2 C C3 using pentaspherical coordinates. He considered the invariant infinitesimal operators of 1,*2 associated with its curvature lines and applied them to construct a canonical frame of V2 and its fundamental invariants in order to study curves on V2 and to single out some special classes of V2 C C3. In the book Blaschke [BI 29), the differential geometry of the conformal space C3 was considered along with the differential geometry of the Laguerre space and the space whose fundamental group is the group of spherical transformations of S. Lie. Blaschke introduced the invariant derivatives associated with the curvature lines of surfaces and used them to find the principal derivational (Weingarten) formulas and to study some other topics of the conformal differential geometry of V2 C C3. Since 1925 T. Takasu published a series of papers in which he studied the same topics as in the book Blaschke [BI 29): the differential geometry of the space C3, of the Laguerre space, and of the space whose fundamental group is the group of spherical

Notes

71

transformations of S. Lie. Takasu considered a sphere as the generating element of a space and curves and surfaces as envelopes of one- and two-parameter families of spheres. Takasu summarized the results of most of his papers in the monographs Takasu [Ta 38, 39).

G. F. Laptev [Lap 49, 50, 53, 58a, b] introduced a new method of differentialgeometric investigations of submanifolds embedded in homogeneous spaces and spaces

with fundamental group connection. This method can be applied to differentialgeometric investigations of submanifolds in spaces with an arbitrary generating element. The method is based on the calculus of exterior differential forms and the general theory of representations of finite-dimensional Lie groups. The core of the method is the general principle of reduction of differential-geometric investigations of embedded submanifolds to the study of fields of geometric objects that are obtained from some original field by means of prolongations and inclusions. The equations of transformation of components of geometric objects are written not in closed form (as was done earlier) but in the form of completely integrable systems of Pfaffian equations. This greatly simplifies the investigations and comprises one of the main advantages of this method. Laptev proved that for any submanifold embedded in a homogeneous space or a space with a connection, there exists a geometric object such that if its components are appropriately chosen, then the submanifold is defined up to constants. The object of the lowest order of this type is called the complete fundamental object of a submanifold.

While tensorial methods usually require a preliminary construction of a certain connection on a submanifold, Laptev's method does not require this. The advantage of Laptev's method can be seen especially in those cases when one does not know how to construct an invariant connection on a submanifold under consideration in differential neighborhoods of low orders. This is exactly the case when one studies a

submanifold V' C C" and V'" C P" . Laptev and his followers demonstrated the fruitfulness of this method by applying the method to various problems of differential geometry (e.g., see Laptev [Lap 53, 58a] and Akivis [A 52a, b; 61a)). The geometry of surfaces V2 C C3 and hypersurfaces V"-1 C C" was also considered in the papers do Carmo, Dajczer, and Mercuri [CDM 85], Finzi [Fi 02, 21, 22, 23],

Haantjes [Haa 42c, 43], Matsumoto [Mat 55], Muto [Mu 40a], Nishikawa [Ni 74], Nishikawa and Y. Maeda [NM 74), Sulanke [Su 82, 88), Thomsen [Tho 23, 25], and Wong [Won 43), among others.

Note that in most of the papers mentioned above the conformal differential geometry of multidimensional submanifolds V'" is constructed by the methods of Riemannian geometry. This makes a geometric interpretation of the obtained results extremely difficult. In all these papers the authors do not go further than to find the derivational (Frenet) equations and to prove the general theorems on determination of the submanifolds V' by means of a system of tensors. The invariant normalization of a hypersurface V"'1 C C", which we presented in Section 2.2, was first constructed in the paper Akivis [A 52b) (its short version is in Akivis [A 52a]). For a more detailed description of the developments in the construction of an invariant normalization of hypersurfaces in different spaces, see the survey papers Laptev [Lap 65) and Lumiste [Lu 75]. 2.3. In local differential geometry the rigidity theorems contain conditions under which two submanifolds of a homogeneous space can differ only by their location in the

72

2. HYPERSURFACES IN CONFORMAL SPACES

space. For hypersurfaces in a projective space, the rigidity problem was considered in Fubini (Fu 16, 18a, b; 20] (see also pp. 605-629 of the book Fubini and tech [FLT 26]), Cartan [Ca 20b], Jensen and Musso [JM 94), and Akivis and Goldberg [AG 93) (§7.4). The problem of conformal rigidity of submanifolds is also of great interest. This problem was studied in Cartan [Ca 17], do Carmo and Dajczer [CD 871, and Sack-

steder [S 621 (see also the paper Sulanke (Su 82], in which the author considered problems close to the rigidity problem). However, in these papers the problem of conformal rigidity was investigated in the framework of Euclidean geometry. In Section 2.3 we present the solution of this problem in the framework of conformal differential geometry. Theorem 2.3.1 is close to Theorem 2.1 in Sulanke [Su 82). The difference is that our proof has been made in the general (not necessarily orthogonal) frame bundle associated with a hypersurface. This is the reason that our proof can be easily transferred to the case of nonisotropic hypersurfaces in a pseudoconformal space. The rigidity problem is closely connected with the problem of deformation of submanifolds. E. Cartan [Ca 171 studied the conformal deformation of hypersurfaces V"-' C C". V. I. Vedernikov (Ved 50a, b; 541 considered the problem of conformal deformation for normalized submanifolds V2 C C3 and V"-' C C", discussed different definitions of conformal deformation and gave their geometric interpretations. 9. Cartan [Ca 37a] presented the philosophy of studying uniqueness, existence, and rigidity questions for submanifolds of a homogeneous space via the use of moving frames and the theory of Lie groups. P. A. Griffiths (Gr 74) gave an updated and clear exposition of this Cartan philosophy together with some applications to geometry.

2.4. Canal surfaces in C3 and canal hypersurfaces in C" were studied in the 1950s in Akivis [A 52b], Geidelman [Ge 49, 50a, b; 57], Matsumoto [Mat 55), and Vedernikov [Ved 571 and later in Chen and Yano (CY 73b] and in Sulanke [Su 92). Dupin's submanifolds were introduced in Dupin [Du 22] in 1822. These submanifolds and their generalizations were studied by many authors. We indicate here some of recent papers on the Dupin cyclides: Cecil [Ce 89, 911, Cecil and Chern (CC 891; Cecil and Ryan [CR 78, 80, 851, Chern [C 911, Miyaoka [Miy 84, 89a, b], Miyaoka and Ozawa [MO 891, Pinkall [Pi 85a, b], Pinkall and Thorbergsson [PT 89), and Thorbergsson [Thor 831. A detailed bibliography of works in this direction can be found in the recently published book Cecil [Ce 921. In addition to this bibliography we mention here the paper Vedernikov [Ved 58), in which another multidimensional analogue of Dupin's cyclides was considered.

The congruences (i.e., (n - 1)-parameter families) of hyperspheres S" in the space C" generalize the congruences of spheres in C3. The latter congruences were studied by G. Darboux and A. Ribaucour. They singled out a special class of such congruences, R-congruences, which are characterized by the fact that the curvature lines correspond on two sheets of the envelope of any such congruence. W. Blaschke in the above-mentioned book Blaschke [BI 291 and some other geometers considered the R-congruences in the framework of conformal geometry. Papers on the theory of congruences of spheres in C3 and of hyperspheres in C" were also published by Backes [Ba 56, 611, Demoulin [Demo 19], Geidelman [Ge 60, 67b], Krivonosov [Kr 62], Tikhonov [Ti 61, 63, 64], and Vedernikov [Ved 62]. Theorems 2.4.1-2.4.3 are due to Akivis (A 651. 2.5. All results of this section are due to Akivis [A 52a, b].

Chapter 3

Submanifolds in Conformal and Pseudoconformal Spaces 3.1

Geometry of a Submanifold in a Conformal Space

1. We consider now a smooth, connected and simply connected submanifold VI of dimension m, m < n - 1, in the conformal space C". In Sections 3.1 and 3.2, we will use the following index ranges:

0<1:,77,(
1
(3.1.1)

In addition we denote (Ai, Aj) = 9ij, (Aa, AA) = gap, 73

(3.1.2)

74

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

and introduce the following normalizing condition: (3.1.3)

(Ao,A"+1) = -1.

Thus the matrix of scalar products of frame elements can be written in the following form: 0

0

-1

0

90

-1

0

900 0

0 0

0

(AE, An) =

(3.1.4)

Since the space C' is proper conformal, the matrices (gij) and (gap) are nonsingular positive definite matrices. We denote by (g'j) and (g°p) their inverse matrices. Since the hyperspheres Aa are tangent to V"', we have (3.1.5)

dAo = wooAo +w'Ai,

where w' = wo are basis forms on V'". This equation shows that the submanifold V'" is determined by the following system of Pfaffian equations:

w° = 0,

(3.1.6)

where w° = wo . It follows from (3.1.5) that

g = (dAo, dAo) = gijw'c,j,

(3.1.7)

and this form determines a conformal structure on V. This form is nondegenerate and positive definite. The equations of infinitesimal displacement of a first-order frame of the submanifold VI have the form (1.2.6) where the 1-forms wf satisfy the structure equations (1.2.10) of the space C". In the same manner as was done for a hypersurface in Subsection 2.2.1, we can prove that on the submanifold VI the matrix (wF) of components of infinitesimal displacement of a first-order frame of V"' takes the following form:

n=

(wE) -

wo

w'

0

0

wi

wi

wi

wo p 0

9jkwk

9ikwk

wi

p

(3.1.8)

WOO

90

4 -woo

and as follows from equations (1.2.8), the forms nected by the conditions

and wa are con-

9ijwe + 9apwR = 0,

(3.1.9)

dgij = 9ikwj + 9kjwi ,

(3.1.10)

dgap = 9oyw0, + g. wa.

(3.1.11)

3.1

Geometry of a Submanifold in a Conformal Space

75

For a fixed point x E V' (i.e., for w' = 0), the forms w' and wQ vanish, since the normal and tangent bundles of hyperspheres are transformed into themselves. By virtue of this, for w` = 0 the matrix (wf) takes the form

a{ -

0

0

0

a rrp

a;

0

0

0

xQ

0

0

9ikirk

9°anp

-nooo

7r0 0

(3.1.12)

Moreover equations (3.1.10)-(3.1.11) become O69ii = 21rog,,, V69al1 = 27roga0,

(3.1.13)

where the operator V6 is defined as in Subsection 2.1.2. The set of frames of first order associated with V' is a fiber bundle R1(VI) whose base is the submanifold V' itself and whose fiber is the collection of frames associated with a point x = Ao E V"'. The forms w' are base forms of this bundle and the forms 7r1 are its fiber forms. Let us consider the subgroup H = (V ') of the group PO(n+2, 1) of conformal transformations that leaves invariant the point AO and the bundle of tangent hyperspheres with the basis Ao and AQ. This subgroup acts intransitively on a fiber of frames of first order associated with the point Ao, since the quantities gi, and gQa are invariants of the group PO(n + 2, 1). This subgroup will act transitively on the subfamily of orthonormal frames defined by the conditions

9il = ail, g.0 = 6«a,

(3.1.14)

by means of which equations (3.1.10)-(3.1.11) take the form w , + w° = 0, _111 + w = 0, wp + W3 = 0,

(3.1.15)

2. If we take exterior derivatives of equations (3.1.6), we obtain the following exterior quadratic equations: w° A w' = 0.

(3.1.16)

Applying Cartan's lemma to equations (3.1.16), we arrive at the equations

A?.wl, a

=A?, A.

(3.1.17)

Exterior differentiation of equations (3.1.17) leads to the following exterior quadratic equations (3.1.18) (VA 13 +9iiwn+l) Aw' = 0, which, by Cartan's lemma, lead to the equations VA + gi)wn+1 = . kwk,

(3.1.19)

76

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

where Ask are symmetric with respect to all lower indices and

VA1 = dal, - A,k8 - AkjB; + Ap 9p

(3.1.20)

In formulas (3.1.20), B' = wj and 0' = w* - gwo. Applying exterior differentiation to equation (3.1.19), we obtain the following exterior quadratic equations: [D,\1 k + 3(A(ij9k)1 - 9(ijAk)t)w' +l

- 3AAjAk)Iw10J A wk

= 0,

(3.1.21)

where

A1jk =

A1jk9i - a lkBj - a 1BIk + Ai kep

and parentheses in the lower indices denote symmetrization with respect to the corresponding indices. From (3.1.21), by Cartan's lemma, we find that 3(A"j9k), - 9(ij,\k),)wn+I - 3AAJAk),wp =

(3.1.22)

where the quantities A .kt are symmetric with respect to all lower indices. Note also that by means of the operator V equations (3.1.10) and (3.1.11) can be written in the form Vgij = 29ijwooo, 0900 = 2900wo.

(3.1.23)

Equations (3.1.17) constitute the first differential prolongation of equations (3.1.6); equations (3.1.17) and (3.1.19) constitute the second differential prolongation of equations (3.1.6); equations (3.1.17), (3.1.19), and (3.1.22) constitute the third differential prolongation of equations (3.1.6), etc. Each of these differential prolongations introduces a new system of quantities: etc.

If w' = 0, then the system of quantities A that arose in the first differential prolongation of equations (3.1.6), satisfies the equations V 8A, = -9ij7rn+1,

(3.1.24)

and thus this system itself is not a geometric object. However, if we consider the system A jointly with the tensor g;j which, by (3.1.23), satisfies the system of equations V69ij = 2g,,ir , (3.1.25) then we obtain a geometric object with Zm(m+1)(n-m+1) components. This geometric object is called the fundamental geometric object of second order. If we continue the process of exterior differentiation and application of Cartan's lemma k- I times, we obtain a system of quantities A, MI-k, ... , A0, i,...i. This system jointly with the tensor gij constitutes the fundamental geometric object of order k. Thus the following theorem is valid:

3.1

Geometry of a Submanifold in a Conformal Space

77

Theorem 3.1.1 The systems of quantities {gii, A }, {gii, A j, A°.k }, {gi a, k+ kt}, etc. constitute geometric objects of the submanifold V ", which are connected with differential neighborhoods of V'" of orders 2,3,4, etc. The sequence of geometric objects we have constructed is called the fundamental sequence of geometric objects of V'. It was proved by Laptev (see Laptev [Lap 531, p. 352) that for a smooth submanifold V'" embedded into any n-dimensional homogeneous space S, the fundamental sequence of geometric objects of V'" entirely exhausts the differential geometry of V"'. In the same paper Laptev proved that there exists a lowest number p such that the fundamental object of order p determines the submanifold V"' up to a motion in the space S. This fundamental object is called complete. 3. As can be seen from the above considerations, the structure of fundamental geometric objects of V'" is rather complex, so it is difficult to find their direct geometric meaning. Using the fundamental geometric objects, we will construct simpler geometric objects whose geometric meaning is easier to establish, and they entirely exhaust the differential geometry of V'. We will single out from the frame bundle 1V (Vm) some subbundles that are defined in differential neighborhoods of second and third order of a point x E V"' and connected more closely with V"'. First, in the same way as we did for a hypersurface, we construct the quantities

A° = mA g(3.1.26)

where gig is the inverse tensor of the tensor gig, that is, gikgkj = 6'. By (3.1.23) this tensor satisfies the following system of differential equations:

Vg'' = -2wogi'.

(3.1.27)

Note that the tensor g°p, which is the inverse of the tensor goo, satisfies the equations similar to equations (3.1.27): V9°a = -2wog°p.

(3.1.28)

If we differentiate equations (3.1.26) with respect to the fiber parameters, we arrive at the following system of equations: V5A° + 2A07r0 + an+i = 0.

(3.1.29)

Equation (3.1.29) shows that the quantities A° constitute a geometric object of the submanifold V"'. Let us define a new object: Ao = g,

A'3.

(3.1.30)

By equations (3.1.23) and (3.1.29), the object A° satisfies the equations

VA° _ -w + A°kwk.

(3.1.31)

78

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

If the point Ao is fixed (i.e., if w' = 0), then the equations (3.1.31) take the form

V6Aa = -7r o

(3.1.32)

We consider now the bundle of tangent hyperspheres with the basis

X. = AQ + x,Ao and find under what condition this bundle is invariant. Since 6X0 = (V 6xa + 7ro )Ao + 7rOXa,

the desired invariance condition has the form V6xa + 7ro = 0.

The quantities xa constitute a geometric object that is called the normalizing object of first kind. Comparing the last equation with (3.1.32), we see that xa = A. is a solution of this equation. Thus the hyperspheres Ca = AQ + AQA0

(3.1.33)

produce a basis of an invariant bundle of tangent hyperspheres. The hyperspheres of this bundle are called the central hyperspheres of the submanifold V"' at the point x = A0. The intersection of these hyperspheres is an invariant tangent m-sphere that is intrinsically connected with the point x E V"'. This m-sphere is called the central m-sphere of the submanifold V"'. Equations (3.1.26) and (3.1.30) show that the object A0 defining an invariant bundle of tangent hyperspheres is determined in a second differential neighborhood of a point Ao of the submanifold V1. 4. Let us clarify the geometric meaning of central m-spheres. If we set 0J = A a1

-

agiie

(3.1.34)

we find that V ja 11 = 0.

(3.1.35)

Equations (3.1.35) show that the quantities a, form an invariant bundle of (0, 2)-tensors. It is easy to verify that this bundle of tensors and the tensor g;j satisfy the apolarity condition: a°O.g'j = 0.

(3.1.36)

The tensors a?, define a bundle of invariant quadratic forms x(21 =

(3.1.37)

3.1

Geometry of a Submanifold in a Conformal Space

79

which are connected with a differential neighborhood of second order of the submanifold V'" and are called the second fundamental forms of the submani-

fold V"'. Denote by ml the number of linearly independent forms 0zi. This number coincides with the number of linearly independent tensors in the bundle of tensors a 13. Let us find the geometric meaning of the tensors a13 and the quadratic forms To this end we consider the central hypersphere

S = x°C°

(3.1.38)

and find the directions along which it has a second-order tangency with the submanifold Vf°. Since d2 A, = w'w,°A° +w'w; +tAn+l

(mod Ao, Ai),

(3.1.39)

then, by (3.1.33) and (3.1.34), we have z

°

(3.1.40)

where x° = goox0. Thus the directions along which the hypersphere S has a second-order tangency with the submanifold V' at the point x = Ao satisfy the equation

x°a,w'wJ = 0

(3.1.41)

and form a cone of second order which is called the characteristic cone of the hypersphere S. The system of characteristic cones at the point x forms the bundle defined by the cones

a,w'wJ = 0,

(3.1.42)

which in turn are determined by the quadratic forms Since the coefficients of equation (3.1.40) form a tensor that is apolar to the tensor g'i, x°a°tj.g'2 = 0,

it is possible to inscribe an orthogonal m-hedron of vectors of the tangent space TZ(Vm) into the characteristic cone (see the reference on p. 38). Thus we arrive at the following result:

Theorem 3.1.2 The fundamental object of second order of a submanifold V' allows us to construct the invariant family of central m-spheres and the bundle of tensors a . At a point Ao E V'" the invariant bundle of the second fundamental forms 4i Z), defined by this bundle of tensors, determines a cone of directions along which the submanifold V"' has a second-order tangency with the central m-sphere. Note that the cone indicated in this theorem is the intersection of the cones of second order defined by the quadratic forms 4 . Thus, if the number of

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

80

independent forms among these forms is greater than or equal to m, then this cone could degenerate to the point Ao. If we set A° = C°, then we single out a subbundle RZ(Vm) of second-order frames from the bundle R' (V'") of first-order frames. Then the geometric object as vanishes, and equations (3.1.31) imply that the forms wQ are expressed in terms of the basis forms wk; the remaining fiber forms are only the forms 7r00, it , a°, and Ire. These forms are invariant forms of the stationary subgroup H= (V'") of second order which keeps invariant not only the point A0 but also the central m-sphere Cm.i.! A ... A C,,. 5. Next we consider the symmetric tensor a°p =

ak!

(3.1.43)

Let us show that the rank of the tensor a°Q is equal to the number ml of linearly In fact, if we take an orthogonal basis independent tensors in the bundle in the bundle of normal hyperspheres of V'°, that is, (Ac1A,,) = 9>> = o,

,

(3.1.44)

then the tensor a°p takes the form m

aao =

a?.aA.

(3.1.45)

We can consider the right-hand side of this equation as the scalar product of the vectors a° and ap in the space of m2 variables z,,. The matrix (a°R) is a Gram matrix of the system of vectors a°. It is well-known that the rank of this matrix is precisely equal to the rank of the corresponding system of vectors. If we differentiate equations (3.1.43) with respect to the fiber parameters and apply equations (3.1.35), we find the system of equations that the tensor a°F satisfies: Vda°p = -4aoa°0 (3.1.46) It follows from equations (3.1.46) that the weight of the tensor a°p is equal to

-4. We now construct another tensor,

sa = 9aya"F,

(3.1.47)

whose rank is the same as the rank of the tensor a°'3. Differentiating equations (3.1.47) and applying equations (3.1.23) and (3.1.46), we find the system of equations that the tensor as satisfies:

Vaa = -2wpaa + /sahwh where Aah = 9ay9ik9j!(ao \,yh1h +a jAklh)

(3.1.48)

3.1

Geometry of a Submanifold in a Conformal Space

81

Since the number of linearly independent tensors in the bundle of tensors av is equal to ml, the rank of the tensor aap and consequently the rank of the tensor aQ is equal to mi. Thus the relative invariant

a= a[0, a02 ... aa' 1,

(3.1.49)

which is the sum of the diagonal minors of order mi of the tensor aa, is different from 0 provided that mi 54 0.

If at a point x E Vm, the number ml = 0, then all tensors a7 = 0. Then, as it is easy to see, the central hypersphere has a second-order tangency with VI at the point x. Points of this kind are called umbilical. This proves the following result:

Theorem 3.1.3 At any non-umbilical point Ao E Vm, one can construct a nonvanishing relative invariant a which can be expressed in terms of the fundamental object of second order of the submanifold V"'.

Note also that in the same manner as we did for a hypersurface, we can prove that if the tensors a. = 0 at all points of the submanifold Vm, then V'" is an m-sphere or its open subset. It is obvious that in this case the central m-sphere coincides with the rn-sphere Vm. Suppose that the number ml is different from 0 and is constant on the submanifold V'°. By equation (3.1.48) the invariant a defined by equation (3.1.49) satisfies the equation

Va = -2mia(wo + piw'), where

µ-

MI

1 -2m'a

,

a, 1a.

a." ...aaT:)

(3.1.50)

(3.1.51)

T=I

and poi are the quantities defined in equations (3.1.48). 6. In the bundle of hyperspheres with the basis AO and Ai, we single out a subbundle with the basis

Xi=Ai - xiAo.

(3.1.52)

The condition for this subbundle to be invariant under transformations of the stationary subgroup Hl(Vm) is 6xi = B; Xi. Differentiating equations (3.1.52) relative to the fiber parameters, we find that

oXi = (Vixi +7r°)Ao +7r X,. Comparing the last two equations, we find that the condition for the subbundle in question to be invariant is Vaxi = -7r9'.

(3.1.53)

82

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

The quantities xi form a geometric object that is called the normalizing object of second kind.

We will now construct a solution of equation (3.1.53) that is intrinsically connected with the submanifold V'". Let us show that such a solution can be found if on the submanifold V' a relative invariant I of weight 1 is given. In fact the differential equation that such an invariant satisfies can be written in the form: dl = I(wo + piw'). (3.1.54) Exterior differentiation of equation (3.1.54) and application of Cartan's lemma leads to the following system of equations that the quantities pi satisfy: Vpi = w° + pi,wf ,

(3.1.55)

where pig = pp. The system (3.1.55) proves that the quantities xi = -pi produce a solution of equation (3.1.53). Therefore the hyperspheres

Ci = A; - piAo

(3.1.56)

form a basis of the bundle of normal hyperspheres, and their intersection, the (n - m)-sphere CI A ... A Cm, is an invariant normal (n - m) -sphere that is intrinsically connected with the point x E V'". Now it is easy to construct a relative invariant I of weight 1. To this end, we consider the relative invariant a defined by (3.1.49). Its weight is -2mt. Thus, if we set

I = a-

,

(3.1.57)

then we obtain an invariant of weight 1. Since the invariant a is determined by a second-order differential neighborhood of the submanifold V'", the geometric object pi is expressed in terms of

the components of the fundamental objects of V' of third order. Thus the invariant bundle of normal hyperspheres with the basis Ci is determined by a third-order neighborhood of a point Ao E V"'. We have proved the following result:

Theorem 3.1.4 The fundamental object of third order of a submanifold V'" allows us to construct, at any point Ao E V"', the normal m-sphere Ct A ... A Cm, which is intrinsically connected with the submanifold Vm.

Note that the tensor as allows us to construct a series of relative invariants necessary for construction of the bundle of normal hyperspheres. For example, the objects aa,a*[oa,,, and all other symmetric functions of this tensor are relative invariants of this kind. However, the invariant a*,aI aC,1,2 ... a"T' l is the only

invariant that certainly is never zero. This is the reason that we will take this invariant as the basis of our construction. 7. The hyperspheres Co and Ci which we have constructed determine a ,inplete invariant normalization of the submanifold V"'. The invariant point

3.1

Geometry of a Submanifold in a Conformal Space

83

Cn+1, in which the hyperspheres C° and C; intersect one another, can be

written now in the form Cn+1 ,+j = An+1 - p

«+

1

2

(µkµ

k

+1a

(3.1.58)

where (3.1.59)

µk = 9k"p1.

The points Co = Ao and Cn+1 and the hyperspheres C; and C,, passing through these points can be taken as the elements of an invariant frame of the submanifold V'n. If we write the equations of infinitesimal displacement of this frame in the form dCC = of C,,,

(3.1.60)

then the forms o{ , which are the components of infinitesimal displacement of this frame, are also of invariant nature. The matrix (o{) has the same form as the matrix (wi ); see equation (3.1.8). If we solve equations Co = Ao,

C;=A;-p;Ao, C°=A°+\°Ao,

(3.1.61)

Cn+1 = An+1 - it kAk + ,\°A° + (pk/lk + A0A°)Ao 2

with respect to & we obtain the following formulas for the inverse transformation of the frames {A(} and {CC}:

Ao=Co, Ai = C; + p1Co,

(3. 1.62)

A°=Ca-\aCo, An+1 = Cn+1 + pkCk -PC. + (pkpk + a°a°)Co. a

If we differentiate equations (3.1.61), apply (3.1.8), and substitute for the elements A their expressions taken from (3.1.62), then we find the following expressions of the forms of in terms of the forms WE of infinitesimal displacement of the original frame: ak = Wk, a° = 0, of +1 = 9ijai,

o°

Woo +

Wk a4= wJ - gik

kw)

- tiwk

as = wR

a; = a,wi, as = b°;W', oo = -cijW".

(3.1.63) (3.1.64) (3.1.65)

where a 13 is the tensor defined in a second-order neighborhood of the point

Ao E V' by equations (3.1.34), the tensors b°k and c;j are defined by the formulas b°1 = Ao( - gooaA pd

(3.1.66)

84

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

and

Cij = liij + 'UiElj + A°A

j

-2

(µkµk + A°A°)9ij,

(3.1.67)

and the quantities A°i and pij are defined by equations (3.1.31) and (3.1.55). It follows from equations (3.1.66) and (3.1.67) that the tensor b°i belongs to a third-order neighborhood of the point Ao E VI and the tensor cij belongs to a fourth-order neighborhood of the point Ao E V'". Moreover the tensor cij is symmetric: cij = eji. Comparing equations (3.1.64) and (3.1.54), we find another expression for the form oo: a0 = d log I. (3.1.68) This expression means that the form o0 is a total differential. 8. Let us distinguish those hyperspheres tangent to the submanifold V"' which have a second order tangency with V'". A tangent hypersphere can be represented in the form: (3.1.69) S = x°Ao + x°A°,

and it has a second-order tangency with V' if and only if (d2Ao, S) = 0.

Since the second differential of the point AO is satisfies is determined by formula

(3.1.39), the last equation can be written as x°wiwi° - x°wiw,"+i = 0,

where x° = gapxa, or in the form

(x°A, - x°9ij)w'wj = 0, from which it follows that x°A 83 - x°gij = 0.

(3.1.70)

Contracting equations (3.1.70) with the tensor gij, we obtain the equation

x°A° - x° = 0,

(3.1.71)

by means of which, relation (3.1.69) takes the form S = x" (A. + A, AO).

By (3.1.33), from the last equation it follows that

S = x°C°,

3.1.72)

where C° are central tangent hyperspheres of the submanifold V"'. Equation (3.1.72) shows that any osculating hypersphere of the submanifold V' is its central hypersphere.

Geometry of a Submanifold in a Conformal Space

3.1

85

By (3.1.71) and (3.1.34), equations (3.1.70) take the form

xoa° = 0.

(3.1.73)

This means that the tensors of the invariant bundle of tensors, defined by the tensors a , corresponding to the osculating hyperspheres (3.1.69), vanish. Suppose now that there are m1 independent tensors in the bundle of tensors 0.. Since the tensors a are symmetric with respect to the indices i and j and are connected with the tensor gig by the apolarity condition (3.1.36), it follows that the number m1 satisfies the inequality

m1<

m(m + 1) 2

-1.

(3.1.74)

If n > m + m1, then it is possible to choose a basis in the bundle of central hyperspheres such that the hyperspheres Cu, u = m+m1 + 1, ... , n, form a basis of the subbundle of osculating hyperspheres. Then the osculating sphere of the submanifold V' is the intersection of all hyperspheres Cu, and its dimension is equal to m + m1. We denote this osculating sphere by C.1+11. We will take the hyperspheres C.,, a' = m+ 1, ... , m+m1, to be orthogonal to the hyperspheres Cu. In the basis we have constructed, the tensors a are linearly independent, and the tensors a result the first equations of (3.1.65) take the form

=0.

(3.1.75)

Now at each point x E V'" the osculating sphere Cz +m' is determined uniquely, and consequently the forms ai, connected with the displacement of this sphere are linear combinations of the basis forms ai: u

u

i

If these forms vanish, then the osculating sphere C= +."' is fixed, and the submanifold V' belongs entirely to this sphere. In this case it is natural to assume that m + m1 = n and C= +m' = C". Then equations (3.1.75) take the form (3.1.76)

where a = m + 1,. .. , n, but unlike the situation in formula (3.1.65), all tensors

a in (3.1.76) are linearly independent. By virtue of this, the matrix (a°p) occurring in formula (3.1.43) is nonsingular. 9. We will now determine a set of tensors that defines an m-dimensional submanifold VI in a conformal space C" up to a conformal transformation of this space.

For simplicity we will assume that the osculating sphere C= +'"' of the submanifold V' coincides with the space C", that is, formulas (3.1.76) hold, and the tensors a,°j . are linearly independent.

86

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

As was already noted in Subsection 2.3.2, to define a submanifold V' up to a conformal transformation, it is necessary to express all components of infinitesimal displacement of a moving frame associated with a point Ao E V'"

in terms of the basis 1-forms. We will look for a submanifold V' along with a family of frames {C{) of third order. As we did earlier, we denote the components of infinitesimal displacement of such frames by on. In addition we suppose that on the submanifold VI curvilinear coordinates u', i = 1, ... , m, are introduced, which are defined in a connected and simply connected domain D of the space R'", and that the basis forms o' are expressed in terms of the differentials du' of these coordinates in the simplest way:

at = du'.

(3.1.77)

Then formulas (3.1.63) and (3.1.65) show that for finding the forms a+1, a°, oQ,

and a°, the tensors gig, a', b°i, and c;, must be given in the domain D, and these tensors must satisfy the conditions indicated earlier. The form ao is expressed by (3.1.68) where the relative invariant I is defined by formula (3.1.57), and a now is simply the determinant of the nonsingular matrix (aoO) defined by relation (3.1.43), a = det(a°p).

Next the forms a, are expressed in terms of the tensor g,i and its partial derivatives 9,j,k = eu{- precisely in the same manner as was indicated in Subsection 2.3.2 for a hypersurface V"-1 C C". To find oF, we consider the system of equations (3.1.35) which the tensor a . satisfies. For w' = du' 36 0, this system takes the form da°3 ,J - a°AJ

a°,kQk + ap q Ra° - b°ijkduk , 7

( 3.1 .78 )

where oi; = of -6 ao, np = op -bpao, and the tensor b k is expressed in terms of the fundamental geometric object of third order. Since the components of the tensor a are given as functions of the variables u', and we have already found the 1-forms a0 and off, equations (3.1.78) can be written as

2.a3 = b,jkduk,

(3.1.79)

where the quantities bk can be computed from equation (3.1.78). As the tensor bk itself, they are determined by the fundamental geometric object of third order. Let us contract equations (3.1.79) with the tensor g'Dg)ganq.

Then, by

(3.1.43), we find that a-y0op

=

bk°duk,

(3.1.80)

where bk° = g'pgigaPgb k. Since det(a''p) 96 0, the forms ap can be easily found from equations (3.1.80).

3.1

Geometry of a Submanifold in a Conformal Space

87

Of course the tensor Elk must also satisfy the conditions of consistency. The latter will arise if one substitutes the forms aF in equations (3.1.79): since

m(m+1) 2

-I>n -m,

the number of these equations is not less then the number of forms op. Thus we have proved the following result:

Theorem 3.1.5 Suppose that in a connected and simply connected domain D of the space R" a symmetric nonsingular positive definite tensor g,,, symmetric

linearly independent tensors a, a = m + 1,.. . , m + ml, a tensor bai, and symmetric tensors b,k and c,, are given and that these tensors satisfy the conditions of consistency following from equations (3.1.79) and the structure equations (3.1.81)

of the conformal space. Then, in the space C", n = m + ml, these tensors determine the submanifold Vm up to a conformal transformation of the space C", and the above tensors have the geometric meaning indicated in Subsection

3.1.1. Note that from all tensors defining the submanifold V' up to a conformal transformation of the space C", only the tensor c, is determined in a differential neighborhood of fourth order. Hence the complete object for a submanifold V' is the fundamental object of fourth order. Note also that it is possible that the consistency conditions (3.1.81) allow us to express some of the tensors defining the submanifold VI in terms of partial derivatives of other tensors, as happened with the tensor cij for a hypersurface (see Subsection 2.3.2). However, we will leave this investigation to the curious reader. If the osculating (m + ml)-sphere does not coincide with the whole space C", then the formulation of the fundamental theorem is more complicated. This theorem was formulated and proved in the paper Akivis [A 61a]. 10. In concluding this section, we will give a projective interpretation of the conformal theory of a submanifold Vm C C". Under the Darboux mapping, a submanifold VI C C" is transformed into a submanifold Um belonging to a hyperquadric Q" of the projective space P"+. Moreover the image of a family of first-order conformal frames of the submanifold Vm under the Darboux mapping is a family of first-order projective frames of the submanifold U"`. Namely, under the Darboux mapping, the image of the point Ao E V.n C Q" C C" is

the point Ao E Um C P"+', the image of the normal hyperspheres Ai of V,n C C" are the points Ai E TAO(Um) C pn+i, and the image of the tangent hypersphere A. C C" and the point An+1 E C" are the points AQ E P"+' and An+1 E P"+'. The points A0 and Ai form a basis of T=(Um), and the points

88

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

A0, A. and A"+I in P"+1 determine the normal N,, (U-) of the submanifold U'" C P"+1. Moreover, since (A°,A1) = (An+1,Ai) = 0,

the normal N=(Um) is polar-conjugate to the tangent subspace T=(Um) with respect to the hyperquadric Q".

As was the case for a hypersurface V" C C", all geometric notions and objects of a submanifold V' C C" can be interpreted in terms of the In projective differential geometry of the submanifold U'" C Q" C particular, under the Darboux mapping, the image of the invariant frame (Co, Ci, C«, Cn+l } of V' C C" constructed in this section, is the invariant P"+1.

frame {Co, Ci, CO, Cn+I } of U'" C P"+1. This frame is characterized by the fact that the (n - m)-dimensional subspace C,"+1 A ... A C,, A Cn+1 C N=(U'") and the (m - 1)-dimensional subspace C1 A ... A C,,, C TT(Um) are defined invariantly.

Let us consider a second-order differential neighborhood of the point Ao E U" C Pn+1 We have d2A0 = wiw°Aa +wiwn+lAn+i

(mod Ao, Ai)

Thus the quadratic forms

w'wI =

and w' w; }1 = gijw'wj

are the second fundamental forms of the submanifold U' C P"+1 at the point A0. Equations (3.1.24) and (3.1.23), which the coefficients A t and gij of these forms satisfy, show that these forms form an invariant bundle. If we transfer to the invariant frame {CE) C Pn+I, we obtain

d2Co - aia°C° + aia°+ICn+I

(mod Co, Ci).

The quadratic forms 4i 2)

= a'a; =

)

and

"+1 = ,ia{+1 = gijaiaj (2)

form also a basis of second fundamental forms of the submanifold U"' C P"+1 at the point Co. But now the forms 4(2) constitute an invariant subbundle, and the form'( )1 is invariant itself. Moreover, by (3.1.7), the form 1 coincides = g. with the first fundamental form of the submanifold V'" C C": '1' By (3.1.63) and (3.1.65) the second differential of the point Co E U"' C P"+1 satisfies the following relation: d2Co

= (aC° + gijCn+1)aa(mod Co, C.

Hence the osculating subspace of the submanifold U"' is defined by the points Co, Ci, and

3.2

Submanifolds Carrying a Net of Curvature Lines

89

Cij = a0ij C. + 9ijCn+1

Since g'ja = 0, we have C"+1 = mgijCij. Since the number of linearly independent tensors among the tensors a23 is equal to m1, and the tensor gij cannot be expressed in terms of the tensors ate, the number of linearly independent points Cij is equal to m1 + 1. This number is equal to the maximal number of linearly independent forms in the bundle (fir) 4)" 1 }. Then the dimension of the osculating subspace of U' is equal to m + m1 + 1.

Under the Darboux mapping, the image of the osculating sphere of the submanifold V"' C C" is the intersection of the hyperquadric Q" with the (m + mi + 1)-dimensional osculating subspace of U"'. Thus the dimension of an osculating sphere of the submanifold V' C C" is one less than the dimension of the corresponding osculating subspace of U' C P"+1. Hence this dimension is equal to m+m1, and this matches the result of Subsection 3.1.8. Using the same method, we can consider osculating subspaces of higher orders of U' C P"+1 and prove that the dimension of each of these subspaces is one higher than the dimension of the corresponding osculating sphere of the submanifold V"' C C".

3.2

Submanifolds Carrying a Net of Curvature Lines

1. In Section 2.4 we naturally introduced the notions of principal directions and curvature lines for a hypersurface V"-' of a conformal space C". We now consider an m-dimensional submanifold V' in the space C". As we have shown in Section 3.1, with a first-order differential neighborhood of a point x E V'", there is associated the quadratic form (3.1.7), and with a secondorder differential neighborhood of a point x E V', there is associated the invariant bundle of second fundamental forms (3.1.37): 9 = 9ijw'w', '(2) = a9Ijw'wJ

(3.2.1)

If m < n-1, then, in general, the quadratic forms (3.2.1) cannot be reduced simultaneously to sums of squares, since the number of these forms exceeds two. Thus, it is not possible to define the principal directions and the curvature line

on a general submanifold V' if m < n - I. However, it is possible to consider a special class of submanifolds Vrn at each point of which the forms g and X121 can be reduced simultaneously to sums of squares. Such submanifolds VI are called submanifolds carrying a net of curvature lines. Submanifolds of this kind have many common properties with hypersurfaces. We will study such submanifolds in this section. For simplicity we suppose in this section that a submanifold V' belongs to a proper conformal space. This implies that the form g is positive definite.

90

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

By means of the Darboux mapping (see Subsection 1.1.5), a submanifold

V' of the space C" can be mapped onto a submanifold U"' lying on a hyperquadric Q" of a projective space P1+1 Moreover the quadratic forms g become the second fundamental forms of the submanifold U1, and to a submanifold V' that carries a net of curvature lines, there corresponds a submanifold U' that carries a net of conjugate lines. The latter submanifolds were studied in Chapter 3 of Akivis and Goldberg [AG 93]. In the present book we will discuss the conformal theory of submanifolds carrying a net of curvature lines that is independent of the projective theory. and

2. As we did in Section 3.1, with each point x E VI we associate a conformal frame whose point A0 coincides with the point x E V', whose hyperspheres Ai, i = 1, ... , m, are orthogonal to VI at the point x, whose hyperspheres A0, a = m + 1, ... , n, are tangent to V'" at the point x, and

whose point An+1 is the second point of intersection of the hyperspheres A; and A0. The equations of infinitesimal displacement of such a frame have the form of (1.2.6), and the forms w{, determining the infinitesimal displacement, satisfy the structure equations (1.2.10) and are connected by a series of relations (3.1.8), one of which has the form n+1

wi

= 9i,iwj,

(3.2.2)

where wi = w01 are basis forms on the submanifold VI and g.j = (A;, A;) is the tensor associated with the quadratic form g defining a conformal structure on the submanifold V"'. As we have shown in Section 3.1, in the moving frame described above, the submanifold V'" is defined by the system of equations (3.1.6), w° = 0,

(3.2.3)

and the differential prolongation of equations (3.2.3) leads to the system (3.1.17),

w; =\ 1) Wj, '\= 13

a in Section 3.1 the system of tensors (3.1.34),

(3.2.4) allowed us to construct

a = a - -\k19kt9ii, m

(3.2.5)

by which means we introduced the invariant bundle of quadratic forms 4;z1; see equation (3.2.1). The tensors 0°- and g;j satisfy the apolarity condition a fig`) = 0.

(3.2.6)

Suppose that the submanifold VI carries a net of curvature lines. If we choose the hyperspheres A, of our conformal frame so that they are orthogonal

3.2

Submanifolds Carrying a Net of Curvature Lines

91

to the corresponding curvature lines of this submanifold, its quadratic forms g and 41(2) become

g = (Wl)2 + ... + (Wm)2, a1 (wl)2 + ... + am(wm)2,

where a° =

(3.2.7)

Then the ith family of curvature lines is defined on the

submanifold Vm by the system of equations

wj =0

if j # i.

Now gij = bij is the Kronecker symbol, and equations (3.1.10) become w; +W." = 0.

(3.2.8)

Moreover it follows from equations (3.2.2), (3.2.5), and (3.2.7) that gij = a = 0 for j 54 i. By setting ) = .1, we obtain Wn+1 = Wi

where now, and in what follows, summations are not taken with respect to the indices i, j, k unless the sign E indicates this. If we take exterior derivatives of equations (3.2.9) and apply equations (3.2.8), we arrive at the following exterior quadratic equations:

Da°AW'-E(a;-A )wi Awj=0,

(3.2.10)

J#i

where Dal = dal - 2A°w;

)i° WO - Wn+1

Let us consider the system of equations (3.2.10). By (3.2.5) the following relations hold:

Eak

(3.2.11)

m k=1

and consequently

As a result equations (3.2.10) can be written in the form

AA? A w' - E(a? - aflwf A wj = 0.

3.2.12)

If in the bundle of quadratic forms '(2), there is at least one form all eigenvalues a° of which are distinct, then it follows from equations (3.2.12) that the forms wi, j 54 i, occurring in them are linear combinations of the basis forms wk:

_

likwk, k

i

j.

(3.2.13)

92

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

By (3.2.8), the coefficients l,k in (3.2.13) are skew-symmetric with respect to the indices i and j: (3.2.14) lik + ljk = 0.

Geometrically, our last statement means that if in the bundle of quadratic forms {s) there is at least one form with distinct eigenvalues, then the submanifold V '" carries a unique net of curvature lines. Another extreme case is the case when at any point x E V'", the eigenvalues of all quadratic forms ',

are equal one to another. Then, by the apolarity condition (3.2.6), all these eigenvalues are equal to 0; that is, all tensors a, are zero-tensors. But then, as we noted in Subsection 3.1.5, the submanifold V'" is an m-dimensional sphere (or its open subset) on which any smooth curve enjoys the properties of curva-

ture lines, and the net of curvature lines on V' is indeterminate. As we will see later, these two extreme cases do not exhaust all possible types of nets of curvature lines on a submanifold V"'. However, if the submanifold V'" carries at least one net of curvature lines, and this net is taken as a coordinate net, then decompositions (3.2.13) hold. If we substitute for the forms wi in equations (3.2.12) their expressions (3.2.13)

and equate to zero the coefficients in the independent exterior products wi A wk, j, k i4 i, then we obtain the relations

(a7 -

(ak - a°)l 13 = 0,

(3.2.15)

where i 76 j, k and j 54 k.

3. As in Section 3.1 we denote the number of linearly independent forms among the quadratic forms 1' by ml. These forms are expressed by formulas (3.2.7), and their coefficients satisfy the apolarity condition (3.2.6) which can now be written in the form

a? =0.

This implies that instead of inequality (3.1.74), the number m1 satisfies the inequality

ml <m-1. Equations (3.2.9) imply that the second differential of the point Ao E V' satisfies the relation d2Ao = >(a°AQ +An+,)(wi)2 (mod Ao, Ai).

(3.2.16)

Consider the hyperspheres

Bi = a°AQ + An+l

(3.2.17)

and write their coordinates in the form of a rectangular matrix with m columns

3.2

Submanifolds Carrying a Net of Curvature Lines

93

and n - m + 1 rows: I Am+1

...

1

Am+1

m

1

A= \ An

An

m

1

Along with the matrix A we consider the matrix a m+1 l

A=

"

m+1 a,,,

..................

a1 an

am an

whose entries a' are defined by formula (3.2.11). The rank of matrix A is equal to the number of linearly independent forms among the quadratic forms g and Since the rank of the system of forms I' is equal to ml and the form g is not a linear combination of the forms the rank of the system of forms g and and together with it the rank of matrix A, is equal to ml + 1. But the rank of matrix A coincides with the rank of matrix A, since these

matrices can be obtained one from another by means of elementary matrix transformations. Hence the rank of matrix A is also equal to m1 + 1. Since the entries of matrix A are coordinates of the hyperspheres B;, the rank of this system of hyperspheres is also equal to m1 + 1. Consider the hyperspheres tangent to the submanifold Vm at the point x = A0 and orthogonal to all hyperspheres B;. By (3.2.16) they are osculating hyperspheres of the submanifold V m at the point x, and their intersection is the

osculating sphere of the submanifold V' at the point x. Since the rank of the system of hyperspheres B, is equal to m1 + 1, the dimension of this osculating sphere is equal to m + m1. As in Section 3.1 we denote this osculating sphere by C=+m' Let us change the basis in the bundle of tangent hyperspheres in such a way that in the new basis the hyperspheres A,,, u = m + m1 + 1,... , n, contain the osculating sphere C= +.n' , and the hyperspheres a' = m + 1, ... , m + m1, are orthogonal to the sphere Ci +.nt . The hyperspheres Aa' together with the point An+1 constitute a basis in the bundle of hyperspheres determined by the hyperspheres Bi. If this specialization has been made, the last n - m - m1 rows in the matrices A and A become rows with all zeros. We will write the new matrices without these zero rows: 1

1m+1 Am+mt

...

...

1

1

m+l Am

_

A= \m+mt m

a1m+ 1

1

a m+l m

...........

a m+mt 1

m

94

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Figure 3.2.1

Figure 3.2.2

Note that the rank of each of these matrices coincides with its number of rows,

m1+1<m. 4. If a submanifold V'" carries a net of curvature lines, then m distributions , of dimension m -1 are connected with this net. Each of these distributions is orthogonal to one family of curvature lines and passes through the tangents to the lines of the remaining m - 1 families. The distribution A, is determined by the equation w' = 0, where i is a fixed index. A net of curvature lines on VI is said to be totally holonoric if each of the m distributions i , is involutive, that is, if each 0, determines on Vm a one-parameter foliation whose leaves Vi'-1 contain the curvature lines not belonging to the ith family. In contrast, a net of curvature lines on V'° is said to be irreducible if each distribution 0; is not involutive. Each pair of families of curvature lines from the net of curvature lines of V' determines a curvilinear two-web in the sense of Blaschke (see Blaschke, z

1131 55), pp. 96-98).

If the distribution 0'" defined on V'" by the pair of families 1' and li of curvature lines is holonomic, then the quadrangles formed by these lines are

not closed (see Figure 3.2.1). On the other hand, if the distribution 0" is holonomic, then these quadrangles are closed (see Figure 3.2.2). In this case, the two-web {l', l' } is called quadrilateral (see Blaschke [Bl 55], pp. 99-100). All two-webs generated by the net of curvature lines on V'" are quadrilateral if and only if this net is totally holonomic. If the net of curvature lines is irreducible, then none of the two-webs generated by this net is quadrilateral. By the Frobenius theorem (see Kobayashi and Nomizu [KN 63], vol. 2, p. 323), equation w' = 0 is completely integrable if and only if the condition

dw'Aw'=0 holds. But by (3.2.13), this condition can be written as This condition is equivalent to the fact that for j, k j6 i, the coefficients symmetric with respect to the lower indices:

k = lk,.

are

(3.2.18)

3.2

Submanifolds Carrying a Net of Curvature Lines

95

Moreover the coefficients lj'k satisfy conditions (3.2.14). All these imply that for i i4 j, k and j 96 k, we have ljk

lik = -lki = lji = l,j = -lki _

lj'k,

that is, all these coefficients are equal to 0. Thus we have proved the following result:

Theorem 3.2.1 A net of curvature lines on a submanifold V' is totally holonomic if and only if the conditions 0 hold for mutually distinct indices i, j, and k. We now find the geometric sufficient condition for a net of curvature lines

on a submanifold V' to be totally holonomic. This condition is connected with the structure of the system of hyperspheres Bi introduced in Subsection 3.2.3.

Theorem 3.2.2 If for any point x of a submanifold V°' carrying a net of curvature lines any three hyperspheres Bi associated with the point x do not belong to a pencil, then the net of curvature lines on V' is totally holonomic. Proof. We consider the system of equations (3.2.15) where the indices i, j, and k are fixed and mutually distinct. The determinant of the matrix of coefficients of each pair of these equations corresponding to different values of o: has the form

a - a'

a*, - a°

-

aa ap and can also be written in the form

ak

- ap

1

1

1

a?

aq

ak

as

ap

ak

System (3.2.15) has only the trivial solution with respect to the coefficients lik

if and only if, for any three mutually distinct indices i, j, and k, at least one of the determinants 0 does not vanish. However, since the columns of A are coordinates of the hyperspheres Bi, Bj, and Bk, this condition is equivalent to the fact that any three such hyperspheres do not belong to a pencil. Suppose further that only one distribution Di associated with a net of curvature lines of V'° is involutive, and consider those curvature lines of VI on integral submanifolds Vi'"-1 of Ai that do not belong to the ith family of curvature lines of V'°. We will prove the following theorem: Theorem 3.2.3 If the distribution Ai is involutive, and among hyperspheres

Bj, j $ i, no two hyperspheres coincide, then the net of lines induced on integral submanifolds Vim-1 of Di by the net of curvature lines on V' is a net of curvature lines on Vim-1.

96

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Proof. Since the distribution w' = 0 is involutive, by (3.2.18), for fixed indices i and j, k 0 i, the conditions l k = lk, hold. It can be easily checked by considering the second differential of the point A0 on the submanifold Vvm'I, that the coefficients 1?k determine on Vim-I one more invariant quadratic form

E

j,k#i Vi'-' will be a net of curvature lines if and only The net of lines induced on if this quadratic form reduces to a sum of squares, that is, if ljk = 0 for j k. Permuting indices in equations (3.2.15), we can write them as follows:

(ak - aj*)I" - (a° - aJ )ljk = 0. Transposing the indices j and k in the last equation, we obtain

(aj - ak )Iki -

(aa

- ak)Ikj = 0.

Subtracting the second equation from the first one and taking into account with respect to the lower indices, (3.2.14) and the symmetry of coefficients we find that (a! - ak)1 k = 0.

If the hyperspheres Bj and Bk do not coincide, then aj 0 ak at least for one value of a. By virtue of this, we have lk = 0. 0 5. Let us find under what conditions a submanifold VI carrying a net of curvature lines belongs to its osculating sphere. Theorem 3.2.4 Suppose that a submanifold V"' C C" carries a net of curvature lines and that at each point x the submanifold V"' possesses an osculating sphere C= +m' of dimension m + ml, where m1 + 1 < m. Suppose also that the system of hyperspheres Bi orthogonal to the sphere C= +" is such that any subsystem consisting of m - 1 hyperspheres has the same rank m1 which the whole system of hyperspheres Bi has. Then the osculating sphere C= +.n' is constant at all points x E V'°, and V' ties entirely in this sphere C'4-m'. Proof. We make the same specialization of frames that has been made at the end of Subsection 3.2.3. Then the matrix A becomes A, and the second system of (3.2.9) splits into two subsystems (cf. formulas (3.1.75)): 0,

(3.2.19)

where a' = m + 1, ... , m + ml, u = m + m1 + 1, ... , n. Taking the exterior derivatives of the equations of the second of these subsystems, we obtain

w' nwa, +w;+I Awn+1 =0.

(3.2.20)

3.2

Submanifolds Carrying a Net of Curvature Lines

97

Since the hyperspheres Au form a basis of the bundle of hyperspheres having the osculating sphere C.1+11 in common, the 1-forms wu, where u # m + ml + 1,. .. , n, and also the forms wu are linearly expressed in terms of the basis forms w'. Thus we can write

wn' =

wntl = µjw .

(3.2.21)

Substituting for the forms w°',wa, and wnt1 in equations (3.2.20) their decompositions (3.2.19) and (3.2.21), and taking into account that in our specialized frame wi +i = w', we obtain µj + A° µQ, j = 0,

(3.2.22)

j. The coefficients in p and pu,j in these equations are the columns of the matrix A which are different from the jth column. Since the theorem hypotheses imply that the rank of the matrix A is not reduced if we discard one of its columns, system (3.2.22) has only the trivial solution for any fixed values of the indices j and u: where i

µu, j = 0, lei = 0.

By virtue of this, from system (3.2.21) it follows that wu, = wu+l = 0. Since

in addition we have wo = wi = 0, it follows also that w°' = 0 for any u u m + ml + 1, ... , n. This implies that also wu = 0 and dAu = wuAv But this result means that the osculating sphere C= ++", , which is the intersection of the hyperspheres A, is fixed and that the submanifold V'n belongs to a conformal space C"'+m, of dimension m + ml. E Theorem 3.2.4 is a conformal analogue of the generalized Segre theorem which is known in projective differential geometry (see Akivis and Goldberg [AG 93], p. 82). We now consider a submanifold VI satisfying the condition opposite to the condition of Theorem 3.2.4; that is, we suppose that mi + 1 = m. In this case all spheres B; defined in a second-order neighborhood of a point x E V'n are linearly independent, and for m > 3 no three of them belong to a pencil. Thus, by Theorem 3.2.2, the net of curvature lines on V'n is totally holonomic. Note

that although for m = 2 we cannot satisfy the hypotheses of Theorem 3.2.2, but a net of curvature lines on V2, if it exists, is always totally holonomic. For ml + 1 = m the osculating spheres of V'n, which can vary from point to point of V'n, have dimension 2m - 1 at any of its points. It is also possible to prove that under the Darboux mapping the image of a submanifold V'n carrying a net of curvature lines, for which ml + 1 = m, is an m-dimensional submanifold U'n of the hyperquadric Q" of the projective space Pn+1 and that U'n carries

98

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

a holonomic net of conjugate lines with an osculating subspace at each point of dimension 2m. Cartan was the first to consider this kind of submanifolds in a projective space (Cartan [Ca 20a], Ch. IV). This is the reason that they are called the Carton varieties. 6. We now derive some consequences of Theorems 3.2.2 and 3.2.4.

Theorem 3.2.5 Only m-spheres and hypersurfaces of a conformal space can possess an irreducible net of curvature lines. Proof. For rn > 2 a net of curvature lines on a submanifold V°' is irreducible if none of the 1-forms w' defining this net is completely integrable, namely, if

for any i, there are values j and k, j 0 i,k $ i, j 96 k, for which there are nonvanishing coefficients ljik. But this can be true only in two cases: either coordinates of any three hyperspheres Bi, Bj, and Bk are linearly dependent and all these hyperspheres belong to a pencil, and no two of them coincide, or the corresponding coordinates of these three hyperspheres are equal one to another, and these hyperspheres coincide. In the first case, we have ml = 1, and by Theorem 3.2.4, the submanifold V1 belongs to a sphere C'"+', which means that V°' is a hypersurface. In the second case, ml = 0, and the submanifold V- is an m-sphere C' or its open subset. We now state without proofs three theorems that are also consequences of theorems proved in this section. Theorem 3.2.6 A submanifold V' that carries a net of curvature lines may be decomposed into a q-parameter family of p-dimensional submanifolds VP (where

p + q = m) carrying an irreducible net of curvature lines in two cases: i.

When p hyperspheres of the system of hyperspheres B, of the submanifold V'" belong to one bundle, but no two of them coincide. In this case the submanifolds VP are hypersurfaces of a conformal space CD+'

ii.

When p hyperspheres of the system of hyperspheres Bi coincide among themselves. In this case the submanifolds VP are p-dimensional spheres.

Theorem 3.2.7 If on a submanifold V' that carries a net of curvature lines, p hyperspheres of the system of hyperspheres B, coincide, then the submanifold V' is the envelope of an (n - p)-parameter family of m-dimensional spheres.

7. We will now prove the existence theorem for submanifolds V' carrying a holonomic net of curvature lines. Theorem 3.2.8 Submanifolds V'" carrying a holonomic net of curvature lines on which the system of hyperspheres Bi satisfy the conditions of Theorem 3.2.4 exist. The solution of the system of equations defining such submanifolds depends on 2m(m - 1) arbitrary functions of two variables.

Submanifolds Carrying a Net of Curvature Lines

3.2

99

Proof. The system defining such submanifolds consists of equations (3.1.8)-

(3.1.11), (3.2.3), (3.2.9), and (3.2.13) in which now gij = dij and lik = 0 for k 0 i, j. We will write these equations as n+l

w0

-

0

0

o+ n+l _

- 0+

WO

wn+1 = wi i

wi = wn+1 = 0,

wn+1 - 0+

n+1 Wa

=0, (3.2.23)

w« = gaown+l+

o i

i

wo

w + w; = 0,

wa = gao,2w, dgao = gaywp + gyoca«,

w° = 0,

X°w',

(3.2.24)

and

wi =

l;jwj.

By means of the latter equations and the conditions w, + wi = 0 of the As a result we have the following system (3.2.23), we find that expressions for the forms wj :

wi = lj;w' -

(3.2.25)

Exterior differentiation of equations (3.2.23) and the first equation of (3.2.24) leads to identities. Exterior differentiation of remaining equations (3.2.24) and equation (3.2.25) gives the following exterior quadratic equations:

Da°Awi=0, i 1,Aw'-1ljAwj=0, where

10

(3.2.26)

Aa° = dA° + a°wo + E 1kIt (A° + Apw° k - a°)wk i , o - w° n+1+ S

k#i

j 0+ Al.j i= dliil+ 111-40

k

kq6i,j

j

j

lk;ilk

k

lii(lkk - lii)w - 2

0

wj W.

k#i,j

The system of equations (3.2.23)-(3.2.26) is closed with respect to the operation of exterior differentiation. We will apply the Cartan test (see Akivis and Goldberg JAG 93), p. 13) to investigate the consistency of this system. The number q of independent unknown functions AX? and A1q. in the exterior quadratic equations is q = (n - m)m + m(m - 1) = m(n - 1). The first character sl of the system under consideration is equal to the number of independent

exterior quadratic equations, s, = (n - m)mZ+ m(m - 1) =1. m(2n -m -

1).

Its second character 82 = q-si = am(m-1), and the third and all subsequent characters are equal to 0: 83 = ... = 8m = 0. This implies that the Cartan number Q = Si + 282 = 1 m(2n + m - 3).

100

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Let us find the number N of parameters on which the most general integral element of the system of equations (3.2.23)-(3.2.26) depends. To find N, we apply the Cartan lemma to equations (3.2.26): AX? =

Alii =

Ali =

l1+14

;;jw' - ljjjwi.

It is easy to see from these equations that the number N of independent coefficients in these equations is

N = m(n - m) + 2 3m(m - 1) = 2m(2n + m - 3). Since N = Q, the system of equations (3.2.23)-(3.2.26) is in involution, and an m-dimensional integral manifold V'", defined by this system, depends on 82 = 1m(m - 1) arbitrary functions of two variables.

3.3

Submanifolds in a Pseudoconformal Space

1. Until now we considered real submanifolds of a proper conformal space C". The conformal structure of such submanifolds is determined by the positive definite form (3.3.1) 9 = gijw'w" i, j = 1,.... m, (see formula (3.1.7) in Section 3.1), and this is the reason that these subman-

ifolds do not carry real isotropic directions. The situation is different for a pseudoconformal space C.". The isotropic cones C. of this space are defined by the equation

9 := 9r,w''w' = 0,

r' s = 1, ... , n,

(3.3.2)

whose left-hand side is a nondegenerate quadratic form of signature (p, q),

p + q = n. For p > 0 and q > 0, this cone is real. Thus the tangent subspace TT(Vm) can have a real intersection with the cone C2, and then the submanifold V' carries real isotropic directions. By a real transformation of coordinates, the left-hand side of equation (3.3.2) can be reduced to the form

g = (w')2 + ... + (w')2 - (wn+t)2 - ... - (w")2.

(3.3.3)

Therefore, on the submanifold V'" C C9", the quadratic form g defined by equation (3.3.1) can have different signatures that depend on the numbers m, p, and q and on the mutual location of the tangent subspace T=(VI) and the isotropic cone C. The form g can also be a degenerate quadratic form- this happens if at each point of the submanifold V'", its tangent subspace T=(V'") is tangent to

3.3

Submanifolds in a Pseudoconformal Space

101

the isotropic cone C. In this case the submanifold V "' is called isotropic or lightlike (e.g., see Kossowski [Kos 89]).

Let us consider, for example, the pseudoconformal space C, for which the quadratic form g is of signature (2, 1). The equation of isotropic cones of this space can be reduced to the form (W1)2 + (W2)2 - (W3)2 = 0.

(3.3.4)

Three different possible mutual locations of the cone C_. and the tangent subspace r. = T=(V 2) to the submanifold V2 C Cl are presented in Figures 3.3.1, 3.3.2, and 3.3.3.

Figure 3.3.2

Figure 3.3.1

Figure 3.3.3

102

3. SU13MANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

In the first case, the quadratic form g on the submanifold V2 is positive definite, and this submanifold carries a proper conformal structure. In the second case, the form g is indefinite, and the submanifold V2 carries a pseudoconformal structure of signature (1, 1). Finally, in the third case, the form g on V2 is a degenerate form of signature (1, 0), and the submanifold V2 is isotropic. In the first case, the submanifold V2 does not carry real isotropic directions. In the second case, it carries a net of isotropic lines, and in the third case, it carries a double family of isotropic lines. A similar situation occurs for a hypersurface Vn-1 in the pseudoconformal space C" or C,_1, which is a space of Lorentzian signature. For any value of n, the isotropic cones of these spaces have the same form, which is represented on Figures 3.3.1, 3.3.2 and 3.3.3, and only three different locations of these cones and the tangent hyperplane T=(V"'1) are possible. If at any point x E Vn-1 this location is of the form of Figure 3.3.1, then the hypersurface V11-1 is called spacelike. If this location is of the form of Figure 3.3.2, then the hypersurface 1,,n-I is called timelike. Finally, if this location is of the form of Figure 3.3.3, then the hypersurface Vn-1 is isotropic or lightlike. This terminology is related to that of general relativity. A space-time in special relativity is a four-dimensional Minkowski space, and in general relativity it is a four-dimensional pseudo-Riemannian manifold of Lorentzian signature. In both cases its metric has the signature (3, 1) (or (1, 3)-this depends on the method of presentation). In general relativity the isotropic cone Cy plays the role of the light cone. This cone divides the tangent space T=(C,) (or space T=(C3)) into two domains-internal and external. Directions belonging to the first domain are called timelike, and directions belonging to the second domain are called spacelike (see Figure 3.3.4). The tangent hyperplane Ts(V3) to a

Figure 3.3.4

3.3

Submanifolds in a Pseudoconformal Space

103

spacelike hypersurface contains only directions located outside of the cone C.', namely spacelike directions. For a timelike hypersurface V3, the tangent hyperplane TZ(V3) contains both spacelike and timelike directions. 2. Let us now consider the geometry of spacelike hypersurfaces of the

pseudoconformal space C". As in Section 2.1 we associate with each point x of the hypersurface V"-I a conformal frame in such a way that Ao = x, the hypersphere A" is tangent to V'-1 at the point x, and the hyperspheres Ai, i = 1, ... , n, are orthogonal to V"-I at this point. Then, as in Section 2.1, the hypersurface V"-I is determined by the equation wo = 0,

(3.3.5)

and the 1-forms wi = wo, i = 1, 2, ... , n - 1, are basis forms on this hypersurface.

The quadratic form g, defining the conformal structure in the space C1 at the point x, can be reduced to the expression 9=9ywiwj

- w n) (

2

(3.3.6)

Moreover the quadratic form

9Iv^-' = 9ijwiwi,

(3.3.7)

defining the conformal structure on the hypersurface V"-1, is positive definite. By virtue of this, there are no real isotropic directions on a spacelike hypersurface. The coefficients of the form g are equal to the scalar products of the basis

hyperspheres Ai and A. This implies that in contrast to (2.1.1), we now have (An, An) _ -1,

(3.3.8)

and three of the formulas (2.1.2) will be changed. These three formulas can be written as 9iiwn - w; = 0, wn+I + won = 0, WO + wn+I = 0.

(3.3.9)

The remaining formulas of (2.1.2) will not be changed. The changes indicated above do not imply essential changes in the subsequent equations, which are obtained from equations (3.3.5) by means of differential prolongations, and in

the construction of the main geometric objects and tensors associated with the hypersurface V"-I. This is the reason that we are not going to consider in detail this construction as well as other topics of the theory of spacelike hypersurfaces that we investigated in Chapter 2 for hypersurfaces of the proper conformal space C".

104

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Suppose now that a hypersurface V"-' C Cl is timelike. Then its tangent hyperplane T=(V"'') is located with respect to the isotropic cone Cs as indicated on Figure 3.3.2. Then at any point x E V"'', the quadratic form (3.3.2) which is of signature (n - 1, 1) reduces to g = 9iiw'w` + (w")2,

(3.3.10)

and the quadratic form g determined by equation (3.3.1) is of signature (n - 2, 1). Thus the last form defines a real isotropic cone C=(V"-') in the tangent hyperplane TZ (V"-' ). Since in the case under consideration, the formula

(An,An)=1 is still valid (see Section 2.1), all formulas and conclusions of Chapter 2 are valid too. Note only that since the isotropic cone C. (V"-') for a timelike hypersurface

is real, its mutual location with the cone a{1c wi = 0, determined by the tensor a;3 and connected with a second-order neighborhood of a point x E V"-', can be more diverse than for a hypersurface of the space C" or for a spacelike hypersurface of the space Cl. It would be interesting to construct a classification of timelike hypersurfaces based on the location of these cones. 3. Finally, we consider isotropic hypersurfaces of the space Cl . For such hypersurfaces the quadratic form g is of signature (n - 2, 0). Our considerations will be simpler if we consider the Darboux mapping of an isotropic hypersurface V"-' C CI and all geometric objects associated with this hypersurface. The hypersurface V"-' will be mapped onto a submanifold U"-' of dimension n - 1 belonging to the Darboux hyperquadric determined in the space P"+' by the equation 9rsxrx' - 2x0xn+1 = 0, r,s = 1,...,n

(3.3.11)

(see Section 1.2).

As usual, we locate the vertex A0 of the moving frame at the variable point x E U"-' and the vertices A,,.. . , An_, in the tangent (n- l)-plane T= (U"-' ) Then in addition to equations (1.2.7) and (1.2.8), which the components of infinitesimal displacement of the moving frame of the space C" and of the projective frame corresponding to the conformal frame under the Darboux mapping satisfy, one more equation w0

=0

(3.3.12)

holds.

But since the hypersurface V"-' is isotropic, the tangent (n - 1)-plane T=(U"-') is tangent to the asymptotic cone of the Darboux hyperquadric. This cone corresponds to the isotropic cone C. of the space Cn. We place the vertex Al on the rectilinear generator along which the cone Cx is tangent to the subspace T=(U"''). We also place the vertex An on the cone Cs but outside of this tangent subspace T=(U"-') (see Figure 3.3.5).

3.3

Submanifolds in a Pseudoconformal Space

105

Figure 3.3.5

Then these points satisfy the following relations: (3.3.13)

(At, At) = (An, An) = (Ao, At) = (Ao, An) = 0,

where as in Chapter 1, the parentheses denote the scalar product defined by formula (1.2.4). In addition we normalize the points Al and An by the condition

(At, An) = -1.

(3.3.14)

By virtue of this, the matrix of scalar products of the elements of the moving frame takes the form

(AC, A,,) =

0

0

0

0

-1

0

0

0

-1

0

0

0

0

0

-1

gij 0

0

0 0

-1

0

0

0

0

,

(3.3.15)

where t:, q = 0,1, ... , n + 1; i, j = 2, ... , n - 1. As a result equation (3.3.11) of the Darboux hyperquadric takes the form

gijx`xj - 22'x' - 2x02^+1 = 0,

(3.3.16)

where gijx'xj is a positive definite quadratic form. It follows that the (n - 3)-dimensional subspace, determined in the space pn+1 by the points Ai, i = 2,... , n - 1, does not have real common points with the Darboux hyperquadric, and the subspace, which is conjugate to the above subspace with respect to this hyperquadric and is determined by the points Ao, A1, An, and An+t, intersects this hyperquadric in the following ruled surface of second order: x1xn + 20x^+1 = 0.

106

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Figure 3.3.6

The above four points are located on this ruled surface as indicated in Figure 3.3.6.

The equation of the asymptotic cone C. at the point x = A0 of the Darboux hyperquadric has the form

g = gijw'w' - 2w'w" = 0,

(3.3.17)

and the equation of the asymptotic cone C, of the submanifold U11-1 as well as the equation of the isotropic cone of the hypersurface V"-1 has the form 9Av'

= 9ijw`w3 = 0,

i, j = 2, ... , n - 1.

(3.3.18)

Hence this cone has at the point x a single rectilinear generator A0A1 along which the subspace TZ (U"-1) is tangent to the asymptotic cone of the Darboux hyperquadric. As in Chapter 1 we write the equations of infinitesimal displacement of the moving frame associated with the point x E U"-I in the form dAE =wjAn ,

,t =0,...,n+1

(3.3.19)

(see formula (1.2.6)). In addition to equations (1.2.7), (1.2.8), and (3.3.12), the 1-forms wf also satisfy the equations obtained if one differentiates equations (3.3.13) and (3.3.14):

w"=0, wI=O+ w"0+w"+1=0+ w10+w"+1=0+ wl+w"=0. n n n I 1

1

(3.3.20)

If we also differentiate the equation 91i = 0, we find that n

Since the tensor gij is nondegenerate, it follows that

wi = 9ijwjn where g'j is the inverse tensor of the tensor gij.

(3.3.21)

3.3

Submanifolds in a Pseudoconformal Space

107

Next, taking the exterior derivatives of equation (3.3.12) and taking into account the first equation of (3.3.20), we obtain w, A wp = 0,

(3.3.22)

where in contrast to (2.1.6), the index i takes the values from 2 to n - 1. Applying Cartan's lemma to equation (3.3.22), we find that

wn =.ijwo,

i,j = 2,...,n - 1,

where a;j = )j;. Taking into account equation (3.3.21), we find that wl = g'kAkjwo = ajiwo,

(3.3.23)

where J1 = g'kAkj is a symmetric nondegenerate affinor. We consider now the differentials of the points AO and A1. By (3.3.12), (3.3.19), and (3.3.20) we obtain

r dAo = woAo +woAI +waAi,

(3.3.24)

dA1

From these equations and (3.3.23) it follows that if wo = 0, then the point AO moves along the isotropic straight line AOA1 belonging to the cone Cs and

describes the entire line AoAI or its part; that is, the submanifold

U"-1

is

a ruled submanifold. Moreover the 1-form wo defines the displacement of the point Ao along the straight line AOA1. Next equations (3.3.24) show that at any point of the straight line AOA1 i the tangent (n - 1)-dimensional subspace is fixed and coincides with the subspace TZ(Un-1) = AO A Al A A2 A ... A A"_1. Thus, the submanifold Un-1

is tangentially degenerate of rank n - 2 (see Chapter 4 of Akivis and Goldberg [AG 93]), since the tangent subspace T=(Un-1) depends precisely on this number of parameters. Let us take an arbitrary point X = Al + xAo on the rectilinear generator ABA1 of the submanifold Un`. Its differential satisfies the following formula: dX == (wi + xwo)A1

(mod Ao, A1).

Since, by (3.3.23) we have

wl + xwo = (a + xb )wa, there are singular points on the straight line AoAI, and their coordinates are determined by the equation det(a + xdj1) = 0.

(3.3.25)

Since the tensor A is symmetric, this equation has n - 2 real roots if we count each root as many times as its multiplicity.

108

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

Thus we have proved the following result:

Theorem 3.3.1 Under the Darboux mapping, to an isotropic hypersurface V"-' of the pseudoconformal space Cl there corresponds a ruled tangentially degenerate submanifold Un=2 of rank n - 2 whose rectilinear generator carries n - 2 real singular points if each of them is counted as many times as its multiplicity. These points are the images of singular points of the isotropic

hypersurface V". The loci of singular points on isotropic hypersurfaces Vn-' are submanifolds whose dimension is less than n - 1. These submanifolds are called focal submanifolds. The dimension of focal submanifolds depends on the multiplicity of their elements-singular points. If x, is a simple root of equation (3.3.25), then to this root there corresponds a family of torses (developable surfaces) on the submanifold U,"=21 defined by the system of equations WI + XIwo = 0.

(3.3.26)

From the well-known theorem of linear algebra on orthogonality of eigendirections of a symmetric linear operator, it follows that to distinct roots of equation (3.3.25) there correspond two mutually orthogonal families of torses on Un=z It is not difficult also to describe submanifolds on Un-12 corresponding to multiple roots of equation (3.3.25). Note that since in theory of relativity, to isotropic straight lines of the space C' there correspond lines of propagation of light, therefore to singular points

on isotropic hypersurfaces there correspond sources of light or points of its absorption, and their focal submanifolds are lighting surfaces or surfaces of light absorption. The further study of isotropic hypersurfaces in the space Cl can be of interest for general relativity.

3.4

Line Submanifolds of a Three-Dimensional Projective Space

1. In this section we will consider some submanifolds of the pseudoconformal space C2. Since the geometry of this space is equivalent to that of the Grassmannian G(1,3) (see Section 1.4), the study of submanifolds of the space CZ is directly connected with the study of the line submanifolds of the projective space P3: ruled surfaces, congruences, and complexes of straight lines. These kinds of manifolds were studied intensively by many authors. However, the relationship between the geometry of submanifolds of the space CZ and the geometry of line manifolds of the space P3 was used substantially only by a few authors. In this section, while studying the geometry of line submanifolds of the space p3, we will use this relationship systematically. In Section 1.4, we have already derived the basic formulas, which we will apply in this section, and considered linear complexes and linear congruences

3.4

Line Submanifolds of a Three-Dimensional Projective Space

109

of straight lines in the space P3, and their images under the Plucker mapping,

namely in the space C. In the current section, we will consider arbitrary three- and two-parameter submanifolds of straight lines in the space P3 which are called complexes and congruences of straight lines, respectively. Their images under the Plucker mapping are smooth three- and two-dimensional submanifolds on the hyperquadric 11(1, 3), namely hypersurfaces V3 and two-dimensional submanifolds

V2 in the space C. 2. First of all, note that under the Plucker mapping, an arbitrary smooth line of the pseudoconformal space CZ is the image of a ruled surface of the projective space P3 and that isotropic curves of the space C2 correspond to developable ruled surfaces (torses) of the space p3. Isotropic curves and torses satisfy the equation g = 0 where the quadratic form g is determined by equation (1.4.16).

Next, we consider a complex ,c of straight lines in the space P3. Under the Plucker mapping, to this complex there corresponds a submanifold V3 C P5, and its geometry is equivalent to the geometry of the complex of straight lines in the space p3. We associate a moving frame with any point x of a hypersurface V3 in such a way that its vertex ao coincides with the point x and denote by T=(V3) the three-dimensional tangent subspace to V3 at the point x. Since the cone C2, determined by the equation g = 0, is associated with any point of the space Ci, we must distinguish two kinds of three-dimensional tangent elements T.(V3) to the hypersurface V3 C C. In the general case, these elements intersect the cone C2, and in the special case, they are tangent to this cone. In the latter case, they are called isotropic. A hypersurface V3, all tangent elements of which are nonisotropic, is called nonisotropic. Its preimage in the space p3 is a complex of straight lines of general type. A hypersurface V3, all tangent elements of which are tangent to isotropic cones, is called isotropic. Its preimage in the space P3 is a special complex of straight lines. We will now prove the following theorem:

Theorem 3.4.1 A special complex of straight lines is a set of tangent straight lines to a two-dimensional surface V2 of the projective space P3. Proof. Since a straight line of a complex is depends on three parameters, it follows that only three out of four 1-forms 03, 8Z, 0', and 82 determining a displacement of the line ao in the space P3 (see Subsection 1.4.3), are independent on the complex re. We will write the condition relating these forms as

81 = oB2 + (iO2 + ryB1.

(3.4.1)

110

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

On the complex rc, the form g defined by equation (1.4.16) can be written as gj. = 2(Q(924)2 + Q9292 + ry9192 - 9291).

The discriminant of this quadratic form is 1

2a 3

Q 0

-1

ry

-1

0

ry

= -2(Qry+a).

If the complex x in question is special, then the rank of its quadratic form 91, does not exceed two, and A = 0. It follows that a = -Qry

(3.4.2)

9i = -Qry02 + 002 + ry91.

(3.4.3)

and

We now consider the point Ml - QM2 on the straight line ao = Ml A M2 and calculate the differential of this point: d(llil - QM2) = (01 - Q91)MI + (92

- dQ - Q92)M2 + (9i - Q92)(M4 +7M3)

(3.4.4)

Hence the point MI -QM2 describes a two-dimensional surface V2 whose tangent plane is the plane MI A M2 A (A14 + 7M3), and the line as is tangent to this surface.

The converse is trivial. Since the image of the complex of tangents to a two-dimensional surface V2 of the space p3 is a submanifold V3 C fl(1,3) carrying a two-parameter family of straight lines, these straight lines are the images of pencils of lines tangent to V2, and hence the submanifold V3 is an isotropic submanifold. 0 Let us study in more detail the structure of an isotropic submanifold V3 C 11(1,3) representing a special complex of straight lines of the space p3. We take the point M1 as the moving point of the surface V2 and the points A12 and M4 on its tangent plane. Then we obtain Q = ry = 0, and equations (3.4.3) and (3.4.4) take the form

0=0

(3.4.5)

dMI=9IMI+91M2+91M4.

(3.4.6)

and

Thus the forms 02 and 04 are basis forms on V2. In the frame described above, the second fundamental form of the surface V2 can be written as 9;92 +9;94

(e.g., see Akivis and Goldberg [AG 93], pp. 38-42), and the equation '1 determines the asymptotic lines on the surface V2.

=0

3.4

Line Submanifolds of a Three-Dimensional Projective Space

111

The image of the pencil of tangents to the surface V2 under the Pliicker mapping is a rectilinear generator of the hyperquadric 0(1,3) which is determined by the points ao = Ml A M2 and a4 = Mt A M4. This proves one more time that V3 is a ruled submanifold. Taking into account equations (3.4.5), we now calculate the differentials of the points ao and a4: dao = (91 +022 )ao - 62a2 - 0',a3 + 92a4, (3.4.7)

1

da4 = 94ao - 943a2 + 02a3 + (9i + 94)a4. 1

This implies that the ruled submanifold V3 is tangentially degenerate (see Akivis and Goldberg [AG 93], pp. 113-120): its tangent 3-plane at any point of its rectilinear generator ao A a4 coincides with the 3-plane ao A a2 A a3 A a4 and depends on two parameters. We can see from equations (3.4.7) that the developable surfaces of this tangentially degenerate submanifold V3 are determined by the equation B2 9 43

- e'02

= 0.

(3.4.8)

1

But this is exactly the equation 0. Thus the developable surfaces of V3 correspond to the asymptotic lines of the surface V2 C P3. Moreover, if the surface V2 C P3 generating a special complex of straight lines consists of elliptic points, then the developable surfaces of V3 are imaginary; if the surface V2 C P3 consists of hyperbolic points, then the ruled submanifold V3 is also hyperbolic (i.e., it carries two real families of developable surfaces);

and if the surface V2 C P3 consists of parabolic points (i.e., it is itself developable), then its image on the hyperquadric 11(1,3) is a tangentially degenerate three-dimensional submanifold V3 carrying a one-parameter family of two-dimensional plane generators. Since the hyperquadric fl(1, 3) is endowed with the structure of the pseudoconformal space C24, the following theorem is valid:

Theorem 3.4.2 Any three-dimensional isotropic submanifold V3 of the pseudoconformal space C24 is a tangentially degenerate ruled submanifold whose rectilinear generators are isotropic straight lines. This submanifold is the image of a special linear complex of straight lines of a projective space p3 which is the

collection of tangents to a two-dimensional submanifold V2. The submanifold V3 is elliptic, hyperbolic, or parabolic whenever the submanifold V2 generating the complex is elliptic, hyperbolic, or parabolic, respectively. The correspondence between surfaces V2 C P3 and ruled submanifolds on the Plucker hyperquadric was first noted by E. Bompiani [Bom 12].

3. We will now study general complexes of straight lines in the space P3. Their equation can be also written in the form (3.4.1) but now without

112

3. SUBMANIFOLDS IN CONFORMAL AND PSEUDOCONFORMAL SPACES

condition (3.4.2). One can easily prove that the equation of a nonisotropic complex of straight lines in Ps can be reduced to the form 9° 2-- 03 1

(3.4.9)

(see Finikov (Fin 56), Ch. 18). By virtue of this and relations (1.4.14) and (1.4.15), the differential of a straight line of the complex can be written as dao = (91 + 922)ao - 923a2 - 03(a, - a4) - 041a3.

(3.4.10)

The quadratic form g, determining a conformal structure on this complex, can be written as g = 2((#13 )2 - B29i) (3.4.11)

and has the signature (1, 2). The equation g = 0 determines a real cone of second order (an isotropic cone) in the three-dimensional space T. (V3) that is tangent to the submanifold V3 representing the complex in question on the hyperquadric f1(1,3). On the complex of straight lines in the space P3, the equation g = 0 determines the set of developable surfaces of this complex. The linear complex al + a4, as well as linear complexes of the pencil c = al + a4 + Aao,

(3.4.12)

are tangent to the complex determined by equation (3.4.9), since the condition (c,dao) = 0 holds for any A. The further theory of nonisotropic complexes of straight lines in the space Ps can be constructed in a way similar to that used in Chapter 2 for construction of the theory of hypersurfaces in a conformal space C":

a. In a second-order differential neighborhood, a central tangent complex is invariantly distinguished from the pencil (3.4.12) of tangent linear complexes.

b. In the same neighborhood, an invariant quadratic form to the quadratic form g, is defined.

which is apolar

c. The equation 4 = 0 determines ruled surfaces on the complex, and along them, the complex has a second-order tangency with the central tangent complexes.

d. Jointly the equations g = 0 and V(Z) = 0 determine the developable surfaces on the complex which have a second-order tangency with all linear complexes of the tangent pencil (3.4.12). In general, four such developable surfaces pass through any straight line of the complex. The points of tangency of their cuspidal edges and a line of the complex are called the inflectional centers of the line of the complex.

3.4

Line Submanifolds of a Three-Dimensional Projective Space

113

e. Using the quadratic forms g and 'I ), we can define on the complex the affinor of Burali-Forti (see Subsection 2.4.1) whose principal directions, and the ruled surfaces enveloping these principal directions correspond to the principal directions and the curvature lines of a hypersurface. The abovementioned affinor allows one to construct a classification of nonisotropic complexes of straight lines (see Kovantsov (Kov 63], pp. 71-87).

f. In the theory of nonisotropic complexes the interpretation of properties of a hypersurface that are connected with the tensors aijk and cij defined in a third- and fourth-order differential neighborhood of the complex can be also given.

4. Finally, we consider a two-parameter family of straight lines in the space P3, which is called a congruence of straight lines. The image of a congruence of straight lines under Pliicker mapping is a two-dimensional surface V2 on the hyperquadric 0(1, 3). We consider the tangent plane T=(V2) to V2 at an arbitrary point x E V2. We can classify congruences of straight lines depending on the location of this plane with respect to the isotropic cone C= defined in the tangent subspace T., (1l(1,3)) by the equation g = 0. The following cases are possible: a. For any point x E V2, the plane T,, (V2) and the cone C,, have only the point x in common. Then the surface V2 does not carry isotropic lines, and the corresponding congruence in p3 does not have real developable surfaces. The congruences of this type are called elliptic.

b. For any point x E V2, the plane T=(V2) intersects the cone CZ; namely T=(V2) and C. have two common rectilinear generators. Then the surface

V2 carries an isotropic net that corresponds to the net of developable surfaces on the congruence in p3; namely two real developable surfaces pass through any straight line of the congruence. The congruences of this type are called hyperbolic.

c. For any point x E V2, the plane Tz(V2) is tangent to the cone C1f namely T1(V2) and CZ have a unique common rectilinear generator. Then the surface V2 carries a unique family of isotropic lines that corresponds to a unique family of developable surfaces on the congruence in p3. The congruences of this type are called parabolic.

d. For any point x E V2, the plane T=(V2) belongs to the cone C=; namely T. (V2) is one of rectilinear generators of C=. Then the surface V2 is totally isotropic since any of its lines is isotropic. The congruences of this type in the space p3 are called degenerate.

We will not investigate the structure of elliptic, hyperbolic, and parabolic congruences, since these congruences are described in great detail in literature (e.g., see the monographs by Finikov [Fi 50, 56]).

114

3. GEOMETRY OF SUBMANIFOLDS

We will rather study the structure of a degenerate congruence of straight lines. We recall that the cone C. carries two families of plane generators. Hence, there are two kinds of degenerate congruences of straight lines. Suppose that the point ao E f)(1, 3) corresponds to the line ao of the congruence in question. For a degenerate congruence of the first kind, the plane Tr (V') coincides with a plane generator of the first family of the cone C. We

place the vertices ao = x, al, and a2 of the moving frame into the plane T. Then from equations (1.4.14) and (1.4.15), we find that

daft = (0; +02 )ao - 0;at - 03a2. Thus the equations of this congruence can be written as

o1 = 0, oa = 0,

(3.4.13)

and the forms 0i and oz are linearly independent on this congruence. Taking the exterior derivatives of equations (3.4.13), we arrive at the following exterior quadratic equations:

ofA3=0, 9zA03=0, from which it follows that 3 = 0.

(3.4.14)

It is easy to see that the system of equations (3.4.13) and (3.4.14) is completely integrable. From this system it follows that the plane llfi AM2AM3 = p remains fixed as the line as = Af, A M2 moves along the congruence. Thus a degenerate congruence of first kind is a two-parameter set of straight lines lying in a fixed plane p. For a degenerate congruence of second kind, the plane T=(V2) coincides

with a plane generator of the second family of the cone Cz. We place the vertices ao = x, aI, and a3 of the moving frame into this plane T. Then from equations (1.4.14) and (1.4.15), we find that the equations of this congruence have the form

o2=0, 02=0,

(3.4.15)

and the forms 0; and 0; are linearly independent on this congruence. Taking the exterior derivatives of equations (3.4.15), we arrive at the following exterior quadratic equations: oz A o; = 0,

oz A of = 0,

from which it follows that 02 = 0.

(3.4.16)

It is easy to see that the system of equations (3.4.15) and (3.4.16) is also completely integrable. From this system it follows that the point M2 remains fixed as the line ao = M1 A Aft moves along the congruence. Thus a degenerate congruence of second kind is a bundle of straight lines with its center at the point M2.

Notes

115

NOTES 3.1. A. P. Norden constructed the theory of normalized m-dimensional submanifolds V'" in n-dimensional projective and other homogeneous spaces whose fundamental group is a subgroup of the group of projective transformations (see Norden [N 47, 48, 49, 50a, b]). On a normalized submanifold V'", a torsion-free affine connection

can be uniquely defined, and as a result, tensorial methods can be applied to the study of normalized submanifolds. Essentially, in this case, the submanifold V' is under study not by itself but along with a supplementary geometric construction-its normalization. This is the reason that most of the results obtained by the method of normalization are connected not only with the submanifold V'" but also with this geometric construction. Only if one succeeds in constructing a normalization that is connected intrinsically with the submanifold V'" will all geometric objects and quantities obtained by means of this normalization be connected intrinsically with the submanifold V"`. As we have shown in Section 3.1, in the space C", the submanifold V' is normalized if through any of its points there passes an (n-m)-dimensional sphere orthogonal to V'". Norden [N 48, 50b] studied conformal differential geometry of a surface V2 in the space C3 and constructed three invariant normalizations of V2 that are defined by three invariant circles orthogonal to V2. Following A. P. Norden, Vedernikov and Tikhonov [VT 54] found a metric characterization of images and quantities of the conformal theory of surfaces V2 C C3.

There are numerous papers where curves in the spaces C2,C3,C", the spaces with a conformal connection, and other spaces close to the conformal space are studied: see, for example, the papers Barner (Bar 611, Fialkow [Fia 42], Haantjes [Haa 41, 42a, b], Kasner and de Cicco [KC 41] Lagrange [LagR 41a, b; 50], Liebmann [Lieb 23], J. Maeda (Ma 421, Pendl [Pen 76], S. Sasaki (SaS 39], Schubarth [Schu 26], Sulanke [Su 81], Takasu [Ta 38], Thomsen [Tho 25], Verbitsky [Ver 59], van der Woude [Won 481, and Yano [Y 40b]. Submanifolds in the space C", the spaces with a conformal connection, and other spaces close to the conformal space are also studied in numerous papers: for example, in Blair [Bla 82], Blaschke [Bl 25), Bryant [Br 88], Bushmanova and Norden [BN 70],

do Carmo and Dajczer [CD 871, Chen [Ch 73a, 74], Deszcz (De 89, 90], Fialkow [Fia 45], Haimovici [Hai 37, 39], Kowalski [Kow 731, Perepelkine [Per 35], Petrescu [Pet 46, 48], S. Sasaki [SaS 40a], Schiemankgk and Sulanke [SSu 801, Sulanke [Su 84], Verheyen and Verstraelen [VV 80], Yano [Y 39b, c; 40a, c; 42, 43a, b), Yano and Chen [YC 71a, b; 73], Yano and S. Ishichara [YI 69], and Yano and Mdto [YM 42a, b).

Rosenfeld's work played an important role in the development of the conformal differential geometry of families of m-dimensional spheres in the space C". Rosenfeld

[Ro 47, 48a] introduced the notion of a symmetry figure in a homogeneous space and studied differential geometry of families of symmetry figures. The symmetry figure is a geometric figure r that is invariant under an involutive transformation J, namely a transformation satisfying the condition J2 = Id where Id is the identity transformation. The transformation J is called a symmetry with respect to r. In the Euclidean space R", the symmetry figures are m-planes of any dimension m; in particular, if m = 0, then they are points. In the projective space P", the symmetry figures are m-pairs consisting of m- and (n - m - 1)-planes. In the conformal space C", the symmetry figures are m-dimensional spheres if m+ > 1 and pairs of points if m = 0. The space of symmetry figures appeared to be a space with an affine con-

116

3. GEOMETRY OF SUBMANIFOLDS

nection and in some instances a Riemannian manifold. As a result tensorial methods can be applied to the study of this space. Since in the space C' the symmetry figures

are m-dimensional spheres (if m = 0, they are pairs of points), the results of the general theory of symmetry figures can be applied for their study. Rosenfeld [Ro 48b] introduced a conformally invariant metric and invariant local parameters in the space of m-dimensional spheres and considered the geometry of congruences and pseudocongruences of m-dimensional spheres (in particular, congruences of pairs of points)

in C". Note that points of conformal and pseudoconformal spaces as well as points of a projective space are not symmetry figures-they are figures of more general type called parabolic figures (see Rosenfeld, Zamakhovskii, and Timoshenko [RZT 88]). This is the reason that in studying point manifolds of conformal and pseudoconformal spaces, one needs to consider geometric objects that are more complex than tensors. In Chapters 2 and 3 we have done precisely this. Similarly subspaces of a projective space are parabolic figures. We will study manifolds of such subspaces in Chapters 6 and 7. Conformal properties of special submanifolds in the Euclidean and Minkowski

spaces R', R", and R4 are studied in many papers: see, for example, Chen [Ch 73aJ, Gheysens, Verheyen, and Verstraelen (GVV 81, 83], Houh (Hou 74], Rosca and Buchner [RB 79), Rouxel [Rou 74, 79, 80, 81a, b; 82), and Verstraelen [Vers 781. Vedernikov [Ved 63] studied the problem of conformal deformation for two normalized

submanifolds V' C C". Verbitsky [Ver 52] (see also Vedernikov [Ved 63]) proved that if m > 4, then a submanifold V' C C" is conformally applicable onto an mdimensional rn-sphere if and only if V' is the envelope of a one-parameter family of spheres. Yano [Ya 39b, c; 40a, 42, 43a] studied the conformal geometry of a submanifold V' in an n-dimensional Riemannian manifold, constructed conformally invariant tensors associated with a second-order differential neighborhood of V', and found the derivational (Frenet) equations and their integrability conditions. Fialkow [Fia 44, 45] found a complete system of conformally invariant tensors of a submanifold

V' and considered some special submanifolds V' in an n-dimensional Riemannian manifold.

Note that in m 1 of the papers mentioned above the conformal differential geometry of multidin. sional submanifolds V' is constructed by the methods of Rie-

mannian geometry. This makes a geometric interpretation of the results obtained extremely difficult. In all these papers the authors do not go further than to find the derivational (Frenet) equations and to prove the general theorems on determination of the submanifolds V' by means of a system of tensors.

An invariant normalization of a submanifold V' C C" was first constructed in Akivis (A 61a]. In our exposition we follow this paper. While determining a

set of tensors that defines a submanifold V' C C" up to a conformal transformation of C", we assumed for simplicity that n = m + ml, namely that C" coincides with the osculating sphere C= +" of V'. Similar considerations in the general case n > m + ml are more complex (see Akivis [A 61a)). For a more detailed description of the developments in the construction of an invariant normalization of submanifolds in different spaces, see the survey papers Laptev [Lap 651 and Lumiste [Lu 75]. 3.2. The results of this section are due to Akivis [A 63a, 64]). As we noted, most of the results for submanifolds V'" carrying a net of curvature lines in the space C" are similar to those for submanifolds U' carrying a net of conjugate lines in the space

Notes

117

Pn+l (see Akivis [A 61b, 63b, 64] and the book Akivis and Goldberg [AG 931, Ch. 3) and can be obtained from them by applying the Darboux mapping. 3.4. The congruences, pairs of congruences, and complexes in three-dimensional Euclidean, affine, and projective spaces were studied intensively in numerous papers and books; see, for example, the classic books Konigs [Ko 95] and Zindler [Zi 02] and more recent monographs Bol [Bo 50), Hlavaty [HI 451, Finikov [Fin 50, 56], and Kovantsov [Kov 631.

Bompiani [Bom 12] showed that a pencil of tangents to V2 C P3 at a point x E V2 is represented by the points of a rectilinear generator of 52(1, 3) C Ps, and the complex of tangents to V2 C P3 by the (quadratic) congruence of rectilinear generators of 52(1,3) C P5. The connection between line submanifolds of the space P3 and point submanifolds of the space C2 was studied in the papers Bompiani [Bom 12), Rosenfeld [Ro 48b], and Akivis [A 65a].

22

1. CONFORMAL AND PSEUDOCONFORMAL SPACES

at the point s, and the intersection a fl 11(1, 3) is a real cone of second order. We denote this cone by C,. In the space CZ this cone is an isotropic cone with its vertex at the point s. This cone is the image of a special linear complex of the space P3 that consists of all straight lines of P3 intersecting the straight line s. Let r and s be two linear complexes in the space P3, whose images are two points in P5 which we denote by the same letters. A linear congruence in p3 is a collection of straight lines belonging simultaneously to both complexes; that is, it is the set r fl s. To find the geometric meaning of a linear congruence, we

consider the straight line r A s determined in P5 by the points r and s. The parametric equation of this line is

t = Ar + is. The location of this line with respect to the hyperquadric 12(1,3) depends on the quadratic trinomial

(t, t)=A2(r,r)+2Aµ(r,s)+µ2(s,s),

(1.4.7)

whose discriminant A is equal to

0 = (r, s)2 - (r, r) (s, s).

If 0 < 0, then the straight line r A s has no common points with the hyperquadric 12(1, 3). The linear congruence r fl s, corresponding to such a line, is called elliptic. If 0 > 0, then the straight line r A s has two common points

p and q with the hyperquadric 12(1, 3). The linear congruence r n s, corresponding to such a line, is called hyperbolic. Such a congruence consists of all

straight lines of the space P3 intersecting two straight lines p and q, which are called the directrices of the linear congruence r n s. If 0 = 0, but not all coefficients of the quadratic trinomial (1.4.7) vanish, then the points p and q coincide, the straight line r n s in P5 is tangent to the hyperquadric 11(1, 3), and the congruence r n s is called parabolic. Finally, if all coefficients of the quadratic trinomial (1.4.7) vanish, then the straight line rAs lies on the hyperquadric 12(1, 3); that is, this line is a rectilinear generator of 12(1, 3). This straight line r A s is the image of a pencil of straight lines of p3 determined by the intersecting lines r and s. The linear congruence r A s degenerates in this case into a two-parameter family of straight lines lying in a 2-plane r of the pencil r A s. Such a degenerate linear congruence is called a plane field of straight lines. The image of such a plane field of straight lines is a two-dimensional plane generator of the hyperquadric 12(1, 3). Since the set of 2-planes r in the space P3 depends on three parameters, the hyperquadric 12(1, 3) carries a three-parameter family of two-dimensional plane generators corresponding to 2-planes of the space p3. Moreover the hyperquadric 12(1,3) carries also a second three-parameter family of two-dimensional plane generators corresponding to the bundles of straight lines of the space P3, since the bundles of straight lines, just as plane fields, are linear images in the space P3.

Chapter 4

Conformal Structures on a Differentiable Manifold 4.1 A Manifold with a Conformal Structure 1. The notion of conformal structure appeared in the study of the problem of conformal transformation of a metric of a Riemannian manifold V = (M, g) where M is a differentiable manifold and g is a metric given on M. The metric can be given on M by means of a nondegenerate quadratic form dsZ = g;jdu'du',

which determines the square of the element of arc length. Here u', i = 1, . . , n, are curvilinear coordinates on the manifold M, and gig are the components of the metric tensor g. A conformal transformation of the metric is defined by the formula .

dsZ = ads2,

where dsZ is a new metric and o is a positive factor defining the conformal transformation of the metric. A conformal structure on a manifold M is the collection of all Riemannian metrics obtained from a fixed Riemannian metric by conformal transformations.

In other words, we can say that a conformal structure on a manifold M is defined by means of a relatively invariant quadratic form

g = gijdu'du'.

We will not assume that the form g is positive definite; that is, we allow the metric to be pseudo-Riemannian. If in a neighborhood of each point x E M, this form can be reduced to a canonical form having p positive and q negative squares, we will say that the form g has the signature (p, q). In this case we will 119

120

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

denote the conformal structure on the manifold M by CO(p, q) where n = p + q is the dimension of the manifold M (see Kobayashi [Ko 72], Ch. 4, §6). The equation g = 0 defines in the tangent space Ts(M) a cone C,, of second order called the isotropic cone. Thus the conformal structure CO(p, q) can be given on the manifold M by a field of cones of second order. The cone CC C T. (M) remains invariant under transformations of the group

G=SO(p,q)xH, p+q=n, where SO(p,q) is the special n-dimensional pseudoorthogonal group of signature (p, q) (the connected component of the unity of the pseudoorthogonal group O(p, 9)), and H is the group of homotheties. Thus the conformal structure CO(p, q) is a G-structure defined on the manifold M by the group G indicated above.

If p = n and q = 0, then the conformal structure is called the proper conformal structure. In this case the quadratic form g is positive definite,

and the group is G = SO(n) x H, where SO(n) is the special orthogonal group. On the other hand, if q > 0, then the structure CO(p, q) is said to he pseudoconformal. In particular, if p = 1 and q = n - 1, then the structure is called conformally Lorentzian.1 For 2 < p < n - 2, the structure CO(p,q)-structure is called ultrahyperbolic (see Barrett et al. (BGPPR 94]). For the proper conformal structure, the isotropic cone is imaginary, while for a pseudoconformal structure, it is real. As far as possible, we will study proper conformal structure and pseudoconformal structure simultaneously. 2. We consider the manifold M, associate with any point x E M its tangent space TZ(M), and define the frame bundle whose base is the manifold M and the fibers are the families of vectorial frames {e1i...,en} in T=(M) defined up to a transformation of the general linear group GL(n). The frames indicated above are called the frames of first order. They form the frame bundle that we will denote by RI (M). Let us denote by {w1, ... , wn} the co-frame dual to the

frame {e1i...,en}: w'(ei) = 6j.

Then an arbitrary vector { E T=(M) can be written as

The forms w' can be considered as differential forms on the manifold M if we assume that t = dx is the differential of the point x E M. Thus the form g can be written as (4.1.1) g = gcjw'w'. Since the manifold M is referred to the curvilinear coordinates u1, ... , un, the differential 1-forms w' are independent linear combinations of the differentials of these coordinates: I Note that the structure for which p = n - 1 and q = 1 is called conformally Lorentzian by some authors.

4.1

A Manifold with a Conformal Structure

121

W' = x,du',

(4.1.2)

where xi are independent variables that are the parameters of the general linear

group GL(n). In particular, if x, = b , then w' = du', and the quadratic form g takes the form indicated on p. 119. The variables u' are base variables for the frame bundle RI(M), and the variables xi, are its fiber variables. If the point x remains fixed on the manifold M, then du' = 0 and consequently w' = 0. Conversely, if W' = 0, then from equations (4.1.2) it follows that u' = const; that is to say, the point x stays fixed on the manifold M. Thus, the forms W' are base forms of the frame bundle W (M). The formulas (4.1.2) show that the forms w' are also defined on the frame bundle 1Z1(M). Exterior differentiation of equations (4.1.2) gives the following exterior quadratic equations: dw`=w'Awj, (4.1.3)

where the forms W., are expressed linearly in terms of the differentials of the parameters x., and also the forms w', and for w' = 0; that is, for a fixed point x E M, the forms w, are invariant forms of the general linear group GL(n). Let us define Wjfl.i=o = Thus the forms wj" are fiber forms of the bundle Rl (M) of frames of first order, but they themselves are defined on the bundle R2(M) of frames of second order, since in their expressions in terms of the differentials du' and dxi, the variables x'k determining the location of a frame of second order, enter (see more details on these in Laptev [Lap 66]). Note that the variables xijk can be considered symmetric with respect to the indices

j and k.

The differential prolongation of equations (4.1.3) gives the second group of structure equations of the manifold M: dw =W A Wk +Wk AWilk,

(dwj'k -w Awjk -Wk Aw +w

/ Wok) A wk = 0,

(4.1.4)

(4.1.5)

where the forms wok are linearly expressed in terms of the differentials dxjk which together with the variables xi define the location of a frame of second order in the fiber bundle RZ(M). The forms wj'k and wj' are fiber forms of the frame bundle 1Z2(M). For w' = 0, they are the invariant forms ask and a, of the group G2(n) of admissible transformations of frames of second order It follows r.k and ink = (Kobayashi [Ko 72], Ch. 1, §8) ; here that the group G2(n) depends on n2 + 1n2(n + 1) = 1n2(n + 3) parameters. 2 2

Note also that the forms Wok are defined on the bundle 1Z3(M) of frames of third

order of the manifold M. Equations (4.1.5) are the conditions of compatibility of equations (4.1.3) and (4.1.4).

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

122

3. Let us denote by d the operator of differentiation with respect to the fiber parameters. Then similarly to what we did in Subsection 2.1.2, we can prove that equations (4.1.3) imply that bwi = -Wjir;.

(4.1.6)

Hence the condition for the form g to be relatively invariant is

b(9ijw'wJ) = 27rgijwV, and this condition leads to the equations

69ij - 9kj7k - giknj = 27rgij.

(4.1.7)

If we denote the left-hand side of this equation by Vsgij, then equation (4.1.7) takes the form (4.1.8) D69ij = 2agij. If the point x moves along the manifold M (i.e., the base parameters are not fixed), then equation (4.1.7) becomes dgij - gkjWk - gikwj = 20gij + gijkwk, where gijk = gjik and 0(b) = 7r. We define new forms W = W; + 7,kWk,

(4.1.9)

(4.1.10)

It is obvious that =o = it , and by the symmetry of the equations (4.1.3) preserve their form where ryk

dw' = wj AC4.

(4.1.11)

Let us substitute for the forms w, in equations (4.1.9) their values taken from equations (4.1.10). As a result we obtain d9ij - 9kjWk - 9ikwk = 20gij + (9ijk - 9ir7jk - 9rj7ik)wk.

(4.1.12)

These equations show that the quantities ryYk can be chosen in such a way that the right-hand sides of these equations vanish. In fact set gir7jk + 9rj7ik = 9ijk

and permute the indices i, j, and k cyclically: 9jr7ki +

9jki, 9kr'Yij + 9a Ykj = 9kij

If we add the last two equations and subtract the preceding one, we find that 29kr?2j = 9jki + 9ikj - 9ijk

4.1 A Manifold with a Conformal Structure

123

This implies the equation k

I

kl

7ij = 29

('9j1, +

gilj - 9ijl),

(4.1.13)

where gkl are the inverse tensor of the tensor gkl. Substituting the values of 7 into equations (4.1.9), we obtain dgij - gikWj - 9kjwk = 29gij.

(4.1.14)

If we suppress the tilde in equations (4.1.11) and (4.1.14) and return to the original notations, then equations (4.1.11) will still have the form (4.1.3), and equations (4.1.14) become

d9ij - 9ikwj -

9kjwk = 29gij.

(4.1.15)

Moreover equations (4.1.4) and (4.1.5) also preserve their form, since they were obtained by prolongation of equations (4.1.3).

Next we define a differential operator V by means of forms w in such a way that, for example, V9ij = dgij - 9ikWJ - 9kjw; .

Let us note that the operator V is not the operator of covariant differentiation, but it has all properties of a differential operator. Making use of this operator, we can rewrite the system of equations (4.1.15) as follows: Vgij = 29gij.

(4.1.16)

By taking the exterior derivatives of equations (4.1.15), we get -wk A

29ijd9.

From these relations it follows that d9 = wk A 9k, wk A

9jlw,k + 29ijek) = 0,

(4.1.17)

where 9k are differential 1-forms in the bundle R2(M) of second-order frames. Applying Cartan's lemma to the second equation of (4.1.17) and setting wi = 0 in the resulting equations, we find that 9il7jk + gjl'n;k + 2gijirk = 0,

(4.1.18)

where irk = 9k(d) are restrictions of the 1-forms 9k to the fiber .7"=(M). By permuting the indices i, j, and k in equations (4.1.18) cyclically, solving the three simultaneous equations obtained with respect to gilailk, and contracting the resulting equation with gt°, we arrive at the equations 7rjk = 9jk9'riq -

6'7r.,.

(4.1.19)

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

124

These relations are consequences of the specialization (4.1.10) where the quantities -Yk are determined by equations (4.1.13). After this specialization the conditions imposed on the fundamental tensor g;j of the conformal structure reduce to (4.1.15). Equations (4.1.19) show that the group G2(n) of transformations of second-order frames reduces to the group leaving equation (4.1.15) invariant. This reduced group depends on n2 + n parameters. We denote this reduced group by G2(n). From (4.1.19) it follows that (4.1.20)

wok = 9jk9i191 - bj'9k - bkej + Ajkjw1.

In view of this, equations (4.1.4) read as follows: dw = w A wk + wk A (9jk9'101 -'5 0k - bk0j) +

Ajklwk Awl

,

(4.1.21)

where A ,'k, = A (kll.

Next we have to compute the exterior differentials of the forms 9,, which occur in equations (4.1.17) and (4.1.21). By differentiating the first group of equations (4.1.17), we get

w'A(d9;-wA9j)=0, and by virtue of the generalized Cartan lemma (see Laptev [Lap 66] or Bryant et al. [BCGGG 91], p. 323), we obtain d9; - w; A 9j = wj A 9;j,

(4.1.22)

where the forms 9;j are defined in the bundle of third order frames and satisfy the equation w' Awj A 9ij = 0. Thus the forms a;j = 9;j (b) are symmetric with respect to the indices i and j: 7r;j = aji. By taking the exterior derivatives of equations (4.1.21), we get (V Ajk - 9jk9'mOm1 + bkOj1) A Wk A w1 = 0,

(4.1.23)

where

A'nklw -

Ajklwm

From (4.1.23) it follows that the forms xjkl := V

9jk9'm7rm1 + 6k7rjl,

where V6Ajk, = VAjk,(b), are symmetric with respect to the indices k and 1. Hence, by alternating them with respect to these indices, we find that V6Aj'k1 +

2(bkajl - bjirjk - gjk9'm7r( +gj19'mlrmk) = 0.

(4.1.24)

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125

By contracting these relations with respect to the indices i and 1, we obtain VsAjkr = 2 ((n - 2)ajk + 9jk9'r7 ,i

(4.1.25)

Since the forms ajk are symmetric with respect to the indices j and k, the forms ajk = (n-2)ajk+gjk9'i ai, occurring on the right-hand sides of equations (4.1.25) are also symmetric with respect to the indices j and k. Let us prove that if n > 3, then these forms are linearly independent. In fact, setting them equal to 0, we obtain (4.1.26) (n - 2)ajk + 9jk9"aar = 0. By contracting these relations with the tensor 9jk, we find that

(2n - 2)9"irii = 0. If n > 3, it follows that g''a;; = 0 and, by virtue of (4.1.26), ajk = 0. Hence, if n > 3, the determinant of the system of forms ajk is nonzero, and these forms are linearly independent, since the forms ajk are linearly independent. Equations (4.1.25) show that the linearly independent forms `ajk enable us to make a specialization of the object Ajiki in such a way that this object will satisfy the condition A'(jk), = 0.

(4.1.27)

With this specialization, we obtain ajk = 0 and ajk = 0. This means that the group of admissible transformations of third-order frames reduces to the group G2(n). Hence, for n > 3, the conformal structure CO(p, q) is a G-structure of finite type 2 (see Sternberg [St 641, Ch. VII, §3). Now from relations (4.1.24) it follows that after this specialization the object Akr becomes a tensor, since now from (4.1.24) it follows that VoAjk, = 0. We

Since the form a = 9(b) does not occur in the denote this tensor by differential equations that the tensor C'kj satisfies, this tensor is not changed under a conformal transformation of the metric. The tensor CCkl is called the tensor of conformal curvature of the structure CO(p, q), or the Weyl tensor. The relations ajk = 0 also imply that 8,, = A;jkwk.

(4.1.28)

This means that if n > 3, then for the conformal structure CO(p, q) in the prolonged structure equations, there will be no independent differential forms connected with the bundles of frames of order greater than two. Thus, when n > 3, a conformal structure CO(p, q) is defined on the manifold M by means of the fundamental relative tensor g;j satisfying equations (4.1.15) and 9;. Furthermore the forms w', 9, and w. satisfy and by 1-forms equations (4.1.3), (4.1.17), and (4.1.21) in which one needs to replace the object by the tensor Cjkr. This tensor satisfies equations (4.1.27), and the forms 8; satisfy the equations dB; = w, A Oj + C;jkw' A wk,

(4.1.29)

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

126

which follow from (4.1.22) and (4.1.28). In equations (4.1.29), Cijk = Ailjk). 4. In the bundle of first order frames over a manifold M with a conformal structure CO(p, q), we consider the 1-forms

B = wil + b0.

(4.1.30)

By means of them, equations (4.1.15) can be written as

dgij - 9ik8 - 9kio; = 0.

(4.1.31)

By means of the same forms, the structure equations (4.1.3), (4.1.17), (4.1.21), and (4.1.29) of the CO(p, q)-structure can be written in the following way:

dw'=9Aw'+wjAOil,

(4.1.32)

dO=w'A9i,

(4.1.33)

dB' = Oj A w' + 9j' A 6k + gjkwk A gi18, +

CCktwk

A wI,

do,=9iA0+O AOI+CijkwjAwk.

(4.1.34)

(4.1.35)

For CCk, = Cijk = 0, up to notations, equations (4.1.31)-(4.1.35) coincide with the structure equations (1.2.8) and (1.2.11) of the conformal space or the pseudoconformal space. For this reason the object {CJk1,Cijk} is called the curvature object of the conformal structure CO(p, q). We attach to a point x E M a local conformal space (Cyn)= of signature (p, q), assuming that x E (CQ )x, and in this space we choose a frame consisting of two points, A0 = x and An+1, and n hyperspheres A; passing through these

points (see Section 1.2). The scalar products of the elements of this frame satisfy the conditions

(Ao,An+1) = -1, (Ao,Ai) = (An+1,Ai) =0, (Ai,Aj) = gij.

(4.1.36)

The infinitesimal displacement of this frame is defined by the equations

dAf = 0(A,I, where 000

n+1 -en+1

00, = w',

, Y1= 0,1, ... , n + 1,

- e e0n+1

-

001 = B;,

(4.1.37)

_

+1 - 0, (4.1.38)

i ij = 9ijwj, i 0n+1 - 9 op The conformal space (Ca )= is the compactification of the tangent space T=(M), which is obtained by joining to T=(M) the point at infinity y with coordinates (0,0,. .. , 0, 1) and the isotropic cone with vertex at this point.

Bn+1

If we fix a point x E M, then equations (4.1.33)-(4.1.35) and (4.1.31) become

dB = 0, d9 = 9 A 9k, d9i = 9, A 9 + 9 A 9j, dgij = 900 + gkj8;`. (4.1.39) Equations (4.1.39) show that the form 9 is an invariant form of the group H of homotheties acting in the tangent space TT(M). The forms which in addition to equations (4.1.39) satisfy equations (4.1.31), are invariant forms

4.1 A Manifold with a Conformal Structure

127

of the group SO(p, q) that leaves invariant the cone C= determined by the equation g = 0 in the space TT(M). To clarify the geometric meaning of the forms Bi, it is necessary to make the compactification of the tangent space T=(M) (see Section 1.3). Here we will describe this compactification in more detail. In the space Ty(M) that carries the structure of a pseudo-Euclidean space R, n' we consider a collection of hyperspheres which, in the Cartesian coordinates x', is determined by the equation kgi3x'x' + 2hix' + 21 = 0.

The numbers k, hi, and I are homogeneous coordinates of these hyperspheres. These numbers can be taken as coordinates of a point in the projective space P=+'. The compactified tangent space (CQ)Z(M) is identified with the set of hyperspheres of zero radius which is given in the space P= +' by the equation

gi,x'x' - 2x°xn+' = 0,

(4.1.40)

where x° = k,xn+' = l and xi = g''hi. Thus, after compactification, the tangent space T=(M) is enlarged by the point at infinity y with coordinates (0, 0, . . . , 0, 1) and by the isotropic cone C. with vertex at this point y whose equation is the same as the equation of the cone C., namely

gijx'x' = 0. For a pseudoconformal structure CO(p, q), the signature of the quadratic form in the left-hand side of equation (4.1.40) is equal to (p + 1, q + 1), and equation (4.1.40) can be considered as the equation of an absolute in the projective space P=+'. This absolute determines a non-Euclidean geometry in the space Pz +' . As mentioned in Section 1.3, the fundamental group of this nonEuclidean geometry is isomorphic to the group PO(n+2, q+1) where n = p+q. On the hyperquadric (4.1.40) itself, the geometry of a pseudoconformal space Cq of signature (p, q) with the same fundamental group is induced. But on the compactified tangent space (Cq)y(M), the point x, at which this space is tangent to the manifold M, is fixed. Thus the forms Bi together with the forms 0 and 01 are invariant forms of the isotropy group of the space CQ that leaves invariant the point x. The forms Bi determine a displacement of the point y in the space (C4 )Z, the forms B, determine the rotation of this space, when the points x and y are fixed, and the form 0 determines the homothety of this space with respect to the same points x and y. If the point x on the space (C9 )= is fixed, then (CQ ),, becomes an n-dimensional pseudo-Euclidean space Rq = (Cq )z \C= of signature (p,q) (we recall that n = p+q). The forms 0,0,', and Bi are invariant forms of the group of motions of this space. The forms 8i determine the subgroup T(n) of translations in this space, the forms 0 determine the subgroup SO(p, q) of rotations, and the form 0 determines the subgroup H of homotheties.

128

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Equations (4.1.39) are the structure equations of the group G' which is the differential prolongation of the group G whose transformations leave the cone

C(p,q) C T. (M) invariant. The group G' is the group of transformations of the local pseudoconformal space (C,','). in which the point x is fixed. It follows from equations (4.1.39) that the group G' is isomorphic to the group of motions and homotheties of the pseudo-Euclidean space RQ :

G' °-° (SO(p,q) x H) K T(n),

where as earlier x denotes the semidirect product. This group G' is the structure group of the prolonged G-structure which is invariantly connected with the pseudoconformal structure CO(p, q). Its subgroups, SO(p, q) and H, act on the bundle of first-order frames, while the subgroup T(n) acts on the bundle of second-order frames. The quantities C kt occurring in equations (4.1.34) form the first structure tensor of the CO(p, q)-structure. We called this tensor the tensor of conformal curvature (see p. 125). It is defined in a third differential neighborhood of this structure. The quantities Cijk occurring in equations (4.1.35) do not form a tensor. They are defined in a fourth differential neighborhood and form a homogeneous geometric object taken together with the tensor C,kl. If this object vanishes, then the system (4.1.37) is completely integrable and defines a frame bundle in the conformal space C'. In this case, equations (4.1.31)(4.1.35) are the structure equations of the space Ca (see Sections 1.2 and 1.3). is not equal to zero, then the system (4.1.37) If the object is not completely integrable, and equations (4.1.31)-(4.1.35) are the structure equations of an invariant conformal connection, associated with the manifold M. This connection is a connection in the principal subbundle E(M, G', p) of the bundle of second-order frames over M whose structure group is G'. Equations (4.1.31)-(4.1.35) are the structure equations of the G-structure mentioned above. The latter structure is a differential-geometric structure of order two (see Kobayashi [Ko 72], p. 9). Cartan [Ca 23] called these equations the structure equations of the normal conformal connection associated with the quadratic differential form (4.1.1). While in Cartan [Ca 23] only proper conformal structures were considered, in Cartan [Ca 22a, b] the pseudoconformal structures were studied as well. Note only the conformal connection we considered is slightly different from Cartan's normal connection since in Cartan [Ca 23], the 1-form 0 is a total differential. Note one more time that as was the case for a conformal structure given on a differentiable manifold M, the structure equations (4.1.31)-(4.1.35) show that if n > 3 the further prolongation of these equations will not require the introduction of bundles of frames of order higher than two, but will lead only to the prolongation of the objects CIO: and C, . Hence, as was already noted earlier, the conformal connection on a manifold is a structure of finite type 2 (see subsection 4.1.3, p. 125). 5. We will now find the conditions that the quantities C'kt and C;1k, occurring in equations (4.1.34) and (4.1.35), satisfy. These conditions are conse-

4.1 A Manifold with a Conformal Structure

129

quences of equations (4.1.31)-(4.1.35). First, we observe that equations (4.1.34) and (4.1.35) show that

Cjkl = -Cjlk, Cijk = -Cikj.

(4.1.41)

Next , differentiatin g (4 . 1 . 31) , we obtain 9imC

kl + 9j-C kl =

0.

(4.1.42)

If we define the tensor Cijkl = 9imCkl+

(4.1.43)

then the above equations become

Cijkl + Cjikl = 0

(4.1.44)

These equations mean that the tensor Cijkl is skew-symmetric not only in the last two indices but in the first two as well. Applying exterior differentiation to equations (4.1.32) and (4.1.33), we get two more conditions: (4.1.45) Cjkl + Cklj + Clik = 0, Cijk + Cjki + Ckij = 0.

(4.1.46)

These relations are similar to the Ricci identities in Riemannian geometry (see Eisenhart [Ei 26], Eq. (8.11) in §8). Moreover relations (4.1.45) are equivalent to the equations (4.1.47) Cot + Cikij + Cil jk = 0, These conditions have the same form as the similar conditions for the curvature tensor in Riemannian geometry. Let us write these conditions for three more combinations of the indices i, j, k, and 1:

Cjilk + Cjlki + Cjkil = 0, Cklij + Ckijl + Ckj:i = 0, Clkji + Cljik + C,iki = 0.

Adding condition (4.1.47) to the first of the three conditions and subtracting the last two of these conditions, we obtain

Cijk = Cklij.

(4.1.48)

Hence, in particular, one can find the conditions Cjiki

-

Ckiji

and thus, in view of this, equations (4.1.27) can be replaced by Cjki = 0.

(4.1.49)

130

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Equations (4.1.41), (4.1.44), (4.1.45), (4.1.47), and (4.1.48) show that the tensor C'kl of conformal curvature of a CO(p,q)-structure satisfies the same conditions as the curvature tensor R'k, of a Riemannian manifold (e.g., see Eisenhart [Ei 26], §8, or Dubrovin,'omenko, and Novikov [DFN 92], §30). Furthermore it satisfies an additional condition (4.1.49) that distinguishes it from the curvature tensor of a Riemannian manifold. Thus, the tensor CJkl is trace-free. Let us find also the differential equations that the quantities CJkl and Cijk satisfy. Applying exterior differentiation to (4.1.34) and (4.1.35), we obtain (VCjkl + (akCjlm - gjkg'JCplm)wm] A w k Awl = 0

(4.1.50)

and

(OCijk + C;,k9l) Awj Awk = 0.

(4.1.51)

We can also obtain equations (4.1.50) from equations (4.1.23) if we substitute the expressions (4.1.28) for the forms 8,j into them. Equations (4.1.50) are equivalent to the following equations: [VCijkl - 2C,jk:8 + (9ikCjlm - 9jkCilm)wm] A Wk Awl = 0

(4.1.52)

which the relative tensor Cijkl satisfies and which are more symmetric than equations (4.1.50). Equations (4.1.50)-(4.1.52) imply that when a point Ao E M is fixed (i.e., when w' = 0), the quantities CCkl , Cijkl and Cijk satisfy the following equations:

VJCj'kl = 0, VoCijkl = 2Cijkly,

(4.1.53)

V6Cijk + C;jkirl = 0.

(4.1.54)

This means that the quantities Cjkl, as we mentioned earlier, constitute the tensor of conformal curvature of the CO(p, q)-structure, and the quantities Cijk, constitute a relative tensor of weight two. This tensor is also called the Weyl tensor. As regards the quantities Cijk, generally speaking, they do not constitute a tensor. However, as one can see from equations (4.1.53) and (4.1.54), taken together with the components Ckl of the tensor of conformal curvature, they constitute a homogeneous geometric object. Equations (4.1.53) are equivalent to the equations OCj'kI = Cjklmwm,

(4.1.55)

(4.1.56) OCijkl - 2CijklO = Cijklmwm, where Cijklm = gipC klm By virtue of equations (4.1.41), (4.1.44), and (4.1.47)(4.1.49), the quantities CJkim satisfy the conditions:

Cijklm = -Cjiklm = -Cijlkm = Cklijm, C0:1)m = 0, Cjkim = 0.

(4.1.57)

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A Manifold with a Conformal Structure

131

Furthermore, substituting decompositions (4.1.56) into equations (4.1.52), we find that Cij(klml + 9ilkCljltml - 9j[kClillm] = 0.

(4.1.58)

Equations (4.1.58) enable us to prove that for n > 3, the quantities Cijk can be expressed in terms of the quantities Cj'klm. Indeed, contracting equations

(4.1.58) with the tensor gwe obtain Cjkmi + (n - 3)Cjkm + gjkCm - gjmCk = 0,

(4.1.59)

where Ck = gijCijk. Contracting equation (4.1.59) one more time with the tensor 9jk and taking into account the last condition of (4.1.57), we get (2n -

0,

whence, since n > 2, we obtain Cm = 0. In view of this, for n > 4, from (4.1.59) it follows that

Cjkt = -n

(4.1.60)

Equations (4.1.60) show that if n > 4 and Cilkl = 0, then the relations Cijk = 0 hold too. Thus, for n > 4, the condition 0 is necessary and sufficient for a manifold M with CO(p, q) -structure to be conformally flat, that

is, to admit a conformal mapping to a hyperquadric QQ C Pn+' of signature (p, q), or a pseudoconformal space Cq . In particular, for n > 4 and the signature (n, 0), the condition CCkl = 0 is necessary and sufficient for a manifold M with CO(n)-structure to be conformally mapped onto an n-dimensional sphere. On the contrary, for n = 3, we can prove that the tensor Cjkl is identically equal to zero. This can be proved if we refer the space T=(M) to an orthogonal coordinate system for which gij = 0, i 34 j, and use condition (4.1.42) and the fact that the tensor is trace-free. Then, according to the system of equations (4.1.54), the geometric object Cijk becomes a tensor. The vanishing of this tensor characterizes three-dimensional conformally flat structures. 6. Note that structure equations (4.1.31)-(4.1.35) of the CO(p, q)-structure can be written in the index-free form, the same way as is usually done in many books on differential geometry (e.g., see Kobayashi and Nomizu [KN 62] or Gardner [Car 89]). To this end, consider the 1-form w = (wi) with its values in the space T=(M) and defined in a first-order frame bundle, a matrix 1-form a = (8) and a scalar 1-form 9 in a second-order frame bundle, and a covector form V = (6i) in the third-order frame bundle. Along with the tensor g = (gij), these forms satisfy the following structure equations:

V9=0, dw=9Aw-aAw,

(4.1.61) (4.1.62)

132

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD dO = -cp A w,

+(gw)A(Wg-I)+9, dip=W A0-W Aa+4i.

(4.1.63) (4.1.64) (4.1.65)

In these formulas Vg = (dgij - gik0j - gk j9k), d is the operator of exterior differentiation, and A is the symbol of exterior multiplication. In addition, in all exterior products of 1-forms occurring in equations (4.1.61)-(4.1.65) multiplication is performed row by column: for example, a detailed writing of equation (4.1.63) has the following form:

dw'=aAw'-9j' Aw). In equations (4.1.61)-(4.1.65), the forms 9 = (9f) and 4' = (4,) are the curvature forms of the conformal structure CO(p, q). The expressions for their components are

9j = Cjklwk Aw', $i = Cjjkw3. Awk. However, in what follows we will apply the index notations since they seem to us more convenient in the study of different types of conformal structures.

4.2

Weyl Connections and Riemannian Metrics Compatible with a Conformal Structure

1. As was indicated in Subsection 4.1.1, the conformal structure CO(p, q) on a manifold M of dimension n = p+q is a set of conformally equivalent Riemannian metrics with the same signature. In this section we consider Riemannian

metrics as well as Weyl connections compatible with a conformal structure given on a manifold M. Each of these metrics or connections has the property that it generates this conformal structure. We recall that the Weyl connection is a torsion-free affine connection on the manifold M such that parallel transport with respect to it preserves the angles between vectors tangent to the manifold M and that the angles are determined by the relatively invariant quadratic form (4.1.1) given on the manifold M (see Norden (N 50a], p. 158) Consider the forms 9i occurring in the structure equations (4.1.32)-(4.1.35) of the normal conformal connection defined on the manifold M by the conformal

structure CO(p, q). These forms do not depend on the basis forms wi of the manifold M. As we indicated earlier, equations (4.1.37) and (4.1.38) show that when w' = 0, the forms ai = 9i(b) determine displacements of the point A"+, of the local conformal space (C")s. Let us now assume that the point A"+1 is fixed in every local conformal space (C")S. Then ai = 0 and 9i = Pijwj.

(4.2.1)

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Weyl Connections and Riemannian Metrics

133

In this case equations (4.1.33) become

dO=pijw'Awi.

(4.2.2)

This means that the form 9 can differ from a linear combination of the basis forms only by a total differential. Thus this form can be written as (4.2.3)

8 = p,w' - d log gyp,

where the quantities pi form a covector on M. Hence equations (4.1.16), which the fundamental tensor gij of a CO(p, q)-structure satisfies, take the form

Vg,j = 2(-dlogW+pkwk)9ij.

(4.2.4)

It follows that O(W29ij) = 2Pkwk(co29ij)

or

dg-ij - 9ikwj - 9kjwk = 29ijPkwk,

4.2.5)

where g`,j =
If we assume that in equations (4.1.21), Ask, = CCk, and that relations (4.1.30) hold, then equations (4.1.21) coincide with equations (4.1.34). Substituting for the forms Oi in equations (4.1.21) their expressions (4.2.1), we obtain dwj = wj Awk + Rjklwk Awl, (4.2.6) where Rjkl = Cjk, + 9j[k9'mPlmjlJ - bjPJkIJ - b['0101-

(4.2.7)

Equations (4.1.3), (4.2.5) and (4.2.6) show that the forms wj' define on the manifold M the affine Weyl connection, and the tensor R,'kl is the curvature tensor of this connection. Its Ricci tensor is Rjk =

2((n - 1)Pjk - Pkj + pgjk),

(4.2.8)

where p = giipij. The scalar curvature of this Weyl connection has the form

R = gjkRjk = (n - 1)p.

(4.2.9)

Conversely, the formulas that we derived above allow us to find the tensor CJk, of conformal curvature in terms of the curvature tensor of the Weyl connection. To this end, we find the tensor pjk from equations (4.2.8) and (4.2.9):

Pjk =

2((n - 1)Rjk + Rkj) n(n - 2)

R9jk

(n - 1)(n - 2)'

(4.2.10)

134

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Substituting these expressions for pjk in formulas (4.2.7), we find the expression for CJki:

Cjkl =

Rjkf + n(nl 2) [(n - 2)(Rkl - R4k)6 - (Rkj + (n - 1)Rjk)61 +(Rrj + (n - 1)Rjt)bk - (Rim + (n - 1)R,nl)9°"9jk +(Rkm + (n - 1)Rmk)g'mgjl] +

R

(n - 2)(n - 1) (9ki6i - 91j6k) (4.2.11)

This expression for the tensor in the case of the nonsymmetric Ricci tensor was obtained in Klekovkin [Klk 81a] (Eq. (26)). Suppose further that the quantities p;j in equations (4.2.1) are symmetric,

p;j = p,,. Then it follows from equations (4.1.33) that dO = 0, and the form 0 is a total differential, 0 = d log W. This implies that Pk = 0. It follows that equations (4.2.5) take the form Vggij = 0, which means that the quadratic form g';jw'wj is an invariant form. Thus the tensor gj defines a Riemannian metric on the manifold Al, and the forms w, define a Riemannian connection on this manifold.

Calculating the curvature tensor and the Ricci tensor of this Riemannian connection by means of (4.2.7) and (4.2.8), we obtain Rj'k1 = Cjk1 + 9jlkg'mplml,l - 6lkPliI1)

(4.2.12)

Rjk = 2((n - 2)Pjk + P9jk).

(4.2.13)

and

It follows that the Ricci tensor of the Riemannian connection is symmetric: Rjk = Rkj as it is supposed to be (e.g., see Eisenhart [Ei 26], §8, or Dubrovin, Fomenko, and Novikov [DFN 92], §30). If we find the quantities pjk from equations (4.2.13) and substitute their values into equations (4.2.12), we find the expression for the tensor of conformal curvature of a Riemannian manifold (see Eisenhart [Ei 26], §28): 1

2 (6

Ri,-6 Rjk+gjlRk-9jkRi)+(n - 1Rn - 2)

(4.2.14)

where Rk = g"Rmk. Note that we can also obtain equations (4.2.14) from equations (4.2.11) taking into account that a Riemannian connection is an equiaffine Weyl connection, and therefore the Ricci tensor R;j of this connection

is symmetric: R,3 = R, . Thus we have shown how the Weyl structures and the Riemannian structures, which are substructures of the conformal CO(p, q)-structure, can be obtained from the CO(p, q)-structure by reduction of the group of admissible transformations of its adapted frames. 2. All the Weyl connections that we have constructed above and that are defined on the conformal CO(p, q)-structure are similar one to another and are

4.2

Weyl Connections and Riemannian Metrics

135

not intrinsically connected with this structure. However, in some cases among these connections it is possible to find an invariant connection by specializing

the object C,,k that satisfies equations (4.1.54) provided that the principal parameters are fixed (i.e., for w` = 0). Now we will show how to find this invariant connection for a conformally

recurrent structure. For n > 4, the CO(p, q)-structure is called conformally recurrent (see Adati and Miyazawa [AdM 68]) if the quantities Cjktm on the right-hand sides of equations (4.1.55) are connected with the quantities CCkt by the following equations: Cjktm = Cjktam.

(4.2.15)

Although the quantities C}ktm do not form a tensor on the manifold M, equations (4.2.15) are invariant and are preserved under admissible transformations of the adapted frames, and the quantities Cjktm are transformed concordantly with the quantities a,,,. For a conformal recurrent structure, from formulas (4.1.60) and (4.2.15) we find that 1 (4.2.16) C,,k C1 fkat.

n-3

Substituting these values for Ci,k into formulas (4.1.55) and taking into account

that OoC;,k = 0, we obtain

C;,k(Vdat - (n - 3)at) = 0.

(4.2.17)

Next consider the rectangular matrix C = (C{tjk}) where I is the row index and {ijk} is the column index. This matrix has n rows and 1n2(n-1) columns. If the rank C = n, then from equations (4.2.17) it follows that

V5at - (n - 3)at = 0.

(4.2.18)

This equation shows that in the case under consideration, a specialization is possible that implies at = 0 and at = 0. As a result of this specialization, the forms Bt become principal forms; that is, they have form (4.2.1). These forms define an invariant Weyl connection associated with a conformally recurrent structure. Since the specialization indicated above also implies that C,ktm = 0, it follows that for this invariant Weyl connection, the tensor Cfkt of conformal curvature satisfies the condition VCJkt = 0;

(4.2.19)

in other words, this tensor is covariantly constant in this connection, so the invariant connection we have constructed is conformally symmetric. Thus we have proved the following result:

136

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Theorem 4.2.1 If on a manifold M, dim M > 4, a conformally recurrent CO(p, q) -structure is given, and the rank of the matrix C formed from the tensor of conformal curvature of this structure is equal to the dimension of the manifold M, then this structure defines on M an invariant conformally symmetric Weyl connection.

3. Let us investigate how a Riemannian connection on the manifold M is changed under a conformal transformation of a Riemannian metric. The solution of this problem can be found in many books (e.g., see Eisenhart [Ei 26], §28, or Norden [N 50a], §44). However, in all these books the authors considered only proper Riemannian metrics. We would like to call our attention to some specific properties arising when conformal transformations of pseudoRiemannian metrics are considered. On the manifold M we consider an invariant quadratic form 9 = gijwiwJ

(4.2.20)

with signature (p,q), p+q = n = dim M defining on M a Riemannian metric. For p = n, q = 0, the metric will be proper Riemannian, and for q > 0, it will be pseudo-Riemannian. A torsion-free of lne connection 7 is defined on the manifold M by basis forms w` and connection forms wr, satisfying the structure equations dw' = wl A w', dwj' = w A w,t + Rfkiwk n wi, (4.2.21) where

is the curvature tensor of this connection; see (4.2.6). This connection is called the Riemannian connection or the Levi-Civita connection if V9ii

dgii - gikwi - gktwi = 0;

(4.2.22)

here the symbol V is the operator of covariant differentiation in the connection 7.

Let us make a conformal transformation of the metric (4.1.1); that is, we transfer from this metric to another metric dal = ads2 = agjiw'ur1,

where a = a(x) is a function of a point x E M and a(x) > 0. The metric tensor corresponding to dal has the form

g.j = agij.

(4.2.23)

While doing this, we assume that the basis forms wi are not changed. Denote by ry the Riemannian connection defined by the metric dal and by V the operator of covariant differentiation in this new connection ry. Then

vgij = 0.

.4.2

Weyl Connections and Riemannian Metrics

137

If we denote the connection forms of the connection ry by Uj, we can write the last equation in the form

d9ij - 9ikwj - 9kjwi = 0.

(4.2.24)

Applying equations (4.2.23), we obtain

gijdlog a+dgij-9ik1' -9kjwi = 0.

(4.2.25)

The forms w and wj determine two affine connections on the manifold M. Thus their difference can be written in the form

wj - w =

(4.2.26)

where Tk is the deformation tensor of the connection (see Norden [N 50a], §34). Since both connections in question are torsion-free, the tensor Tj'k is symmetric: T, r`k = Tk j . Since the quantity or is a function of a point x E M, or = a(x), and a > 0, we have d log a = akwk.

(4.2.27)

Substituting expressions (4.2.26) for i and expression (4.2.27) into equation (4.2.25), taking into account equations (4.2.22), and then equating to zero the coefficients of the linearly independent forms wk, we obtain 9jtT;k = 9ijak.

(4.2.28)

Cycling these relations with respect to the indices i, j, and k, we obtain two more equations:

9klTji = 9jkai, (4.2.29) 9kiTij + gilTkj = 9kiaj Adding equations (4.2.29) and subtracting equation (4.2.28) from the result, we find that 29k1T j = gikaj + 9jkai - 9ijak, from which it follows that Tip =

2(6kaj +bjai -9ijak),

(4.2.30)

where ok = 9kiai. Now the connection forms of the Riemannian connection ¶ defined by the metric ds2 can be expressed as

i = wf + 2(bj'akwk +a,w' - a'w1),

(4.2.31)

where wj = 9jkwk. This formula gives the law of transformation of a Riemannian connection under conformal transformation of a Riemannian metric.

4. We next consider the geodesics of a Riemannian manifold. Locally such curves are the shortest and the straightest curves. We derive equations of

138

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Figure 4.2.1

geodesics applying their second definition. This is especially convenient since this second definition is more appropriate for isotropic geodesics, which will be considered later. Let x = x(t) be a smooth curve on a Riemannian manifold M endowed with the metric defined by formula (4.2.20). Its tangent vector di is a linear combination of the basis vectors e; of the space TT(M):

d or

dx = w'e;, where w' = a'dt. Since along the curve x = x(t),

de, = w; ej,

where wl are the connection forms of the Riemannian connection y, we have d2x = (dw'

The curve x = x(t) is the straightest if the vector d2x is collinear to the tangent vector dx (see Figure 4.2.1). Hence the equations of geodesics of the Riemannian connection y have the form dw' + w'wj., = Bw'.

(4.2.32)

In a Riemannian manifold V,", q > 0; namely, in a pseudo-Riemannian manifold, isotropic geodesics-geodesics that are tangent to the isotropic cone

C,, at each of their points x-are of special interest. In addition to equation (4.2.22), such curves also satisfy the equation

g,,w'w' = 0.

(4.2.33)

We will now prove the following result which confirms the existence of isotropic geodesics:

4.2

Weyl Connections and Riemannian Metrics

139

Figure 4.2.2

Theorem 4.2.2 If a geodesic of the manifold VQ is tangent to the isotropic cone at one of its points xo, then this curve is tangent to the isotropic cones at any other of its points x; that is, this curve is an isotropic geodesic. Proof. The geodesic x = x(t) in question is uniquely defined by the system of differential equations (4.2.32) and initial conditions x(to) = xo and ee It=," = ao. Since by hypothesis the geodesic is tangent to the isotropic cone at the point xo, we have

9' 7

9t. aoao = 0,

where by g° we denote the values of the components of the metric tensor g,j at the point xo (see Figure 4.2.2). The last condition can be rewritten in the form

(9,,w'w'r)jx=xa = 0,

(4.2.34)

since we have we = a'dt along the curve x = x(t). Differentiating the left-hand side of equation (4.2.33) and taking into account equations (4.2.22) and (4.2.32), we find that d(g=jw'w') = 29yw'w'9,

where the differentiation is carried out along the curve x = x(t). Along this curve, the 1-form a is a total differential and can be written in the form 0 = dcp,

where cp = c(t). Thus the last equation can be written in the form d(9tjw'w3) = 2dW g;1w'w'.

Integrating this equation, we find that 9iiw'wi = Ce2''.

But since for t = to condition (4.2.34) holds, we find that C = 0 and that 9I.iw'wj = 0

140

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

everywhere along the curve x = x(t), so this curve is an isotropic geodesic. It follows from Theorem 4.2.2 that in the Riemannian manifold VQ,q > 0, through any point x and along any isotropic direction emanating from this point, there passes one and only one isotropic geodesic.

Note that as we already stated several times earlier, the usual model of space-time in general relativity is a four-dimensional Riemannian manifold with signature (1, 3) (Lorentzian signature) (e.g.. see Chandrasekhar (Cha 83), Ch. 2, §11). Isotropic geodesics of this manifold are curves of propagation of light

impulses. Hence they are important in this theory. 5. Finally we will find how geodesics of a Riemannian manifold are changed under conformal transformation of a Riemannian metric. Under such transformation, the metric tensor undergoes the transformation defined by formula (4.2.23), and the connection forms undergo the transformation defined by formula (4.2.31). Thus the equations of geodesics in the Riemannian connection y defined by the metric ds2 take the form

dw'+uri(w + 2(6 o'kwk+o;w' -(7'W,)) =Bw'. This equation can be reduced to the form dw' + wrwf - 2 r1'w'wj _ (N - (rkwk )w'

and by means of relation (4.2.27) and the condition wj = gfkwk, it can also be written as 1(gik.rwk)a' _ (B-dlogo)w'. dw`+u'1ww(4.2.35)

Comparing equations (4.2.32) and (4.2.35), we see that in the general case, under conformal transformation of a Riemannian metric, geodesics do not remain invariant. The reason for this is the third term on the left-hand side of equation (4.2.35) containing o'. However, there are two cases where equation (4.2.35) defines the same curves as equation (4.2.32). First of all, this happens if o' = 0, that is, if a = const. In this case equation (4.2.35) coincides with equation (4.2.32) with B = 0, and all geodesics are transformed into geodesics. But this case is not so interesting, since the conformal transformation has a very special form if or = const. Second, equation (4.2.35) defines the same curves as equation (4.2.32) if g,kwjwk = 0, that is, if the geodesic is isotropic. In this case equations (4.2.35) and (4.2.32) coincide if B = 0 + dlog jol. Thus we have proved the following result:

Theorem 4.2.3 Under the general conformal transformation of a Riemannian metric on a manifold M, isotropic geodesics and only such geodesics remain invariant. This theorem is important in general relativity. The invariance of isotropic geodesics of four-dimensional space-time under a conformal transformation of

4.3

A Conformal Structure on Submanifolds of a Conformal Space

141

a Riemannian metric demonstrates the importance of the theory of conformal structures in general relativity. Theorem 4.2.3 shows that isotropic geodesics can be considered not only in Riemannian manifold Vo for q > 0 but also on a manifold endowed with a conformal structure CO(p, q), p + q = n, q > 0. In particular, in a conformal space C.", isotropic geodesics are rectilinear generators of isotropic cones that are transformed into rectilinear generators of a hyperquadric QQ of the space Pn+1

4.3 A Conformal Structure on Submanifolds of a Conformal Space 1. In Chapter 3 we studied the geometry of submanifolds V"' of the conformal and pseudoconformal spaces. In this section we will consider the conformal structure induced on a submanifold VI by the geometry of the ambient conformal space. We will also study affine connections arising on V' when it is normalized. As an ambient space, we will consider not only the proper conformal space C" but also pseudoconformal spaces C, of signature (n - r, r). Let V'" be a nonisotropic m-dimensional submanifold of a conformal space C, n. The tangent subspace T=(Vm) intersects the isotropic cone Cs of the space

C, along a cone of second order C. (V'n) = T. (VI) fl C. Moreover the cone C=(VI) is a nondegenerate cone of second order. Thus, on the submanifold V'", a conformal structure is induced, and its signature depends on the mutual location of the tangent subspace T=(Vm) and the isotropic cone C. Since the submanifold V'" is nonisotropic, the space TZ(Vm) is not tangent to the isotropic cone C. If TT(V'n) has only one point x common with the cone C=, then a conformal structure arising on V' is proper conformal; that is, it is of signature (m, 0). On the other hand, if the cone Cz (V'n) is of signature (p, q), p + q = m, q < r, p < n - r, then a conformal structure arising on V "' is the conformal structure CO(p, q) (see Section 3.3). We will assume that the signature of the cone CC(VI) is the same at all points x E V'". We associate with the submanifold V'" the bundle 1 (VI) of conformal frames {Ao, Ai, Aa, An+1 } of first order in the same manner as in Section 3.1. In this section we will use the same range of indices that we used in Chapter 3:

1
142

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

We will find the expressions of the invariant forms of this conformal strucand the object C'ijk in terms of ture, its tensor of conformal curvature the quantities related to the submanifold V'n. First, we write the equations of infinitesimal displacement of first-order frames of the submanifold V': dAo = w$Ao + w'A;, dAi = w°Ao + w; Aj

+W°Aa

dAo. = wOA0 + wQAj

+wQAp,

dAn+1 =

+W; +' An+1, (4.3.2)

Wn+1A1 +wn+1A0 -WOAn+1,

where as in Section 3.1, wa 0 = O,

Wn+I =0 0

wn+1 a =O

wa

w°+1 = 9ijwj,

Wn+1 = 9`iW9 ,

Wa =

W0n+1

Wn+1 = 9apwA

= O,

d9ij = gikWj +9kjWk,

(4.3.3)

(4.3.4)

d9ap = gatiWp + 97pWa,

and as in Section 3.1, gap = (Ac,A0) (cf. formulas (3.1.8), (3.1.10), (3.1.11), and (3.1.17)). Next we will write the structure equations that the forms Wool Wi,Wi,w° wQ and wQ satisfy and that follow from equations (1.2.11): dW00 =woA

dw'=WOO Awo+woAW;,

dw, =w°Awo+wk AWk+w AwQ+wn+' Aw' dw°=w°Awoo --wi Awl +w°Awa°,

(4.3.5)

dw0 = w0 A WO +W' A wo + wQ A wa, dwoo = wo A woo + w' A wP + wO A w,.

Let us find dependencies between the forms w', 8, 9 and Oil occurring in structure equations (4.1.31)-(4.1.35) of the pseudoconformal structure CO(p, q), and the forms wf occurring in equations (4.3.2). We will assume that the forms w' in these two groups of equations are the

same. By virtue of this, from equations (4.1.32) and the second equation of (4.3.5) it follows that

(0-wp)Aw'+wj A(9 -w)=0, and this implies that

Oj-Wi

ajk=akj.

(4.3.6)

4.3 A Conformal Structure on Submanifolds of a Conformal Space

143

But the values of the tensor gij on the submanifold V'" and on the desired conformal structure also coincide. Hence subtracting the first equation of (4.3.4) from equation (4.1.31), we obtain

9ikv' -w;) +9kjY7 - wi) = 0. Substituting for the forms B - w, their values from (4.3.6), we find that 2gij(0 - woo) + (gik 4, + gkja ')w' = 0.

(4.3.7)

It follows that the 1-form 0 - wo is a linear combination of the basis forms wk: 0 - woo = akwk.

(4.3.8)

Substituting this expression into equations (4.3.7) and equating to 0 the coefficients of independent forms wi, we find that

29ijai +

gk ja = 0.

Proceeding in the same way as before for finding the forms (4.1.18), we find the quantities ask from equations (4.3.9):

(4.3.9)

from equations

ask = 9ik9'iar - 6 Qk - bkaj.

(4.3.10)

Exterior differentiation of equation (4.3.8) by means of (4.1.33) and the first equation of (4.3.5) leads to the equation

(Vc,-w°+B,)Aw'=0, where Va, = dai - ak(wk - MM). Applying Cartan's lemma to this exterior quadratic equation, we find that 0

We will write these equations assuming that the point x is fixed on the submanifold V°' (i.e., for wi = 0):

V6ai-7r?+rr,=0. These relations prove that the geometric object a; can be reduced to 0 by means of the fiber parameters. Geometrically, this means making a special choice of

hyperspheres A; which are orthogonal to the submanifold V'". Having done this, from equations (4.3.10) we obtain ak = 0, and

9=woo, 0=w, 0,-w°(4.3.11) Next, by means of (4.3.11), we write the third group of equations (4.3.5) in the form

dw, =0jAw'+wjk Awk+gjkwkAg"0, +(atQ;k + gjk9"" Qml - 9Q09'mAjkApr,i)wk Awi.

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

144

Comparing this equation with equation (4.1.34) and taking into account that wl = 8 , we find the following expression for the components C, kl of the tensor of conformal curvature: -9°p9""(Xjk'\ml

2CCkl =

- AjlAmk)

blQjk - dkQjl

(4.3.12)

+9""(9jkfml - 9jiQmk) Since the tensor Ckl is trace-free, by contracting equation (4.3.12) with respect to the indices i and 1, we find that 0 = -9°p9'm(AjkAmi - AjiAmk)

(m - 2)0j,, + 9jk/,

(4.3.13)

where ,0 = gijfij. If we contract the last equation with the tensor gjk, we obtain

Q-

1

2m-29°p(

m2A°AA

ij kl)(

- 9' k9i1A° Ap

4.3.14 )

(see (3.1.26)). If m > 2, then from (4.3.13) and (4.3.14) where A° = we find the values of Qjk for which C'kl = 0: Qjk =

m _-2 ga 1g$m(AjkAml - Aji'mk)

2(m - 1)(m -

(4.3.15)

2)9Jkt9ap(m2A°AR _ g°99uA PA9I).

Now it follows from formula (4.3.12) that the tensor of conformal curvature Ckl is determined in a second-order differential neighborhood of the submanifold V'" and is expressed in terms of the quantities A. Although the quantities A? themselves do not form a tensor, the quantities (4.3.12) constructed by means of A form the tensor of conformal curvature. This tensor is determined by the submanifold V"' itself and is not dependent on the choice of the frame bundle 1V (V"') associated with V'°. Thus we have proved the following result:

Theorem 4.3.1 The tensor of conformal curvature CJkl of the conformal structure associated with a submanifold VI of the conformal space Cr is determined in a second-order differential neighborhood of the submanifold V'". This tensor has the form (4.3.12), where the quantities Qjk are determined by equations (4.3.15) and does not depend on the choice of the frame bundle RI (Vm) associated with V"'. 2. As we have proved in Subsection 3.1.5, the vanishing of the tensor a 13 on the submanifold V', which is equivalent to the condition A91. 1) = gijA°, characterizes m-spheres or their open subsets. But as equation (4.3.12) shows, this condition leads to the vanishing of the tensor of conformal curvature C'kl. Thus, as one can expect, m-spheres are conformally flat submanifolds. Now we will give an example of an m-dimensional submanifold that is different from the m-sphere S' and carries a locally flat conformal structure. We will

4.3 A Conformal Structure on Submanifolds of a Conformal Space

145

prove that the 1-canal submanifolds Vm possess such a structure. Suppose that a submanifold V'" is the envelope of a one-parameter family of m-dimensional spheres Sm. In the bundle of frames of first order, we will take a subbundle

in such a way that at any point x E VI , the tangent hyperspheres A. pass through the m-sphere S"`. Since by virtue of equations (4.3.2) we have dA0 = w° Ao + wQ Aj + wOAo,

the displacement of the m-sphere S'" is determined by the forms wa and w«, and these forms must be expressed in terms of an 1-form on which the m-sphere S"' depends. Next we will further specialize first order frames, taking the m-sphere Sm to be fixed if w' = 0. Then the forms wo and wQ will be expressed in terms of the form w' only. Thus the form w° = -gijg°owp will also be expressed in terms of the form w' . By means of equations (4.3.3), we find that a l wk wiQ = altw , o = 0,

k = 2,...,m.

Thus the matrix (a'.) takes the form

a 0 0 0

...

0

0

...

0

0

...

0

........

(4.3.16)

Now we will compute the tensor of conformal curvature CCkl of the conformal structure associated with the 1-canal submanifold V'". This tensor is determined by the formula (4.3.12). A simple calculation shows that by equation (4.3.16), we have \jl'\mk 0+ xjk'\ml

-

-

and as a result we find from (4.3.15) and (4.3.12) that /3jk = 0 and Ckl = 0, that is, 1-canal submanifolds possess a locally flat conformal structure. Thus such submanifolds admit a conformal mapping onto an m-dimensional sphere. 3. To compute the object Cijk, it turns out to be necessary to separate from the bundle R' (V'") of first-order frames a subbundle in which at any point x E Vm, a tangent m-sphere Am+l A ... A An is fixed. The simplest way to arrange this is to take the central m-sphere Cm+l A... A Cn constructed in Section 3.1 as such a tangent m-sphere. In view of this, on the submanifold V m we will take the subbundle RZ(V-) of second-order frames which are defined

in a second-order neighborhood of a point of VI and have the hyperspheres, Co, = Aa + A0Ao,

with as = mgaog'jaQ, as their tangent hyperspheres. In the frames of this coincide with the corresponding components of subbundle, the quantities

146

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

the tensor 0°., and as a result the quantities )a will vanish: as = 0. Moreover the form wQ becomes a linear combination of the basis forms w': ° = b,,' W0,

(4.3.17)

(cf. formulas (3.1.65)). Thus, in the frame bundle 7Z2(Vm), formulas (4.3.15) take the form /3jk =

-ml 2(Ajk - 2gjkA),

(4.3.18)

where

Ajk = gal3gima .4 , A= m

1

1

Ajkgjk

are, respectively, a symmetric tensor and an invariant of the submanifold V' that are connected with a second-order differential neighborhood of 171. This implies that the quantities Qjk also form a symmetric tensor defined in a second-

order differential neighborhood of V'. Taking the exterior derivatives of equations (4.3.11) and using the bundle of second-order frames, we find that [VQij + (a fbak - Cijk)wk] A wi = 0,

(4.3.19)

where VQij = d/3ij - QkjBk - /.3ik0 . Since the quantities /3ij form a symmetric tensor, we have qq k VOij = Nijkw

where Qijk = /3jik are determined in a third order differential neighborhood of the submanifold V'°. Substituting for V/3ij in equations (4.3.19) their values from the last equations, we obtain (Qijk + aa,bok - Cijk)wk h wi = 0.

Then Oijk - Qikj +

a kbaj - 2Cijk = 0,

since Cijk = -Cikj. This implies that Cijk = 1 (/3ijk - l3ikj + a bak - a kbaj).

(4.3.20)

Thus the quantities Cijk are completely determined in a third-order differential neighborhood of the submanifold V'". The results of these calculations can be formulated in the form of the following theorem:

Theorem 4.3.2 The normalization of a submanifold V' by means of the family of central m-spheres determines on I'm a normal conformal connection corresponding to the conformal structure CO(p, q) which is induced on vm by the

4.3 A Conformal Structure on Submanifolds of a Conformal Space

147

geometry of the ambient space. Moreover the tensor of conformal curvature CkI of this connection can be expressed in terms of elements of a second order neighborhood of the submanifold V°', and the quantities C;;k can be expressed in terms of elements of a third-order neighborhood of the submanifold V'°. We remind the reader (see Subsection 4.1.4) that for the conformal structure CO(p, q) of general type, the tensor C'jki is determined in a third order neighborhood, and the quantities C;jk are determined in a fourth-order neighborhood. 4. In Section 4.2 we already mentioned the notions of an affine connection and the Weyl connection on a differentiable manifold M. Now we will consider in more detail how these notions can be introduced in a frame bundle.

The forms w, in equations (4.1.3) are defined invariantly in the bundle R2(M) of second-order frames. An affine connection ry on the manifold M is defined in the bundle R2(M) by means of an invariant horizontal distribution 0 given by a system of invariant forms

oil = w -

(4.3.21)

which vanish on the distribution A. The distribution 0 is invariant with respect

to the group of affine transformations acting in a fiber R'(M) of the frame bundle R` (M) (see [Lic 55], Ch. II, §3). Using equations (4.3.21), we eliminate the forms wj' from the second equation of (4.1.3). As a result we obtain dw' = wl A B,' + R'Jk.,' A wk,

(4.3.22)

where Rk = r(Jkl. The condition for the distribution 0 to be invariant leads to the following equations: dO = Bk A ok +

A

(4.3.23)

The form 0 = (off) is called the connection form of the connection ry. The and Rk, are tensors called the torsion tensor and the curvature quantities tensor of this connection. Conversely, one can prove that if in the frame bundle IZ2(M) the forms w' and o satisfy equations (4.3.22) and (4.3.23), then the forms oj' define an affine connection ry on M, and the tensors Rijk and Rj'k, are, respectively, its torsion and curvature tensors. If the torsion tensor Rjk = 0, then the connection 7 is called a torsion-free affine connection. For such a connection r k = 1' , and structure equations (4.3.22) and (4.3.23) take the form dw' = w3 A off, dog = o A 0k + R'k,wk A

(4.3.24)

(see Kobayashi and Nomizu [KN 63], vol. 1, Ch. II, §§2 and 3, or Laptev [Lap 53], p. 324, and formulas (4.3.2)). In what follows we will only be interested in torsion-free affine connections.

148

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

An affine connection allows us to define the covariant differentiation on the manifold M. Consider a vector field f = f iei on M. For a fixed point x E M, the coordinates f' satisfy the equation

bfi + fjrr = 0,

(4.3.25)

which is the condition for the vector f to be invariant with respect to admissible transformations of first-order frames in the tangent space TT(M). As earlier, in (4.3.25), b is the symbol of differentiation with respect to fiber parameters and

xjl = wp(b). As the point x moves along the manifold M, equations (4.3.25) become

V ' := df' + i;jOl =

(4.3.26)

If an affine connection is present, then the quantities f form a tensor. In fact, taking the exterior derivatives of equations (4.3.26) and applying (4.3.24), we obtain ji (V

where v{j = do + { Bk - CkB . It follows from this equation that i

k

vfj - fjRw

and as a result va f, = 0. This equation proves that the quantities

are transformed according to the tensor law under admissible transformations of first order frames (see Subsection 2.1.2). The quantities f, are called the covariant derivatives of the vector field £ with respect to the affine connection -y. The vector field f is called parallel if the following condition holds:

vfi=0.

(4.3.27)

This condition is a system of Pfaffian differential equations which, in general, is not completely integrable. However, we can integrate this system along any

curve x = x(t) on the manifold M. This will define a parallel vector field { along the curve x = x(t). If the system of equations (4.3.27) is completely integrable, then we will say that the manifold M possesses an absolute parallelism. For such manifolds M, the curvature tensor vanishes: Rki = 0. Now suppose that on the manifold Al a conformal structure is given by a relatively invariant quadratic form 9 = 9ijf'fi.

A torsion-free affine connection ry defined on the manifold M Is said to be the Weyl connection if the equation g = 0 is preserved under parallel translation with respect to the connection ry; that is, under parallel translation the form g

4.3 A Conformal Structure on Submanifolds of a Conformal Space

149

can only be multiplied by a factor. In differential form this condition can be written as follows: (4.3.28)

dg = 29g.

It follows from (4.3.28) that the tensor gij satisfies the equation Vg,, = 29g;3, or

(4.3.29) dg;j - 9kjO, - gikO, = 20g;j. These conditions characterize the affine Weyl connection defined on the manifold M by the conformal structure g. Next we consider a nonisotropic submanifold V' in the conformal space C. For such a submanifold, we have (see formulas (4.3.4) and (4.3.5))

dgij = gkjw;k+ gikwjk, (4.3.30)

dw' = woo A wo + wo A wi,

dw'-woAJ+wkAw'+w°Aw' 0 j a+0+1 Aw'n+1 k ) J Let us define w. = 9 + 6'wo. Then by (4.3.3), equations (4.3.30) become dgij - gkjek ; - g,kejk = 29>>woo, dw' = w3 A 99,

d9j = B A Bk + (6kw +

gilgjkw, )wk -

9

gad

A w'.

(4.3.31)

Comparing the first group of equations (4.3.31) with equations (4.3.29), we see that they coincide if 6=WOO.

The second group of equations (4.3.31) completely coincides with the first group

of equations (4.3.24). As to the third of equations (4.3.31), they contain the forms wk which are not linear combinations of the basis forms w' in the frame bundle R' (V'"). Thus in this case the forms {w', 9 } do not determine the Weyl connection on the submanifold V'n. Let us associate a normal (n - m)-sphere Sn-n' with each point of the submanifold Vn'. This (n - m)-sphere is called the normalizing (n - m) -sphere, and the submanifold V m is called normalized in the sense of A. P. Norden (cf. Subsection 3.1.6 and also Norden [N 50a], p. 326). Next we choose the normal hyperspheres A, in a first-order frame in such a way that they pass through the normalizing (n - m)-sphere Sn-n` introduced above. Then for a fixed point x E V'n the hyperspheres A; satisfy equations

Mi =7r A so a° = 0, and the forms w° are linear combinations of the basis forms wj: wo = µijwp.

(4.3.32)

150

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Substituting for the forms w9 in the third group of equations (4.3.31) their values from (4.3.32), we obtain

d9j = Bj A9k + (6kµj1 + djµkl - 9'm9jkllmI -

AJ.

Then the forms {w', 9; } determine the Weyl connection on the submanifold V'", and the curvature tensor of this connection has the form 2Rj'k1 - akl[jl - b%iljk + bjl'kl - bjlflk - 9'm9jklAml + 9'm9jllmk _9imgofA.kxml

(4.3.33)

We have proved the following result:

Theorem 4.3.3 If a nonisotropic submanifold V'" of the conformal space C" is normalized in the sense of Norden by means of a family of normal (n - m)-spheres, then an affine Weyl connection is induced on V'", and the curvature tensor of this connection is determined by the formula (4.3.33).

4.4 A Conformal Structure on a Hypersurface of a Projective Space 1. A conformal structure on a hypersurface V" in a Euclidean space R"+1 is induced by the Euclidean metric of R"+' (e.g., see Eisenhart [Ei 26], Ch. V). In the same way a conformal structure on submanifolds of a conformal space C" is induced by the geometry of C" (e.g., see Cartan [Ca 17) and Section 4.3). As far as affine and projective spaces are concerned, if we restrict ourselves to the frame bundle R' (V"), they have no metric and do not generate a conformal structure on submanifolds. However, a tangentially nondegenerate hypersurface of a projective space possesses a relatively invariant second fundamental form, and hence one can consider the conformal structure induced by this form. We will call this structure the asymptotic conformal structure. In this section we will study in detail this asymptotic conformal structure on a hypersurface. Let P"+1 be an (n + 1)-dimensional projective space over the field of real numbers. We will use the following range of indices:

0<{,,1,(
AoAAl A...AA"+1 =1.

(4.4.1)

We will denote the set of such frames by R(P"+' ). It depends on n2 + 4n + 3 parameters, namely on the same number of parameters on which the fundamental group PSL(n + 2) of transformations of the space P"+' depends. This

A Conformal Structure on a Hypersurface of a Projective Space

4.4

151

group is locally isomorphic to the group SL(n + 2) (see Rosenfeld [Ro 96], §0.8.8).

The equations of infinitesimal displacement of the frame {Af} can be represented in the form dAt = w( A,7.

(4.4.2)

The differential 1-forms wE are left-invariant forms of the Lie group SL(n + 2), and they satisfy the Maurer-Cartan structure equations: dwf = wf A w4,

(4.4.3)

which are obtained by exterior differentiation of equations (4.4.2). By exterior differentiation of equations (4.4.1), we obtain the following linear conditions on the forms w{ : wo + WI +

... + n+i = 0.

Let us consider a smooth hypersurface V" in the space

(4.4.4) P"+'.

With any

point x E V", we associate a family of moving frames such that the point AO of each of these frames coincides with the current point x, and the points A, lie in its tangent hyperplaneTx(Vn) (see Figure 4.4.1). As the point x moves along the hypersurface V", all frames associated with the hypersurface V" make up the bundle 1Z' (V n) of frames of first order. Since the differential dAo = wo Ao + woA; + wo +' An+1

of the point Ao must belong to the tangent hyperplane T=(V") of the hypersurface V" at the point x, the frame R' (V") under discussion is given by the differential equation: wo+i = 0,

(4.4.5)

and the forms w` = wo are linearly independent base forms on the hypersurface Vn.

T, (V")

Figure 4.4.1

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

152

Taking exterior derivatives of (4.4.5) and making use of the structure equations (4.4.3), we get w'Aw!'+1 = 0.

Applying the Cartan lemma to this equation, we obtain w7+1 = b;jw',

(4.4.6)

where b;j = bj;. Prolonging equations (4.4.6) twice, we obtain the following systems of differential equations (e.g., see Akivis and Goldberg LAG 93], pp. 211-212): dbij

- bkjwk - bikwj + bi) (wo + n+i) = b$Jkwk

(4.4.7)

and

db;jk - bijkw; - bilkwj- b;jlw,'t + b;jk(wo + bijklw', + 3b(;j(wk) -

448

which are associated with the hypersurface V". In equations (4.4.7) and (4.4.8) the quantities bijk and b;jk, are symmetric with respect to all indices. Equations (4.4.7) show that the quantities b;j constitute a relative tensor.

This tensor is called the second fundamental tensor of the hypersurface V". The relatively invariant quadratic form 4?(2) = b;jw'w'

(4.4.9)

is called the second fundamental form (or the asymptotic form of second order) of the hypersurface V". It plays the role of relatively invariant form (4.1.1).

If the form 't(2) is nonsingular (i.e., det(b;j) i4 0), then the set of tangent hyperplanes to V" depends essentially on n parameters. Hypersurfaces that possess the given property are called tangentially nondegenerate. In what follows we will only consider tangentially nondegenerate hypersurfaces V" of the space p"+' As one can see from (4.4.8), the symmetric quantities bijk occurring in (4.4.7) constitute together with b;j a homogeneous geometric object. This object allows us to construct the following symmetric relative (0, 3)-tensor:

B,,k = bijk -

3

n3

byjbkl,

(4.4.10)

where b; = Wkb;jk and bjk is the inverse tensor of the tensor b;j. The tensor B;jk is called the Darbour tensor of the hypersurface V". In view of formula (4.4.10), the tensor b;j and the Darboux tensor Bijk are connected by the apolarity condition b''B;jk = 0. (4.4.11) The relatively invariant cubic form

`p(3) = Bijkwi'wk

(4.4.12)

4.4

A Conformal Structure on a Hypersurface of a Projective Space

153

is called the cubic Darboux form of the hypersurface V". G. F. Laptev ([Lap 531, p.

371) has proved (see also Akivis and Goldberg [AG 93], p. 219) that

vanishing of this form everywhere is a necessary and sufficient condition for a hypersurface V' to be reduced to a hyperquadric of the space P"+' 2. The relatively invariant nondegenerate quadratic form 1'(2) defines a conformal structure on the hypersurface V". It plays the role of the relatively invariant form (4.1.1). We will call this structure the asymptotic conformal structure. If the second fundamental form 4t(2) is of signature (p, q), then the structure defined by this form is a CO(p, q)-structure. In particular, if the form 41(2) is positive definite, then this structure is the proper conformal structure,

that is, a CO(n)-structure. In Section 4.1 we introduced invariant forms of a conformal structure. Now we will evaluate these forms for the asymptotic conformal structure on the hypersurface V". These forms must satisfy structure equations (4.1.31)-(4.1.35) if the second fundamental tensor bij of V" is taken as the fundamental tensor 9ij

Equations (4.1.31) and (4.4.7) imply that

2d; (wa +)n+i)+ 2bjkwk,

(4.4.13)

where bj'k = b"bijk. If we differentiate the forms wk, using the structure equa-

tions (4.4.3) of the space P"+', and then compare the result with equations (4.1.32), we find that

9=

(w0 - n+1)

(4.4.14)

2 By exterior differentiation of equation (4.4.14), we find the following expressions for the forms 9i: (4.4.15) 9i = 2(w° + bijwn+i) +7ijwi, where 7ij = 7ji are still unknown quantities. To find them, we will differentiate equation (4.4.13). We obtain

d9' - 9j A w' - 9' A 0' - bjkwk A b"9i 7jkdj - bjk7imb"'')Wk A wi

_

(4.4.16)

We observe that the quantities bj'ki = b'mbmjki that arise on differentiating the relations (4.4.13) will not occur in (4.4.16), since they are symmetric with respect to the indices k and I and they vanish while being contracted with the skew-symmetric form wk A wi.

Comparing equations (4.1.34) and (4.4.16), we see that 4bj[kb[)m - Yjlk61l -

bj[k'Yilmbm',

(4.4.17)

where Cki is the tensor of conformal curvature of the asymptotic conformal structure defined on the hypersurface V". Contracting these equations with

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

154

respect to the indices i and l and taking into account that the tensor C'Jkl is trace-free (see (4.1.49)), we arrive at the following expressions for the quantities

4(n - 2)

bj1 bk]k

-

bijb[°bklmbP9.

1 4(n - 1)(n - 2)

(4.4.18)

One can easily check that the right-hand side of this equation is symmetric with respect to the indices i and j. Hence the invariant forms of the asymptotic conformal structure on the hypersurface V" of the space P"+l can be written as 1

O = ,j' - -6)(w6 +wn+i) + 2b'.kWk,

9i = (W°+bijw'n+1)+ 4(n1 2)(6[ibklm - (n 1

1)bijbp(gbk],"bPq)

(4.4.19)

By virtue of (4.4.17) and (4.4.18) the tensor of conformal curvature can be evaluated as follows: Cijk1 =

bD°bpjlkblliq + 4(nl 2) (bajbv(kblli + bj(kbllpbyi

-bpbP(kblli - bpbj(kbli) -

4(n - 1)(n - 2)

(bPbp - bPqbPq)bj(kbgi,

(4.4.20)

where b' = bijbj and bk = bi1WPblpk

The tensor (4.4.20) can be expressed in terms of the Darboux tensor Bijk of the hypersurface V". In fact it follows from equations (4.4.10) that 1

b''k = B''k+ n+2(b'jbk+bikbi+bkibi) Substituting the expression for the object bijk into formula (4.4.20), we get

Cijkl =

1

1

bP°Bpj[kBl]iq + 4(n --21 (Bj[kblli + bj(kBgi) 11

4(n - 1)(n -

(4.4.21)

2) Bbj(kblli,

where Bii = bk1 bP°BkP,Blgj and B = bijBij. Thus we have proved the following result:

Theorem 4.4.1 With a tangentially nondegenerate hypersurface V" of a projective space P"+', there is invariantly associated an asymptotic conformal structure whose tensor of conformal curvature is expressed in terms of the Darboux tensor of the hypersurface V" by formulas (4.4.21).

4.4

A Conformal Structure on a Hypersurface of a Projective Space

155

We observe that on a Riemannian manifold with a metric gij the tensor of conformal curvature is evaluated by formula (4.2.11) which has a structure similar to that of formula (4.4.21). Let us also note that the formula (4.4.21)

does not make sense when n = 2, and as we indicated in Section 4.3, when n = 3, the formula (4.4.21) becomes an identity. Hence in what follows we will

suppose that n > 4. 3. The study of hypersurfaces V" of the space Pn}1 with flat asymptotic conformal structure is especially interesting. When n > 4, such hypersurfaces are characterized by the condition Cijkl = 0. By virtue of (4.4.21), this condition leads to the equations lP"Bpj(kBUiv+n 12(Bjlkb1li+bjlkB1li)-

(n - 1)(n

- 2)Bbjlkbili = 0. (4.4.22)

The simplest solution of this system is Bijk = 0. But this condition describes nondegenerate hyperquadrics in the space P"+1 Thus the asymptotic conformal structure on a nondegenerate hyperquadrics is flat. This corresponds to the fact that a nondegenerate hyperquadric of the space P"+1 is a model of an n-dimensional conformal or pseudoconformal space. However, the hyperquadrics are not the only hypersurfaces with flat conformal structure. We will prove now that envelopes of a special kind of oneparameter families of hyperquadrics also possess a flat conformal structure. Let us consider a one-parameter family Q(t) of nondegenerate hyperquadrics in the projective space P"+'. In the general case, its envelope is foliated into a one-parameter family of (n-1)-dimensional characteristics X(t), each of which is a biquadratic algebraic submanifold, determined by the following system of equations: Q(t) = 0, Q1(t) = 0.

In the case when the biquadratic submanifold X(t) represents a double quadric, this envelope has second-order tangency with each hyperquadric of the family. Such an envelope is called a second-order envelope.

Theorem 4.4.2 If a hypersurface V" of the projective space Pn+1, n > 4, is the second-order envelope of a one-parameter family of nondegenerate hyperquadrics, then the asymptotic conformal structure on this hypersurface is flat.

Proof. Consider a subbundle of the bundle 1V(V") of first-order frames {A() such that the point Ao coincides with the current point x on the envelope V", the point An+1 belongs to that hyperquadric Q(t) of the family that osculates the hypersurface V" at the point Ao, and the remaining points Ai lie in the intersection of tangent hyperplanes TA0(Q(t)) = TAO(V") and

to the hyperquadric Q(t) at the points Ao and An+1, respectively. In these frames the hyperquadric Q(t) has the following equation:

Pijx'xi - 2xoxn}1 = 0.

(4.4.23)

156

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

Since the hypersurface V" and the hyperquadrlc Q(t) have the second-order tangency at the point Ao, the coordinates of the point Ao+dA0+ zd&A0 must satisfy the equation (4.4.23) up to infinitesimals of second order. This implies that Pij = bij, and that the equation (4.4.23) of the hyperquadric Q(t) takes the form bijx'xj - 2xox"+i = 0, (4.4.24) where bij is the second fundamental tensor of the hypersurface V". The characteristics X (t) are determined on the envelope V" by the equation t = const or by the differential equation dt = 0. Then the condition for the hyperquadric Q(t) to be invariant has the form (see Laptev [Lap 53], p. 359, or Akivis and Goldberg [AG 93], p. 219) dPEn

- P
(mod (dt = 0)),

(4.4.25)

where

(

0

0...0

0

(Pen) =

-1) 0

(bij)

.....:.............. 0 -1

0. .0

0

and w is a linear differential form. One of the differential consequences of the system (4.4.25) is the following relation: dbij - bkjwk - bikwjA: + bij(wo + wn+i) = 0 (mod (dt = 0)).

(4.4.26)

Comparing the equations (4.4.7) and (4.4.26), we see that bijkwk = 0 (mod (dt = 0)).

(4.4.27)

Since the differential dt can be expressed linearly in terms of the basis forms, we have

dt = tiw'.

(4.4.28)

Therefore

bijk = hijtk. (4.4.29) where hi j = hji. Alternating this equation first with respect to the indices j and k and second with respect to the indices i and j and taking into consideration the fact that the quantities bijk are symmetric, we obtain

hijtk = hiktj.

(4.4.30)

Since the quantities tk are not simultaneously equal to zero, the general solution of system (4.4.30) has the form

hij = c titj.

(4.4.31)

4.4

A Conformal Structure on a Hypersurface of a Projective Space

157

As a result equation (4.4.29) can be rewritten as

bi,k = c titjtk = hihjhk,

(4.4.32)

where hi = citi. By virtue of (4.4.32) the Darboux tensor of the envelope V" takes the form Bqk

h`hjhk

n

+2 bpghphq(biihk + bikhi + bkihi).

(4.4.33)

It can be immediately verified that the expression (4.4.33) for the Darboux tensor Bijk satisfies system (4.4.22). Consequently the asymptotic conformal structure on the envelope V" is flat. 4. Let us study in more detail the structure of the envelope of second order of a one-parameter family of hyperquadrics in the projective space P"+1, n > 4. Let a hypersurface V" be the second-order envelope of a one-parameter family of hyperquadrics Q(t) := PP,,(t)x{x'n = 0.

(4.4.34)

Then the characteristics X (t) of the envelope V" are double quadrics, which are determined by the following system of equations:

Q(t) = 0, Q1(t) = P,,(t)xEx', = 0.

(4.4.35)

The pencil of all hyperquadrics passing through the intersection of the hyperquadrics Q(t) and Q'(t) is defined by the equation

Q'(t) +.\Q(t) = 0.

(4.4.36)

This pencil contains a two-fold hyperplane in which the characteristic X(t) lies. Hence there exists a function .\(t) such that Q1(t)

+ \(t)Q(t) = r2(t).

(4.4.37)

In this equation

r(t) := lt(t)x{ = 0

(4.4.38)

is the equation of the hyperplane containing the characteristic X(t). Let us find closed form equations for all one-parameter families of hyperquadrics in P"+1 that have second-order envelopes. Since the equations Q(t) = 0 and ,I,(t)Q(t) = 0, where O(t) 34 0, determine the same family of hyperquadrics, equation (4.4.34) can be replaced by an equivalent one Q(t) :=

e.fo

Q(t) = 0,

(4.4.39)

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

158

where the integrand A(s) is taken from equation (4.4.37). Let us differentiate this equation with respect to t: fo Q'(t) = efo \(')di a(t) Q(t) + e

= e fo b(e)ds efo a(e)de

Q'(t)

A(t) Q(t) + (-A(t) Q(t) + i'(t))efo

a(s)ds

R2(t)

Hence

Q'(t) ='s'(t), where ii(t) = e') f'A(')d' . 7r (t). Thus we have shown that by means of a suitable normalization of (4.4.34) condition (4.4.37) can be reduced to Q1(t) = aa(t).

(4.4.40)

Equation (4.4.40) is equivalent to the following system of first-order ordinary differential equations: (4.4.41) F (t) = IC(t)l"(t). Its general solution has the form P07(t) = cEn + J elE(s)ln(s)ds,

(4.4.42)

0

where cc,, are arbitrary constants.

Hence, if a one-parameter family of hyperquadrics (4.4.34) possesses a second-order envelope, then this family can be represented by the equation (

rt

(cc" + J lt(s)l,,(s)ds)x(x" = 0.

(4.4.43)

0

The functions 14(t) in it are chosen so that the hyperplane (4.4.38) will not remain fixed when the parameter t varies: lE(t) # ctl(t),

(4.4.44)

where q are constants. Thus we have proved the following result: Theorem 4.4.3 A one-parameter family of hyperquadrics Q(t) in the projective space p"+i has a second-order envelope if and only if the equation of this family can be reduced to (4.4.43), and condition (4.4.44) holds. If n > 4 and the hyperquadrics of the family (4.4.43) are nondegenerate, then the envelope of this family carries a flat asymptotic conformal structure.

A Conformal Structure on a Hypersurface of a Projective Space

4.4

159

Figure 4.4.2

5. We consider the following construction as an example. Let Rn+1 be an (n + 1)-dimensional Euclidean space in which a Cartesian system of coordinates (xl, x2, , xn+1) is fixed. We consider the equation n

E(x')2 = f2(xn}1),

(4.4.45)

i=1

where f is a thrice differentiable function. This equation defines a hypersurface in Rn+1 which is a hypersurface of revolution of the curve

xi = x2 = ... = n-l = 0; xn+1 = t; xn = f(t)

(4.4.46)

around the axis xn+1 (see Figure 4.4.2). It is easy to check that the hypersurface (4.4.45) is the envelope of second order of the one-parameter family of nondegenerate hyperquadrics of revolution

E(xi)2 = 2(f2)II(xn+1)2 -

(t(f2)',

-

(f2),)xn+l

-t(f2)'+ f2 + 5t2(f2)".

i=1

(4.4.47)

Under a projective transformation of the space P"+', the hypersurface (4.4.45) will become the second-order envelope of a one-parameter family of nondegenerate hyperquadrics. Hence, if n > 4, then the projective images Vn of the given hypersurfaces of revolution carry a flat asymptotic conformal structure. 6. We have invariantly defined an asymptotic conformal structure on a hypersurface in a projective space. In the same way one can define an asymptotic

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

160

conformal structure on hypersurfaces in those spaces whose fundamental group is a subgroup of the projective group. In particular, this can be done on hyper-

surfaces in a Euclidean space R"+'. However, on a hypersurface V" C Rn+I, one can also define a conformal structure departing from the first fundamental form of the V", induced by the metric of the space R"+' (see Subsection 4.1.1). Such a conformal structure on Vn is said to be a metric conformal structure. Cartan [Ca 17] and Verbitsky [Ver 52] have proved that a metric conformal structure on a hypersurface V" C R"+I is flat if and only if the hypersurface Vn is the envelope of a one-parameter family of hyperspheres (see also Subsection 4.3.2). Since the metric quadratic form is proportional to the second fundamental form only on a hypersphere, the following theorem is true:

Theorem 4.4.4 The metric and asymptotic conformal structures on a hypersurface V" in a Euclidean space Rn+' coincide if and only if the hypersurface V" is a hypersphere. These conformal structures on Vn are flat.

NOTES 4.1. The theory of conformal structures arose in studying those properties of Riemannian and pseudo-Riemannian manifolds that remain invariant under conformal transformations of the metric (e.g., see Eisenhart [Ei 26)). Cartan [Ca 231 introduced and studied a manifold M with a conformal connection (see also Cartan (Ca 37b]). Later on, conformal connections were considered by many authors (e.g., see the books Kobayashi [Ko 72), S. Sasaki [SaS 48], and T. Y. Thomas [ThT 4]).

In the 1930s E. Cartan, J. M. Thomas, T. Y. Thomas, and other geometers developed intensively the theory of spaces with a conformal connection. These spaces and different aspects of the theory of conformal connections were studied, for example, in Cartan [Ca 23, 37b], Kulkarni [Kul 70, 88], Muta [Mu 40b, 42], Ogiue [Og 67], S. Sasaki [SaS 48), Schouten and Haantjes (SH 36), J. M. Thomas [ThJ 6), T. Y. Thomas (ThT 34], Vranceanu (Vr 40, 43, 51], Yano [Y 39d, 43b, 47, 74, 76], and Yano and Muto [YM 38, 39, 41a, b)). In some of these papers, the theory of curves and submanifolds of different dimensions in a space with a conformal connection was developed.

For the conformal transformation of a connection and the fundamental tensor of a Riemannian manifold, see Fubini (Fu 09], Schouten [S 18], and Weyl [We 18, 211 (see also the books Schouten [S 241, Ch. VI, §5, and Schouten and Struik [SS 38), §19).

More details on G-structures of finite type can be found in the book Sternberg [St 64] (Ch. VII, §3).

The fact that G' °-' (SO(p,q) x H) x T(n) was observed by E. Cartan [Ca 22b] who noted that the conformal transformations, leaving a given point x E M invariant,

consist of combinations of a dilation about x, a rotation around x, and an elation. Cartan (Ca 22b] also noted that for n > 4, the vanishing of the rotational curvature

Notes

161

implies the vanishing of the elational curvature which is equivalent to the fact that for n > 4, Cfkj = 0 = C;Jk = 0 (see p. 131). 4.2. The tensor of conformal curvature of a Riemannian manifold was first defined by H. Weyl [We 18] (p. 404) who proved that this tensor vanishes for n = 3 and that for n > 3, its vanishing is a necessary condition for V" to be conformally Euclidean. Weyl [We 21] also gave necessary and sufficient conditions for the flatness of affine, projective, and conformal spaces. Weyl [W 18] proved that C;ki = 0 in a conformal space C". Finzi [Fi 21, 221 found necessary and sufficient conditions for the Riemannian manifold V" to be a conformal space C". Different types of this condition were found for n = 4 by Cartan [Ca 22a], for n > 3 by Schouten [S 21, 27] (see also the books Eisenhart [Ei 26], §28, and Schouten and Struik [SS 38], S 19) and by others. The expression (4.2.11) of the tensor C;k, in terms of the tensor of Riemannian curvature is well-known (e.g. see equation (28.12) in Eisenhart [Ei 261). Formula (4.2.31) giving the transformation law for the connection forms under a conformal transformation is valid for both the proper Riemannian and pseudoRiemannian metrics. For the proper Riemannian metric, the transformation law for its curvature tensor can be found in many books (e.g., see Eisenhart [Ei 261, §28, or Norden (N 50a), §44). The fact that the isotropic geodesics are conformally invariant appears in this book for the first time. 4.3. As we mentioned above, E. Cartan introduced an n-dimensional space with a conformal connection in Cartan [Ca 23]. In the same paper he considered an mdimensional submanifold V' in that space, conformal connections induced on V' by the connection of the ambient space, and problems of conformal mapping and conformal deformation of such submanifolds. V. Hlavaty 1111 361 studied conformally invariant properties of n-dimensional Riemannian manifolds, constructed the Weyl connection and developed the theory of curves and the theory of nonholonomic (n-1)dimensional submanifolds in these spaces. S. Sasaki [SaS 39, 40] studied the theory of curves and hypersurfaces in an n-dimensional space with a conformal connection. A. P. Norden [N 49] investigated the differential geometry of a submanifold V' in the space C", showed that the intrinsic geometry of a normalized submanifold V' is the Weyl geometry with an angular metric induced by the metric of the ambient space, and found the Weingarten equations for V' and their integrability conditions. Most of the results presented in this section can be found in Akivis [A 85] and in §1 of Akivis and Konnov [AK 93]. The example which we considered in Subsection 4.3.2 is equivalent to the result in Vedernikov [Ved 631. The converse was proved in Verbitsky [Ver 52] for a hypersurface

provided that n > 5. Cartan [Ca 171 proved that in general the hypersurfaces in C", n > 5, do not admit a conformal deformation except in six cases which he indicated, the first of which coincides with our example. Note that both E. Cartan and L. L. Verbitsky considered the conformal structure induced on a hypersurface by the metric of the ambient Euclidean space. 4.4. The existence of asymptotic conformal structure on a hypersurface of a projective space was observed in Grif iths and Harris [GH 79] (p. 445). The results of this section are due to Akivis and Konnov [AK 93] (§2); see also Konnov [Kon 92a, b]. Note that Konnov [Kon 92b] investigated also hypersurfaces with flat asymptotic conformal structure.

162

4. CONFORMAL STRUCTURES ON A DIFFERENTIABLE MANIFOLD

The formula (4.4.21) for calculation of the tensor of conformal curvature C;,ki was found by T. Sasaki [SaT 88] and independently by Konnov (Kon 92a). From a global point of view, compact hypersurfaces with a flat metric conformal structure were described in do Carmo, Dajczer, and Mercuri [CDM 851.

Chapter 5

The Four-Dimensional Conformal Structures 5.1

Structure Equations of the CO(2, 2)-Structure

1. The study of four-dimensional conformal structures is of particular interest because of their close relation to the theory of gravitation (e.g., see Atiyah, Hitchin, and Singer [AHS 78], Gindikin [Gin 82] and Penrose [P 68]). The metric of Einstein's space has the signature (1, 3). This is the reason that the study of the pseudoconformal CO(1, 3)-structure on a four-dimensional manifold M is most interesting. As was recently shown in Barrett et al. [BGPPR 94], the CO(2, 2)-structures find important applications in the theory of superstrings. Thus a pure geometric investigation of this structure is also of interest. Along with the CO(1, 3)- and CO(2, 2)-structures, we will also study the proper conformal structure CO(4,0) = CO(4). We will consider both the general properties of these structures and the differences between them which are related to the fact that these structures are considered on a real manifold. We will start from the study of pseudoconformal structures CO(2, 2), which are called ultrahyperbolic (see Subsection 4.1.1). By means of a real transformation of coordinates the quadratic form g defining this structure on a real manifold M can be reduced to the form g = 9iiw'w3 =

2(w1W4

- w2w3).

In fact, in a pseudoorthogonal frame, this form can be written as 9=

(w1)2 - (W2)2 + (W3)2 - (W4)2.

163

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

164

Figure 5.1.1

It is easy to see that by means of the real transformation

f

w1 + w4

f

wl - w4

- w'

f

w2 + w3

4

'

w2 - w3

w2

3

'

we obtain the expression of g indicated above.

After this transformation the equation of the isotropic cone C. of the CO(2, 2)-structure can be written in the form g = 2(w'w4 - w2w3) = 0.

(5.1.1)

The frames lei) in the tangent space T=(M) in which the equation of the isotropic cone C. has the form (5.1.1) form the bundle of adapted frames of the structure CO(2, 2). All vectors of these frames are isotropic, that is, they belong to the isotropic cones C. The coordinate bivectors el A e2 i el A e3, e2 Ae4

and e3 A e4 also belong to these cones (see Figure 5.1.1). This is the reason that these frames are called isotropic. The 1-forms wi occurring in equation (5.1.1) form the conjugate bundle of isotropic co-frames. Equations (5.1.1) of the isotropic cone C. can be written in two different ways: 1

w2

W3 = W4 =

w'

w3

-a and ;2 = a;4-

Hence this cone carries two families of real two-dimensional plane generators, which determine two-dimensional isotropic directions on the manifold M. The first of these families is determined by the system of equations W2 + AW4 = 0,

(5.1.2)

W'+µw2=0, w3+µw4=0,

(5.1.3)

La '+aw3=0, and the second one is determined by

5.1

Structure Equations of the CO(2, 2) -Structure

165

Figure 5.1.2

where A and p are nonhomogeneous projective coordinates on two real projective lines RPQ and RPp. Following R. Penrose (see Penrose and Rindler [PR 861, Ch. 6, §2), we will call 2-planes, determined on M by equations (5.1.2) in TZ(M), a-planes, and those determined by equations (5.1.3) /3-planes.

Figure 5.1.2 represents the projectivization of the isotropic cone Cx of the CO(2, 2)-structure and of its a- and /3-generators. On the manifold M the a-planes and /3-planes form two fiber bundles EQ and E0 with common base M and the plane generators of the first and the second family of the cones C= as their fibers. These fibers are isomorphic to the projective lines RPQ and RPp, respectively. The fiber bundles E. = (M, RPQ) and Ep = (M, RPp) are called isotropic fiber bundles. From our considerations it follows that on the pseudoconformal CO(2, 2)-structure the isotropic fiber bundles are real. In the adapted frame only the following components of the tensor g,j will be nonzero: 914 = 941 = 1, 923 = 932 = -1. In view of this, equations (4.1.31) imply that the forms 9j' satisfy the conditions

I 94=92=93=94=0, 82 = 03, 84 = e3, 83 = 8j, 84 = 82,

(5.1.4)

01+84=0, 82+93=0. Now equations (4.1.32) take the form

dw' =

(9-91')nw'+w2A0'+w3n93,

dw2=(9-62)nw2+w' n8; +w4n9;, (5.1.5)

dw3 = (8 + 02) n w3 +W' A 0 + w4 n 92,

dw4=(9+9i)nw4+w2n81+w3A02. By virtue of equations (5.1.4), among the forms 8. only the forms 02, 92, 0i, 03, 01, and 02 are independent. If w' = 0, these forms together with the 1-form

166

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

0 are the invariant forms of a seven-parameter group G C GL(4) that preserves the cone C. determined by (5.1.1). The group G is the structure group of the conformal CO(2, 2)-structure. To clarify the structure of the group G, we will write equations (4.1.33) and (4.1.34) for nonvanishing forms 8 assuming that the point x E M is fixed (i.e., for w' = 0). As a result we find that d6 = 0,

{

d8; = 81 A 8z + 0i A 03, d82 = -0i A 82 + 0; A 031

{ {

d8?

= (8i - 82) A 8i,

(5.1.6)

(5.1.7)

(5.1.8)

d82 ' = 82 1 A (0' - 82), d8i = (01 + 02) A 8i,

d0 =0gA(01+82).

(5.1.9)

If we add and subtract equations (5.1.7), we find that d(B1 + 9) = 29; A 83

(5.1.10)

d(8l - e2) = 2e; AO' .

(5.1.11)

and

Equation (5.1.6) shows that the form 8 is a total differential and an invariant form of the one-parameter group H of homotheties that sends each plane generator of the cone Cx into itself. Equations (5.1.9) and (5.1.10) show that the forms 01, 8 and 8i + 82 are invariant forms of the three-parameter group GQ that is isomorphic to the group SL(2), which sends the family of a-planes of the cone C. into itself and keeps its ,0-planes fixed. Similarly it follows from equations (5.1.8) and (5.1.11) that the forms 021, 82 and 01 - 02 are invariant forms of the three-parameter group Ga; the latter group is also isomorphic to the group SL(2), which sends the family of #-planes of the cone C. into itself and keeps its a-planes fixed. The groups G. x H and Go x H are the structure groups of the fiber bundles EQ and E0. Therefore the structure group G of the conformal CO(2, 2)-structure is

decomposed into the direct product of the groups H, G. and Ga, that is, G S, H x SL(2) x SL(2). As in the case of the general conformal structure CO(p,q), if we prolong the group G, we arrive at the group G' G x T(4), where T(4) is the group of translations of the pseudo-Euclidean space R. 3. To write structure equations (4.1.34) for the CO(2, 2)-structure, we consider its tensor of conformal curvature C{,kl. In view of conditions (4.1.41), (4.1.44), and (4.1.48), this tensor has 21 essential nonvanishing components

Structure Equations of the CO(2, 2)-Structure

5.1

167

that satisfy 11 independent conditions arising from (4.1.47) and (4.1.49): C1234

- C1324 + C1423 = 0,

C1224

= C1334 = C1213 = C2434 = 0,

C1314

- C1323 = C1424 - C2324 = 0,

C1214

+ C1223 = C1434 + C2334 = 0,

(5.1.12)

C1414 = C2323 = C1234 + C1324

Hence the tensor Cijkl has 10 independent components in all. We denote them as follows: C1212 = a0, C1214 = al, C1234 = a2, C1434 = a3, C3434 = a4, (5.1-13)

C1313 = b0, C1314 = b1, C1324 = b2, C1424 = b3, C2424 = b4-

The remaining components of the tensor of conformal curvature C;3kl are ex-

pressible in terms of the above components (5.1.13) by means of relations (4.1.41), (4.1.44), (4.1.48), and (5.1.12). Now we can write equations (4.1.33) and (4.1.34) for the CO(2, 2)-structure in more detail. The former can be written as

d9=W1 A01+W2A82+W3A93+W4A94,

(5.1.14)

and by (5.1.4) and (5.1.13), the latter has the form

d9l_ 91 AWl -04 AW4+91 A0' +01 A93 -2[alwl A W2 + a2 (W1 A W4 - W2 A W3) + a3W3 A W4

(5.1.15)

+blwl A W3 + b2(Wl AW4 +W2 AW3) + b3W2 A w4],

M=

92 Aw2 -93 AW3 -01 A92 +91 A93 +2[-a1Wl A W2 - a2(Wl A W4 - W2 A w3) - a3W3 A W4

(5.1.16)

+blwl A w3 + b2(wl A W4 + w2 A W3) + b3W2 A w4],

d9; _ 91 Aw2+93 A W4 + (9] -92) A0; (5.1.17)

+2[bowl A W3 + bl (wl A W4 + w2 A W3) + b2W2 A w4], d92 =

02AW1+94 AW3+9z A(01 -92) (5.1.18)

-2[b2wl A W3 + b3(Wl A W4 + W2 A W3) + b4W2 A w4], dBl =

91 Aw3+02 Aw4 +(9, +02) A93 (5.1.19)

+2[aowl A W2 + a1(Wl A W4 - W2 A W3) + a2W3 AW4], and

d03 = 93 A W1 + 04 A W2 + 93 A (91 + 02) -2[a2W1 A W2 + a3(W1 A W4 - W2 A W3) + a4W3 A W4].

(5.1.20)

168

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

It follows from equations (5.1.15) and (5.1.16) that

d(01+B2)=281 AB3+0 Awl+02Aw2-03AW3-04Aw4 -4[aiw1 A w2 + a2(w' A w4 - w2 A w3) + a3w3 A w4]

(5.1.21)

and

d(Bf -02)=202A01+01Aw1-02Aw2+03Aw3-04Aw4 -4[biw1 A W3 + b2(w' A W4 + W2 A W3) + b3W2 A W4I.

(5.1.22)

Let us rewrite now in terms of the new notations (5.1.13) those 10 differential equations (4.1.56) that the independent components of the tensor of conformal curvature Cijkl satisfy: dao + 2ao(0 - 01 - 02) - 4a1B? = ao,w',

dal + a,(20 - 01 -02) - ao03 - 3a203 = a,iw',

dal + 2x26 - 2x183 - 2a3B2 =

(5.1,23)

day + as(20 + 01 + 8)--3a163 - 44191 = as,w', da4 + 2x4(0 + 011 + 02) - 4a383 = a4,W',

dbo + 2bo(B - 01 + 02) - 4b102 = bo,W', db1 + 51(20 - 01 + 192) - bo02 - 3b201 = b1,w',

db2 + 2628 - 251192 - 2b30 = b2,w',

(5.1.24)

db3 + b3(20 + 0' - 02) - 3b28; - b40 = b3,w', db4 + 2b4(0 + Oil + 02) - 46302 = b4;w'.

We can see from (5.1.23) and (5.1.24) that when w' = 0, the differentials of the components a,,, u = 0, 1, 2, 3, 4, of the tensor of conformal curvature are expressible only in terms of these components, and by the same token the same is true for the components bu. In view of this the tensor of conformal curvature of the structure CO(2, 2) is decomposed into two subtensors C. and CO with the components au and bu, respectively. Equations (5.1.17)-(5.1.22) allow us to establish a geometric meaning of the

subtensors Ca and Co of the tensor of conformal curvature of the CO(2, 2)-structure. If we compare these equations with equations (5.1.8)-

(5.1.11), we can easily see that the quantities au are the components of the curvature tensor of the fiber bundle Ea formed by the first family of plane generators of the cones C=, while the quantities bu are the components of the curvature tensor of the fiber bundle E0 formed by the second family of plane generators of the cones C.

The CO(1,3)-Structure and the CO(4,0)-Structure

5.2

169

The exterior quadratic forms

ej =

2 [aowl A w2 + al (Wl A w4 - w2 A w3) + a2W3 A w4],

e; + e2 = -4 [alwl A w2 + a2 (w' A w4 - W2 A w3) + a3W3 A W4], 03 = -2 [a2W1 /WW2 +a3(wl Aw4 -W2 Aw3) +a4W3 A W4], (5.1.25)

occurring on the right-hand sides of equations (5.1.19), (5.1.21), and (5.1.20) are the curvature forms of the fiber bundle E0, and the forms

01 =

2 [bowl A W3 + bl (Wl A W4 + W2 A W3) + b2W2 A W4],

e; - 92 = -4 [blwl A w2 + b2(wl A w4 + w2 A w3) + b3w2 A W4],

e2 = -2 [b2w' AW3+b3(wl AW4 -w2AW3)+b4W2AW4], (5.1.26)

occurring on the right-hand sides of equations (5.1.17), (5.1.22), and (5.1.18) are the curvature forms of the fiber bundle E. If both subtensors CQ and Cp vanish, then the tensor of conformal curvature of the CO(2, 2)-structure also vanishes, and the CO(2, 2) -structure itself becomes conformally flat. In this case the manifold M with such a structure is diffeomorphic to the pseudoconformal space C2, that is, to the hyperquadric of signature (2,2) in a five-dimensional projective space. If one of the subtensors CQ and CO vanishes, then the CO(2, 2)-structure is called conformally semifiat. Examples of such structures will be considered in Section 5.5.

5.2

The CO(1, 3)-Structure and the CO(4, 0)-Structure

1. We now consider the pseudoconformal CO(1,3)-structure. doorthonormal frame its fundamental form g becomes

In a pseu-

g = -(wl)2 - (W2)2 - (w3)2 + (w4)2. Transformations of the tangent subspace T. (M) preserving the form g make up

the pseudoorthogonal group SO(1, 3) which is called the Lorentz group. The isotropic cone C. C T2(M) which is determined by the equation g = 0 remains invariant under transformations of the group G = SO(1, 3) x H where H is the group of homotheties. By means of the real transformation w1

±

4

-W± W 4 1

-! Wl,

-1 W4, W2 -9 W2, W3 -+ W3

the form g can be reduced to the form 9 = 2w1w4 - (w2)2 - (W3)2.

170

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Figure 5.2.1

It is easy to see now that the cone g = 0 carries real one-dimensional generators (the straight lines w' = w2 = W3 = 0 and c = W2 = w3 = 0 are examples of such generators) but does not carry two-dimensional generators. Next, by means of the complex transformation

W -9W , 1

1

W2 + iW3

VL

2

-1W ,

W2 - i(,)3

V`

-)W3 , W4 -)W 4,

we again reduce the form g to the form (5.1.1): g = 2(W'W4 - W2W3),

where the forms wi and w4 are real, and the forms w2 and w3 are complex conjugate forms

W' =W'

---4 =W4 u =w2.

(5.2.1)

It follows that the isotropic cone g = 0 carries two-dimensional complex conjugate plane generators. The complex frame transformations that we used above take place in the complexified tangent space CT=(M) = (see Figure 5.2.1). Moreover, in CTX(M), we will consider only such transformations that preserve its real subspace T=(M), and also we considered the symmetry correspondence (the complex conjugacy) with respect to T=(M).

A vectorial frame in the space CT,, (M), in which the form g on the CO(1, 3)-structure reduces to form (5.1.1), satisfies the conditions i31 =e1, e4 = e4, e2=e3.

(5.2.2)

Such a frame is called a Newman-Penrose tetrad (see Newman and Penrose [NP 62] and Chandrasekhar [Cha 83], Ch. 1, §8). In such a frame the vectors el and e4 are real, and the vectors e2 and e3 are complex conjugate.

The CO(1, 3) -Structure and the CO(4, 0) -Structure

5.2

171

As in the case CO(2, 2)-structure, equations (5.1.2) and (5.1.3) determine two families of two-dimensional plane generators on the cone C, but now these plane generators are complex conjugate. On the C0(1, 3)-structure, the parameters A and p in equations (5.1.2) and (5.1.3) are complex coordinates on the projective lines CPa and CPp. These equations determine two families of complex conjugate plane generators on the isotropic cone C= lying in the space CT=(M). These plane generators are, respectively, the a-planes and the /3-planes of the CO(1,3)-structure. If in equations (5.1.2) we replace all quantities by their conjugates, we obtain equations (5.1.3), where p =1 Thus there is a one-to-one correspondence between a-planes and /3-planes of these two families of plane generators of the cone CZ, and this correspondence is determined by the condition p = A. Since to each point x E M of a real manifold M carrying a CO(1, 3)-structure there correspond two families of 2-planes, the family of a-planes and the family of 0-planes, determined by complex parameters A and p, two bundles,

E. = (M, CPa) and Ep = (M, CPp), arise on M, and these two bundles have the manifold M as their common base and the families of complex plane generators of the cone C. as their fibers. These bundles are called the isotropic bundles of the CO(1, 3)-structure. Since p = A, the isotropic bundles EQ = (M, CPQ) and E0 = (M, CPp) are complex conjugates: -`p = Ea. On the cone C= of the CO(1, 3)-structure, there is a bijective correspondence between its a- and /3-generators, and this correspondence is determined by the condition p = A. Moreover two complex conjugate generators of the cone CZ intersect one another along its real rectilinear generator. The equation of this generator can be found from equations (5.1.2) and (5.1.3) provided that it = A. Solving these equations, we find that W1 = AAw4, W2 = -AW4, W3 = -XW4.

Hence the directional vector of the rectilinear generator can be written in the form

= AAel - Ae2 - Ae3 + e4.

(5.2.3)

Since the basis vectors of the complexified space CT= satisfy relations (5.2.2),

the vector l: is real. It depends on one complex parameter or two real parameters. Equation (5.2.3) can be considered as the equation of the director two-dimensional surface of the three-dimensional cone CZ in the real space T=(M).

Since the isotropic fiber bundles Ea and Ep of the Lorentzian structure CO(1, 3) are complex conjugates, its structural group G admits two isomorphic representations:

G'-° SL(2, C) x H SL(2, C) x H on the fiber bundles Ea and Ea. Moreover the groups SL(2, C) and SL(2, C) act concordantly on Ea and E0. The group G depends on seven real parameters.

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

172

2. Consider now the proper conformal structure CO(4, 0) = CO(4). For this structure the fundamental form g can be reduced to the form g = (WI)2 + (W2)2 + (w3)2 + (w4)2.

This form is invariant under transformations of the group 0(4) of coordinates in the tangent space T2 (M), and the cone C. determined by the equation g = 0 remains invariant under transformations of the group G °_I SO(4) x H. By means of the complex transformation WI + 1W4

y

-1 W,

r

WI - .

4

W2 + tw3

4 -1 W,

2-

-) W

2

W2 - 2W3 2

3

we again reduce the form g to the form (5.,1.11). To be able to do this, we must of course assume again that the tangent space T1(M) is complexified and is

the product: CT,(M) = T=(M) ® C. Now all basis forms are complex, and they are connected by the conditions

w = W1, w = -W2.

(5.2.4)

For the CO(4, 0)-structure the notions of a- and p-planes and the isotropic bundles En and E0 can be defined in the same way as for the CO(2, 2)-structure

and CO(1, 3)-structure. Furthermore in equations (5.1.2) and (5.1.3) the parameters A and p are complex, and thus the fibers of the bundles E and Ep are isomorphic to the complex projective lines CPa and CPp exactly as for the CO(1, 3)-structure. From equations (5.2.4) it follows that for the proper conformal structure CO(4), each of the systems of equations (5.1.2) and (5.1.3) remains invariant under passage to the complex conjugate values if the parameters A and p undergo the following transformation:

A-t-

,

p-4---.

In view of this, the isotropic fiber bundles E and Ep are self-conjugate: Ea = Ea, Ep = E0. This implies that for the CO(4)-structure, the structure group G can be represented in the form: G = Ga x Go x H where Ga and Go are the groups acting on the fiber bundles E. and Ep, respectively. But in view of Subsection 4.1.1, Ga x Go = SO(4), and the group SO(4) can be represented as the direct product SO(4) = SU(2) x SU(2) where SU(2) is the two-dimensional special unitary group. As a result we find that for the C0(4)-structure, G = SU(2) x SU(2) x H, and two groups SU(2) act independently on the families of a- and 0-planes of the isotropic cone C1.

3. We will return now to the CO(1,3)-structure. For this structure all equations (5.1.4), (5.1.5), and (5.1.12)-(5.1.20) are still valid, but not all quantities occurring in these equations are real. In particular, as we noted earlier, the basis forms w` satisfy the equations (5.2.1).

The CO(1, 3) -Structure and the CO(4, 0) -Structure

5.2

173

The forms 99 occurring in equations (5.1.5) are invariant forms of a complex

representation of the real six-parameter Lorentz group SO(1, 3) that leaves invariant the cone C, determined by the equation g = 0 in the tangent space T=(M). The form 0 is real, 8 = 9, since this form is an invariant form of the one-parameter group H of real homotheties which also leaves invariant the cone C. The structure group G' of the pseudoconformal structure CO(1, 3) is isomorphic to the semidirect product: G' ?° (H x SO(1, 3)) x T(4), where T(4) is the four-dimensional group of translations with the invariant forms 9i. The group G' is obtained as the first prolongation of the group C - H x SO(1, 3). We will prove now the following theorem:

Theorem 5.2.1 On the CO(1, 3)-structure, the complex forms 99 occurring in equations (5.1.5) satisfy the following relations:

91=0i, 92=-02, 9i=02, 93=01;

(5.2.5)

the forms 9i satisfy the relations 91 = 01, 92 = 93, 03 = 02, 04 = 04;

(5.2.6)

and the components a and b,,, u = 0, 1, 2, 3, 4, of the curvature tensors Ca and Cp of the isotropic fiber bundles Ea and E0 satisfy the relations b = a,,.

(5.2.7)

Proof. To prove (5.2.5), we write equation (5.1.5) for conjugate quantities and subtract the result from corresponding equations (5.1.5). Taking into account equations (5.2.1) and 0 = 9, we arrive at the following system of equations: W1 A (9j - 91) + W2 A (93 -

)+

A ( 0 3 - 02) = 0,

W1 n (91 - 9i) + W2 A (92 + 92) + W4 A (93 - 02) = 0,

/x(91 -91)+W4A(92 -93)=0,

(5.2.8)

-91)+W3A(0 -91)=0. Each expression with the forms 0 -B in parentheses occurs in equations (5.2.8) twice. Applying Cartan's lemma to equations (5.2.8) and comparing the same expressions in the left-hand sides, we obtain relations (5.2.5). To prove (5.2.6) and (5.2.7), we write equation (5.1.14) for conjugate quantities. Taking into account that 9 = 9, we find from this equation and equations (5.1.14) that W1 A (01 - 91) + W2 A (82 - 93) + W3 A (93 - 92) + W4 A (04 - 94) = 0.

174

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Applying Cartan's lemma to the last equation, we obtain e1 - 01 = P11W1 +P12W2 +P13W3 +P14W4, 02 - 03 = P21 W' + P22 W2 + P23 W3 + P24 W4, (5.2.9)

03 - 02 = P31W1 + P32W2 +P33W3 +P34W4,

04 -04 =P41W1 +P42W2 + P43W3 +

P44W4,

where pij = pji. Finally, if we write equations (5.1.15)-(5.1.20) for conjugate quantities and take into account relations (5.2.5) proved earlier, we find that (B1 - 61) A W' - (04 - e4) A W2

-2[(al - b1)w' A W2 + (a2 - b2)(w' A W4 - W2 A W3) + (a3 - ;3 )W3 A w4

+(b1 -Cl1)W' AW3+(b2-a2)(W' AW4+W2AW3)+(b3-a3)W2AW4]=0, (5.2.10)

(02-93)AW2-(03-62)AW3 -2[(al - b1)w' A W2 + (a2 - L2)(w'

A W4 - W2 A W3) + (a3

- b3 )W3 A W4

-(b1 - al )W1 A W3 - (b2 - a2)(W' A W4 + W2 A W3) - (b3 - U3 )W2 A W4] = 0, (5.2.11)

(01 -Bl)AW3+(02-93)AW4 +2[(ao - bo)Wl A W2 + (a1

A W4 - W2 A W3) + (a2 - b2)W3 A W41 = 0,

(5.2.12) and

(03-42)Awl+(04-e4)AW2 -2[(a2 - b2)w' A W2 + (a3 - b3)(W' A w4 -W2 A w3) + (a4 - b4)W3 A W4) = 0. (5.2.13)

If we substitute for the differences of the forms Bi and 4i into equations (5.2.10)-(5.2.13) their expressions (5.2.9) and equate to zero the coefficients in independent exterior products of the basis forms W', we find that pig = 0, and this immediately implies relations (5.2.6) and (5.2.7). Let us state some consequences of relations (5.2.5)-(5.2.7) that were proved in Theorem 5.2.1.

Equations (5.2.5) show that the complex forms 6, occurring in them are expressed in terms of precisely six linearly independent forms. This number is equal to the number of parameters on which the Lorentz group depends. These six forms are real invariant forms of the group SO(1,3). Equations (5.2.6) show that among the forms 01 there are two real forms and two complex conjugate forms, and all four forms Bi are expressed in terms of four linearly independent real forms.

Finally, equations (5.2.7) show that the curvature tensors Ca and Co of the isotropic fiber bundles E. and Eo of the C0(1,3)-structure are complex

5.2

The CO(1,3)-Structure and the CO(4,0)-Structure

175

conjugates: Co = C. This matches the fact proved in Subsection 5.2.1 that the isotropic fiber bundles Ea and E0 of the CO(1, 3)-structure are complex conjugates themselves: E0 = Ea. Similarly the curvature forms ea and 80 of this structure are also complex conjugates: e0 = e0. It follows that if one of the tensors C. or C0 of the CO(1, 3) -structure vanishes, the other one vanishes too. This implies that the CO(1,3)-structure cannot be conformally semiflat without being conformally flat.

4. Now consider again the CO(4)-structure. For this structure a theorem similar to Theorem 5.2.1 is valid.

Theorem 5.2.2 On the CO(4) -structure, the complex forms 00 occurring in equations (5.1.5) satisfy the following relations: -1 -1 9 +91=0, 9z+02=0, 012 +92=0, 03 +93=0,

(5.2.14)

the forms 0, satisfy the relations 94 = 01, 93 = -02,

(5.2.15)

and the components au and bu, u = 0, 1,2,3,4, of the tensors Ca and CO, into which the tensor of conformal curvature splits, satisfy the relations ao = a4, a1 = -a3, a2 = a2,

bo=b4, bi=-b3, b2=b2

(5.2.16)

The proof is similar to that of Theorem 5.2.1. Note that the structure equations (5.1.5) of the CO(4)-structure, relations (5.2.4) among the forms w'

for this structure, and the fact that the form 0 is real must be used in this proof. Relations (5.2.14) show that the 1-forms 01 and 02 are pure imaginary. These forms along with the complex forms 02 and 91 determine a complex isotropic representation of the six-parameter group 0(4). Relations (5.2.15) show that there are two independent forms among the forms O. For example,

the forms 91 and 02 can be taken as independent forms. They determine a complex representation of the group T(4) of translations in the local space SS(M) of the conformal structure CO(4). From relations (5.2.16) it follows that the curvature tensors C. and CO of the isotropic fiber bundles E. and E0 of the conformal structure CO(4) are independent of one another but satisfy the conditions Ca = Ca and C0 = CO. Of course this corresponds to the self-conjugacy of the isotropic fiber bundles Ea and E0 of the proper conformal structure CO(4): Ea = Ea and E0 = E0 noted in Subsection 5.2.2. The following theorem combines results of Section 5.1 and Theorems 5.2.1 and 5.2.2:

Theorem 5.2.3 The isotropic fiber bundles Ea and EO are naturally defined on four-dimensional conformal structures CO(2, 2), CO(1, 3), and CO(4). The

176

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

base of these fiber bundles is the four-dimensional manifold M. On a CO(2, 2)-structure, these fiber bundles as well as their curvature tensors Ca and Cp are real. On a CO(1, 3)-structure, these fiber bundles as well as their curvature tensors Ca and Cp are complex conjugate: Ep = Ea and Cp = Ca. On a CO(4) -structure, these fiber bundles as well as their curvature tensors Ca

and Cp are self-conjugate: P. = Ea, Ep = Ep and Ca = Ca, Cp = Co.

5.3

The Hodge Operator

1. In the theory of four-dimensional conformal structures, the Hodge operator plays an important role. It can be defined as follows: Consider a nondegenerate relatively invariant quadratic form 9 = 9iiwiwi,

i, j = 1,2,3,4,

(5.3.1)

that defines a conformal structure on a four-dimensional differentiable manifold M. The differential equations which the coefficients of the form g satisfy can be reduced to the form (4.1.15):

dgii - 9ikwj - 9kiw, = 2g. 0.

(5.3.2)

It follows that the inverse tensor gii of the tensor gii satisfies the equations dgii + gikwk + gkiwk = -2giiO.

(5.3.3)

Since the quadratic form g is nondegenerate, its discriminant det (gii) is different from zero and preserves its sign on the manifold M in question, which

of course we assume to be oriented. Moreover, as the canonical expressions of the quadratic form g given in Sections 5.1 and 5.2 show, for the conformal structures CO(4) and CO(2,2), this discriminant is positive, and for the conformal structure CO(1, 3), it is negative. Let us calculate the differential of the discriminant g` of the quadratic form g. Applying the well-known formula for differentiation of determinants, we find

that dg = 9'9''d9ii.

Substituting for dgii in the last equation their values (5.3.2), we obtain dg = 2g'(w; + 40).

If a point x of the manifold M is fixed, then this equation takes the form

bg` = 2g(r + 4a),

(5.3.4)

where, as in Section 4.1, 5 is the operator of differentiation with respect to fiber parameters of the frame bundle R(M) associated with the manifold M;

The Hodge Operator

5.3

177

7r, = we(d) and 7r = 8(5) are 1-forms defining admissible transformations of frames in the tangent space T=(M). Consider further the basis forms of the manifold M. If a point x E M is fixed, then the wi satisfy equations (4.1.6):

5wi = -w'7r

(5.3.5)

.1

In view of this, for the exterior product w' A w2 A w3 A w4, we have 5(w' A w2 A w3 A w4) = -7ri (W' A W2 A W3 A W4 ).

(5.3.6)

For a four-dimensional Riemannian manifold M, the exterior form

dV =Vg w'AW2Aw3AW4 is the volume element (e.g., see Eisenhart [Ei 26], §52, or Wells [Wel 80], Ch.

4, §1). But for a manifold M with a conformal structure, this form is only relatively invariant, since by (5.3.4) and by (5.3.6) we have

6(/W'AW2AW3Aw4)=47r(/ w' A W2 A W3 A W4). Hence, on a manifold M with a conformal structure, it is impossible to define a volume element in an invariant manner. Let us find a simpler expression for the form dV. To this end, we define the discriminant tensor eijkl = V_19=1 Eijkl,

(5.3.7)

where 1

eijkl =

-1 0

if i, j, k, 1 is an even permutation of the indices 1, 2, 3, 4; if i, j, k, I is an odd permutation of the indices 1, 2, 3, 4; if at least one pair of these indices coincides.

By means of this tensor the form dV can be written as dV = 4 eijkl wi A w3 A wk A

w.

Next we calculate the expression V6eijkl := 5eijkl - emjkl1rm - eimkt7r - eijml'nk - eijkmlrI m I

Since 5eijkl = 0, then by (5.3.4) and (5.3.7), it follows that V6eijkl = 47r eijkl;

(5.3.8)

that is, the discriminant tensor is a relative tensor of weight 4 (cf. formula (2.1.20)).

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

178

The tensor

hijki

- esjpq gpkggt is called the Hodge tensor. Since equation (5.3.3) implies that

(5.3.9)

V6g" = -2g"a, by differentiating the tensor hijkt and taking into account equation (5.3.8), we find that V6hijki = 0, so the Hodge tensor is conforrnally invariant. 2. We will now point out some properties of the Hodge tensor. First of all, it is easy to see that this tensor is skew-symmetric not only with respect to the lower indices i and j but also with respect to the upper indices k and 1. As in preceding chapters, we denote by ei, i = 1,2,3,4, a basis of the fourdimensional tangent space T=(M) of the manifold M, and by w' its co-basis. Let 14' be a bivector space whose elements are bivectors i P=2P"e;Aej,

Pij

i

=-Pr,

and let A2 be the space of exterior quadratic forms

a = 2aijw'Awj,

aij= -aji,

over the manifold Al. For a fixed point x E M, each of these two spaces is six-dimensional.

Since the Hodge tensor is skew-symmetric with respect to the lower and upper indices, it defines the linear operators h: IV -a TV and h: A2 -> A2

whose coordinate representation is

qij = hktijpkt and

P13 = hifkt akl,

(5.3.10)

where q = Zq'jei A ej = h(p) and p = z p;jw' Awj = h(a). These operators are called the Hodge operators or *-operators.

Let us find the square of the Hodge operator. To this end, we take an orthogonal basis in the tangent space T=(M) in which gij = 0 for i $ j. In this basis, formula (5.3.9) becomes hijkt (5.3.11) = eijkt gkkgu

Further we apply the operator h2 to the form or. By means of the second formula of (5.3.10), we have (h2(a)),j - 2hijP°hpgktaki-

5.3

179

The Hodge Operator

The reason that the factor i appeared in this formula is that when we take the sum with respect to the skew-symmetric pair of indices p and q, each term appears twice. If in the last formula we substitute for hit their values (5.3.11), we obtain 1

(h2(a))ij =

2

eijpq 9PP9ggepqkl 9kk911 akl P'q

The quantities eijpq and epgkl in this equation are different from zero only if all their lower indices are distinct. But this is possible only if k = i, l = j or k = j, l = i. Thus the right-hand side of the last equation reduces to the form (h2(o))ij =

2

[tee jpg

9PPggg9iigjjyij.

P.g

Y

Moreover the indices p and q can take only two distinct values that are different from i and j. If we denote these two values again by k and 1, we find that

(h2(a))ij = e jkl giIgii9kk9ll yij,

where on the right-hand side there is no summation with respect to any of the indices i, j, k, and 1. But e? kl = 9I and gIIgL gkkgll = g-l Hence

(h2(a))ij = sign g aij. This means that

h2(o) = sign g Id,

(5.3.12)

where Id is the identity operator: Id: A2 -+ A2. We now enumerate the properties of the Hodge operator that were established above as well as some additional properties that follow from the proven properties. 1. The Hodge operator defines the linear mappings

h: W -4 W and h: A2 -a A2. 2. The Hodge operator is symmetric, since the discriminant tensor generating the components of the Hodge operator by raising the indices is symmetric with respect to the bivector indices: eijkl = eklij-

3. The Hodge operator is conformally invariant. 4. It follows from equation (5.3.12) that for the conformal structures CO(2, 2) and CO(4), the Hodge operator satisfies the condition h2(y) = Id,

(5.3.13)

180

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

that is, it is involutive, and for the conformal structure C0(1, 3) the Hodge operator satisfies the condition

h2(o) = -Id,

(5.3.14)

that is, it is anti-involutive. 5. It follows from equation (5.3.13) that for the conformal structures CO(2, 2) and CO(4), the Hodge operator has two real triple eigenvalues Al = 1 and A2 = -1 to which there correspond two real three-dimensional eigensub-

spaces in the bivector space W and the space of exterior 2-forms A2. Moreover the following splittings take place:

W_ and A2 = A+ ®A? ,

W = W..

(5.3.15)

where W± and A2 are the eigensubspaces corresponding to these eigenvalues and the symbol ® denotes the direct sum. 6. It follows from equation (5.3.14) that for the CO(1, 3)-structure the Hodge

operator has two complex conjugate eigenvalues Al = i and A2 = -i to which there correspond two complex conjugate three-dimensional eigensubspaces in the complexified spaces CW and CA2. Formula (5.3.15) is still valid, but the subspaces W+, W_ and A+, A? are complex conjugate. For four-dimensional conformal structures of all three possible types, the eigensubspaces of the Hodge operator corresponding to the eigenvalue Al are called self-dual, and the eigensubspaces of the Hodge operator corresponding to the eigenvalue A2 are called anti-self-dual (see Atiyah, Hitchin, and Singer [AHS 78]).

3. We will now compute the components of the Hodge tensor for the CO(2, 2)-structure, assuming that the tangent space T1(M) is referred to an isotropic frame in which its fundamental form g has form (5.1.1). In this frame the matrix of coefficients of the form g takes the form 0

(90 =(9ij)=

0 0

0 0

-1

0

-1

0

0 0

1

0

0

0

1

The determinant of this matrix g = 1. The components of the Hodge tensor are computed by means of formula (5.3.9) where the discriminant tensor etfkl is determined by formulas (5.3.7). Thus, after some computations, we find that

(h' k, )

1

0

0 0

0

0 0

0

-1

0 0 0

0 0 0

0

0

-1

-1

0 0

0

0

0

0

0

-1

0

0

0

0 0 0

0 0 1

5.3

The Hodge Operator

181

where the bivector indices (i, j) are ordered as follows: (1, 2), (2, 3), (3,1), (2, 4), (1, 4), (3,4)

(cf. Subsection 1.4.1). As can be expected, the matrix of the operator h is symmetric. Let us find the eigenvalues of the Hodge operator in the frame considered. The characteristic polynomial of this operator can be written in the form

1-A 0

det (h-A Id) =

0

0

0

0

0

-A

0

0

-1

0

0

0 0

0

-A

0

0

0 -1-A

0

0

0

-1-A

0

-1 0

0

0

0

0

0

= (1-A)3(1+a)3.

0

0 1-A

Thus the Hodge operator has two real triple eigenvalues Al = 1 and \2 = -1. Next we will find eigendirections corresponding to these eigenvalues in the space A2. If a = zo;jw' Awj E A2, then p = h(a) =

w' Awi.

Thus in our frame we have P12 = 0`12, P24 = -0'14,

P34 = 0'34,

P31 = -0'31,

P14 = -a23, P24 = -a24-

As usual, we write equations for finding eigendirections in the form (h13kl

_ A Ifkl)akl = 0.

By means of the previous relations the last equations imply that (1 - \)o12 = 0,

0'14 + Aa23 = 0,

(1 + A)0`31 = 0,

(1 - A)a34 = 0,

0`23 + A0`14 = 0,

(1 + A)a24 = 0.

Hence to the eigenvalue Al = 1 there corresponds the eigensubspace A2, defined by the equations a31 = 0, 0`14 + 0`23 = 0, 0'24 = 0,

(5.3.16)

and to the eigenvalue A2 = -1 there corresponds the eigensubspace A. defined by the equations 0`12 = 0,

0`14 - a23 = 0,

0`34 = 0-

(5-3.17)

Thus the forms WI A w2, W1 A w4 - W2 A W3, W3 A w4

(5.3.18)

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

182

form a basis of the space A', and the forms W' AW3, WI AW4 +W2 A W3, W2 A W4

(5.3.19)

form a basis of the space A2 .

Consider further the conformal structure CO(4). As we have shown in Subsection 5.2.2, its fundamental form g also can be reduced to form (5.1.1), but its basis forms are complex and satisfy conditions (5.2.2): W1

= w1, w3 = -w2.

(5.3.20)

Thus the basis forms of the eigensubspaces A2 and A2 of the Hodge tensor are also complex forms satisfying the conditions W 1 A w2 = W3 A w4, W1 AW4 - W2 A W3 = -(W 1 A W4 - W2 A W3)

(5.3.21)

and W1 A W3 = w2 A w, W1 A W4 + w2 A W3 = -(W1 A W4 + W2 A W3).

(5.3.22)

This means that the self-dual and anti-self-dual eigensubspaces A2 and A2 of the Hodge tensor of the C0(4)-structure satisfy the conditions 1T+ = A. and A? = A2 ; in other words, they are self-conjugate. For the conformal structure CO(1,3), the fundamental form g can he reduced to form (5.1.1) in a complex frame satisfying conditions (5.2.1):

wl = w1, w = W4, w = w2.

(5.3.23)

Thus the complex basis forms of the eigensubspaces A2 and A2 of the Hodge tensor satisfy the conditions W1 AW2 =W1 AW3, Wu3AW4 =W2Aw4,

w1nw4-w2AW3=w'AW4+w2Aw3,

(5.3.24)

which means that they are complex conjugate. Thus the eigensubspaces A{. and A2 of the Hodge tensor of the C0(1,3)-structure are complex conjugates themselves.

Of course the last two results agree with the properties 5 and 6 of Subsection 5.3.2. 4. Now we will return to the study of the curvature forms of four-dimensional conformal structures. As we saw in Section 5.1, for the CO(2, 2)-structure, these forms decompose into two subsystems (5.1.25) and (5.1.26). The first of these subsystems is composed from the curvature forms of the isotropic fiber bundle E., and the second of these subsystems is composed from the curvature forms of the isotropic fiber bundle E0. Comparing these forms with the basis forms (5.3.18) and (5.3.19) of the eigensubspaces A A. and A2 of the Hodge operator, we arrive at the following result:

Completely Isotropic Submanifolds

5..4

183

Theorem 5.3.1 The curvature forms of the isotropic fiber bundle Etr of the CO(2,2)-structure belong to the eigensubspace Ai. of the Hodge operator (i.e., they are self-dual), and the curvature forms of the isotropic fiber bundle Eo belong to the eigensubspace A? (i.e., they are anti-self-dual).

It is easy to see that the curvature forms of the isotropic fiber bundles Ea and Eo of the conformal structures CO(4) and CO (1, 3) enjoy similar properties

since in the appropriate complex frames they have the same forms (5.1.25) and (5.1.26). Moreover, to the relations T+ = A+ and A? = A2 between eigensubspaces of the Hodge operator of the CO(4)-structure there correspond the relations Ea = E0, Eo = E0 and C. = CO, Co = Co between its isotropic fiber bundles and their curvature tensors (see Theorem 5.2.3). Similarly, to the relation X. = AZ between eigensubspaces of the Hodge operator of the

CO(1,3)-structure there correspond the relations EQ = E0 and Ca = Co between its isotropic fiber bundles and their curvature tensors. Finally, we consider semifiat four-dimensional conformal structures. If the curvature tensor Co of the fiber bundle E0 vanishes (i.e., the conformal structure is Q-semifiat), then its curvature form belongs to the self-dual eigensubspace A+ of the Hodge operator. This is the reason that such structures are called self-dual. On the other hand, if the curvature tensor C', of the fiber bundle E,, vanishes (i.e., the conformal structure is a-semifiat), then its curvature form belongs to the anti-self-dual eigensubspace Az of the Hodge operator. This is the reason that such structures are called anti-self-dual. Note that the CO(1,3)-structure cannot be self-dual or anti-self-dual (i.e. semiflat) without being conformatly flat. This result immediately follows from the fact that for the CO(1,3)-structure, Co = Ca.

5.4

Completely Isotropic Submanifolds of Four-Dimensional Conformal Structures

1. We will now give a geometric interpretation for the subtensors C0 and Co of the curvature tensor of the CO(2, 2)-structure. Let 1' = {'e,, and let g = p'ei be two vectors in the tangent space T1(M), and l: ng be the bivector defined by these two vectors. Consider two bilinear forms associated with this bivector:

C(t; Ar1) = C>>k1= C1,klei,i and

9(i; Arl) = (9ik9t1 -

111 = (g,k9,i - 9ugik)t('71J1&n11

Their ratio K(f A n) =

C(f A n) 9( A rl)

184

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

is the conformal curvature of the bivector which is called the conformal sectional curvature.

Since a-planes and /3-planes are isotropic bivectors, for them we have g(t; A q) = 0, and thus the expression K(f A q) does not make sense for them. Therefore we will consider for them only the numerator C(r; A q) of this expression and will call it the relative conformal curvature of two-dimensional isotropic direction. Let us denote the bivector t; A q by p: p = r; A q, and compute C(p) taking into account equations (5.1.12) and (5.1.13): C(p) = ao(p12)2 + 2a,p12(p14 - p23) + a2[2p12p34 + (p14 - p23)21 p231)

+ a4(p34)2 +2a3p34(p'4 +bo(pl3)2 +2b,p13(p14 +p23) +b2[-2p13p42 + (p14 +p23)2) 1

1t'

+p23)llll+

1

b4(p42)2.

-2b3p42(p14

(5.4.1)

By (5.1.2), the a-plane a(A) is determined by the vectors

fa = e3 - Ae, and

77A = e4 - Ae2.

Hence the coordinates of the bivector pa = fa Aq,, are the minors of the matrix A

0

1

( 0 -A 0

0 1

'

they are

p12

= A2, p13 = 0, p14 = -A, p23 = A, p94 = 1, p42 = 0.

Substituting these expressions into equations (5.4.1), we find that C(pa) = aoA4 - 4a,A3 + 6a2A2 - 4a3A + a4 := C0(A).

(5.4.2)

In exactly the same way, by virtue of (5.1.3), the Q-plane 3(p) is determined by the vectors fm = e2 - pel and r]l, = e4 - pea

This implies that the coordinates of the bivector p =1; A 77,A are

P12 = 0, p13 = p2, p14 = -p, p23 = -p p34 = 0, p42 = -1, and the following formula holds:

4C(p,) = bop' - 4b,µ3 + 6b2µ2 - 4b3µ + b4 := C0(µ)

(5.4.3)

Thus the components of the subtensors Ca and Cp of the tensor of conformal curvature of a CO(2, 2) -structure are the coefficients of the polynomials CQ(A)

Completely Isotropic Submanifolds

5.4

185

and C0(µ), by means of which we can evaluate the relative curvature of the a-planes a(A) and (3-planes /3(A), respectively.

Those isotropic 2-planes of the structure CO(2,2) for which C(A) = 0 or Cp(p) = 0 are called the principal a-planes or principal /3-planes of the isotropic bundles EQ and E0, respectively. Since polynomials (5.4.2) and (5.4.3) are of

fourth degree, it follows that, in general, the isotropic cone C. carries four principal a-planes and the same quantity of principal ,Q-planes if we count each of these planes as many times as its multiplicity. From the definition of self-dual and anti-self-dual CO(2, 2)-structures (see Subsection 5.3.4) we obtain the following result:

Theorem 5.4.1 A CO(2, 2) -structure is self-dual if and only if all its /3-planes

are principal planes. Such a structure is anti-self-dual if and only if all its a-planes are principal a-planes. A CO(2, 2) -structure is conformally flat if and only if all its a-planes and /3-planes are principal planes. 2. A two-dimensional submanifold V of the manifold M endowed with a pseudoconformal CO(2, 2)-structure is called completely isotropic if its tangent subspaces T=(V) belong to one of the two isotropic fiber bundles EQ or E0 of isotropic 2-planes of the CO(2, 2)-structure. In accordance with this definition, we may have two types of completely isotropic submanifolds on M. We denote them by V. and Vp, respectively.

The submanifolds VQ are determined on M by the system of equations (5.1.2). On these submanifolds the 1-forms w3 and w4 are independent. Taking the exterior derivatives of (5.1.2), we obtain the system of equations 9,\ A w3 = 0, 9,\ A w4 = 0,

(5.4.4)

9a := dA + X(9 + 92) - B3 + A203

(5.4.5)

where

From (5.4.4) it follows that on the submanifolds VQ,

9a = 0.

(5.4.6)

By taking the exterior derivative of this equation by means of (5.1.19)-(5.1.21), excluding dA, and setting the coefficient of the product w3 A w4 equal to zero,

we arrive at an equation of fourth degree in A which can be reduced to the equation aoA4 - 4ai A3 + 6a2A2 - 4a3 A + a4 = 0,

(5.4.7)

whose left-hand side coincides with the polynomial CQ(A) defined by formula (5.4.2). In exactly the same way, taking the exterior derivatives of equations (5.1.3) that determine the submanifolds V0, we arrive at the Pfaffian equation: 9N := du + µ(91 l - 92) - 92 1 + p292 = 0.

(5.4.8)

186

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

From this equation, just as above, we obtain the following algebraic equation: boµ4 - 4blp3 + 6b2p2 - 4b3µ + b4 = 0,

(5.4.9)

whose left-hand side coincides with the polynomial Co(p) defined by the formula (5.4.3). The Pfaffian equations (5.1.2) and (5.4.6) determine a distribution A(a) of two-dimensional elements on the five-dimensional fiber bundle EQ. If this distribution is involutive, then It has a three-parameter family of integral surfaces, which are projected onto the manifold M as completely isotropic twodimensional submanifolds Va. Consequently the involutory property of the distribution 0(a) is equivalent to the existence on the manifold Al of a threeparameter family of completely isotropic submanifolds Va. Moreover, since exactly one integral surface of the distribution A(a) passes through each element of the manifold EQ, exactly one completely isotropic submanifold VQ is tangent to each a-plane of the CO(2, 2)-structure. The structure CO(2, 2) in this case is a-semiintegroble.

The involutory property of the distribution A(a) is equivalent to complete integrability of the Pfaffian system (5.1.2), (5.4.6). In view of (5.4.7), a necessary and sufficient condition for complete integrability of this system is the vanishing of the subtensor Ca. Analogous arguments apply to the fiber bundle Eo. Therefore anti-self-duality of the C0(2,2)-structure is equivalent to its a-semiintegrability, and self-duality in its turn is equivalent to its ,0scmiintegrability. Thus we have proved the following result:

Theorem 5.4.2 A four-dimensional manifold Al endowed with a semiflat CO(2, 2) -structure is characterized by the property that it carries a three-parameter family of completely isotropic two-dimensional submanifolds and a one-pa-

rameter family of such submanifolds passes through each point of M. If the manifold M caries two three-parameter families of completely isotropic twodimensional submanifolds, then the CO(2, 2) -structure is flat. 3. Let us assume now that equation (5.4.7) does not vanish identically. Then it has four roots Ap, p = 1,2,3,4, each of which determines a cross-section sp(a) : Al -a EQ of the bundle E0. To this cross-section there corresponds the isotropic distribution Ap(a) on M, which is generated by one of the families of principal a-planes. We will call these distributions the principal distributions of the fiber bundle EQ. The principal distributions O1,(a) are, generally speaking, not involutive, because the root Ap of equation (5.4.7) may not satisfy the equation (5.4.6), by the differentiation of which we obtained equation (5.4.7). However, if this

root satisfies equation (5.4.6), then the distribution Op(a), defined by it, is involutive and determines an isotropic foliation FF(a) on the manifold M. To find an integrability condition for the principal distribution Op(a), we will suppose that A = Ap in (5.4.7), and we will differentiate the resulting

5.4

Completely Isotropic Submanifolds

187

identity. Replacing dA with the help of (5.4.6) and making use of (5.1.23), we obtain dCa(A) + 2Ca(A)(6 + 9 + 9) + 4ACa(A)B?

_ (ao1A4 -

6a2,A2 - 4a3,A + a4s)Wt = 0,

i = 1, 2, 3, 4.

Therefore, on the manifold M the system of equations C.(A) = 0, dC,(A) = 0 is equivalent to the system consisting of equation (5.4.7) and the equations ao,A4 - 4a1,A3 + 6a2,A2 - 4a3,.\ + a4, = 0,

(5.4.10)

where a,, and further but, u = 0, 1, 2, 3, 4, are the covariant derivatives of the components of the tensors Ca and CO, respectively. In exactly the same way, the distribution A,(Q), determined by a root pu of equation (5.4.9), is involutive if and only if this root also satisfies the equations boip4 - 4b11 3 + 6b2ip2 - 4b3;1e + b4i = 0.

(5.4.11)

In view of this, the following theorem is true.

Theorem 5.4.3 A root Au of equation (5.4.7) determines an isotropic foliation

F,(a) on the manifold M if and only if it satisfies equation (5.4.10). A root pu of equation (5.4.9) determines an isotropic foliation F,(#) on the manifold M if and only if it satisfies equation (5.4.11). We deduce some consequences of this theorem. A pseudoconformal CO(2, 2)-structure is called a-semirecurrent if its tensor Ca satisfies the condition

au, = k,au

(5.4.12)

and Q-semirecurrent if its tensor CO satisfies the condition but = l{bu;

(5.4.13)

we recall that here u = 0,1,2,3,4 and i = 1,2,3,4. A pseudoconformal CO(2, 2)-structure is called recurrent if VC = VC,

(5.4.14)

where C is the tensor of conformal curvature, and cp is an 1-form (cf. Subsection 4.2.2 and Adati and Miyazawa [AdM 67)). Now we obviously have the following corollary:

Corollary 5.4.4 All four principal distributions Ap(a) are involutive if and only if the CO(2, 2)-structure is a-semirecurrent. Four foliations in general position on a manifold M make up a four-web on M. Such four-webs were studied by some authors (see Goldberg [Go 881, Ch. VII). Thus, if the roots of equation (5.4.7) are real and distinct, then the manifold M carries an isotropic four-web formed by the foliations FF(a). An analogous statement is true for the distributions Op(Q) determined by the roots of equation (5.4.9).

188

5. THE FOUR-DIMENSIONAL. CONFORMAL STRUCTURES

Corollary 5.4.5 If the CO(2, 2)-structure is recurrent, then all eight principal distributions O(a) and Op(#) are involutive, and the manifold M carries two isotropic 4-webs. Proof. For a recurrent structure, by condition (5.4.14), we have CijkIm = WmCijkl, and consequently conditions (5.4.12) and (5.4.13) hold.

Corollary 5.4.6 Every multiple root of equation (5.4.7) or (5.4.9) determines an isotropic foliation on the manifold M. Proof. For example, let A be a multiple root of equation (5.4.7). Using an admissible transformation of the adapted frame, we can set this root equal to zero, A = 0. Then from (5.4.7) we conclude that a3 = a4 = 0. In view of this, from (5.1.23) we obtain a4i = 0. But then the root A = 0 satisfies equations (5.4.10), and the distribution defined by it is involutive. N 4. Consider now a CO(1, 3)-structure. As we have proved earlier (see equations (5.2.7)), for such a structure the coefficients of the polynomials C,(A) and C0(µ) are complex conjugate. By virtue of this, the roots of these polynomials are also complex conjugates. Thus the principal two-dimensional directions on the isotropic bundles Ea and Ep (which also satisfy the condition Ep = E.) are also complex conjugate. Moreover two complex conjugate two-dimensional principal directions of the bundles EQ and Ea determined by the roots Ap and µp = Ap of equations (5.4.7) and (5.4.9) intersect one another along a real generator of the cone C. This generator has the same direction as the vector cn, p = 1, 2, 3, 4, defined by formula (5.2.3) for A = Ap. Thus the isotropic cone CZ of the CO(1,3)-structure carries four real principal isotropic directions. Now we will prove the following result:

Theorem 5.4.7 The integral curves of each of four families of real principal directions on a manifold M endowed with a CO(1,3)-structure are isotropic geodesics of the manifold M.

Proof. Note first that in general, the geodesics of conformally equivalent Riemannian metrics generating a conformal structure on the manifold M are not conformally invariant. However, it is possible to prove that the isotropic geodesics on M enjoy this property. The equations of geodesics on a Riemannian manifold M can be written in the form (5.4.15) dt;1 + t'Bj = where t;' are coordinates of vectors tangent to the geodesics. For four-dimensional conformal structures, in the isotropic frame bundle the forms 0 satisfy relations (5.1.4), and by (5.2.3), the coordinates of isotropic vectors on the CO(1,3)-structure have the form

L = AA, C2 = -A, C3 = -A, 4 = 1,

(5.4.16)

5.4

Completely Isotropic Submanifolds

189

where A is a complex parameter on the cone C. By virtue of (5.4.16), equations (5.4.15) of isotropic geodesics on the CO(1,3)-structure can be written as follows:

d(J) - \01 - A03 = aa(n - B;), -da+AX8 +01 = -A(r -B2),

-d+A39 +62=-a(k+82 ), -AB; - AB? = is + 8

(5.4.17)

.

By relations (5.2.5), which the forms 9, of the CO(1, 3)-structure satisfy, only two of equations (5.4.17), for example, the second and the fourth, are independent. Excluding the 1-form K from the second equation by means of the fourth equations, we find that dA + A(8 + B2) - 03 + A29 = 0.

(5.4.18)

But this equation precisely coincides with equation (5.4.6) which the complex parameters \p determining the principal directions on the isotropic cones C. satisfy.

Note also that the integral curves of the principal isotropic directions of the CO(1, 3)-structure form isotropic geodesic congruences on the manifold M. In general, the manifold M carries four such congruences. As we will see further, the real principal directions on the isotropic cones C. of the CO(1, 3)-structure play an important role in the Petrov classification (see Chandrasekhar [Cha 83] or Penrose and Rindler [PR 86]) of Riemannian metrics in general relativity. 5. In conclusion we consider a CO(4)-structure. By (5.2.16), equation (5.4.7) takes the form aoA4 - 4a1 A3 + 6a2A2 + 4-a1 A + do = 0,

(5.4.19)

where a2 is a real number. Let us investigate this equation. If we take the complex conjugate values of all terms of (5.4.19), we obtain aoA -

4a1a3 + 6a2A + 4a1A + ao = 0.

Comparing this equation with equation (5.4.19), we see that if Al is a root of equation (5.4.19), then the number A2 = - a, is also its root. Thus in the complex plane these two roots are located as shown in Figure 5.4.1. It follows that the roots Al and A2 cannot coincide. Furthermore, if Al = A3, then \2 = A4. Thus we have proved the following result: equation (5.4.19) has either four distinct roots or two pairs of double roots; in the latter case the isotropic fiber bundle EQ carries two double principal distributions. However, since these distributions are complex, they do not define foliations on the real manifold M.

190

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Figure 5.4.1

By (5.2.16), for the isotropic fiber bundle E0(14) the equation Cf(µ) = 0 can be written in the form boµ° - 4b,µ3 + 6b2µ2 + 4b1µ. + bo = 0,

(5.4.20)

where b2 is a real number. Applying this equation, we can prove the results on the principal distributions of the isotropic fiber bundle Ep similar to those we proved above for the principal distributions of the isotropic fiber bundle E0. 6. For the CO(1,3)-structure, equations (5.4.7) and (5.4.9), which by (5.2.1) are complex conjugates of one another, are connected with A. Z. Petrov's classification of Einstein spaces. We remind the reader that an Einstein space is a four-dimensional pseudoRiemannian manifold of signature (1, 3) whose curvature tensor satisfies the condition 1

Rjk-Z9,kR=-

8aG c4

TA:,

(5.4.21)

where Rjk = RRki is the Ricci tensor, R = gikR,,k is the scalar curvature of the Riemannian manifold, Ti f is the energy-momentum tensor, G is the gravitational constant, and c is the speed of light. Equation (5.4.21) is called the Einstein equation. In empty space, that is, in a region of space-time in which T,j = 0, the Einstein equation can be reduced to the form Rid = 0.

This implies that the curvature tensor of this space coincides with its Weyl tensor: Ckl. This follows from the expression of the tensor in terms of RJkI, R,k, and R (see formula (4.2.14)).

5.4

Completely Isotropic Submanifolds

191

The classification of Einstein spaces is connected with the structure of its tensor of conformal curvature. Hence this classification is of a conformal nature.

This classification was first constructed by in Petrov [Pe 54] (see also Pirani [Pi 57]).

To give a geometric characterization of Einstein spaces of different types, we will also apply the isotropic geodesics on the manifolds endowed with a CO(1, 3)-structure which we considered in Subsection 5.4.4. Since for the CO(1, 3)-structure, equations (5.4.7) and (5.4.9) are complex conjugates, for classification of Einstein spaces it is sufficient to consider only one of these equations, for example, the first one. By means of this equation, this classification can be conducted as follows:

1. Type I of Petrov (we use the Penrose notation for types; see Chandrasekhar [Cha 83], Ch. 1, §9, or Penrose and Rindler [PR 86], Ch. 8) is characterized by the fact that all roots of equation (5.4.7) are distinct, and as a result the isotropic fiber bundle Ea, as well as the fiber bundle Ea, has four distinct principal isotropic distributions. Since the principal distributions of the fiber bundles E and Ea are complex conjugates, they determine four real principal directions on each isotropic cone C. In view of this, the manifold M carries four congruences of isotropic geodesics. In the case under consideration, equation (5.4.7) admits the following specialization. If we combine the basic distributions e4 A e3 and el A e2 with the principal distributions of the fiber bundle EQ determined by the values A = A, and A = A2 of the roots of equation (5.4.7), then the latter equation takes the form 2a1.A3 - 3a2A2 + 2a3A = 0.

Now aI = 0 and Al = oo, and the remaining three nonvanishing components a,, a2, and a3 of the Weyl tensor (and also their complex conjugate components bl, b2, and b3) are relative invariants of the Einstein space of type I.

2. Type II of Petrov is characterized by the fact that equation (5.4.7) has one double root and two simple roots. As a result the isotropic fiber bundle EQ has one double principal isotropic distribution defined by the double root of equation (5.4.7) and two principal isotropic distributions of general type. The fiber bundle Ea has the same kind of principal isotropic distributions. On the isotropic cones CZ, these distributions give rise to three real one-dimensional principal directions, one of which is double and the two others are simple. In view of this, the manifold M carries three congruences of isotropic geodesics, one of which is double.

If we combine the basic distribution e4 A e3 with the integrable principal distribution of fiber bundle EQ corresponding to the double root ai = A2 of equation (5.4.7) and combine the basic distribution el A e2 with the

192

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

principal distribution corresponding to the simple root '\3 of equation (5.4.7), then the latter equation takes the form 2ai.A3 - 3a2A2 = 0,

and the remaining two nonvanishing components ai and a2 of the Weyl tensor (and also their complex conjugate components bl and b2) are relative invariants of the Einstein space of type II.

3. Type D of Petrov is characterized by the fact that equation (5.4.7) has two distinct double roots. Hence the isotropic fiber bundle E, as well as the fiber bundle Ep, has two double principal isotropic distributions. On the isotropic cones C., these distributions give rise to two double principal directions. In view of this, the manifold M carries two double congruences of isotropic geodesics.

If we combine the basic distributions e4 Ae3 and el Ae2 with these double distributions, then equation (5.4.7) takes the form a2

A2

= 0,

where a2 is the only nonvanishing relative invariant of the Einstein space of type D.

4. Type III of Petrov is characterized by the fact that equation (5.4.7) has one triple root and one simple root. As a result, each of the isotropic fiber bundles Ea and Ep has one triple principal isotropic distribution and one principal isotropic distribution of general type. In view of this, the isotropic cones Cx carry one real triple and one simple principal direction, and the manifold M carries one congruence of triple isotropic geodesics and one congruence of simple isotropic geodesics.

If we combine the basic distribution e4 A e3 with the triple principal distribution and combine the basic distribution el A e2 with the principal

distribution corresponding to the simple root of equation (5.4.7), then the latter equation takes the form aIA3 = 0,

where al is the only nonvanishing relative invariant of the Einstein space of type III.

5. Type N of Petrov is characterized by the fact that all four roots of equa-

tion (5.4.7) coincide. Then each of the isotropic fiber bundles E. and Ep has one quadruple principal isotropic distribution. In view of this, the isotropic cones C. carry one quadruple principal direction, and the manifold M carries one congruence of quadruple isotropic geodesics.

Four-Dimensional Webs and CO(2, 2)-Structures

5.5

193

Table 5.4.1

Petrov's type I

II

Roots of the equation

Characterization of principal

Characterization of

C0(A) = 0

distributions

congruences

4 simple

.1D 0 Aq, p # q,

4 different of

p,q = 1,2,3,4

general type

Al = A2

1 double and 2 of general type 2 double 1 triple and I of general type 1 quadruple

A3

D

A3, A4,

A4

III

Al = A2 A3 = A4 Al = A2 = 1\3 1\4

N

Al = A2 = A3 = A4

isotropic geodesic

1 double and 2 simple 2 double 1 triple and 1 simple 1 quadruple

If we combine the basic distribution e4 A e3 with this quadruple isotropic distribution, then the equation (6.7) takes the form aoA4 = 0,

where ao is the only nonvanishing relative invariant of the Einstein space of type N.

The Petrov classification is briefly represented in the Table 5.4.1.

5.5

Four-Dimensional Webs and CO(2, 2)-Structures

1. Before we start to study CO(2, 2)-structures connected with the theory of four-webs, we will consider a simpler example of such a structure that arises when one studies a manifold of null-pairs in a real projective plane RP2.

Let RP2 be a real projective plane. A pair (x,l) consisting of a point x and a straight line 1 is called a null-pair. If the point x does not lie on the straight line 1, then the null-pair is called nondegenerate. Otherwise, it is called degenerate.

With a nondegenerate null-pair we associate a moving projective frame in such a way that its vertex Ao coincides with the point x: Ao = x, and the points Aa, a = 1, 2, lie on the line 1 : A1, A2 E 1. The equations of infinitesimal displacement of this frame have the form dAo = wo°Ao +w°Aa, dAo = woAo +w;Ab.

(5.5.1)

194

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

The 1-forms occurring in these equations satisfy the structure equations of the plane RP2: dwo = wo A wo, dw0 = woo A woo + woo A Wb

(5.5.2)

dwo =woAwoo +wQAWo dwb = wb A w00 + ws A wC , a, b, c = 1, 2.

The forms woo and wo are basis forms on the four-dimensional manifold of null-

pairs. The quadratic form (5.5.3)

g = 2w0awa

determines a pseudo-Riemannian metric on this manifold such that the manifold carries the pseudo-Riemannian structure 0(2, 2), and along with this struc-

ture, it carries also the pseudoconformal CO(2, 2)-structure. The equation g = 0 determines the isotropic cone Cz of this CO(2, 2)-structure. If we set w1 = W10+ W2 = w2 w3 = -w0 2, w4 = w0 1

then the quadratic form (5.5.3) takes the form (5.1.1). We now compare equations (5.1.5) and (5.5.2) and choose one possible solution of the resulting system of equations: B, = w1 - wo, 02 = w2 WOO, B = 0,

-

Ba=wa, 0 =w1,

83=B1=0.

We write out the systems of equations determining the distributions 0(a) and 0((3) on the isotropic fiber bundles EQ and E0: wo - Aw4 = 0, wo + Awp = 0, (5.5.4)

1 Ba = dA + A(w1 + w2 - 2wo) = 0;

wa + µwo = 0, w2 - Rw° = 0, BN = dµ + µ(w1 - w2) - W21 + µ2w2 = 0.

(5.5.5)

By taking the exterior derivatives of the last equations of systems (5.5.4) and (5.5.5), we find the polynomials CQ(A) and C,s(µ): CQ(A) = A2 and Co($I) = 0. Hence it follows that the tensor of conformal curvature of the CO(2, 2)-structure under consideration has one nonzero independent component: a2 = C1234 = 1. In view of this, this structure is not conformally flat. However, since the tensor

CO(p) = 0, the CO(2,2)-structure under consideration is self-dual and consequently 0-semiintegrable. The equation CQ(A) = 0 has two pairs of multiple roots: Al = A2 = 0 and A3 = A4 = oo, which determine two isotropic fiber bundles on M. It is not difficult to give a geometric description of a set of completely isotropic submanifolds of the manifold of nondegenerate null-pairs on the plane RP2. Since the condition Al = A2 = 0 implies that wo = wo = 0, the isotropic

5.5

Four-Dimensional Webs and CO(2,2) -Structures

195

a-submanifold corresponding to these roots of the equation C0(A) = 0 is the set of null-pairs with fixed point Ao and variable line Al A A2. In exactly the same way, the isotropic a-submanifold corresponding to the roots A3 = A4 = 00

of the equation C0(A) = 0 is the set of null-pairs with fixed line Al A A2 and variable point Ao. furthermore equations (5.5.5), determining isotropic /3-submanifolds, are conditions for the point K = A2 - µA1 and the line I = AO A (A2 - µA1) to be fixed. Therefore an isotropic ,0-submanifold is determined by a null-pair (K, I) and consists of all null-pairs formed by the points AO of the line I and the lines Al A A2 passing through the point K. The family of these isotropic (3-submanifolds depends on three parameters. 2. Now we consider the relationship between four-dimensional webs and pseudoconformal CO(2, 2)-structures. Let W(3, 2, 2) be a three-web (see Bol [Bo 35], Akivis [A 69] or Akivis and Shelekhov [AS 92]), formed by three foliations W a = 1, 2, 3, of codimension two on a four-dimensional manifold M. Suppose that F. are leaves of these foliations passing through the point x E M. The two-dimensional tangent subspaces T. (F.) generate a cone of second order C. of signature (2, 2) in the space TT(M). In fact let I be a one-dimensional subspace in Tt(F1). Consider the linear spans 1AT=(F2) and IATz(F3). They have a two-dimensional common subspace Tra(I) passing through the line 1. Changing the subspace I in Tr(F1), we obtain a one-parameter family of subspaces ap(I). In the same way any three subspaces 7rp(I) generate a two-parameter family of subspaces zr,, containing the tangent subspaces TT(F0). Thus these two families irp and Ira are two families of plane generators of the second order cone Cz of signature (2, 2) in the tangent space T=(M). The field of these cones determines a pseudoconformal CO(2, 2)-structure on M. Two-dimensional plane generators 7r0 and as of these cones form the isotropic fiber bundles E,, and Ep on the manifold M. The systems of differential equations defining the web W(3, 2, 2) on the manifold M can be reduced to the following form: W° = 0, w° = 0, w° + 0 ° = 0, a = 1, 2. 2

1

(5.5.6)

1

1

The left-hand sides of equations of these three systems are completely integrable systems of 1-forms. The same systems determine the subspaces T,,(F0) in the tangent space T=(M). Let us find the equation of the cone Cx containing the subspaces indicated above. We will look for this equation in the form uabW°Wb 1

+ 2vabw°Wb + UJabW°Wb = 0.

1

1

2

2

1

Since the equations w° = 0 and w° = 0 must satisfy the above equation, we 2 have uab = Wab = 0. Substituting b = -wb into the remaining part, we find 1

that v11 = V22 = 0 and v21 = -v12. s

196

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Therefore the equation of the cone C. is written as I

W W

2 - W2W1 = 0.

2

1

1

(5.5.7)

2

Comparing this equation with equation (5.1.1), we obtain wI = w1 1

2 = W2

'

1

W3

'

= W1 2

'

W4 = W2.

(5.5.8)

2

In view of (5.5.8), equations (5.1.2) and (5.1.3) which determine the a-planes and the Q-planes of the CO(2, 2)-structure can be written as

wi+AWI=0, w2+Aw2=0 2

1

2

1

and

w1+µw2=0, w1+µw2=0. 2

1

2

Since the a-planes contain the tangent subspaces TT(F,), they are called isoclinic. On the other hand, the /3-planes intersect all three tangent subspaces T,(F,), and this is the reason that they are called transversal (cf. Akivis (A 74] or Akivis and Shelekhov [AS 92], Section 1.3). The conditions of integrability of the foliations W, determined by systems (5.5.6) can be reduced to the form dw° = Wb A Wb + abWJ A w°, 1

1

I

1

1

(5.5.9)

1 dW° =WbA WS -abWbAWa, 2

2

2

2

where abc = aibd l is the torsion tensor of the web W(3, 2, 2) (see Akivis [A 74] or Akivis and Shelekhov [AS 92], p. 12). If we prolong equations (5.5.9), we obtain dwa = we A wa + becewc A We, 1

Vab =

2

(5.5.10)

9bcs`,

where wb are the forms of the affine connection -y defined on the manifold M of the web W(3, 2, 2), beCe is the curvature tensor of this web, and as usual Vab = dab - acwb. If we compare systems (5.5.9) and (5.1.5), we can see that one of the solutions of the resulting system of equations is Bs = B1 = 0, B2 = Ws, 8 = w19

(wt + W2 - al (wl

0 a

W3) - a2(W2 - w4)),

01 = 1 Pi - w2 - al (WI +w3 ) 02

= 2[w2 - w1 - a,

aa(W2

(5.5.11)

+ W4)),

/Wl + w3) - a2 (W2 + W4)].

5.5

Four-Dimensional Webs and CO(2, 2)-Structures

197

We will take this solution and write out the systems of equations determining the distributions O(a) and O(,Q) on the fiber bundles Ea and Ep: wl + Awl = 0, W2 + Awe = 0, 1

2

2

1

(5.5.12)

0A = dA - A[a1w1 \1 + w1) + a2(w2 + w2)] = 0; 2

1

2

5 wl + pw2 = 0, W1 +;2 = 0, 2 2 1

OA = d'a + µ(w1 - 4Z) - w2 +'U2 w2 = 0.

(5.5.13)

Exterior differentiation of the last equation of (5.5.12) leads us to the equation

Ca(A)=0.A4+PA3-(p+q)A2+qA+0=0, where p = P12 - p21 and q = q12 - q21. Hence it is clear that besides the roots Al = 0, A2 = oo, and A3 = 1, which determine three leaves F, forming the web W(3, 2, 2), this equation has a fourth root that is determined by the equation

p A = q. This root determines a fourth invariant distribution A, associated with the web W(3, 2, 2) which, in general, is nonintegrable. It is easy to see that if p = 0, q # 0 or p 54 0, q = 0 or p = q 0 0, then the fourth distribution A coincides with one of the foliations gyp, forming the three-web W(3, 2, 2). The webs W(3,2,2), for which this is the case, are called special. If p = q = 0, then the polynomial C0(A) is identically zero, Ca(A) _- 0, the CO(2,2)-structure is anti-self-dual, and the web W(3,2,2) is isoclinic. If p 54 O, q 54 0 and p 54 q,

then the fourth distribution A is different from the foliations

qw' +p2' = 0.

(5.5.14)

If p 54 0, q 54 0 and p j4 q, then equations (5.5.14) are equivalent to the equations w' 1

' w' _ -qw', '

(5.5.15)

2

where w' are basis forms on the integral submanifolds of equations (5.5.14). Taking the exterior derivatives of equations (5.5.14) and eliminating the forms w' and w' from the exterior quadratic equations by means of (5.5.15), we find 2 that the condition of integrability of the distribution A has the form 1

[Pdq - qdp+pq(P - q)a1w'] Ad = 0. Since the forms w' are linearly independent, it follows from this equation that

pdq - qdp = pq(q - p)ajw'.

(5.5.16)

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

198

Exterior quadratic equations obtained by exterior differentiation of equations (5.5.14) imply the following equations: dp = p.

+ Pew'. + P,w', dq = qwi + q,tt + q,w{. 1

2 2

1

1

1

(5.5.17)

2 2

Substituting these decompositions into equations (5.5.16), we find the following form for the integrability conditions of the distribution A: P(Pgi - qpi) - q(pq1 - qp,) = pq(q - P)a1. 2

(5.5.18)

2

In exactly the same manner as we found the polynomial CQ(A), by taking the exterior derivative of the last equation of (5.5.13), we find the polynomial CO(p)

= 3l11µ° - (36112 - 3l11)µ3 - 3(3122 - 3i12)µ2 - (3222 - 38i22)µ - 3222,

Four roots of the polynomial Cp(p) determine four invariant (transversally geodesic) distributions which are invariantly associated with the web W(3, 2, 2) and which, in general, are not integrable. If some root is a multiple root, then the corresponding transversally geodesic distributions coincide and become integrable. The polynomial C0(µ) is identically zero if and only if the CO(2, 2)-structure is self-dual, and the web W(3, 2, 2) is transversally geodesic. In this case a one-parameter family of transversally geodesic submanifolds passes through every point x E M. This will be the case if and only if all coefficients of the polynomial Ca(p) vanish: where sbce =

t

8i11 = 0,

{

1

3 222 2

1

8111 = 33112,

= 0, 2

31112 = 3122,

2

1

3222 = 38122

3. The web theory allows us to construct some explicit examples of semiintegrable CO(2, 2) -structures.

Example 5.5.1 Consider the three-web W(3,2,2) defined on the space R° of variables xI,x2,yt and y2 by the closed form equations

z' = x' + y', z2 = (x2 + y2)(y' - x').

(5.5.19)

(see Goldberg [Go 85, 86) or Goldberg [Go 88), pp. 422-425). The foliations A1, A2 and A3 of this three-web are determined in R° by equations x° = const, y° = const, and z° = const,

respectively.

To find the invariant forms w° and 1

equations (5.5.19):

s ° of this three-web, we differentiate

dzt = dxt + dy', dz2 = -(x2 + y2)dx' + (y' - x')dx2 + (x2 + y2)dy' + (y' - x')dy2.

Four-Dimensional Webs and CO(2, 2)-Structures

5.5

199

By virtue of this, we obtain w' = dx',

W2 = - (x2 + y2)dx' + (y' - x')dx2, (5.5.20)

Z' = dy',

22

=

(x2 + y2)dy' + (y' - x')dy2.

These equations must be solvable with respect to dx', dx2 and dy', dy2. Thus the following two determinants must be nonvanishing: 1

'

1 0, 12

0

-I -(x2+y2) y'-x'

1

0

x2+y2 yl - xl

0,

This condition holds if and only if

y' - x' # 0. Hence we will consider, for example, the half-space of R' in which y' > x'. By means of (5.5.7) and (5.5.20), the equation of the isotropic cone Cx of the CO(2, 2)-structure associated with the given three-web is 2(x2 + y2)dx'dy' + (y1 - x')(dx'dy2 - dx2dy') = 0.

To find the tensors C. and CO of the CO(2, 2)-structure determined by the quadratic form on the left-hand side of this equation, we will differentiate the forms 11 ° and w a in (5.5.20) and find consequently the forms wb and the components of the torsion tensor abc = d(baC; next we take exterior derivatives of the forms wb and find the covariant differentials Dab and the components bbce of the curvature tensor of the given three-web. For the three-web (5.5.19) this work has been done in Goldberg [Co 85, 86[ (see also Goldberg [Go 881, pp.

422-425). We will write the values of those quantities indicated above which we will need for the computation the tensors Ca and CO, 2

p2i=921=0, p1I=-91i(xlyt)2; b2

= -b121 = (x1

2?y1)2,

2 bill = -(a;' y1)z,

and the remaining components of the curvature tensor are equal to 0. This implies that all components of the tensor Ca vanish and that the tensor CO has only one nonvanishing component: 8(x2 + y2)

bo = - (x1 -

y1)2'

Hence the CO(2, 2) -structure associated with the given three-web is anti-self-

dual, the three-web itself is isoclinic, and the equation C0(p) = 0 takes the form

bop' = 0.

200

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Thus this equation has a quadruple root at p = 0, and as a result the isotropic fiber bundle E0 has only one integrable distribution Op(0). By (5.5.13), the equations of this distribution can be written as

W' =dx' =0, Z' = dy' =0, Put differently, this distribution is determined by the equations x' = const, y' = const, y1 > x'.

The system (5.5.12) for the given three-web takes the form

dx' + ady' = 0, (x2

+ y2)(-dx' + Ady') + (y' - x')(dx2 + Ady2) = 0,

dA + yJ2Ax,(dx' + dy') = 0.

This system is completely integrable and determines a three-parameter family of integral surfaces of the isotropic fiber bundle Ea. Example 5.5.2 Consider the three-web W(3, 2, 2) defined on the manifold R4 of variables x1, z2, y', and y2 by the closed form equations

z' = x' + y', z2 = -x'y2 + x2y'

(5.5.21)

(see Goldberg [Go 85, 86] or Goldberg [Go 88], pp. 425-428).

From (5.5.21) we find that dz' = dx' + dy',

dz2 = -yzdx' + y'dx2 -XI dy2 + x2dy',

and hence W1

= dx',

j2 = -yzdx' + y'dx2,

2' = dy',

Z2 = -x'dy2 + x2dy'.

5522

1

The conditions of solvability of these equations with respect to dx', dx2 and dy', dy2 are

Di =

_1Y2 Y

A2=1X2 _x'

1=--100-

These conditions are satisfied, for example, if x' > 0 and y' > 0. All further constructions we will make in this open domain of the space R4. By means of (5.5.7) and (5.5.22), the equation of the isotropic cone Cx of the CO(2, 2)-structure associated with the given three-web is (x2 +Y 2)dxIdYI - x'dx'dy2 - y'dx2dy' = 0.

5.5

Four-Dimensional Webs and CO(2,2)-Structures

201

For the three-web (5.5.21) computations show (see Goldberg [Go 85, 86] or

Goldberg [Go 88], pp. 425-428) that in this case p = q = 0. As a result the tensor C,, = 0, the CO(2, 2)-structure is anti-self-dual, and the three-web itself is isoclinic. The nonvanishing components of the tensor CO are (X2

y2

x1

bo

=

y1

-x

461

1

(x 1)2

y1

(yl)2

Hence we have

Cp(l)=

\y1

x1/(x1

y2)p+y1

+x1J14 3

Thus the equation C0(p) = 0 has a triple root at p = 0, and this root determines the integrable distribution Op(0) on the isotropic fiber bundle Ep. In addition the fourth root p -Z Y of the equation C0(µ) = 0 determines 1-1- 1+ 1

another distribution of Ep whici, in general, is not necessarily integrable.

Example 5.5.3 Consider the three-web W(3, 2, 2) defined on the manifold R4 of variables x1, x2, y1, and y2 by the closed form equations

z1 = x1 +y' + (x1)2y2, z2 = x2 +y2 - 2x1(y2)2

(5.5.23)

(see Goldberg [Go 87] or Goldberg [Go 88], pp. 431-432).

From (5.5.23) we find that

dz' = Odx1 + dy' + I (x1)2dy2,

dz2 = - (y2)2dx1 + dx2 + (2 - O)dy2, where 0 = 1 +x1y2, and consequently

i1 = Mdx',

w2 = dx2 - 2(y2)2dx1,

(5.5.24)

Z1 = dy1 + 1(x1)2dy2,

w2 2

= (2 - O)dy2.

The condition of solvability of these equations with respect to dx1, dx2 and dy', dy2 is 0 i4 0, 2. As a domain in which these conditions are satisfied we take an open domain of the space R4 defined by the inequalities:

-1<xIy2<1. By means of (5.5.7) and (5.5.24), the equation of the isotropic cone C. of the CO(2, 2)-structure associated with the given three-web is: (y2)2dxldyl +2(1 - 3(x1y2)2)dx1dy2 - 2dx2dyl - (x1)2dx2dy2 = 0.

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

202

For the three-web (5.5.23) computations show (see Goldberg [Go 87] or Goldberg [Go 88], pp. 431-432) that in this case all components b of the tensor Co vanish. That is, the CO(2, 2)-structure associated with the web (5.5.23) is self-dual, and the three-web itself is transversally geodesic. However, this three-web is not isoclinic, since we have

A2-20+2 P

A2-2A+2

03(2 - A)2 , 9 = 202(2 - 0)3'

and in the domain under consideration the common numerator of these two fractions does not vanish,

&2-20+2=(0-1)2+1=(xly2)2 40. The fourth isotropic distribution &,(X1) of the isotropic fiber bundle EQ is determined by the following value of A: A

-E=A-2 q

0

by means of which the differential equations of this distribution have the form

(A-2)w'+Ow' =0. 2 1

It can be proved (see Goldberg [Go 87) or Goldberg [Go 88), pp. 431-432) that integrability conditions (5.5.18) are satisfied for this distribution, and hence it determines the fourth isotropic foliation which together with three foliations of the three-web compose a transversally geodesic four-web.

5.6

Conformal Structures of Some Metrics in General Relativity

1. Examples of pseudoconformal CO(1, 3)-structures arise in the study of conformal structures of certain Riemannian metrics in general relativity. We will

write these metrics in the form in which they are given in the book Chandrasekhar [Cha 831. First we consider the axially symmetric metric

ds2 = e2"(dt)2 - e2`'(dcp - kdt)2 - e2µ'(dx2)2 - e2pj(dx3)2

(5.6.1)

(see Chandrasekhar [Cha 83], Ch. 2, §11, Eq. (17)) where t, gyp, x2, x3 are coordinates on a four-dimensional space-time manifold Al and v, 0, µ2, µ3i k are functions of coordinates x2 and x3. Since we are interested only in the conformal structure of the metric (5.6.1), we will divide the left- and righthand sides of (5.6.1) by the function e21+3, which is not zero on the manifold M. As a result we obtain g = e2E(dt)2 - e2' (dcp - kdt)2 - e2((dx2)2 - (dx3)2,

(5.6.2)

Conformal Structures of Some Metrics in General Relativity

5.6

203

where f = v -µ3,p =10 -113 and (= µ2 - µ3 depend only on x2 and x3. Let us reduce the quadratic form (5.6.2) to the form (5.1.1). To this end, we set

= - ((ef + ke")dt - e"dcp), 4 = I ((e4 - ke")dt + e"dcp), (5.6.3)

2 = I (e4d22 - idx3), 3 = I` (e(dx2 + idx3).

In (5.6.3) the forms wi and w4 are real, and the forms w2 and w3 are complex conjugates. Taking the exterior derivatives of (5.6.3) and eliminating dt, dcp, dx2, and dx3, we find that dW1 =nW2AW'+1LW3Aw' +mw2AW4+fffW3AW4, d1R14 = PUJ2 A Wl + pW3 Awl + qW2 A w4 + QW3 A w4, (5.6.4)

dw2 = SW2 A W3, dw3 = sw2 A W3.

By (5.6.3), all exterior differentials of the forms w' are real. It follows that the functions m, n, p, and q are complex functions of x2 and x3 and that s is a pure imaginary function of x2 and x3. Comparing equations (5.6.4) and (5.1.5), we find that (9i - 9) Awl + (0 + nWl + mw4) A w2 + (63 1 + Y ' + i iw4) A w3 = 0,

0 Awl+(02 -0-8W3)AW2+0' Aw4 = 0, 0 AW' + (-9 - 62 + SW2) AW3 + 02' AW4 = 0,

(0 +pw1 +gW4)nw2+(Bi +pwl +'gW4)A w3- (0 + 01) AW4=0. (5.6.5)

Applying Cartan's lemma to equations (5.6.5), we obtain four groups of decompositions (with symmetric coefficients) of the forms 0 and B, with respect to the basis forms w'. Since each of the forms 0 and 6i occurs in these decompositions at least two times, the coefficients can be expressed in terms of four independent coefficients. As a result we arrive at the following decompositions

204

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

of the forms 0 and 0 = -C1W1 - (C2 - 2 1(n+q))W2 - (C3

61 =ciW' -

- 2(n+(j))W3 -C4W4,

2(n-q))W2- I(i-4)W3-C4W4,

e2 = (c2 - (n + q - 2s))W2 + (-c3 +'z (n + 9 - 21))w3, 2

92' = C2W' + C4W3 + mW4, 03

=

(5.6.6)

C3 W 1 + C4W2 + i-w4,

01 =pW1 +ClW2+C3W4, BI = pWi + CI W3 + C2W4,

where ci are independent coefficients. Equations (5.6.6) and (5.2.3) show that the coefficients c1 and c4 are real and that the coefficients C2 and c3 are complex conjugate: C3 = C2.

Equations (5.6.6) also show that if the point x is fixed on the manifold M (i.e., if w' = 0), then on the CO(1,3)-structure, associated with the axially symmetric metric (5.6.1), the forms 0 and 0 vanish. This means that in the case under consideration we have a cross-section in the bundle of isotropic frames that is associated with the CO(1, 3)-structure in the general case. This cross-section associates a unique isotropic frame to each point x E M. Taking the exterior derivatives of equations (5.6.6) and applying equations (5.1.14)-(5.1.20), we can prove that the 1-forms dci -Bi are linear combinations of the basis forms w'. We will write these decompositions in the form

dc,-0,=-Cijwi. If we fix the point x (i.e., if we set w' = 0) and denote the operator of differentiation with respect to the fiber parameters by 5, we find that bcj - a,(6) = 0.

Moreover the real coefficients c1 and c4 correspond to the real forms 01 and 04, and the complex conjugate coefficients c2 and c3 correspond to the complex conjugate forms 02 and 03. It follows from the above equations that by means of variation of fiber parameters determining the location of a frame of second order, we can reduce all coefficients ci to zero. As a result the forms 0i become linear combinations of the basis forms W': 9, = C,1Wl.

(5.6.7)

Geometrically this means that on each hyperquadric Q=, which represents compactification of the tangent space T=(M), the point at infinity y is fixed, and as a result the conformal connection associated with the CO(1, 3)-structure

5.6

Conformal Structures of Some Metrics in General Relativity

205

becomes the Weyl connection (see Subsection 4.2.1). If we perform the abovementioned specialization, the equations (5.6.6) take the form B=

2

(n + q)w2 + (ii + q)w3, 2

Bl = - 2 (n - q)w2 - 2(n - q)w3, Bz = -2(n + q - 2s)w2 + (n + q - 2s)w3,

(5.6.8)

Ba = mw4, 03 = mw4, Bi = Pw1

Bl = Pw1.

Taking the exterior derivatives of (5.6.8), we obtain

= 2(-(n3+q3)+(n+q)s+I3 E 93 (n+q)s)w2Aw3, d0 = (n3 - Q3 - n3 + q3 - (n - q + n - q)s)w2 A w3, 692 =

2(n3+q3+U3+q3+(n+q-n-q+4s)s)w2Aw3,

dBz = mpw2 A wl + mpw3 A wl + (mq + m2)w2 A w4 + (mq + m3)w3 A w4, dB1 = (pn + p2)w2 A wl + (pr+ + p3)w3 A wl + pmw2 A w4 + pmw3 A w4, (5.6.9)

where the functions mi, ni, pi, qi and si, i = 2, 3, are linear combinations of partial derivatives of the functions m, n, p, q, and s, respectively, with respect to the parameters z2 and z3. Since 03 = BZ and Bi = 9i, the exterior differentials d93 and d02 are the complex conjugates of dOz and dB?, respectively. Let us find the curvature tensor of the pseudoconformal structure associated

with the metric (5.6.1). Substituting in structure equations (5.1.14)-(5.1.20) for the forms B, Bi and dB, M their expressions from (5.6.7)-(5.6.8), taking into account relations (5.1.12), and equating to zero coefficients of independent exterior 2-forms, we obtain C1212 = ao = -p(2n - s) - p2, C1214 = al = 0,

C3434 = a4 = -fn (2-q - s) - m2, C1434 = a3 = 0,

C1234 =a2 = g(Z(n3+q3+n3+q3+(n+q-n-q+4s)s)+6fl1p1. (5.6.10)

As we have proved in Theorem 5.2.1, the components b of the tensor Cp are the complex conjugates of the corresponding components a of the tensor C. For the relative conformal curvature of the isotropic fiber bundles E. and Ep associated with the metric (5.6.1), we obtain the following expressions: C.(A) = aoa4 + 6a2 '\2 + a4, Cf(tl) = aop4 + 6a2,i2 + a4.

(5.6.11)

These polynomials are biquadratic, and their discriminants are, respectively, equal to D = 9(a2)2 - aoa4, D = 9(a2)2 - aoa4.

206

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

Since in the general case each of the polynomials (5.6.11) has four distinct

roots, the axially symmetric metric (5.6.1) belongs to type I of the Petrov classification (see Subsection 5.4.6). On the other hand, if the discriminant D vanishes, then each of these polynomials has two pairs of double roots, the metric (5.6.1) belongs to type D of the Petrov classification, and each of the isotropic fiber bundles Ea and Ep has two pairs of double principal distributions. If the coefficient ao or a4 of polynomials (5.6.11) vanishes, then each of these polynomials has a pair of double roots, and the metric (5.6.1)

belongs to type II of the Petrov classification. Then each of the isotropic fiber bundles E, and Ep has a double principal distribution. Finally, if both coefficients ao and a4 vanish, then each of the polynomials (5.6.11) again has two pairs of double roots, and the metric (5.6.1) belongs to type D of the Petrov classification.

2. The Kerr metric is a particular case of the axially symmetric metric (5.6.1). The Kerr metric is expressed by ds2

z

26

=-

(dt)e -

5(d_ 2 az rdt)

sine B - - (dr)z - pz(dW)z

(see Chandrasekhar [Cha 83], Ch. 6, §54, Eq. (133)), where p, gyp, and 9 are the spherical coordinates; here

p2 = r2 + a2 sin2 B, 6 = r2 - 2Mr + a2, a2 = (r2 + a2)2 - a26 sin2 where a and M are constants. The Kerr metric is a solution of Einstein's equation and describes space-time in a neighborhood of a black hole. Computations show that the tensor of relative conformal curvature of each isotropic fiber bundle determined by this metric has the same structure as for the metric (5.6.1). Thus, for the Kerr metric, all results that we obtained for the metric (5.6.1) are still valid. We consider now the conformal structure of the Schwarzschild metric and the Reissner-Nordstrom metric which are also particular cases of the axially symmetric metric (5.6.1). The Schwarzschild metric

ds2 = (1 - r9) r (dt)2 - 1 1_ s (dr)2 - r2 (sin2 B(dW)2 + (dB)2)

(5.6.12)

r

(see Chandrasekhar [Cha 83], Ch. 3, §17, Eq. (60)) describes the geometry of space-time outside of a spherically symmetric body whose gravitational radius is equal to r9 so that r > r9. This metric is obtained from (5.6.1) if we set

e2' = I - rgr ,

e2"

= r2 sin2 B, k = 0, e2 *2 =

1-I r9, r

e2" = r2.

5.6

Conformal Structures of Some Metrics in General Relativity

207

Then equations (5.6.3), (5.6.8), and (5.6.9) take, respectively, the form w1 = r - r9

4=

1

T2

r

dt - dr,

w2 = sin 9 dp - idO', (5.6.13)

dt +

1

T2 - r9r

dr,

w3 = sin B dip + idB,

B=0e=83=B1 =BI =0,

01= lw1-r9 r

r

(5.6.14)

4

'

2(U)2 + W3) Cot

02 = and

dB1_r-3rgw1Aw4

I 1

2r

(5.6.15)

dB2 = 2 w2 A w3.

The components of the tensor of conformal curvature C. of the isotropic fiber bundle E,, of the Schwarzschild metric take the values ao = 0, a1 = 0, a2 =

T9

4r' a3 = 0, a4 = 0,

and the components b of the tensor of conformal curvature CO have the same values. Hence the Schwarzschild metric belongs to type D of the Petrov classification (see Subsection 5.4.6). Geometry of space-time outside of a spherically symmetric body with an electrical charge Q is described by the Reissner-Nordstrom metric b2

ds2 = 2(dt)2 -

(dr)2 -

(5.6.16)

(dB2))

(see Chandrasekhar [Cha 831, Ch. 5, §38, Eq. (48)), where 6 = r2 - r9r + Q2 and Q = const. This metric is obtained from the axially symmetric metric (5.6.1) if we set z

e2v =

2,

e2'1'

= r2 sin2 9, k = 0, e2P2 = a ,

e211a

= r2.

Equations (5.6.10) for the components of the tensor CQ of the isotropic fiber bundle Ea of the Reissner-Nordstrom metric take the form

ryr - Q2 ao = 0, a1 = 0, a2 =

4r2

,

a3 = 0, a4 = 0,

and the components b of the tensor of conformal curvature Cp have the same values. Hence the Reissner-Nordstrom metric also belongs to type D of the Petrov classification.

208

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

For the Schwarzschild metric and the Reissner-Nordstrbm metric, the relative conformal curvatures of the a- and p-planes are calculated according to the following formulas:

C. (,\) = 6a2A2, CO(p) = 6a2µ2. Thus, for the flat isotropic distributions, we have a2

A2

a2µ2 = 0,

= 0.

(5.6.17)

Each of equations (5.6.17) has two double roots:

Al=A2=0, A3=1\4=0o and µl={12=0, µ3=µ4= 00. Hence, on a manifold endowed with a conformal structure defined by the metrics (5.6.12) and (5.6.16), through every point there pass two pairs of totally isotropic submanifolds corresponding to the following values of the parameters A and p:

A=p=0 and A=p=oo.

For the Schwarzschild metric, integrability of isotropic distributions on the isotropic fiber bundle Ea, which corresponds to the values A = 0 and A = 00 of the parameter A, can be immediately verified by applying formulas (5.6.13). In the same way one can verify integrability of isotropic distributions on the isotropic fiber bundle Ep.

5.7

Conformal Structures on a Four-Dimensional Hypersurface

1. As we have shown in Section 4.3, on a tangentially nondegenerate hypersurface V^ of a projective space P1+1, a conformal structure which is called the asymptotic pseudoconformal structure is induced by its relatively invariant quadratic form 'P(2) = bijw'w1,

(5.7.1)

which is defined in a second-order neighborhood of V'. If n = 4, the quadratic form f(2) can have the signatures (2, 2), (1, 3), and (4, 0). Suppose that the signature is constant at all points x E V4. Then, in the first case, a conformal structure induced on V4 is the CO(2, 2)-structure; in the second case, it is the CO(1, 3)-structure; and in the third case, it is the CO(4)structure. Geometrically hypersurfaces of these three types are distinguished from one another by the fact that in the first case, their asymptotic cones C= carry two one-parameter families of two-dimensional plane generators, in the second case, they carry a two-parameter family of rectilinear generators, and in the third case, these cones are imaginary. Hypersurfaces V4 of these three kinds are called ultrahyperbolic, hyperbolic, and elliptic, respectively.

5.7

Conformal Structures on a Four-Dimensional Hypersurface

209

In this section, we will study ultrahyperbolic hypersurfaces V4 of the space P5 and the asymptotic CO(2, 2)-structure associated with them in more detail, find conditions of their conformal semiflatness and flatness, investigate some properties of their principal isotropic distributions, and construct examples of hypersurfaces V4 carrying a conformally semiflat and conformally flat CO(2,2)-structure. Let us refer a hypersurface V4 C P5 to the family of isotropic cones relative

to which the equation of the family of asymptotic cones C= takes the form (5.1.1): g = 2(W1w4 - W2W3) = 0.

Consequently the tensor b,j has the following components: 0

(bi3) =

0

0

1

-1 0

-1

1

0

0

0 0

(5.7.2)

Using formulas (4.4.21), (5.1.13), and (5.7.2), we find the components of the tensor of conformal curvature of the CO(2, 2)-structure: ao = (B222BI13 - 3B122B123 + 38112B223 - B111 B224), g

al = ,1 (B222B133 - 3B122B233 + 3B112B234 - B111B244), a2 = 24 (B222B333 - 38122B334 + 3BI12B344 - B111B444),

(5.7.3)

a3 = 1(B224B333 - 3B223B334 +3B123B344 - B113B444), a4 = 18'(B244B333 - 3B234B334 + 3B233B344 - B133B444), bO = 18'(B333B112 - 3B133B123 + 3B113B233 - B111 B334),

b1 = lg(B333B122 - 3B133B223 + 3B113B234 - B111 B344),

b2 = 2(B333B222 - 3B133B224 + 3B113B244 - B111B444),

(5.7.4)

b3 = g (B334 B222 - 3B233B224 + 3B123B244 - B112B444),

b4 =

g

(B344 B222 - 3B234 B224 + 3B223B244 - B122 B444)

are the independent components The quantities a and b, u = 0,0,1,2,3,4, 1, of the tensors C. and Cp, and the quantities B,Jk are the components of the Darboux tensor of the hypersurface V4 defined by equations (4.4.10). Let us consider the (4 x 4)-matrix B112

BI22 B222

B113 B123 B223 B224

B133 B233 B234 B244

B333 B334 B344 B444

(5.7.5)

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

210

which is composed of 16 essential components of the Darboux tensor B{jk of the hypersurface V4. Let us denote the ith column-vector of the matrix 13 by Li and its jth row-vector by M,,. In addition we consider the skew-symmetric bilinear form K(X, Y) = KijX'Y' determined by the matrix

(Ki j ) =

0 0 0 1

0 3

-1 0

-3 0

0

0

0

0

0

0

(5.7.6)

In accordance with these notations, formulas (5.7.3) and (5.7.4) for the components of the subtensors Ca and Co take, respectively, the following forms: ao = 1 K(LI, L2),

at = 18K(L1,L3),

a2 = 2K(LI,L4),

(5.7.7)

a3 = 16 K(L2, L4),

a4 = I K(L3, L4) and

bo = ! K(MI, M2),

bI = 18K(MI,M3), b2 = 2K (MI, M4),

(5.7.8)

b3 = 16K(M2,M4

b4 = 8K(M3,M4) Hence the asymptotic pseudoconformal CO(2, 2) -structure on the hypersur-

face V4 is semiflat if and only if one of the following two sets of conditions holds:

K(LI, L2) = 0, K(LI, L3) = 0, K(LI, L4) = 0, K(L2, L4) = 0, K(L3, L4) = 0 or

K(MI, M2) = 0, K(MI, M3) = 0, K(MI, M4) = 0,

(5.7.10)

K(M2, M4) = 0, K(M3, M4) = 0.

This structure is flat if and only if conditions (5.7.9) and (5.7.10) hold simultaneously. 2. We will find the geometric structure of completely isotropic submanifolds

of the CO(2, 2)-structure induced on a ultrahyperbolic hypersurface V4 C P5 by its second fundamental form 0121.

5.7

Conformal Structures on a Four-Dimensional Hypersurface

211

Suppose that on V4 there exists a completely isotropic a-submanifold V2(\) of this CO(2, 2)-structure. This submanifold is determined by the system of differential equations (5.1.2): WI + AW1=0, W2 + AW4 = O.

(5.7.11)

As we noted in Section 5.4, the parameter A satisfies differential equation (5.4.6), which by virtue of (5.4.5) and (4.4.19) takes the following form on V4: da +.\(W4 + W3) - W3 + \2W - I (B112)13 - 3B123\2 + 3B233 \ - B334)W3 (B122A3 - 3B223A2 + 3B234A - B344)W4 = 0.

(5.7.12)

Let x = A0 be a point of the hypersurface V4. We calculate the differentials dAOi d2A0, and d3A0 of this point as it moves along the completely isotropic a-submanifold V2(A). Since equations (5.7.11) hold on the submanifold V2(A), we have dA0 =w0Ao+W3(A3-.\A1)+W4 (A4 - AA2) =W4A0+W3B3+W4B4. (5.7.13) The forms W3 and w4 are linearly independent on the submanifold V2(A), and

the points B3 = A3 - AAl and B4 = A4 - .A2 lie in its tangent 2-plane. Differentiating twice (5.7.13) and applying (5.7.12), we see that d2 A0 = 2(S2(W3)2 +2S3W3W4 +b4( .4)2)A1

I

(w3)2

2

+ 2C2W3W4 + b (w4)2) A2

(mod TAo(V2(A))), (5.7.14)

d3A0 =

2

(S1 (W3)3 +

+313W3(W4)2

(5.7.15)

S4( 4)3)A5

(mod Ti2i(V2(A)))r

where T, (V2 (A)) is the tangent 2-plane to the submanifold V2(A), T(2)(V2(A))

is its osculating subspace at the point x = Ao, and the quantities l;; are computed in the following way: ttl = B111)13 - 3B113A2 + 3B133A - B333,

S2 = B112\3 - 3BI23A2 + 3B233I\ - B334,

(5.7.16)

y3 = B122\3 - 3B223\2 + 3B234A - B344, S4 = B222

A3

- 38224\2 + 3B244A - 8444

If we denote the column-vector with the components t, by {(A), then the last formulas can be written as (A) = L1)3 - 3L2A2 + 3L3A - L4.

(5.7.17)

212

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

It follows from relations (5.7.14) and (5.7.15) that the osculating subspace T.(2)(V2(,\)) coincides with or belongs to the four-dimensional tangent subspace T (V4) of the hypersurface V4: T(2)(V2(A)) C T (V4). Relations (5.7.14) and (5.7.15) also imply that the submanifold V2(A) has two second fundamental forms, 1

0(2)(A)

=

0(2)(A)

_ -2(Sl(w3)2

+2e3W3W4 +1;4(W4)2), (5.7.18)

and one third fundamental form, 0(3)(A) =

(5.7.19)

It is easy to verify that the form 1&(3)(A) is proportional to the restriction of the Darboux form P(3) of the hypersurface V4, defined by equation (4.3.12), to the submanifold V2(A). Similar results can be obtained when one considers completely isotropic , 3-submanifolds V2 (µ) determined on V4 by system (5.1.3): wt + µw2 = 0, w3 + µw4 = 0.

(5.7.20)

The 1-forms w2 and w4 are basis forms on V2(p), and the points BZ = A2-pA1 and B4 = A4 - pA3 lie in its tangent 2-plane. The second fundamental forms of such submanifolds have the form 'Y(2)(p) = 2 (rl2(w2)2 + 27)3w2w4 + 174(W4)2)203

(5.7.21)

(1,) = -2(111(W2)2+2172w2W4+113(W4)2),

and their third fundamental form G(3)(µ)

=

1

W4 3 w4 + 3,73W2(W4)2 + l4( ) ) (0l(,d2)3 + 3rl2(W2)2

(5.7.22)

is proportional to the restriction of the Darboux form x+(3) of the hypersurface V4, defined by equation (4.4.12), to the submanifold V2(p). The quantities rl; occurring in equations (5.7.21) are the components of the vector 17(tt) = M1µ3 - 3M2p2 + 3M3p - M4.

(5.7.23)

The following theorem describes the geometrical structure of completely isotropic submanifolds V2 of a hypersurface V4:

Theorem 5.7.1 If a hypersurface V' carries two-dimensional completely isotropic submanifolds V2, then they have one of the following structures: 1. V2 is a two-dimensional plane.

5.7

Conformal Structures on a Four-Dimensional Hypersurface

213

2. V2 is a two-dimensional developable surface, that is, a tangentially degenerate submanifold carrying a family of rectilinear generators, along each of which the tangent subspace to V2 is constant. The osculating subspaces of such submanifolds are three-dimensional.

3. V2 is a tangentially nondegenerate submanifold carrying a family of rectilinear generators. The four-dimensional osculating subspace of such a V2 coincides with the tangent hyperplane of V4. 4. V2 is a tangentially nondegenerate submanifold having four-dimensional osculating subspaces and carrying a conjugate net. (Such submanifolds are called the Carton varieties.) Proof. Let V2(A) be a completely isotropic a-submanifold of a hypersurface V4, and let 0(2) (A), 0(2) (A), and 0(3)(A) be its second and third fundamental forms, respectively. If the forms 0(2) (A) and ¢(2) (A) are identically equal to zero, then by virtue of the relation (5.7.14), V2(A) is a plane generator of the hypersurface V4, so we arrive at the case 1. Suppose that the forms 0(2)(A) and 0(2)(A) are not equal to zero but are proportional. Then from (5.7.18) it follows that l;z S1

=3 = l a 1;2

=k

1;3

By virtue of this expression, we have

0(2)(A) = 2k.i(w3 +kw4)2 and 0(2)(A) = -21:1(w3 +kw4)2, which shows that each of these forms is proportional to the perfect square of the same linear form. Hence V2(A) is a developable surface. Moreover, according to (5.7.14), the osculating subspace of the submanifold V2(A) is three-dimensional, so we arrive at the case 2. If the forms 0('2)(A) and ¢(22) (A) are not proportional but have a common

linear factor, then by virtue of the equation 0(3) (A) =

(A) - w34'(2) (A)

this factor will be a divisor of the third fundamental form 10(3) (A) too. One can prove that this implies that the submanifold V2(A) carries a family of rectilinear generators (e.g., see Akivis and Goldberg [AG 93], p. 235). Thus we arrive at the case 3. Finally, if the forms 0(2)(A) and 4(2)(A) have no linear common factor, then they can be simultaneously reduced to sums of squares, which means that the

submanifold V2(A) carries a conjugate net. Thus the osculating subspace of V2(A) is four-dimensional (see Akivis and Goldberg [AG 93], p. 75), so we arrive at the case 4. Similar arguments hold for completely isotropic submanifolds V2(µ).

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

214

Corollary 5.7.2 A completely isotropic submanifold V2 of a hypersurface V4

lies in its osculating subspace if and only if V2 is a plane generator of the hypersurface V4.

Proof. If V2(A) lies in its osculating subspace, then, according to (5.7.15), we have 0131(A) = 0. Then r;; = 0, i = 1,2,3,4, and 0(2)(A) = 0'(2)(A) = 0; that is, V2(A) is a 2-plane. Similar arguments hold for completely isotropic submanifolds V2(µ). U

Corollary 5.7.3 Let V4 be a hypersurface of a projective space P5 whose second fundamental form reduces to 4D(2) = 2(wlw4 _ w2w3), and let t:(A) be the polynomial defined by formula (5.7.17). Then the following conditions are equivalent:

1. The parameter A in equation (5.7.11) is a root of the algebraic equation VA) = 0.

(5.7.24)

2. The hypersurface V4 carries a two-parameter family of two-dimensional plane generators, which are determined by the system of differential equations (5.7.11).

3. The restriction of the cubic Darboux form W(3) of the hypersurface V4, defined by equation (4.4.12), to the two-dimensional distribution (5.7.11) is the null form. Proof. Each of three conditions of this theorem is equivalent to the fact that on the submanifold V2 (A) the quantities r;;, i = 1,2,3,4, defined by (5.7.16), vanish.

Corollary 5.7.4 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there passes a two-dimensional plane generator, then one of the polynomials C, (A) or C,(p) of the asymptotic CO(2,2)-structure on V4 has a triple root. Proof. Suppose that the family of two-dimensional plane generators on the hypersurface V4 is determined by the system of equations (5.7.11). We consider an admissible transformation of the adapted frame, determined by the relations: to)

-4 W' + \W3, W2 --1 W2 + .\W4, W3 -1 W3, W4 - W4.

In the new frame the parameter A vanishes, and condition (5.7.24) becomes L4 = 0. Taking this equation into account, we deduce from formulas (5.7.7) that a2 = a3 = a4 = 0. Therefore the equation (5.4.7) has a triple root A = 0.

5.7

Conformal Structures on a Four-Dimensional Hypersurface

215

One can easily see that a hypersurface of the kind described in Corollary 5.7.4 can be given by the parametric equation: M(u1, u2, u3, u4) = Mo(ul, u2) + u3M1 (ul, u2) + u4M2(ul, u2),

(5.7.25)

where M, M0, M1 and M2 are points of the projective space P5. The twodimensional plane generator u1 = c1, u2 = c2 , where cl and c2 are constants, pass through every point of this hypersurface. 3. Next we will prove the following result:

Theorem 5.7.5 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there pass two two-dimensional plane generators that are in general position, then the semiflat asymptotic CO(2,2) -structure is induced on the hypersurface V4.

Proof. Since the hypersurface V4 is tangentially nondegenerate and carries a family of two-dimensional plane generators, the second fundamental tensor of V4 has signature (2, 2). Moreover, since through any point of V4 there pass two of its plane generators that are in general position, both generators belong to the same family of isotropic planes of the isotropic cone C.; both of them are either a-planes or Q-planes. But then by Corollary 5.7.4, one of the polynomials Ca(\) or Cf(p) must have two distinct triple roots. Thus, since these polynomials are of fourth degree, one of them is identically equal to 0. Therefore, if these plane generators are a-planes, then Ca(J1) = 0, and the asymptotic CO(2, 2)-structure on V4 is anti-self-dual. If these plane generators are

13-planes, then C0(µ) = 0, and the asymptotic CO(2, 2)-structure on V4 is self-dual. Hence the hypersurface V4 carries a semiflat asymptotic CO(2, 2)structure. As we have proved in Theorem 5.4.2, a four-dimensional manifold with a semiflat CO(2, 2)-structure carries a three-parameter family of completely isotropic two-dimensional submanifolds. These submanifolds are a-submanifolds if the CO(2, 2)-structure is anti-self-dual, and they are Q-submanifolds if the CO(2, 2)-structure is self-dual. Thus, through any point of the hypersurface V4 described in Theorem 5.7.5, there passes a one-parameter family of completely isotropic two-dimensional submanifolds. The plane generators of the hypersurface V4 belong to this family. It follows from Theorem 5.7.1 that other completely isotropic two-dimensional submanifolds are developable surfaces or ruled surfaces or two-dimensional Cartan varieties. If all isotropic a-submanifolds are planes, then t (A) - 0, and by (5.7.16), all components B,Jk = 0; consequently the hypersurface V4 is a hyperquadric. Then all 13-submanifolds are planes, and the CO(2, 2)-structure on V4 is conformally flat. Theorem 5.7.5 allows us to describe some classes of hypersurfaces in the space P5 that carry a semiflat conformal CO(2, 2)-structure. We will find now the closed form equations of hypersurfaces V4 C P5 carrying two families of two-dimensional plane generators that are in general posi-

216

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

tion. According to Theorem 5.7.5, such hypersurfaces carry semiflat conformal CO(2,2)-structures.

Let us consider in the space P5 nine points Mfg, f,g = 0,1,2, in general position, and let (u°, u1, u2) and (v°, VI, v2) be two sets of variables, each of which is defined up to a common factor. The points Mfg determine a hypersurface

pM = ufvgMfg,

f, g = 0,1, 2.

(5.7.26)

This equation can be written in two ways: pM = v°(u°Moo + u' M10 + u2M2o)

+v'(u°MD1 +u1M11 +u2M21) +v2(u°Mo2 + u'M12 +U2 M22), pM = u°(v°M00 + v' Mlo + v2M2o) +u' (v°Moi + v' Mi i + v2M21) +u2(v°M02 + v' M12 + v2M22).

(5.7.27)

This means that the hypersurface V4 carries two families of plane generators

in general position: of = cft and vg = c9s where cf and cg are arbitrary 2 2 1

1

constants. Since these generators have only one common point, they belong to the same family of the isotropic cone C.. Thus the hypersurface V4 carries a semiflat asymptotic pseudoconformal CO(2, 2)-structure. If we assume that the nine points Mfg occurring in equations (5.7.26) lie in the space Ps and are linearly independent, then equation (5.7.26) determines a Segre variety in Ps which is the embedding of the direct product p2 X P2 of two projective planes. The hypersurface V4 constructed above is a central projection of the Segre variety p2 X P2 C Ps onto a five-dimensional space p5 from a two-dimensional projection center Z. In addition, the points Mfg and a two-dimensional center Z of projection must be in general position. 4. Theorem 5.7.5 implies another important result.

Theorem 5.7.6 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there pass four two-dimensional plane generators, and two of them belong to one of the families of isotropic planes of the asymptotic cone of V4, while the other two belong to its second family of isotropic planes, then the flat asymptotic CO(2,2) -structure is induced on V4. 0 Applying this theorem, we will construct an example of a hypersurface with a flat CO(2, 2)-structure. The equation of such hypersurfaces is obtained from (5.7.26) in the case where the terms containing the points MI, and M22 are absent. Indeed, in this case pM = u°v°Moo + u°v1Mo1 + u°v2M02 + u'v°M1o +ui v2M12 + u2v°M20 + u2v' M21,

(5.7.28)

Notes

217

and each of the four pairs of equations ul : u° = Cl, u2 : u° = c2i V1 : V 0 = c3, v2 : v° = C4; ut : u° = cl, vi : v° = 62; and u2 : u° = c3,v2 : v° = E4, where ci and c';, i = 1, 2, 3, 4, are constants, determines a family of two-dimensional plane generators on V4.

The hypersurface (5.7.28) can also be obtained from the Segre variety P2 X P2 C Ps by projecting onto a five-dimensional space P5 if the twodimensional projection center Z contains the points M1I and M22. Let us note that the conformally flat hypersurface (5.7.28) is not a second order envelope of a one-parameter family of hyperquadrics considered in Subsection 4.4.3, and is not a hyperquadric. 5. Finally let us consider the hyperquadric defined in a projective coordinate system by the equation

X°X5+X'X4+X2X3=0.

(5.7.29)

This hyperquadric can also be obtained from the Segre variety p2 X P2 C P8 by projecting onto a five-dimensional space P5 if the two-dimensional projection center Z contains the points M11 and M22 and the straight line M11 A M22 intersects the center Z. As mentioned above, a flat asymptotic pseudoconformal CO(2, 2)-structure can be realized on the hyperquadric (5.7.29). The hyperquadric (5.7.29) is the unique conformally flat hypersurface, all completely isotropic submanifolds of which are two-dimensional plane generators.

NOTES 5.1-5.2. The study of conformal structures on a four-dimensional manifold is of special interest because of their close relation with the theory of gravitation. Spacetime in general relativity is a four-dimensional Riemannian manifold of signature (1, 3). Since many features of general relativity are of a conformally invariant nature, it is interesting to study pseudoconformal structures of signature (1,3). Along with these kinds of conformal structures, one also can consider conformal structures of signatures (4, 0) and (2, 2). By means of complexification of the manifold M, all

these structures can be reduced to one of them, for example, to the structure of signature (4, 0) or (2, 2). This has been done in many investigations (e.g., see Atiyah, Hitchin, and Singer (AHS 78], Gindikin [Gin 82, 83], Manin [Man 84], and Penrose [P 66, 77]).

The contents of Sections 5.1 and 5.2 are connected with the contents of the well-known paper by Atiyah, Hitchin, and Singer [AHS 78] who considered the decomposition of the Weyl tensor into its self-dual and anti-self-dual parts.

There is also a close relation between these sections and the twistor theory. Twistors were introduced in Penrose (P 67, 68a] (see also Penrose [P 77]). The twistor approach is based on associating a complex manifold Z ("the space of twistors") with a real manifold M endowed with a certain geometric structure and reformulating geometric problems posed for M in terms of the holomorphic geometry of the space Z. This approach has been effectively applied to the solution of a number of problems in

218

5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES

geometry and mathematical physics (e.g., see Atiyah, Hitchin, and Singer (AHS 781 and Manin [Ma 84]). There are different ways to define the space of twistors. R. Penrose defined twistors by developing a rather complicated algebraic apparatus in which he used a system of abstract indices, the spinor calculus, and other objects (see Penrose (P 68b, 76, 77) and Penrose and Rindler (PR 86]). Another method is to define twistors using the theory of C-structures (e.g., see Alekseevskii and Graev [AGr 92a, b; 93], Rawnsley (Raw 871, and Wells [Wet 79]). The isotropic fibrations which we consider are essentially twistor fibrations. How-

ever, we refrain from using this term since it is used in geometry and physics with many different meanings (see the references in the above paragraph) while on fourdimensional conformal and pseudoconformal structures, the term "the isotropic fibrations" has a unique meaning. In twistor theory the a- and /3-planes are defined as the proportionality class of a twistor. If one defines the projective twistor space PZ as the projective version of the twistor space Z, then the points of PZ correspond to a-planes. In a similar manner the points of PZ' correspond to 3-planes (e.g., see the book Huggett and Tod [HD 85], p. 56). In physics the adapted frames are called light tetrads or Newman-Penrose tetrads (see Newman and Penrose (NP 62], Penrose [P 68], Penrose and Rindler (PR 86], or Chandrasekhar (Cha 83]). Note one more time that while for the CO(2, 2)- and CO(4)-structures the tensor C of conformal curvature splits into two subtensors C. and Cs, for the

C(1, 3)-structures, there is no such splitting since for these structures we have Co = C,,, and the subtensors Co and Co are two different complex representations of the same real tensor of conformal curvature. Note also that the real theory of four-dimensional Riemannian and pseudo-Riemannian metrics of different signatures and its applications to general relativity and the theory of superstrings were considered in the recent paper Barrett, Gibbons, Perry, Pope, and Ruback [BGPPR 94]. However, in this book we do not consider systematically physical applications of the geometric theories we have constructed. We only note that such applications are possible and supply some references on this matter. The results of these sections are due to Akivis and Konnov [AK 93], §3, and Akivis (A 96).

5.3. The Hodge operator, or the *-operator, is well-known in the theory of harmonic functions on Riemannian manifolds (see Hodge [Hod 41] and also de Rham [Rh 55], §§24--25; Wells [Wel 80], Ch. 5, §1). The Hodge operator on four-dimensional Riemannian manifolds was studied in detail by Atiyah, Hitchin, and Singer [AHS 78)

who introduced the notions of self-dual and anti-self-dual subspaces defined by the Hodge operator. The fact that the Hodge operator is conformally invariant was noted in LeBrun [LeB 82]. In the last paper as well as in Atiyah, Hitchin, and Singer [AHS 781, the Hodge operator was considered in an even-dimensional complex Riemannian manifold whose metric tensor can be reduced to the form g(e;, e,) = 5,,. To our knowledge, the theory of Hodge operator on four-dimensional conformal structures of different real types was considered in this book for the first time (see the forthcoming paper Akivis [A 96]). In particular, it is true for properties of the spectrum of the Hodge operator for conformal structures of different real types and also

for the fact that unlike the CO(2, 2, )- and CO(4)-structure, the CO(1, 3)-structure

Notes

219

cannot be self-dual or anti-self-dual, that is, semiflat without being conformally flat; this fact is of great importance for general relativity. 5.4. The results of this section are due to Akivis and Konnov (AK 93], §2, for CO(2, 2)-structures (see also Akivis [A 83a] and Konnov [Kon 92a]) and Akivis and Zayatuev [AZ 95]) for CO(1, 3)- and CO(4, 0)-structures. See also the forthcoming paper Akivis [A 96]. The classification of the Einstein spaces, which was first found by Petrov (Pe 54] (see also Pirani [Pir 57]), is given in detail in many books in general relativity (e.g., see the books Petrov [Pe 69], §§18-20, Chandrasekhar (Cha 831, Ch. 1, §9, and Penrose

and Rindler [PR 86], Ch. 8). Note that Petrov included types I, D, and 0 (for the last one all coefficients ai = 0) in type 1 of his classification, types II and N in type 2, and type III in type 3. The isotropic directions, which following Chandrasekhar [Cha 83] we called prin-

cipal, were already considered by It. Cartan (Ca 22b] who called them the optical directions. Cartan also indicated that for the Schwarzschild metric these four optical directions are reduced to two double directions. The importance of these directions was recognized in the 1950s after F. Pirani [Pir 57) indicated the physical significance of the Petrov classification. The connection of principal isotropic directions with the Petrov classification was also noted in the book Penrose and Rindler (PR 86], Ch. 8. 5.5. A CO(2, 2)-structure is associated with a three-web W(3, 2, 2) of codimension two given on a four-dimensional manifold. This fact was noted by Akivis in [A 83a] who also considered the CO(2, 2)-structure associated with the set of nondegenerate null-pairs in the real projective plane RP2. G. A. Klekovkin applied the properties of the CO(2, 2)-structure to study webs W(3, 2, 2) (see Klekovkin [Klk 81b, 83, 841). In Goldberg [Go 85, 86, 87, 88], the CO(2, 2)-structures connected with threewebs were used in the construction of maximum rank four-webs W(4, 2, 2) on a fourdimensional manifold obtained as an extension of a three-web W(3, 2, 2). 5.6. As we noted in the text, the Kerr metric is a solution of Einstein's equation. This solution was found by R. P. Kerr (see Kerr [Ke 63] and Kerr and Schild [KS 65])

and was studied in many papers. A significant part of the book Chandrasekhar [Cha 83] is devoted to the investigation of this metric. The Schwarzschild metric is a spherically symmetric solution of the Einstein equation in empty space. A derivation of this solution was first published by K. Schwarzschild [Schw 16a, 16b] shortly after A. Einstein found the fundamental equations (equations (5.4.21)) of general relativity in Einstein [Ein 161. A. Einstein highly praised Schwarzschild's results. The Reissner-Nordstrom metric is a spherically symmetric solution of the system of Maxwell-Einstein equations. It describes a black hole with a mass m and a charge Q. This solution was first presented in the papers Reissner (Re 16] and Nordstrom (No 18] and was independently derived by these two authors. The results of this section are due to Akivis and Zayatuev (AZ 95]. 5.7. The results of this section are due to Akivis and Konnov [AK 93] (§4) (see also Konnov [Kon 92a]).

Chapter 6

Geometry of the Grassmann Manifold The Grassmann manifold, or the Grassmannian, is the set of m-dimensional subspaces of an n-dimensional projective space P". It is denoted by the symbol G(m, n)' . The geometry of Grassmannians was studied in many books and papers (e.g., see Klein [KI 26b] and Hodge and Pedoe [HP 47, 52]). However, the differential geometry of Grassmannians has been considered only in a few papers (see Akivis [A 82b], Leichtweiss [Le 61], and Wong [Won 67]).

The geometry of Grassmannians is closely connected with the geometry of conformal and pseudoconformal spaces. As we have already seen in Section 1.4, the Grassmannian G(1, 3) is endowed with the structure of the pseudoconformal space C. In the same way as one can naturally pass from the geometry of conformal and pseudoconformal spaces to the study of conformal and pseudo-

conformal structures, the geometry of Grassmannians can be extended to the study of almost Grassmann structures. Moreover the same methods that were used in conformal differential geometry can be applied to the study Grassmann and almost Grassmann structures. On the other hand, the theory of Grassmannians is connected with many other branches of geometry and mathematics. This theory finds applications in the theory of multidimensional webs, integral geometry, the theory of hypergeometric functions, mathematical physics, and so on. For this reason we devote a separate chapter to the study of differential geometry of Grassmannians. I Note that sometimes the Grasemannian Gr(m, n) is defined as a set of m-dimensional subspaces of an n-dimensional vector space. The Graasmannian G(m, n) we have defined can

be obtained by the projectivization of that Grassmannian: G(m,n) = PGr(m+1,n+1). The only difference between these two Grassmannians is that the dimensions of their generating elements differ by one. 221

222

6. GEOMETRY OF THE GRASSMANNIAN

6.1

Analytic Geometry of the Grassmannian and the Grassmann Mapping

1. Consider a set of m-dimensional subspaces Pm of an n-dimensional projective space P". We will prove in this section that this set is a differentiable manifold. This is the reason that this set is called the Grossmann manifold or, in short, the Grassmannian. We will denote it by G(m, n). Let us define coordinates on the Grassmann manifold. An element of the Grassmann manifold G(m, n), a subspace Pm, can be given by means of m + 1 linearly independent points xo, x1 , ... , x,,,. We will assume that a projective frame {Ao, A1, ... , A. } is fixed in the space P". In Chapters 6 and 7 we will use the following index ranges:

0 < a, A, 7, d, e, a < m;

m+1
xa = xaAt, where x , are the coordinates of these points. These coordinates form an (m + 1) x (n + 1) matrix xo0

xo1

X

XI

xo xI

°

xo... yi......

M

X..

"/

m

m

Consider the minors of order m + 1 of this matrix: a0

x0

pa0al ...am

a,

x0

...

a x0

ao

xl xlal ... xoam ......................

xa0 in

xal

m

... xam

The subspace Pm can be given by means of another system of m + 1 linearly independent points ya = aaxp, a = det(aQ) 96 0.

The matrix of the coordinates of the points ya also determines the minors gaoal...am of order m + 1. Using the rule of multiplication of determinants, it is easy to prove that gaoa,...am = apaoal...am.

In view of this, the numbers paoal...am can be taken as homogeneous coordinates of the subspace P1. They are called the Grassmann coordinates of

6.1

Analytic Geometry of Grassmannian and Grassmann Mapping

223

the subspace P1. The number of these coordinates is equal to the number of minors of order m + I in matrix (6.1.1); that is, it is equal to (m+i). Thus the Grassmann coordinates can be considered as coordinates of a point of the projective space PN, where N = ( +i) - 1. The mapping of the Grassmann manifold G(m, n) into the projective space PN defined by means of the Grassmann coordinates is called Grassmann mapping. However, the Grassmann coordinates are not independent. They are connected by a series of quadratic relations of the form m+1

p0I0z...0-0apaoai...aa-103+I0h+I =0,

(6.1.2)

a=o

where al, a2, .... am are m distinct numbers chosen from the set 0, 1, ... , n, and ,0o, X131 i ... , Qm+1 are m + 2 distinct numbers taken from the same set (see Hodge and Pedoe [HP 47], Ch. VII, §6, Eq. (2)). The above quadratic relations determine an algebraic variety fl(m, n) in the space PN. In view of this, the Grassmann mapping establishes a one-to-one correspon-

dence between subspaces Pm C Pn and points of the variety fl(m,n) C PN.

We denote the subspace that is spanned by the points xo,x1,...,xm by xo A xl A ... A xm = p, and by the same letter p we will denote the point on fl(m, n) corresponding to this subspace. Note that if n = 3 and m = 1, then the variety fl(m, n) becomes the Pliicker hyperquadric in the space p5. We have already considered this hyperquadric in Section 1.4. 2. Let us prove now that the manifold G(m, n) is a differentiable manifold of dimension (m+ 1)(n-m). Consider a fixed coordinate system (Ao, A1,. .. , An} indicated above, the subspace AO A ... A Am, and all subspaces p that do not intersect the complementary (n - m -1)-dimensional subspace Am+1 A ... A An. A subspace p intersects each of the subspaces Aa A Am+1 A ... A An at a unique point xQ V Am+1 A ... A An: x, = p n (A,, A Am+1 A ... A An). Hence xQ = xQAa + x' Ai,

where there is no summation with respect to the index a and x4 i4 0 since xo Am+1 A... A An. Thus, since the coordinates are homogeneous, we can set x4 = 1. The subspace p is uniquely determined by the points x0, and the matrix of coordinates of these points becomes 1

0

...

0

1

...

0

0 xo +1 0 xi +1

..0 ...... i x'+1

... ...

xo xn

...

xm

The numbers xa are nonhomogeneous coordinates of the subspace p not inter-

secting the subspace Am+l A...AA.. Their number is equal to (m+1)(n-m).

224

6. GEOMETRY OF THE GRASSMANNIAN

Note that the subspace Ao A ... A An has the coordinates xQ vanishing: xQ = 0.

The set of subspaces not intersecting the subspace Am+l A ... A A. admits a one-to-one mapping onto an open neighborhood of 0 of the space of dimension (m + 1)(n - m). We denote this neighborhood by U01...m. In the same manner we can construct a neighborhood Uao«,...a for any subspace A00 A ... A A._ belonging to the coordinate simplex. These neighborhoods cover the entire set G(m, n). Hence this set is a manifold of dimension p = (m + 1)(n - m). It is possible to prove that the formulas relating nonhomogeneous coordinates of any two intersecting neighborhoods U0001 ..Q,, and Upop,...p, are defined by differentiable functions; that is, the manifold G(m, n) is differenR('"+t)(n-m)

tiable.

In algebraic geometry it is proved (e.g., see Severi (Sev 151, or Harris [Har 92), p. 247) that the degree of the algebraic variety il(m, n) in the space PI on which the Grassmann manifold is mapped is equal to deg fl(m, n) = p! U

a!

(6.1.3)

=o

Note that we can always assume that m < n21 since the geometry of the Grassmann manifold G(m, n) is equivalent to the geometry of the Grassmann manifold G(n - m - 1, n). The passage from one of these Grassmannians to the other one is realized by means of a correlation of the space P". 3. Let us study now the structure of the Grassmannian G(m, n) in more detail. Let p and q be two m-dimensional subspaces in P" meeting in the (m - I)-subspace P"`-1. They generate a linear pencil of m-subspaces Ap+µq. A rectilinear generator of the algebraic variety fl(m, n) corresponds to this pencil. All m-dimensional subspaces of this pencil belong to an (m+1)-subspace

pm+t and the pair of subspaces P'"-1 C pm+t completely determines this pencil and consequently a straight line on the variety il(m,n). We consider the set of all m-subspaces P'" C P" passing through a fixed (m-1)-subspace P'"-1. These subspaces form an (n - m)-bundle. On the variety 1l(m, n), there corresponds to this bundle an (n-m)-dimensional plane gen-

erator an-,. The subspaces P'"-1 C P" form the Grassmannian G(m - 1, n) whose dimension is equal to m(n - m + 1). Hence the variety fl(m, n) carries a family of generators a"-"n that depends on m(n - m + 1) parameters. Let Pni+t be a fixed subspace in P". We consider all m-dimensional subspaces p in Pm+1. They form the Grassmannian G(m,m + 1) of dimension m + 1 which is the projective space (P"'+1)* dual to On the variety P'"+l.

fl(m, n), to G(m, m + 1) there corresponds an (m + 1)-dimensional plane generator 0'"+1. Since dim G(m + 1, n) = (m + 2)(n - m -1), the variety fl(m, n) carries a family of plane generators /3'"+1 which depends on (m + 2)(n - m -1) parameters. If the subspaces P'"-1 and Pm+t of the space P" are incident (i.e., they satisfy the condition PI-1 C Pm+1), then the corresponding plane generators

6.1

Analytic Geometry of Grassmannian and Grassmann Mapping

225

an-m and ,61+1 of the variety 11(m, n) meet in a straight line. If they are not incident, then an-m and pm+l have no common points. Let us consider a fixed subspace p of dimension m in Pn. It contains an m-parameter family of subspaces Pm-1.. Therefore the family of generators an-m depending on m parameters passes through the point p E fl(m, n) corresponding to it. In addition an (n - m -1)-parameter family of subspaces Pm+1

passes through the same subspace p in Pn. Consequently an (n - m + 1)parameter family of generators (jm+l passes through the point p E 11(m, n). Furthermore any two generators an-m and ,6m+1 passing through p E fl(m, n) meet in a straight line. It follows that all the plane generators an-m and (fm+1 passing through p E SZ(m, n) are plane generators of some cone Cp(n-m, m+1) with vertex at p, and this cone lies on fl(m, n). The cone Cp is the intersection of the variety fZ(m, n) and its tangent space Tp(fZ), dim Tp(fZ) = p, where p = (m + 1)(n - m). Consider the projectivization of the cone C. with center at the point p, namely the projection of Cp from p onto a projective space PP-1 of dimension p -1. Under this projectivization the image of the cone Cp(n - m, m + 1) is the algebraic variety S(n - m -1, m) of dimension n -1 which carries two families of plane generators of dimensions n - m -1 and m. The variety S(n - m -1, m) is called the Segre variety, or shortly, the Segrean. The Segre variety S(k, l) is an embedding of the direct product Pk x P1 of projective spaces Pk and PI of dimensions k and I into a projective space of dimension (k + 1)(I + 1) - 1 = kl + k + 1. Analytically this embedding can be written by means of the following equations:

4 = tps0,

a = 0,1,...,k; p= 0,1,...,l,

(6.1.4)

where tp and as are homogeneous coordinates in the spaces Pk and P', respectively. These equations are equivalent to the condition rank (x4) = 1.

(6.1.5)

The Segre variety S(k, 1) has the dimension k + 1: dim S(k,1) = k + 1.

It is proved in algebraic geometry that the degree of this variety is deg S(k,l) =

The cone Cp(n - m,m + 1) whose projectivization is the Segre variety S(n - m - 1, m) is called the Segre cone. In the space Pn the set of all m-dimensional subspaces intersecting a fixed subspace p of dimension m in (m - 1)-dimensional subspaces corresponds to the cone C,,. Each of the m-subspaces together with the subspace p belong to an (m + 1)-subspace. The dimension of the cone C,, is equal to n + 1.

6. GEOMETRY OF THE GRASSMANNIAN

226

4. As we saw in Subsection 6.1.1, an m-dimensional subspace p of the space P" is completely determined by matrix (6.1.1) composed from the coordinates of the linearly independent points xa, a = 0,1, ... , m, belonging to the subspace p. Let us transpose matrix (6.1.1) and denote the transposed matrix by the letter X: X_

.o

,_o

-o

x0

x1

x,'"

.................

n X1

xon

=(x{) a

n X.

where { = 0,1, ... , n and a = 0,1, ... , m. This matrix is called the projective matrix coordinate of the subspace p (see Rosenfeld [Ro 961, §2.4.1). The projective matrix coordinate X is defined up to multiplication from the right by a nondegenerate square matrix A = (ap) of order in + 1: X - X A. The subspace p C P" can be also defined dually as the intersection of n - m linearly independent hyperplanes u', i = m + 1,. .. , n, containing p. The equations of these hyperplanes can be written as 0

+ um+lxl + ... + um+lxn =O n

tt0x0

+

,um+1x0

1

uixl

+

... +

unxn

= 0,

where xf are coordinates of a variable point x. The matrix m+1

u0

U =

m+1

m+1

un

U'1

......................... u0"

u1

...

_ (u'),

i = m + 1,...,n,

un

composed from the coefficients of equations of these hyperplanes is called the tangential matrix coordinate of the subspace p (see Rosenfeld [Ro 96], §2.4.1). The tangential matrix coordinate U is defined up to multiplication from the

left by a nondegenerate square matrix B = (b) of order n - m: U - BU. Since all points xp E p belong to the hyperplanes u' D p, we have

UX = 0,

(6.1.6)

where the right-hand side is the rectangular zero (n - m) x (m + 1) matrix.

Consider further a subspace p C P" of dimension m < 1(n - 1) and a subspace p' C P" of complementary dimension n - m - 1. Denote the matrix

coordinate of the subspace p by X: X = (x«), and the tangential matrix coordinate of the subspace p' by U: U = (u{ ). Consider the square matrix UX = C = (ca")

(6.1.7)

of order m + 1. The rank of this matrix depends on the dimension of the intersection p f1 p'. This rank is maximal if p and p' do not have common points. The rank is equal to m - k if dim(p (1 p') = k. (Note that if the

6.1

Analytic Geometry of Grassmannian and Grassmann Mapping

227

subspaces p and p' do not have common points, then k is defined to be equal

to -1: k = -1.) A pair of subspaces p and p' is said to be an m-pair. If k = -1, then the m-pair is called nondegenerate. If k > 0, then the m-pair is called degenerate.

Let (p,p") be a nondegenerate m-pair, and let x be a point of the space P" not lying in the subspaces p or p'. Then there exists a unique straight line l passing through x and intersecting p and p'. In fact the point x and the subspace p determine the linear span x A p of dimension m + 1, and the point x and the subspace p' determine the linear span x A p' of dimension n - m. Since the sum of dimensions of the subspaces x A p and x A p' is equal to n + 1 and (x A p) U (x A p') = P", these subspaces have a unique common straight line I passing through x and intersecting p and p' (see Figure 6.1.1). Denote the points of intersection of the straight line I with the subspaces

p and p' by y and z, respectively. Since the points x, y, and z lie on the same straight line, they are linearly dependent. The fact that the projective coordinates of a point can be multiplied by the same nonzero number, allows us to write this linear dependence in the form x ='y + Z.

(6.1.8)

Since y E p, this point can be represented in the form y = X t, where X is the matrix coordinate of the subspace p, and t = (ta) are the coefficients of the decomposition of the point y with respect to the linearly independent points

x0 E p: y = taxa. Since z E p', the point z satisfies the equation Uz = 0 where U is the tangential matrix coordinate of the subspace p'. Substituting the expression for the point z from relation (6.1.8) into the last equation, we

Figure 6.1.1

228

6. GEOMETRY OF THE GRASSMANNIAN

Figure 6.1.2

obtain

Ux = Uy = (UX)t.

(6.1.9)

Since the pair (p, p') is nondegenerate, the square matrix UX is also nondegenerate, and from equation (6.1.9) we find that

t = (UX)-'Ux. Substituting this value of t in the expression for the point y, we obtain

y = X(UX)-'Ux.

(6.1.10)

5. We will now define the cross-ratio of two nondegenerate m-pairs (p, p' ) and (q, q*) which are in general position (see Figure 6.1.2).

Suppose that x is an arbitrary point of the subspace p, x E p, I is the straight line passing through x and intersecting the subspaces q and q', and y = In q. There exists also a unique straight line !' passing through the point y and intersecting the subspaces p and p', and this line intersects p at the point

Analytic Geometry of Grassmannian and Grassmann Mapping

6.1

229

x' = 1' fl p. Thus, on the subspace p, a mapping w: p -i p arises such that w(x) = x'. Let us prove that the mapping w is linear and find its matrix. To this end, we denote the matrix coordinates of the subspaces p and q by X and Y and the

tangential matrix coordinates of the subspaces p' and q' by U and V. Then, by formula (6.1.10), the point y is determined by the equation

y = Y(VY)-'Vx. The point x' is determined by a similar formula:

X, = X(UX)-'Uy. Eliminating y from the last two equations, we find that

x' = X(UX)-'(UY)(VY)-'Vx.

(6.1.11)

Formula (6.1.11) proves that the mapping w is linear and its matrix has the form

W = X(UX)-'(UY)(VY)-'V.

(6.1.12)

If m-pairs (p, p') and (q, q') are nondegenerate and in general position, then the matrix W is a nondegenerate matrix of order m + 1, and the mapping w: p -+ p is a nondegenerate projective transformation of the subspace p. The matrix W is called the cross-ratio of two m-pairs (p, p') and (q, q'), since for m = 0, the matrix W is the number equal to the cross-ratio of two

points x and y and two hyperplanes u and v. The latter cross-ratio is equal to the cross-ratio of four points x, y, (x A y) fl u and (x A y) fl v (see Rosenfeld [Ro 96], §2.4.6). Consider the points of the subspace p that are invariant with respect to the projective transformation w. Such points are determined by the condition

i = w(x) = Ax. Hence for this fixed point, the straight lines I and I' defining the mapping w coincide, and this double line intersects all four subspaces p, p', q, and q'. Such straight lines are called the transversals of two m-pairs (p, p*) and (q, q*). To the invariant point x there corresponds the eigenvalue \ of the linear operator w. This number A is equal to the cross-ratio of four points in which the line I e l' meets the subspaces p, p', q, and q'. (For the proof see Rosenfeld [Ro 96], §2.4.6).

If the operator w has m + 1 distinct eigenvalues, then its invariant points corresponding to these eigenvalues are points of the subspace p that are in general position. If we take these points as a basis of the subspace p, then the matrix W of the operator w is reduced to diagonal form 0

W=

0

'0

0

...

0

230

6. GEOMETRY OF THE GRASSMANNIAN

In this case, the m-pairs (p,p') and (q,q') have m + 1 transversals. If two eigenvalues of the operator w are complex conjugates, then the corresponding transversals of the m-pairs (p, p') and (q, q') are also complex conjugates, and the real three-dimensional subspace determined by these transversals is invariant. Finally, if some of the eigenvalues of the operator w coincide but its matrix is still diagonalizable, then two m-pairs have infinitely many transversals.

6. Consider the case when the dimension n of the space P" is equal to 2m + 1: n = 2m + 1. Then an m-pair (p, p') consists of two subspaces of the same dimension m, and all four subspaces p, p', q and q' of two m-pairs (p, p*) and (q, q') have equal status. The cross-ratio of m-pairs (p, p') and (q, q') in the space P2m+1 is defined in the same manner as it was defined above for arbitrary m-pairs in P". However,

since in the space P2m+1, the subspaces p,p',q, and q' of two m-pairs (p,p') and (q, q') have equal status, we can compose 24 cross-ratios of this quadruple of subspaces in the same way as it is usually done for a quadruple of points of a projective line (e.g., see Klein [Kl 28], Ch. 1, §7). If the quadruple of subspaces in question is in general position, then these 24 cross-ratios can be expressed in terms of the cross-ratio W as follows:

(P,P*;q,q*) = W,

(P,P';q*,q) = IF-1 ,

(P,q;P*,q*) = I - W,

(P,q;q',p) _ (I (P,q*;q,P*) = I -W-1, (P,q';P',q) _ (I -W-1)-1, W)-

where I is the identity matrix. The remaining 18 cross-ratios each coincide with one of these six, since for any quadruple of subspaces pa, a = 1, 2, 3, 4, of the space P2m+1 the following relations hold: (Pa,Pb;Pc,Pd) = (Pc,Pd;Pa,Pb) =

(Pd,PC;Pb,Pa),

which can be verified immediately. Consider three m-dimensional subspaces p, p', and q of a quadruple of sub-

spaces p, p', q and q'. As was proved earlier, through any point x E p, there passes a unique straight line I intersecting the subspaces p' and q. When the point x describes the subspace p, the straight line I describes an m-parameter family belonging to an (n + 1)-dimensional Segre variety S(1, m). In the space P11+1, the parametric equations of this variety can be written in the form (cf. equations (6.1.4)): z"a = sE t°,

f=0,1; a = 0, 1, ... , m.

This variety carries rectilinear generators defined by the values of parameters t° = to T, r E R. In addition it carries m-dimensional plane generators defined by the values of parameters sE = soa, a E R. A quadruple of m-dimensional plane generators pa of the Segre variety S(1,m) defined by the parameters sa, a = 1, 2, 3, 4, cuts on the rectilinear generators of S(1, m) quadruples of

6.1

Analytic Geometry of Grassmannian and Grassmann Mapping

231

points with a constant cross-ratio: 833 ((

=

I

82 3

31

33

0 81

32

4

31

34

32 404

0

Hence the cross-ratio of a quadruple of m-dimensional plane generators pa C S(1, m) is a scalar linear operator to = A Id. A quadruple of subspaces p, p', q, and q' in the space Pent+' determines four Segre varieties S6(1, m) in this space. Moreover these varieties have common rectilinear generators passing through fixed points of the linear operator w which is defined by this quadruple of subspaces on the subspace p. If the operator w has distinct eigenvalues, then it has exactly m + 1 fixed points, and the varieties S0(1,m) have m + I common rectilinear generators. On the other hand, if the operator w is scalar (i.e., it has the form w = A Id), then all rectilinear generators of the varieties S. (1, m) coincide. As a result the subspaces p, p', q, and q' are m-dimensional plane generators of the same Segre variety S(1,m). Thus we have arrived at the following result:

Theorem 6.1.1 The cross-ratio of a quadruple of m-dimensional subspaces pe of the space p2m+1 is a scalar operator if and only if these four subspaces belong to the same Segre variety S(1, m). Theorem 6.1.1 is analogous to the well-known result from theory of complex numbers which states that a quadruple of points in the complex plane lies on a circle if and only if their cross-ratio is a real number. In view of this, the Segrean S(1, m) in the geometry of the Grassmannian G(m, 2m + 1) is an analogue of a circle in theory of complex numbers. 7. In conclusion, we consider again two m-pairs (p, p') and (q, q') of gen-

eral position in the space Pn, where n > 2m + 1, dimp = dimq = m, and dim e' = dim q' = n - m - 1. The subspaces p and q determine in Pn a subspace r of dimension 2m + 1 -the linear span of p and q, and the subspaces p' and q* intersect one another along a subspace s of dimension n - 2m - 2. All transversals of the m-pairs (p, p') and (q, q*) belong to the subspace r, and the cross-ratio of these m-pairs is equal to the cross-ratio of the m-pairs (p, pl ) and (q, qj) where p1 = p' fl r and ql = q' fl r are also m-dimensional subspaces

of the space r. Thus, by Theorem 6.1.1, the cross-ratio W of m-pairs (p,p') and (q, q') is a scalar matrix if and only if four subspaces p, pl, q, and ql of the spacer belong to the same Segrean S(1,m). Consider further the (n - m - 1)-dimensional subspaces p*1 = p A s and qj* = q A s containing the subspace s; here A denotes the linear span of the corresponding subspaces. They, as well as the subspaces p' and q', belong to the bundle with center at s. If the cross-ratio W = (p, p'; q, q') is a scalar matrix, then the subspaces pj, p', qi , and q' belong to the same Segre cone

232

6. GEOMETRY OF THE GRASSMANN MANIFOLD

whose vertex is the center s of the bundle and whose directrix is the Segre variety S(1,m). This cone has two families of plane generators of dimension n - 2m and n - m - 1, respectively.

6.2

Geometry of the Grassmannian G(1, 4)

1. As an example we consider the Grassmannian G(1,4) of straight lines in a four-dimensional projective space p4. A straight line p of this space is determined by points x and y whose coordinates form the matrix

/ x° 1 y0

x2

x1

y2

yI

x3 y3

x4 y4

(6.2.1)

The minors of this matrix xi

P$j = ly i

xj yj

,

i,j = 0,1,2,3,4,

are the Grassmann (Plucker) coordinates of the straight line p. Since (z) = 10, these coordinates define a point p in a projective space P9. But the dimension of the manifold G(1, 4) is equal to six: p = 2.3 = 6. Thus the algebraic variety 11(1, 4) C P9 also has dimension six.

By equations (6.1.2), the variety fl(1,4) is determined by the equations p01p23 + p°2p31 + p°3p12 = 0, p01p24 + p02p41 + p04p12

-0,

p01p34 + p03p41 + p04p13 = 0,

(6.2.2)

pO2p34 + p°3p42 + p04p23 = 0, p12p34 + p13p42 + p14p23 = 0.

However, since p = 6, only three of equations (6.2.2) are independent. It follows from formula (6.1.3) that the variety 1(1, 4) is of degree five. The set of all straight lines intersecting a fixed straight line p C P4 is a four-

dimensional manifold with a singular straight line p. On the variety f2(1,3), there corresponds to this set a cone C,, with vertex at the point corresponding to the straight line p under the Grassmann mapping G(1, 3) -a Ps. The cone Cp carries a one-parameter family of three-dimensional plane generators, corresponding to the bundles of straight lines whose centers lie on the straight line p, and a two-parameter family of two-dimensional plane generators that correspond to the plane fields of straight lines belonging to 2-planes passing through p. Thus the cone Cy is a Segre cone C,,(2,3), and its projectivization PCp(2,3) is a Segre variety S(1,2) = P' X P2 C P6 = PT(')(f2(1,4)). 2. Next, in the space P9, we consider a three-dimensional subspace P3 that is in general position with the variety 11(1,4). The subspace p3 meets

6.2

Geometry of the Grassmannian G(1,4)

233

12(1, 4) in five points. But since the subspace p3 itself is defined in P9 by four points in general position, the four common points of P3 and 12(1,4) uniquely

determine the fifth point of intersection of p3 and 12(1,4). Thus, any four straight lines of the space p4 uniquely determine the fifth straight line of this space. This configuration of five straight lines in P4 corresponding to five points

of intersection of a subspace p3 with the variety 12(1,4) is called the Vlasov configuration (see Vlasov [Vl 10] and Karapetyan [Kar 62a]). Let us study the Vlasov configuration in P4 and some geometric objects connected with this configuration in more detail. Let p l, p2, p3, and p4 be four straight lines in P4 in general position. Each pair of these straight lines determines a three-dimensional subspace (a hyperplane) in p4. There are six such hyperplanes, and we can take five of them as coordinate hyperplanes a', i = 0, 1, 2, 3, 4, and the sixth one as the unit hyperplane

e=a°+a'+a2+a3+a4. We will assume that

piAP2=a°, PiAP3=a',

PiAP4=a2,

P2AP3 =a3, P2AP4 =a4, P3AP4 =e In these formulas pa A pp denotes the linear span of the points Pa and pp, a,/3 = 1,2,3,4. In the space P4, consider the point frame {A1} which is dual to the tangential frame (a'). Then we have

(a',At)=5, and the straight lines p, a = 1, 2, 3, 4, are represented as follows:

p1= a0Aa'Aa2=A3AA4i P2=a°A a3 A a4 =A1 A A2,

P3=a'Aa3Ae=a'Aa3A(a°+a2+a4)=(A2-Ao)A(A4-Ao), p4 =a2Aa4Ae=a2Aa4A(a°+al+a3)=(A, -Ao)A(A3-Ao). (6.2.3)

Every triple of straight lines pa has a unique secant. Consider, for example, the straight lines pl,p2i and p3. In pairs, they define three hyperplanes a°, a',

and a3 that meet in the straight line a° A al A a3. Hence the straight lines p1, p2, and p3 determine a unique secant

q4=a°Aa'Aa3=A2AA4.

(6.2.4)

Similarly {Pi, P2, P4 }

q3

= a° A a2 A a4 = A, A A3,

{Pi,Ps,P4}q2=a'Aa2Ae=(A3-Ao)A(A4-Ao), {P2,P3,P4} qi =a3Aa4Ac=(A, -Ao)A(A2-A°).

(6.2.5)

6. GEOMETRY OF THE GRASSMANN MANIFOLD

234

In pairs, the straight lines pa and qa define four hyperplanes 0°. Let us prove that these hyperplanes meet in the straight line p5. We have

0'=pIAq,=A3AA4A(A,-Ao)A(A2-Ao), Q2=P2Ag2=AlAA2A(A3-Ao)A(A4-A0), Q3=P3Ag3=AlAA3A(A2-Ao)A(A4-Ao), p4

=P4Ag4 =A2AA4A(A, - Ao) A (A3 - Ao).

In point coordinates, with respect to the frame {A;}, the equations of these hyperplanes can be written as r x0 + XI + x2 = 0, x° + x3 + x4 = 0,

Sl x°+x2+x4 =0, x°+x' +x3 =0. Equations of this system are linearly dependent, and a basis of the solution space of this system is

(-1,0,1,1,0), (-1,1,0,0,1). Thus the hyperplanes (i° meet in the straight line

p5 =(A2+A3-A0)A(A,+A4-A0).

(6.2.6)

We will prove now that the straight line p5 is the fifth straight line of the Vlasov configuration defined by the straight lines p, , p2i P3 and p4. To this end, in the space P9 we will find equations of the subspace p3 defined by the points

pa, a = 1,2,3,4. We will look for these equations in the form aiiP'I = 0.

Since the coordinates of the straight lines pa must satisfy these equations, by (6.2.3), we obtain the following six equations: P14 = 0,

P23 = 0,

P01 - p13 = 0,

P02 - p24 = 0, P°3 + p'3 = 0, P°4 + p24 = 0,

which determine the subspace p3 in P9. However, it is easy to check that the Grassmann coordinates of the straight line ps also satisfy these equations. Thus this straight line p5 along with the straight lines pi, p2i p3, and p4 form the Vlasov configuration. Moreover each of the straight lines PI,P2,P3,P4, and Ps have equal status in the Vlasov configuration, and every four of them determine the fifth one. Note that the straight lines qa and p5 also form the Vlasov configuration defined by the subspace P3 C P9. The subspace P3 is defined by the following system of equations: P14 = 0,

P23 = 0,

001 - P12 = 0,

p03 - p34 = 0, p02 + p12 = 0, p04 + p34 = 0.

6.2

Geometry of the Grassmannian G(1, 4)

235

The point ps is the only common point of the subspaces p3 and 3. Consider two-dimensional planes intersecting all five straight lines of the Vlasov configuration. We will represent such 2-planes in the form P.

a=AAp= \ipja'nay = Iaija'Aaj, where A = \iai and p = pia' are hyperplanes in P4 and pi

a,j -

I

Ai

pj

(6.2.7)

are their Grassmann coordinates. The condition that the 2-plane a and the straight line p with coordinates p'3 intersect one another can be written as

(a, P) = aii p'j = 0.

(6.2.8)

To "see" the Grassmann coordinates of the straight lines composing the Vlasov configuration, we will rewrite equations (6.2.3) and (6.2.6) in the form Pi = A3 A A4, P2 = Al A A2,

p3=AoAA2-A0AA4+A2AA4i p4 = AoAA, -AoAA3+Al AA3, ps = -AoAA, + AoAA2+AoAA3 -AoAA4 -Al AA2-A,AA3+A2AA4+A3AA4. By (6.2.8), the condition that the 2-plane a and the straight line p intersect

one another can be written as a34 = 0, a12 = 0,

Opt - a04 + a24 = 0, a0l - a03 + a13 = 0,

(6.2.9)

-a01 + 0`02 + a03 - a04 - a12 - a13 + a24 + a34 = 0.

As one can expect, the last equation is a linear combination of the first four equations, and as a result the 2-planes a are defined by the first four equations of this system. Consider the Grassmannian G(2, 4) of two-dimensional planes in the space P4. As in the case of the Grassmannian G(1,4), its dimension is equal to six. By means of the Grassmann coordinates aii, the Grassmannian G(2, 4) can be mapped onto the algebraic variety 11(2,4) of the projective space (P9)' which is dual to the space P9 containing the variety 11(1,4). As is the case for the variety 12(1,4), the degree of the variety 11(2,4) is equal to five. Equations (6.2.9) define in (P9)' a subspace of dimension five that meets the variety Q(2,4) in a two-dimensional surface. In view of this, in the space P4 there exists a two-parameter family of 2-planes a intersecting all five straight lines

6. GEOMETRY OF THE GRASSMANN MANIFOLD

236

of the Vlasou configuration. The set of these 2-planes form a congruence called the Vlasov congruence. Let us find the equation of the Vlasov congruence in tangential coordinates.

To this end, we substitute for the Grassmann coordinates a, in equations (6.2.9) their values (6.2.7). We find that A3p4 - A4A3 = 0, A1 /p2 - 1\2/41 = 0, AO/12 - 1\2/10

1\oµ4 + A4/W + 1\2/44 - 1\4/12 = 0,

(6.2.10)

A0141 - AI/10 -A0143 + A3p0 + \1/13 - 1\301 = 0.

All 2-planes determined by the system of equations (6.2.9) intersect the straight lines of the Vlasov configuration. Hence the hyperplane is can be chosen in such

a way that it passes through the straight line p1. Then, by (6.2.3), we obtain 143 = p4 = 0. As a result the system of equations (6.2.10) takes the form AI/12-A2111=0, 1\0112 - A2/W + A4p0 -,\4/12 = 0,

Ao/h - AI/1o + Alpo - A3p = 0. Excluding the quantities µo, p1 and P2 from these equations, we arrive at the following cubic equation: AI (A0 - A3)(A2 - A4) - A2(AO - A4)(A1 -- A3) = 0.

(6.2.11)

This equation determines the Vlasov congruence in tangential coordinates in the space P4: any hyperplane A whose coordinates satisfy equation (6.2.11) contains at least one 2-plane of the Vlasov congruence. Since this equation is of the third degree, the Vlasov congruence is a congruence of the third class. In other words, every pencil of hyperplanes of the space p4 contains three hyperplanes whose coordinates satisfy equation (6.2.11). Note that Karapetyan (Kar 62a] considered the dual Vlasov configuration formed by five 2-planes of the space p4 corresponding to five points of intersection of the variety l(2, 4) with the subspace (P3)' of the space (P9)' which is dual to the space P9 and in general position with f2(2, 4). A two-parameter family of straight lines intersecting five 2-planes of the dual Vlasov configuration forms a hypersurface called the Vlasov hypersurface. It is a hypersurface of third order.

6.3

Differential Geometry of the Grassmannian

1. We will pass now to the study of differential geometry of the Grassmannian. In the space P" we consider the family R(P") of projective point frames {A(}

6.3

Differential Geometry of the Grassmannian

237

and the family R'(P") of tangential frames {af} formed by the hyperplanes at. The condition of duality of these frames has the form (A(, a") = 6k,

(6.3.1)

where the parentheses denote the convolution of the corresponding elements, and the condition (A, a) = 0 is the incidence condition of the point A and the hyperplane a. Let us write the equations of infinitesimal displacement of the point and tangential frames of the space P" (cf. Section 4.3): dAt = w" A,,, dat = 86a",

where " and 0 are 1-forms. By differentiating (6.3.1), we can easily find that 6n + wt = 0. In view of this, the equations of infinitesimal displacement take the form: dAt = w' A,r, dac = (6.3.2) Relations (6.3.1) also imply that the forms WE are related by the condition

Wp +Wj +...+wn =0.

Thus, each of the families of frames R(P") and 7Z* (P") depends on (n + 1)Z - 1 = nZ + 2n parameters. In addition the forms w, satisfy the structure equations of the space P" (cf. Section 4.4): (6.3.3) dw(=wnAWC. Let p be an m-dimensional subspace in P", or briefly an m-plane. We will assume that m < i (n - 1), since the case m > (n - 1) is dual to the first one z and can be studied in a similar manner. Moreover we will assume that m > I

and n - m > 2, since if m = 0 and m = n - 1, the Grassmannian G(m, n) becomes a projective space. We associate with the plane p a subfamily R(p) of projective frames of the space P" such that the vertices A" of its frames lie in the plane p. We denote by R' (p) the subfamily of projective frames that are dual to the frames of the subbundle RZ(p). The hyperplanes a' of the frames of R' (p) pass through the m-plane p. By virtue of this, the plane p can be determined in two ways:

p=AO AA, A...AAm= am+' A...Aa",

(6.3.4)

where as earlier, the expression Ao A A, A ... A A,,, is the linear span of the points A,, and a'+1 A ... A an is the intersection of the hyperplanes a'. The manifold of m-planes p generates the two fiberings in the frame man-

ifolds R(P") and R'(P") whose fibers are the subfamilies R(p) and 1'(p). These fiberings are determined as the projections

n : R(P") -a G(m, n), a' : R' (P") - G(m, n),

(6.3.5)

6. GEOMETRY OF THE GRASSMANN MANIFOLD

238

where 7t-1(p) = R(p), (n')-1(p) = R'(p). From equations (6.3.2) it follows A an that the condition for the plane p = A0 A Al A ... A Am = am+' A to be fixed can be written as wa = 0. So the 1-forms w, are horizontal (base) forms for the fiberings (6.3.5). These forms are linearly independent on the Grassmannian G(m, n), and their number is equal to the dimension p = (m + 1) (n - m) of G(m,n). 0, equations (6.3.2) take the form For Gap = -1rpa'V - apa{,

6A,, = rrpAp,

(6.3.6)

6Ai = 7r°A° + 7r? Aj, 6a' _

where 6 = dI,< =0 and of = we(d). The 1-forms 7rO, 7ri and ni are invariant forms of the stationary subgroup of the m-plane p in the space Pn. They are the fiber forms of the fiberings (6.3.5). 2. We consider now the Grassmann mapping y of the Grassmannian

G(m, n) onto the algebraic variety f l(m, n) of the space P", where +1 y : G(m,n) -a fl(m,n). Suppose that TM (0) is the tangent bundle of the variety It (m, n). Let us find the differential of a point p E fl(m, n) that corresponds to an m-plane p in Pn. By differentiating (6.3.4), we find that dp = wp + w'pi,

(6.3.7)

where w = wo + ... + wm and

p° = AOA...AA°_1 AA, AA0+1 A...AAm = am+' A...Aa'-' Aa° Aa'+l Aan are linearly independent points in P^' that also belong to the variety f2(m, n). The points p° together with the point p determine the tangent subspace TP1) to

the variety fl(m, n), and the dimension of this subspace

is

equal to

pi = p = (m + 1)(n - m). Formulas (6.3.7) show that the space TP1)(fl) is isomorphic to the vector space of rectangular (m + 1) x (n - m) matrices. Next we will find the tangent bundle Ty2)(fl) of second order of 1(m, n) whose element is an osculating subspace of the variety Sl(m, n). To this end, we calculate the second differential of the point p: d2p =

a
Qapp

(mod Tn11

(6.3.8)

where ap = wpwp - 004w,3 are quadratic differential forms and

p A =AOA...AAo_1 A

A

=am+1 A...Aa'-1 Aa°Aa'+'

A

A

A...Aaj-1 Aa0Aaa+1 A...Aan

Differential Geometry of the Grassmannian

6.3

239

are points of the space PN lying on the variety fl(m, n). Since i < j and a < Q, all these points are linearly independent and their number is: P2 = (m21) . (n 2m) We denote by T,, the osculating subspace of second order to the variety fl(m, n) at a point p and will call it the tangent subspace of second order of fl(m, n). This subspace is determined by the points p, p°, and p Via. Thus its dimension is p1 + p2.

Continuing the consecutive differentiation of a point p E fl(m,n), we find

that tliq...$k a'1 ay...ak ('1a2 ... akpll t2...ik

dkp of
Tpk-1)),

(mod

where Wtlt2... ik 0102... ak - k!W(IIL)i2 al az p0102... ak 102 ... ik

Wikl.

(6.3.9)

ak 1

= AoA...AAa,-1 AAil AAa,+l A...AAa, AAt A ail-1 A aal Aa/l+l A ... A au-1 A aa" A aia+1 A ... A an. A

A

A

(6.3.10)

In formulas (6.3.9) and (6.3.10) we used the following ranges of indices:

k=1,...,m+1;

aa=m+1,...,n; A=1,...,k.Thediffer-

ential forms (6.3.9) of degree k are the system of fundamental forms of order k of

the variety fl(m,n). The points p°;z2.i4 k belonging to the variety fl(m,n) are linearly independent and together with the points p, p°, . . . , pa 202"'a , ] form a basis of the osculating subspace of order k to the variety fl(m, n) at a (n k"'), and the dimension point p. The number of these points is pk = of the subspace Tyk) is (mk1)

(n

dim Tyk) = Cm 1

1/

(n 1

m) + \m 2 1/

2

m) +...+ Cm k 1/ \n k m/

In particular, we have dimT(m+1)=

(m+11 (n-m)+...+(m+1)(nm\ 1

)(

1

(\m+l

m+1J

=

\n+11-1=N. m+1)

The last equation show that if m < (n - 1), then the osculating subspace of s order m + 1 of the variety fl(m, n) coincides with the space PN in which the variety fl(m, n) is embedded. 3. Next we will find the asymptotic directions of second order (see Griffiths and Harris (GH 791 or Akivis and Goldberg [AG 93], §2.4) on the variety fl(m, n)

emanating from a point p E fl(m, n). These directions are determined by the condition d2p =_ 0

(mod TP1)).

240

6. GEOMETRY OF THE GRASSMANN MANIFOLD

Thus from (6.3.8) it follows that the equations of the cone of asymptotic directions of second order have the form 0.

w00 = WaWO

(6.3.11)

The quadratic forms QQ constitute the system of fundamental forms of second order of the variety fl(m, n). Equations (6.3.11) are equivalent to the condition

rank (w') = 1.

(6.3.12)

Let us denote by Cpl) the asymptotic cone determined by equations (6.3.11) or (6.3.12). The asymptotic directions of order k on the variety fl(m, n) emanating from a point p E G(m, n) are determined by the condition (mod Tpk-1)).

dkp =_ 0

By virtue of the above congruence for dkp, the equations of the cone of these asymptotic directions have the form 1112... t. 01102 ...Q.

-0

(6.3.13)

Equations (6.3.13) determine the asymptotic cone C(k) in Tp11. The asymptotic cones of different orders are connected by the relation Cpl) C Cp3) C

... Cpm+l)

C Tpl);

that is, they form a filtration. By (6.3.9), equations (6.3.13) can be written as

rank (wo) < k - 1,

(6.3.14)

and thus the cone Cpk) is a determinantal variety. Let us study the geometric structure of the cones Cpl) in more detail. From equation (6.3.12) of this cone, it follows that 1

(6.3.15)

where the parameters atr and r' are determined up to multiplication by reciprocals. Equation (6.3.15) means that the cone C(l) is a Segre cone Co(m+1, n-m) carrying two families of plane generators of dimensions m + I and n - m. We

will denote these generators by 2pl)(a) and A(2) (P), respectively. Note that the generators of different families meet in straight lines. This implies that dim Cpl) = n. The projectivization of the Segre cone Cpl) is the Segre variety S(m, n - m - 1) = P'" x P"-m-1 C p°-1 = PTn1)(f). On the Grassmannian G(m, n), to the Segre cone Cpl) there corresponds a family of m-dimensional

6.3

Differential Geometry of the Grassmannian

241

submanifolds intersecting a given m-dimensional subspace p along (m - 1)-dimensional submanifolds. Hence the cone Cps) is the intersection of the variety f2(m, n) n Tp'1(f2). fl(m., n) and its tangent subspace TP')(f)) : Since for the asymptotic cone Cps) of third order we have

rank (w,J < 2, its equations can be written in the following form:

wa = 71oa +T2oa,

(6.3.16)

where the parameters oa and ra, A = 1, 2, are determined up to multiplication by 2 x 2 matrices that are inverse of each other:

where pKPa = ba. The dimension of the cone Cp3l is equal to the number of independent parameters as an d ra: dim Cy3) = 2(m + 1) + 2(n - m) = 2(n - 1).

To find a geometric meaning of the cone CP3), we consider its projectiviza-

tion PC( ). This manifold is defined in the space P°-1 = P(Tp' (Q)) by the same equations (6.3.16). The points with coordinates 'WQ = r, ,, and "Wi = Tr'cr

belong to the Segre variety S(m,n - m - 1) = PCn21. Since they are defined up to a factor, their linear combination (6.3.16) describes a straight line joining these points. Thus PCa31 is a bisecant variety of the manifold PC('). On the other hand, the manifold PC,21 is the set of singular points of the manifold PCP3). In fact the condition rank (w') < 2 means that the minors M3 of third order of the matrix (Wa) are equal to 0: M3 = 0. Singular points of this manifold are defined by the condition e = 0. The left-hand side of this relation, if it is not trivial, is a minor of second order of the same matrix, and the condition OM-1 = 0 means that all these minors of second order vanish. Hence rank (w.) = 1, and this is the equation of the manifold From equation (6.3.16) it follows that if we fix the matrix (oa) (i.e., if we set ax = ca) and vary entries of the matrix (r2), then a point q E Cy3i will describe

a subspace iy31(a) in the tangent subspace T,'> determined by the points p and cap°. The dimension of the subspace Dy3)(a) is dim Dy3)(a) = 2(n - m). The subspaces Dy31(a) form the first family of plane generators of the cone

242

6. GEOMETRY OF THE GRASSMANN MANIFOLD

C. In the same manner, if we set TI = ci\ and vary entries of the matrix (oa ), then a point q E CP3) will describe a subspace Op3> ((3) in the tangent subspace T11 determined by the points p and cap°. The dimension of the subspace Ay31(Q) is dim Op3)(Ji) = 2(m + 1). The subspaces 43j(3) form the second family of plane generators of the cone Cp31 The cone Cp3l is connected also with the following construction on the Grassmannian G(m, n). Let p be a fixed m-dimensional subspace of the space

P". Consider the set of m-dimensional subspaces intersecting the subspace p at a subspace q of dimension m - 2. This set is a submanifold of the Grassmannian G(m, n), and the subspace p is singular on UD31. On the variety 1(m, n), to the submanifold UP31 there corresponds a submanifold with a singular point p which we will denote also by U. The asymptotic cone CD3' C Tp) is a tangent cone to the submanifold UP(3) C fl(m, n) at the point P.

The submanifold Up3l in G(m, n) is stratified into two families of Grassmann manifolds. In fact, if we fix an (m - 2)-subspace q C p, then the set of rn-dimensional subspaces p passing through q is equivalent to the Grassmannian G(1, n - m + 1), since every subspace p' meets the subspace q*, which is complementary to q, in a straight line. On the other hand, if we fix an (m + 2)-dimensional subspace r passing through p, then the set of all m-dimensional subspaces p` lying in r and intersecting p at an (m - 2)-subspace forms the Grassmannian G(m,m + 2). Under the Grassmann mapping, to the Grassmannians G(1, n - m + 1) and G(m, m + 2) there correspond the algebraic varieties fl(1, n - m + 1) and UP(3) fl(m, m + 2) belonging to and passing through the point p. The dimensions of these varieties are dim Q(1, n - m + 1) = dim G(1, n - m + 1) = 2(n - m) and dim fl(m, m + 2) = dim G(m, m + 2) = 2(m + 1). The plane generators 43)(a) and Di,3)(f3) of the cone Cn3) are the tangent subspaces to the varieties It(1, n - m + 1) and !i(m, m + 2), respectively.

On the Grassmannian G(m, n), to the intersection of the varieties S?(1,n. - m + 1) and S1(m,m + 2) there corresponds a set of m-subspaces p passing through the (m - 2)-subspace q and belonging to the (m + 2)-subspace r. This set is equivalent to the Grassmannian G(1, 3) and is of dimension four. Hence two generators A ,3) (a) and Op3) (A) of the cone Cp3' intersect one another at a four-dimensional subspace of the space TP'I.

The structure of the asymptotic cone CLkl of order k < m + 1 can be investigated in a similar manner. The cone CD k) is a determinantal variety of dimension (n - k + 2)(k - 1) and carries two families of plane generators D,,ki(a) and A ($) of dimension (n - m)(k - 1) and (m + 1)(k - 1), respectively. Moreover two generators belonging to different families have in common a subspace of dimension (k -1)2. We will call the generators of the first family the a-subspaces of the asymptotic cone and the generators of the second

6.3

Differential Geometry of the Grassmannian

243

family its f3-subspaces.

The projectivization PCP(k) of the cone Cpk) is an algebraic variety in the space Tell (fl). The latter space is a collection of (k-1)-secants of the Segre vari-

ety S(m, n-m - 1) = PC,21. Thus all asymptotic cones Cpkl, k = 2.... ,m+1, are determined by the cone C,2) of second order. In the same way as above, we can prove that the submanifold PCP(k) is the set of singular points of the submanifold Cpk+>

Note one more time that for k = 2 the asymptotic cone CP21 lies on the variety fl(m,n), while for k > 2 the asymptotic cones Cpki lie in the subspace Tpll(fl) but not in fl(m,n) itself. 4. The plane generators of each of two families of asymptotic cones Cnk) form a fiber bundle over the variety fl(m, n). Let us consider, for example, the fiber bundle E( k) of generators 4Dk)(a) of dimension (n - m)(k - 1). Its fiber has the dimension (m - k + 2)(k - 1). An integral submanifold of the fiber bundle E,(, k) is called a manifold VQkI

of dimension (n - m)(k - 1) whose all tangent subspaces belong to E. The integral submanifolds are asymptotic a-submanifolds of order k of the variety 11(m, n).

If k = 2, then the asymptotic a-submanifolds are a-subspaces of dimension

n - m of the variety fl(m, n). On the Grassmannian G(m, n), to these asubmanifolds there correspond bundles of m-subspaces with an (m - 1)-dimensional center.

If k = 3, then the asymptotic a-submanifolds are submanifolds VQ31 of dimension 2(n - m) to which on the Grassmannian G(m, n) there correspond bundles of m-subspaces with (m - 2)-dimensional centers. The asymptotic a-submanifolds of any order k can be defined in a similar manner. Consider next the fiber bundle Eokl of generators OP(k) (/3) of the asymptotic cones Cpk) and its integral submanifolds Vpk), dim VAk) = (m+ 1)(k- 1). These integral submanifolds are asymptotic 13-submanifolds of order k of the variety f] (m, n).

If k = 2, then these asymptotic f3-submanifolds are f3-subspaces of dimen-

sion m + I of the variety f2(m, n). On the Grassmannian G(m, n), to these f3-submanifolds there correspond bundles of m-subspaces belonging to a subspace of dimension m + 1.

If k = 3, then the asymptotic f3-submanifolds are submanifolds V(3) of dimension 2(m + 1). On the Grassmannian G(m, n), to these f3-submanifolds there correspond bundles of m-subspaces belonging to a subspace of dimension

m+2. Thus, for each k = 2, ... , m + 1, the variety f? (m, n) carries two families

of plane generators z$ (a) and o, (j3) of dimension (n - m)(k - 1) and (m + 1)(k - 1), respectively. If asymptotic submanifolds of different families intersect each other, then the dimension of their intersection is equal to (k-1)2.

244

6. GEOMETRY OF THE GRASSMANN MANIFOLD

Finally, consider the asymptotic lines of order k on Sl(m, n). Any such line at any of its points is tangent to an asymptotic direction of order k. In the space P", to these asymptotic lines there correspond one-parameter families of enplanes in each of which any two infinitesimally close planes have an (m-k+1)plane in common and belong to an (m+k-1)-plane. Such families of m-planes

are called (m - k + 1)-focal. In particular, in the space P", to the asymptotic lines of second order there correspond (m - 1)-focal (or torsal) families of enplanes, and to the asymptotic lines of order in + 1 there correspond 0-focal families of m-planes.

6.4

Submanifolds of the Grassmannian G(m, n)

1. Families of m-dimensional subspaces in a projective space P" were studied

rather intensively. This started in the nineteenth century from the study of different families of straight lines in a three-dimensional projective space Ps (see Section 3.4). Later these investigations were generalized to the projective

space P". We will consider only smooth families F of subspaces. If an m-dimensional

subspace p of the family 7 depends on r parameters, we will denote such a

family by P. If r < n - m, then the family P forms a point submanifold of dimension m + r in the space P". However, this submanifold can have singular points (e.g., see Akivis [A 57, 87]). If r = n - m, then a family Tr is called a congruence. A congruence of subspaces is characterized by the fact

that there passes a finite number of subspaces of Fr through any point of general position in P". If r > n - m, then a family P is called a complex. If r = (m + 1)(n - m) - 1, then a family .7'' is called a hypercomplex. The latter family is a submanifold of codimension one on the Grassmannian G(m, n). Different types of families of subspaces were investigated by R. M. Geidelman, Moscow, S. E. Karapetyan, Erevan, K. I. Grincevicius, Vilnius, L. Z. Kruglya-

kov, and R. N. Shcherbakov, both Tomsk, and their students and colleagues (see the book Kruglyakov [Kru 80) and the survey papers Geidelman [Ge 67a) and Shcherbakov [Sh 67]).

However, in most of these works the Grassmann mapping of families of m-dimensional subspaces onto the algebraic variety !l(m, n) C PN, N = ( +j) - 1 was not used at all or used occasionally; there are some exceptions; see, for example, the papers Karapetyan [Kar 62a, b, c] and the recent book Mizin, Chupakhin, and Shcherbakov [MCS 911 in which the Grassmann mapping was used as the main tool of investigation. The application of the Grassmann mapping allows one to see many facts of the geometry of families of m-dimensional subspaces from another point of view and to find many new results. We will conduct such an investigation for families of two-dimensional subspaces (2-subspaces) in the space P5. The choice of dimensions 2 and 5 is motivated by the fact that in the space P5, a complementary subspace to a

6.4

Submanifolds of the Grassmannian G(m, n)

245

2-subspace again is a 2-subspace (as in the space p3 a complementary subspace to a straight line is again a straight line).

We will consider families F? depending on r = 2, 3, and 5 parameters. In the first case, a family F2 constitutes a planar hypersurface in the space P5, in the second, a family F3 is a congruence of 2-subspaces, and in the third case, a family F5 is a complex of 2-subspaces. 2. A family of 2-subspaces in the space P5 is the Grassmannian G(2, 5). Its dimension is p = 9, and the Grassmann mapping sends it bijectively onto the

algebraic variety f)(2,5) C PN, N = (3) - 1 = 19. The degree of the variety 11(2, 5) is calculated by formula (6.1.3) and is equal to 42.

We will write for the Grassmannian G(2,5) some formulas from Section 6.3. Since m = 2, then the variety 0(2,5) has at any of its points only two asymptotic cones Cpl) and CP31, CP21 C CP3). In notations of Section 6.2, the equations of the asymptotic cone CP21 of the variety 11(2,5) can be written in the form: rank (m,`,) = 1, a= 0, 1, 2; i = 3A5, 5, (6.4.1)

and the equations of the asymptotic cone CP3) in the form

det (w,) = 0.

(6.4.2)

It follows that the cone CP3> is a hypercone of third order in the tangent subspace TDII(fl) whose dimension is equal to 9. The cone f'-(2) is the Segre cone of dimension 5 whose projectivization is the Segre variety S(2,2) C Ps. The degree of this Segre variety S(2, 2) is equal to (a) = 6. The Segre cone carries two two-parameter families of three-dimensional plane generators Opal(a) and Op21(f3), and the cubic cone -P( 2) carries two two-parameter families of six-dimensional plane generators A(3) (a) and AP(3) Consider next a two-parameter family F of 2-subspaces in the space P5. On the variety f2(2, 5) to such a family there corresponds a two-dimensional smooth submanifold V2. The classification of points of the submanifold V2

depends on the mutual location of its tangent subspaces Tyll (V2) and the cones Cpl) and Cp3).

In the general case, the subspace T,')(V2) has no common straight lines with the cone CP2) and meets the cone C,? in three straight lines. Thus, on V2, there are no asymptotic directions of first order, and there are asymptotic directions of second order. If V2 is a submanifold of general type, then the described situation holds at any point of V2. By virtue of this, a three-web, formed by three families of asymptotic lines of second order, arises on V2. This relationship between the geometry of families and the theory of twodimensional three-webs was considered in Zhogova [Zh 78, 79). In the space P5, to the asymptotic lines of second order of the submanifold V2 there correspond one-parameter subfamilies F1 of 2-subspaces, and these

246

6. GEOMETRY OF THE GRASSMANN MANIFOLD

subfamilies have one-dimensional envelopes. In any 2-subspace of the family .P2, there are three focal points of second order in which this subspace is tangent to the envelopes of three subfamilies passing through this 2-subspace. Note also that a two-parameter family F2 of 2-subspaces in the space P5 can be considered as a hypersurface with two-dimensional plane generators

in P5. In this case, three focal points in a 2-subspace are singular points of this hypersurface-in these points the tangent subspace to a hypersurface is of dimension three.

3. Consider a three-parameter family F3 of 2-subspaces in the space P5. It is a congruence. On the variety fl(2, 5), to such a family there corresponds a three-dimensional submanifold V3. The tangent subspace Tpll(V3) meets the asymptotic cone '"P(3) of third order in a two-dimensional cone of third order. Each generator of this cone determines asymptotic directions of third order at a point p E V3 that are tangent to the asymptotic lines of third order of V3 passing through the point p. In the space P5, to these asymptotic lines there correspond 0-focal families of 2-subspaces containing the subspace p. This implies that the focal points of the subspace p, that is, the points at which 0-focal families are tangent to their envelopes, form a cubic curve W3 in the 2-subspace p. To find an equation of the curve W3, we will write the equation of the congruence .F3 in the form wa = pupBa

(6.4.3)

where a, ,C = 0, 1, 2; i = 3, 4, 5, and the I-forms 00 are linearly independent basis forms on the congruence y3. In the tangent subspace TPII(f1), equations (6.4.3) represent parametric equations of the three-dimensional subspace Tptl(V3). Suppose that x = xaA,, is an arbitrary point of the subspace p C P5. The differential of this point has the form dx = (dxa + x0w, )Aa + xawQA;.

The focal points of the subspace p and the focal directions on the congruence are determined by the conditions xaWa = 0.

By (6.4.3), these conditions can be written as

xap.pBa = 0. Since at the focal points this system must have a nontrivial solution with respect to the forms 0a, the locus of focal points is determined by the following equation: det (xap;,Q) = 0. The latter equation is of third degree with respect to xa, that is, it determines a cubic curve W3 in the subspace p.

6.4

Submanifolds of the Grassmannian G(m, n)

247

On the other hand, the equations of the cone Ty')(V3) n C(3) of focal directions have the form det (paaB0) = 0,

which also immediately follow from equation (6.4.2) of the cone C;31

Consider now the projectivization PTp' 1(12) of the variety D = Q(2,5) and the projectivization PTnll (V3) of the submanifold V3. Denote by 03 the cubic curve in which the projectivizations PTpll (V3) and PC,3) meet: PT$1)(V3) n PCP(3) = 03- If the 2-subspace PTp')(V3) does not meet the Segrean PCP(2) = S(2, 2), then the above cubic curve does not carry singular points. This corresponds to a congruence F3 of general type. If the 2-subspace PTyll (V3) does meet the Segrean S(2, 2), then to each intersection point there corresponds a 1-focal direction on a congruence F3. Then through a 2-subspace p there passes a torse (i.e. a developable submanifold), formed by 2-subspaces of F3. Furthermore, the 2-subspace p contains the characteristic line I of this torse, and the 2-subspace p itself belongs to a three-dimensional subspace L that is tangent to the torse along p. If the curve 03 has one singular point, then the focal cubic W3 decomposes

into a straight line I and a conic C (see Figure 6.4.1). If the curve 03 has two singular points, it necessarily has a third one and thus the focal cubic W3 decomposes into three straight lines that are in general position (see Figure 6.4.2). Congruences F3 of this kind are called totally focal. We will consider the latter case in more detail. Let us choose a moving frame in such a way that the points A0, A,, and A2 are the intersection points of three characteristic straight lines of a subspace p, and that the points A3, A4,

Figure 6.4.1

Figure 6.4.2

248

6. GEOMETRY OF THE GRASSMANN MANIFOLD

and A5 are located in the subspaces Lo, LI and L2, where L. is a 3-subspace tangent to the torse whose characteristic in the subspace p is the straight line 1,, (see Figure 6.4.2). Let 90, 91, and 92 be basis forms of the congruences J chosen in such a way that they are basis forms of the torses indicated above. Then the torse To is determined by equations 91 = 92 = 0. Since the straight line 10 = Al A A2 is the characteristic straight line of the subspace p, then NO C p, from which we obtain w1 w2 0 (mod 91,92).

Since the 3-subspace Lo = p A A3 is tangent to the torse described by the subspace p, we also have wo, wo = 0

(mod 91,92).

In a similar manner we obtain wa, w2 M 0

(mod 90,92),

wi, wi =- 0

(mod 90, 92)

wo, wi

(mod 9°, 91), (mod 90, 9').

and

0

W232 wz = 0

We recall that in these equations i = 3,4,5. Comparing all these relations, we find that w30 =

Wp90r

wi = 0, wz = 0,

w4 0

= 0+

5

w0 =

0+

,) = go" wl = 0, w2 = 0,

w25 =

(6.4.4) r02.

The coefficients p, q and r in these equations can be reduced to 1 by normalizing

the points A3, A4, and A5. Taking exterior derivatives of equations (6.4.4), where p = q = r = 1, we arrive at the following exterior quadratic equations:

won91 -w3n9°=0, won92-wyA9°=0, w; A 9° - w3 A W =0, w2 A 0° - w5 A 92 = 0,

wIA92-w5A0'=0,

w2

(6.4.5)

A0'-w4A92=0.

Consider the submanifold described by the point A0 of intersection of the characteristic straight lines 11 and 12. By (6.4.4), the differential of the point A0 can be written as dAo = woA0 +woA, +w0A2 +w0A3.

(6.4.6)

6.4

Submonifolds of the Grassmannian G(m, n)

249

Thus the subspace L° is tangent to the three-dimensional submanifold (Ao). The 1-forms w0 and wo occurring in formula (6.4.6) can be found from equations (6.4.5) by means of Cartan's lemma: W02 =10262

100 =10101 + 101000,

+ lone°,

(6.4.7)

-w3 =10082 -1308°, -w3 =10082 -13000.

As to the 1-forms w03, this form can be found from equations (6.4.4) by taking p = 1. Substituting the values of all these forms into equation (6.4.6), we find

that dAo = woAo + 1018' Al + 10202A2 + (100A1 + 100A2 + A3)9°.

We further specialize our moving frame by placing the point A3 on the tangent to the line 91 = 92 = 0 of the submanifold (Ao). Then we obtain

100=loo=0 and

dAo =woAo+1018'A1 +10292A2+8°A3. Equations (6.4.7) now become w0 =1010'+

42

= 10202,

(6.4.8)

w3 = 1300°, w3 = 1308°

We need to prove that the coordinate lines on the submanifold (Ao) form a conjugate net. To this end, we find d2Ao = [130(0°)2 + 1011(01)2 ]A4 + [153 0(90)2 + 12 02 (82)2JA5

(mod Lo).

It follows that the second fundamental forms of the submanifold (Ao) are 4(2) = 130()2 + 101(0')2, '

(2)

= 130(90)2 + 102(82)2

The fact that these forms are sums of squares proves that the coordinate lines on the submanifold (Ao) form a conjugate net (see Section 3.2, and for more detail, see Akivis and Goldberg [AG 931, Ch. 3). Moreover the three-dimensional submanifold (Ao) is stratified into three families of two-dimensional surfaces carrying conjugate nets. In fact, taking the exterior derivatives of the basis forms 8° and applying equations (6.4.4), we find that

2+aA0" dO' = PaQ - w2+o)

where there is no summation with respect to a. By the Frobenius theorem, each of the equations 00 = 0 is completely integrable, and this proves that the submanifold (Ao) is stratified as indicated above. Similar conclusions are true for the submanifolds described by the points Al and A2.

6. GEOMETRY OF THE GRASSMANN MANIFOLD

250

Since the totally focal congruences considered above are congruences of special kind, we must prove their existence. This can be done by applying the Cartan test to the system of equations (6.4.4) in the same way as this was done in Section 3.2 for submanifolds carrying a net of curvature lines. It turns out that a totally focal congruence of 2-subspaces in the space P5 exists, and the solution of the system defining such a congruence depends on six functions of two variables.

4. Consider finally a five-parameter family (a complex) F5 of 2-subspaces

in the space P5. The number of parameters, on which a 2-subspace of Y5 depends, coincides with the dimension of the space P5. This is the reason that such complexes are of interest for integral geometry in the sense of 1. M. Gelfand (see Gelfand and Graev [GG 68]).

Under the Grassmann mapping, to a complex FS there corresponds a fivedimensional submanifold V5 C 0(2, 5). At any point p, the submanifold V5 has the five-dimensional subspace Tpll(VS) whose projectivization is the fourdimensional projective space PTP11(V5). To different cases of mutual location of the subspace Tpl1(V5) and the asymptotic cones ($2) and Cp3) there correspond different classes of complexes P. It is convenient to conduct the investigation of these cases by considering the projectivization of the tangent subspace Tpl) (52) under which PCP(') = S(2,2) and PC P31 = VT C Ps, where VT is a cubic hypersurface in Ps. In the general case, the subspace PTp11(V5) meets the variety S(2, 2) at six points which we will call the characteristic points. On the submanifold V5, to these points there correspond six fields of asymptotic directions of first order. They are tangent to six families of asymptotic lines of first

order on V5. In the space P5, to these families there correspond six families of torses (developable surfaces) on the complex P. Through any 2-subspace p C .PS, there pass six torses, one from each family. Thus six characteristic straight lines arise on the 2-subspace p, and six three-dimensional subspaces tangent to these torses pass through the 2-subspace p. Moreover the subspace PTp'1(VS) meets the hypercubic V3 in a threedimensional cubic submanifold. Since the hypercubic V3 C P8 carries two two-parameter families of five-dimensional plane generators, the submanifold

V3 = V3 n PT, (V5) carries two two-parameter families of rectilinear a-generators and fl-generators. In view of this, the submanifold- V5 carries two fiber bundles El?) and F(2) of two-dimensional asymptotic directions of second order. The asymptotic lines of second order on V5 are tangent to twodimensional directions belonging to these fiber bundles. In the space P5, to these lines there correspond 0-focal one-parameter families of 2-subspaces belonging to the complex P. If we fix a point m in a 2-subspace p C .PS, then through this point there passes a two-parameter family of 2-subspaces of the complex Y$ which contains

the 2-subspace p. This family is a four-dimensional cone C,,, with vertex at a point m. On the submanifold V5 C 52(2,5), to the cone C,,, there corre-

6.4

Submanifolds of the Grassmannian G(m, n)

251

sponds an integral submanifold of the distribution E.( 2). On the other hand, if we consider a four-dimensional subspace p passing through a 2-subspace p, p D p, then a two-parameter family of 2-subspaces of the complex F5 belongs to this subspace p. On the submanifold V5 C Q(2,5), to the subspace p there corresponds an integral submanifold of the distribution Ep . A projectivization PCm of the cone C,,, is a two-parameter family of straight lines in a four-dimensional projective space p4. If the vertex m of this cone does not belong to any of six characteristic straight lines, then the family of straight lines indicated above constitutes a ruled hypersurface without singular points. If the vertex m lies on one of the characteristic straight lines, then there is one singular point on each straight line of the family PC,,,, and the family PCm is semifocal. If the vertex m is the intersection point of two characteristic straight lines, then there are two singular points on each straight line of the family PC,,,, and the family PCm is focal. In this case the ruled hypersurface indicated above is tangentially degenerate (see Subsection 3.3.3). In the general case six characteristic straight lines in the 2-subspace p and six three-dimensional tangent subspaces to the torses passing through p are in general position. If they are in a special (not general) position, then the corresponding complexes F belong to special classes of complexes. Such special classes of complexes were considered in Bubyakin [Bub 90, 91]. We will describe briefly results of these papers.

Theorem 6.4.1 If three characteristic straight lines of each 2-subspace p c F5 belong to a pencil with center at a point m, and the corresponding threedimensional characteristic subspaces are in general position, then the point m describes a hypersurface in the space P5 to which the 2-subspaces of the complex .F5 are tangent (see Figure 6.4.3). The tangent three-dimensional subspaces of three other torses belong to the tangent subspace of this hypersurface.

Figure 6.4.3

Figure 6.4.4

6. GEOMETRY OF THE GRASSMANN MANIFOLD

252

Figure 6.4.5

Figure 6.4.6

Theorem 6.4.2 If two triplets of characteristic straight lines of each 2-subspace p C J belong to two pencils with centers at points m1 and m2, then the straight line m1m2 is characteristic, and the points m1 and m2 describe hypersurfaces in the space P5 to which the 2-subspaces of the complex 25 are tangent (see Figure 6.4.4).

In addition to the two cases indicated in Theorems 6.4.1 and 6.4.2, there are only two more configurations of characteristic straight lines of a complex .FS (see Figures 6.4.5 and 6.4.6). For these additional configurations the points m1 i m2 and m3 (or m1, m2, m3, and m4) describe hypersurfaces to which the 2-subspaces of the complex F5 are tangent. As it follows from Theorem 6.4.1, all four classes of complexes indicated above are self-dual; that is, each of them is transformed into itself under any correlative transformation of the space P5. One can also consider special classes of complexes .F5 characterized by a pairwise congruence of characteristic straight lines or characteristic three-dimensional characteristic subspaces of 2-subspaces

of the complex P.

6.5

Normalization of the Grassmann Manifold

1. In this section on the Grassmann manifold G(m,n) of m-dimensional subspaces of an n-dimensional projective space P", we consider a certain supplementary construction called the normalization. By means of this normalization, one can construct the structure of a Riemannian or semi-Riemannian manifold or an affine connection on G(m, n). Let U be an open domain of the Grassmann manifold G(m, n) of dimension p = (m + 1)(n - in) coinciding with the dimension of G(m, n). This domain

6.5

Normalization of the Grassmann Manifold

253

can coincide with the entire manifold G(m, n) or can be its proper subset. The domain U is said to be normalized if to each its m-dimensional subspaces p there corresponds a chosen subspace p' of dimension n - m - 1 in the projective space P", such that p' does not have common points with p. The subspace p' is called the normalizing subspace for the subspace p. We will denote a normalized domain U by U". If U = G(m, n), then we will denote the normalized Grassmann manifold G(m, n) by G" (m, n).

Since the subspace p' belongs to the Grassmannian G(n - in - 1, n), a normalization of the manifold G(m, n) is defined by a normalizing mapping v : G(m, n)

G(n - in - 1, n)

(6.5.1)

given in the domain U C G(m, n) and having a submanifold U' of the Grassmannian G(n - in - 1, n) as its image. Of course we assume that the mapping v is differentiable. Let r be the dimension of the submanifold U'. The number r coincides with the rank of the mapping v. Since dim G(n - in - 1, n) = p = (m + 1)(n - m),

we have 0 < r < p. If r = p, then U' is an open domain of the manifold G(n-m-1, n). If 0 < r < p, then U' is a proper submanifold of G(n-m-1, n). If r = 0, then U' consists of one fixed subspace p' of dimension n - in - 1 in the projective space P". If r = p, the normalization is called nondegenerate. In this case there is a one-to-one differentiable correspondence between the domains U and U'. If 0 < r < p, then the complete preimage v-1(p') of the normalizing subspace p' is a differentiable submanifold of dimension p-r on the Grassmannian G(m, n). If r = 0, then the complete preimage v-' (p') coincides with the entire domain U.

Consider also the case in = 0. In this case the manifold G(0, n) coincides with the projective space P", and the manifold G(n - 1, n) coincides with the dual projective space (P")'. Thus, to a point x E U C P", the normalization v sets in correspondence a hyperplane t; not passing through the point x. 2. Let us write the equations of the normalizing mapping v using differential forms. To this end, with the pair of subspaces p and p' we associate a family of point frames (A(} in such a way that A. E p and A, E p'. For each frame of this family, we have

dAQ=w Ap+w,A,, dAi=w°A,+wAj.

(6.5.2)

As in Section 6.3, the 1-forms w , are basis forms of the frame bundle associated

with the Grassmannian G(m, n). As to the I-forms w° defining displacements of the subspace p', they are no longer fiber forms. They are expressed in terms of the basis forms w' by relations w°

(6.5.3)

These relations are differential equations of the normalizing mapping (6.5.1). The coefficients VP form a square matrix of order p = (m + 1)(n - m), whose rank r is equal to the rank of the mapping v: rank (a A) = r.

254

6. GEOMETRY OF THE GRASSMANN MANIFOLD

The 1-forms wo and w; are fiber forms of the frame bundle associated with the normalized Grassmannian G"(m, n). As in Section 8.3, we set 7rg = wQ(6) and ir; = w? (6), where 6 = dl,,i =o. For the frame bundle in question, the forms aQ are invariant forms of the group GL(m + 1), and the forms 7r are invariant

forms of the group GL(n - m). The quotients of these two groups modulo the subgroup H of homotheties are isomorphic to the groups of projective transformations of the subspaces p and p' of the space P^, respectively. The quotient of group G = GL(m + 1) x GL(n - m) modulo the subgroup H is the stationary subgroup of the m-pair (p, p*). Taking the exterior derivatives of the basis forms wa by means of the structure equations (6.3.3), we find that

This implies the relations (6.5.4)

6W,° = -("J. Tr + wp7rQ,

which describe the law of transformation of the basis forms w, under admissible

transformations of frames associated with a nondegenerate m-pair (p, p') by means of differential forms. These formulas are analogous to formulas (2.1.17). Next, making use of equations (6.3.3), we take exterior derivatives of equations (6.5.3). This leads to the equations (dA,p

- A,k W - \QOWk +

A, WQ +

A,Pw7) A A

0,

or

VapAwp=0,

(6.5.5)

where, as usual, VA ?P denotes the expression occurring in parentheses in the previous formula. Applying Cartan's lemma to equation (6.5.5), we find that DA°a =

k-,Wy,

(6.5.6)

where \"3' - ) 113 If we fix an m-pair (p, p' ), then equations (6.5.6) take the form

Oaa'p = 0,

where as in Subsection 2.1.2, V5A f = VA,"a(6). The last relations show that the coefficients) a form a tensor, which is called the fundamental tensor of the normalized domain U" C G(m, n). This tensor is connected with a first-order differential neighborhood of the m-pair (p, Since the stationary subgroup of an m-pair is the product of the general linear groups GL(m+ 1) and GL(n-m), any geometric object of the normalized Grassmann manifold is a tensor. In particular, the object )'ijk occurring in

6.5

Normalization of the Grassmann Manifold

255

equations (6.5.6) is a tensor. This tensor is connected with a second order differential neighborhood of the normalized Grassmann manifold. In the normalized domain U" C G(m,n), we consider the quadratic differential form

9=w;wa Substituting the values (6.5.3) of the forms w° into this form, we obtain (6.5.7)

9=

siwhere the coefficients are obtained if one symmetrizes the tensor aQ multaneously with respect to both vertical pairs of indices:

g?P = 2(a A + ap°).

Hence the quantities g,? themselves form a tensor that is symmetric with respect to these pairs of indices. In view of this, the quadratic differential form g is invariant in the domain U". Denote the rank of the matrix of coefficients of the quadratic form g by F. If i= = p, then the quadratic form g is nondegenerate and defines a Riemannian

(or pseudo-Riemannian metric) in the domain U". If r < p, then the form g defines a semi-Riemannian metric in the domain U" for which the equation g°pw'a = 0

defines an isotropic distribution of dimension p-r. If the rank r of the mapping v vanishes, then r` = 0, and the form g vanishes. The normalization v is said to be harmonic if the coefficients in equations (6.5.3) are symmetric with respect to the vertical pairs of indices:

a°a = A.

(6.5.8)

VP and r = r. If r < p, and the normalization v If this is the case, then is harmonic, then the isotropic iistribution defined by the form g is integrable, and its integral manifolds coincide with the complete preimages v-1(p') of the normalizing subspaces p'. 3. Now we will establish a geometric meaning for the quadratic form (6.5.7). To this end, we find the matrix coordinates X and Y of the subspaces p = Ao A A, A ... A A, and p' = (Ao + dAo) A (Al + dA1) A ... A (Am + dAm )

with respect to the frame R = {Ao, Al , ... , A,,). We have

X = 1

1

0

...

0%

0

1

...

0

0 0

0

...

1

0

...

0

0..0.......0

_

Im+l

( 0(n-m) x (m+1)

6. GEOMETRY OF THE GRASSMANN MANIFOLD

256

where I,,,+l is the identity matrix of order m + 1 and °(n_,n) x (,n+1) is the zero (n - m) x (m + 1) matrix. By (6.5.2), we also find that

Yap+Wp As was indicated in Subsection 6.1.4, the matrix coordinate of an mdimensional subspace is determined up to multiplication from the right by a nondegenerate square matrix of order m + 1. As such a matrix, we take the matrix (b; +woo)-1 - (6000 -wp).

(6.5.9)

In the last and the following formulas, we assume that two matrices are equiva-

lent if they differ by second-order terms with respect to the elements of the matrix (wi ). Multiplying the matrix Y from the right by matrix (6.5.9), we find that WI WI

(6.5.10)

.

P,

Consider further the normalizing subspaces p' = An,+1 A ... A An and p" = (Am+1 + dAm+1) A ... A (An + dAn). The tangential matrix coordinate U of the first of these subspaces can be easily found: 1

...

0

0

...

0

1

0

...

0

O(m+1)x(n-m)) , 0

where O(m+l)x(n_m) is the zero (m + 1) x (n - m) matrix. The tangential matrix coordinate V of the subspace p" can be found from condition (6.1.6) which in the case in question can be written in the following form:

v°(A;+dA;)=v7(d +w;)+vpwp=0.

(6.5.11)

To find a basis of the solution space of this system, we take vp = dp. Multiplying (6.5.11) from the right by the matrix

(6 + Wk)-1 ' (a - 4), we find that

ilk ^. - 4"(6L -wk) - -Wk. Thus the tangential matrix coordinate V of the subspace p'' has the form

V=(do, -wk).

(6.5.12)

Let us find the cross-ratio W of two such m-pairs (p,p*) and (p', p"') by applying formula (6.1.12). To this end first we compute the products of the matrices occurring in formula (6.1.12): Ux = UY = (ap), VY = (6000 - Wi wp.

(6.5.13)

6.5

Normalization of the Grassmann Manifold

257

Thus we have (VY)-1

=K

+ w°wp)

and

W=

bR + w°wp

(6.5.14)

- Wk l

O(n-m) x (n+1)

/

.

(6.5.15)

In expressions (6.5.13)-(6.5.15), we retain the terms of second order with respect to the elements of the matrix (w{ ). Since such terms are principal, we discard the terms of order higher than two. The formula (6.5.15) determines the matrix W which is the cross-ratio of m-pairs (p, p') and (p', p"). To compute the quadratic form (6.5.7), we find the trace of the matrix W:

tr W =m+l+w°w,, Since for small x we have log(1 + x)

x,

it follows that

log(1+m+lw°w°) ^ m+lw'w°' and as a result we find that

g=w°w,

(m+l)log (l+m+ltr W).

(6.5.16)

The last formula gives the expression of the quadratic form g in terms of the cross-ratio of two infinitesimally close m-pairs (p, p') and (p', p"). 4. A normalization of the Grassmann manifold G(m, n) defines an affine connection on it. In fact, taking the exterior derivatives of the basis forms 4 of the manifold G(rn, n) and applying structure equations (6.3.3), we obtain (6.5.17)

Consider the 1-forms w'Q =bQw;

-54.

(6.5.18)

These forms are expressed in terms of the fiber forms wQ and w. of the frame bundle associated with a normalized Grassmann manifold G"(m,n) (or a domain U" of this manifold). In the tangent space Tp(fl) to the manifold fl(m, n), which is the image of the manifold G(m, n) under the Grassmann mapping, these forms define a subgroup of the general linear group whose transformations preserve the cone CD determined by equations (6.3.11). Exterior differentiation of equations (6.5.18) leads to the following exterior quadratic equations:

°j

k°

dw'p = b° AY°wl °j - bpwk n w' + b{wry n wR ry a n wl, jl e n w'ry - b'Aa`wk 7 kl

(6.5.19)

It is essential that the right-hand sides of equations (6.5.19) are expressed only in terms of the basis forms w, of the normalized Grassmann manifold G" (m, n).

258

6. GEOMETRY OF THE GRASSMANN MANIFOLD

By the facts from the general theory of spaces with of ine connection (e.g., see Kobayashi and Nomizu [KN 63], Ch. III, or Lichnerowicz [Lie 55], Ch. III, or Laptev [Lap 66]), these equations show that the forms wa define an affine

connection on G"(m,n), and the forms occurring in the right-hand sides of equations (6.5.19) are the curvature forms of this connection. Denote this affine connection by r". The connection r" is uniquely determined by the normalization v. Let us write the curvature forms of the connection 1' in the form WQ

= (64 , i + d;b Aki )w' A wk.

(6.5.20)

The alternated coefficients occurring in the right-hand sides of the last equations form the curvature tensor of the constructed connection. Equations (6.5.20) imply that this tensor has the following form:

Ra kl = 2 (baokAj"l + ba 5 ), - b J Ajt -

(6.5.21)

namely this tensor is expressed only in terms of the components of the fundamental tensor of the normalized Grassmann manifold G" (m, n). Equations (6.5.17) show that the affine connection r, is torsion-free. In view of this, the following theorem holds:

Theorem 6.5.1 The normalization v of a normalized domain U" C G(m, n) uniquely determines a torsion-free affine connection r,, with the connection forms (6.5.18) on it. The curvature tensor of this connection is expressed in terms of the fundamental tensor of the normalization v according to formulas (6.5.21).

We will find also the Ricci tensor of the connection r,. Contracting the tensor (6.5.20) with respect to the indices i, l and a, e, we obtain the following expression for the Ricci tensor of the connection F":

Rjk = R"jk; =

2

Ajk + Akj - (n + 1)Ajk

).

(6.5.22)

From these relations it follows immediately that the Ricci tensor of the connection r, is symmetric if and only if the normalization v of the Grassmann manifold G"(m,n) is harmonic. 5. Suppose that the normalization v of the Grassmann manifold G" (m, n) is harmonic (i.e., conditions (6.5.8) hold) and that its fundamental tensor A, is of maximal rank. Then the quadratic form g can be expressed as 9 = A°aw' ij aw'p,

and it defines a Riemannian (or pseudo-Riemannian) metric on the normalized Grassmann manifold G"(m,n).

6.5

Normalization of the Grassmann Manifold

259

Since the left-hand side of relation (6.5.6) is the covariant differential of the tensor A°Q with respect to the connection r, this relation shows that the connection r" is not the Levi-Civita connection of the metric defined by the form g. Nevertheless, we still can construct from the curvature tensor of the connection r" defined by formulas (6.5.21) a tensor that is covariant with respect to the indices i, j, k, and 1. This tensor is defined by the following formula:

Rap'Ye - 1 /A"Rmo

2l im

ijkl

pjkl

- App jm

pikt

Substituting the expression (6.5.21) for the tensor Rpfki c in the above formula, we obtain

R 3kl -

4

(Aiijk -

ik

Aji + A jf Alk - Aij Aki

( 6.5.23 )

Af3'YAac ik + ApaA'c jk it - ApcAar ji lk ,+ ji kl )'

-ApaAc7

it

One can immediately verify that this tensor satisfies the following relations: (6.5.24) R jkic = -R ik-y _ -Rj klc - Rktijp which are the standard relations for the curvature tensor of a Itiemannian

manifold. Next we define the following tensor: (6.5.25)

9ljkl c - AO Aji - A tcARk

Since, in the case in question, the fundamental tensor of the normalization v satisfies the symmetry relations (6.5.8), the tensor (6.5.25) also satisfies the conditions of type (6.5.24). The tensors ski c and g ski c allow us to define a sectional curvature on the normalized Grassmann manifold G"(m,n). To this end, in the tangent space T. (G(m, n) ), we consider two vectors { _ ({Q) and i = (711) and the bivector p = A rl defined by and rl. The coordinates of the bivector p are ij

pap =

1

i

j

i

The sectional curvature of the manifold G"(m, n) at a point x is defined as the ratio

K(p) = R(p,p)

(6.5.26)

9(p, p)

of two quadratic forms defined by the tensors (6.5.21) and (6.5.23): R

p) -

apryc

ij

kl

apryc

ij

ki

- Rijkl papp'tc and 9(p p) = 9ijkt It is possible to find the principal bivectors of the space TT(G(m,n)) for which the sectional curvature takes stationary values. However, for the general case this involves tedious calculations.

papp7e.

6. GEOMETRY OF THE GRASSMANN MANIFOLD

260

6.6

Homogeneous Normalization of the Grassmann Manifold

1. As was indicated earlier, a Grassmann manifold G(m, n) is a homogeneous space. However, in general, a normalized Grassmann manifold G"(m,n) is

not a homogeneous space. In fact even two m-pairs (p, p') and (q, q') have a matrix invariant W-their cross-ratio. In view of this, in general, there is no projective transformation superposing two neighborhoods U(p, p') and &(p, j r) of two m-pairs belonging to a normalized Grassmann manifold GM (m, n) (or its open domain U°). On the other hand, if a normalized Grassmann manifold G° (m, n) is homogeneous, then its fundamental tensor determining the location of an m-pair (p', p" ), which is infinitesimally close to the m-pair (p, p' ), must be covariantly constant; that is, it must satisfy the condition

V) A=0,

(6.6.1)

where V is the operator of covariant differentiation with respect to the affine connection F. Taking the exterior derivatives of the system of equations (6.6.1) by means of structure equations (6.3.3) of the projective space and excluding the differentials dA Q, we arrive at the system of relations

a ij kl jl + kj f it + ij k1 + 7!j 0/7 7 a, Q7 _ c(3 my ik - iii xij \Ik = 0.

op ye ik

a0 Y

aQ 7

1

ary

Oe

(6.6.2)

\Ik

Conditions (6.6.1) and (6.6.2) are necessary and sufficient for the normalization v of the normalized Grassmann manifold G°(m, n) with the fundamental tensor .\,°Q to be homogeneous.

2. Let us find some solutions of the system of equations (6.6.1) and (6.6.2). To this end, first we consider a polar normalization, namely a normalization of the Grassmann manifold G(m, n) by means of a nondegenerate hyperquadric

Q of the space P". Let po be an m-dimensional subspace of the space P" which is not tangent to the hyperquadric Q, and let po be an (n - m - 1)-dimensional subspace of P" which is polar-conjugate to po with respect to this hyperquadric. The subspaces po and po form a nondegenerate m-pair (po,po). The set of subspaces p, located in the same manner with respect to the hyperquadric Q as po (we will clarify below the meaning of the expression "in the same manner"), form an open domain U, and the subspaces p' polar-conjugate to the subspaces p with respect to the hyperquadric Q define the polar normalization of this domain. If the hyperquadric Q is imaginary, then the domain U coincides with the entire Grassmann manifold G(m, n). Let us associate a family of projective frames {AC} with an m-pair (p, p')

in such a way that the points A. E p and Ai E p'. As we did in Chapter 1,

6.6

Homogeneous Normalization of the Grassmann Manifold

261

we denote by (AF, An) the scalar product of the points A( and A, with respect to the hyperquadric Q. Since the points A. and Ai are polar-conjugate with respect to this hyperquadric, we have

gi. = (Ai, A.) = 0.

(6.6.3)

(Ai, Aj) = gij and (Ao, A0) = gall

(6.6.4)

The scalar products

form nondegenerate symmetric matrices (goo) and (gij). With respect to any chosen frame, the equation of the hyperquadric Q can be written as

900xox0 + 9ijxixj = 0.

(6.6.5)

Moreover the signature of each of the quadratic forms g0pxox0 and gijxixj is not changed when the subspace p moves in the domain U" C G(m, n). This condition clarifies the meaning of the expression "in the same manner" which we used above to characterize the domain U. Taking derivatives of equations (6.6.3) and (6.6.4) by means of equations (6.5.2), we find that

9ijwj = 0, dgij = 9ikLJ + 9kj(

,

d900 = 9oywp + 970w0

The first relation implies that

w° = -go0gij 4,

(6.6.6)

and the last two relations can be written as

Vgij = 0, VgoO = 0,

(6.6.7)

where, as earlier, V is the symbol of covariant differentiation with respect to the connection r l. Comparing equations (6.6.6) and (6.5.3), we obtain the following expression for the fundamental tensor of the polar normalization:

00 - a0 gij - -9 gij.

(6.6.8)

Since the tensors g°0 and gij are symmetric, this fundamental tensor satisfies condition (6.5.8), and the polar normalization is harmonic. Since the tensors goo and gij are nondegenerate, the fundamental tensor of the polar normalization is also nondegenerate. From relations (6.6.7) it follows that for the polar normalization we have V.1,oF = 0,

(6.6.9)

262

6. GEOMETRY OF THE GRASSMANN MANIFOLD

which shows that its fundamental tensor is covariantly constant with respect to the connection [''. This can also be immediately verified by checking that by (6.6.8), conditions (6.6.2) are satisfied identically. Hence the polar normalization of the Grassmann manifold is homogeneous. We will give another geometric proof of this. The projective space P", in which a fixed hyperquadric Q is given, is a pseudoelliptic space SQ of signature

q, where the number q is one unit less than the number of minuses in the canonical form of the quadratic form occurring in the left-hand side of equation (6.6.5). But the motions of this space transfer the normalized domain U" into itself. Thus this domain is a homogeneous domain of the space Sa .

For the polar normalization, the quadratic form (6.5.7) can be written as follows-

g = -9°p9ijwuwp. Thus it is nondegenerate and defines a Riemannian (or pseudo-Riemannian) metric on the Grassmann manifold G" with a polar normalization v. By relation (6.6.9), the connection F" is the Levi-Civita connection defined by this metric.

Substituting values (6.6.8) of the fundamental tensor of the polar normalization into expressions (6.5.21), we obtain the following expression for the curvature tensor: Raki f =

2

(9009"`(9ii9jk - gikgjl) + (9Q`g0" -

9Q"g0')9ij9ki).

(6.6.10)

It is also not so difficult to find the expression for the Ricci tensor of the connection F" defined by the polar normalization v. Substituting the values (6.6.8) of the components of the fundamental tensor of the polar normalization v into (6.5.22), we find that ROk = n

1

2

0"9jk;

9

in other words, the Ricci tensor of a polar-normalized Grassmann manifold G"(m,n) is proportional to its metric tensor. But this means that such a polarnormalized Grossmann manifold is an Einstein space (see Subsection 5.4.6 and also Petrov [Pe 69]).

Now we calculate the sectional curvature of the polar-normalized Grassmann manifold G"(m,n). To this end, we compute first the components of the tensor (6.5.25) for the polar normalization: Qo"e = Q" ae

9ijki

9

9

9ik9ji - 9Q"90" 9i:9,k

Hence the quadratic form g(p,p) can be written as 9(p, p) = (E,0 (77,17) - (C' 17)"

(6.6.11)

where we denote by (£, q) the scalar product of the vectors l: and p with respect

to the metric tensor g = (-g'Ogij):

6.6

Homogeneous Normalization of the Grassmann Manifold

263

The expression for the quadratic form R(p,p) is more complicated. By (6.6.10), we have

R(p,p) = 2[g"'g7`r1li) +gijgkt(W,?lt)W,

(6.6.12)

) - (,i,Ck)(,J,rlt))],

where (t., Y1#) = gijtor?p and

tjt) =

The sectional curvature K(p) of the polar-normalized Grassmann manifold G"(m,n) can be found by formula (6.5.26) where the quadratic forms R(p,p) and g(p,p) are expressed by formulas (6.6.12) and (6.6.11), respectively. From

the above formulas it follows that for m > 0 and n - m > 1, the sectional curvature K(p) is variable at a fixed point x E G(m, n), and that for m = 0 or n-rn = 1, the sectional curvature K(p) is constant at a fixed point x E G(m, n). This is natural because G(0, n) = S9" and G(n - 1, n) = (Sq)*. 3. In conclusion, we consider the case when the normalizing mapping v

has zero rank: r = 0. Then the set of normalizing subspaces consists of a single subspace p' of dimension n - m - 1, and the normalized domain U" of the Grassmann manifold G(m, n) consists of the m-dimensional subspaces p not intersecting the normalizing subspace p'. It follows from subsection 6.1.2 that the domain U" is diffeomorphic to the affine space A" of dimension

p = (m + 1)(n - m). A projective space P", in which a subspace p' of dimension n - m - 1 is fixed, is called the m-quasiafine space (see Dobromyslov [Dob 88]). The reason for this name is that if m = 0, this space reduces to an n-dimensional affine space An. We will denote the m-quasiaffine space by A. If we take an m-dimensional subspace p E Ate, as the basic element of the space AM,, then the

space Am coincides with the domain U" of the Grassmann manifold G(m, n) which we considered above.

If we associate a family of point frames with the subspace p E U" in the manner indicated in Subsection 6.5.2, then since the normalizing subspace p' is fixed, we find that dA,, = wOA0 +w' Ai, dAi = w; Aj.

(6.6.13)

Thus equations (6.5.3) take the form W° = 0,

(6.6.14)

and the tensor at for the case in question vanishes:

aQ = 0.

(6.6.15)

Hence the quadratic form g defined by equation (6.5.7) also vanishes, and it defines no metric in the domain U".

264

6. GEOMETRY OF THE GRASSMANN MANIFOLD

The forms w'Q defined by formulas (6.5.18) determine the affine connection r' in the domain U". But by (6.5.21) and (6.6.15), the curvature tensor pi ow of this connection vanishes, and the connection r" is flat. Thus, the domain U" is endowed with the structure of the affine space AP of dimension p = (m + 1)(n - m). However, since the forms w'Q are not linearly independent and expressed by formulas (6.5.18) in terms of the forms w. and w.0, the isotropy group (the stationary group) H of this space is the direct product

H = GL(m + 1) x GL(n - m).

(6.6.16)

This group preserves the Segre cone C; (m + 1, n - m) with vertex at the point x. Moreover all these cones are parallel with respect to the connection r,,

that is, with respect to a parallel transport in the space AP. The common generatrix of all these Segre cones is the Segre variety S(m, n - m - 1) lying in the hyperplane at infinity of the space AP. The variety S(m, n - m - 1) is the embedding of the direct product PI X Pn-m-1 of the projective spaces P"' and Pn-m-1 into the hyperplane PAP = Pp-1. The variety S(m, n - m - 1) carries two families of plane generators of dimensions m and n - m - 1. The

equations of the variety S(m,n - m - 1) in the space PAP can be written in the form (6.1.4) where a = 0, 1,-, m and p = m + 1, ... , n. The affine space AP with the absolute S(m, n - m - 1) and the isotropy group (6.6.16) is said to be the Segre-qlline space and is denoted by SAP. Thus we have proved the following result:

Theorem 6.6.1 Let U" be the domain of the Grassmann manifold G(m,n) formed by its m-dimensional subspaces p not having common points with a fixed subspace p' of dimension n - m - 1 (the normalizing subspace). Then the domain U" admits a mapping onto a Segre-affine space SAP that preserves the, structure of U".

The Segre-affine space SAP is a homogeneous space, and its fundamental group

G == GL(m + 1) x GL(n - m) x T(p), where T(p) is the group of parallel translations of this space, and the symbol x as earlier is the symbol of the semidirect product. Therefore the normalization

of the domain U" C G(m, n) by means of a fixed subspace p' of dimension n - m - 1 is really a homogeneous normalization. This matches the fact that by relations (6.6.15), equations (6.6.1) and (6.6.2), which are conditions for a normalization v to be homogeneous, are satisfied identically. The mapping s: U" -4 SAP described in Theorem 6.6.1 is called the stereographic projection of the Grassmann manifold G(m, n). This mapping is analogous to the stereographic projection of the conformal space C" onto the Euclidean space R" and of the pseudoconformal space Ce onto the pseudoEuclidean space Rq described in Chapter 1. Since the Grassmann manifold

Notes

265

G(1, 3) is equivalent to the pseudoconformal space C2, it admits the stereographic projection onto the pseudo-Euclidean space RZ which is equivalent to the Segre-affine space SA4.

NOTES 6.1. The projective matrix coordinates were introduced in Hua Lo-gen and Rosenfeld [HR 57] (for a more detailed and systematic treatment, see Rosenfeld [Ro 58]). The cross-ratio of two m-pairs was defined in Fuhrman [Fuhr 55]. For formula (6.1.12) for the cross-ratio of two m-pairs and its derivation, see Rosenfeld [Ro 96], §2.4.4.

The cross-ratio of four m-dimensional subspaces in a space of dimension n = 2m + I was considered in Kaplenko and Ponomarev (KP 81). The cross-ratio of four m-dimensional subspaces in (2m)-dimensional space was considered in Goldberg [Go 77, 80) (see also the book Goldberg [Go 88], p. 303). 6.2. The Vlasov configuration was introduced by Vlasov [VI 10] and was studied by Karapetyan [Kar 62a]). 6.3. Algebraic geometry of the Grassmannian was studied in detail starting from the paper Severi [Sev 15] (see also the books Hodge and Pedoe [HP 47, 52], Chapters VII and XIV). The asymptotic cones of second order on the Grassmannian G(m, n) were considered in Karapetyan [Kar 63a, b]. The determinantal varieties were studied in detail in the book Room (Roo 38). On the Segre varieties, see Griffiths and Harris [GH 791, and on the Segre cones, see Akivis [A 80]. The focal families of m-planes were studied by Korovin [Kor 50] and Geidelman (see the paper Geidelman [Ge 67a] and the book Finikov [Fin 561, Ch. 25). However, as far as the authors know, prior to the paper Akivis [A 82b] there was no detailed study of the differential geometry of the Grassmannian. In our exposition we follow that paper. 6.4. More details on the theory of two-parameter families and complexes of twodimensional subspaces in the space P5 can be found in Zhogova [Zh 78, 79] and Bubyakin [Bub 90, 91).

6.5. The normalization v which to a point r E P" sets in correspondence a hyperplane t not passing through the point x was considered in the book Norden [N 50a], §60.

For more detail on a semi-Riemannian metric, see the paper Akivis and Chebysheva [AC 811 in which with an invariant framing of a semi-Riemannian manifold was constructed. 6.6. The polar normalization was considered in the book Norden [N 50a], §572-73.

Essentially, the case where the hyperquadric Q is imaginary and consequently the domain U coincides with the entire Grassmann manifold G(m, n), was studied in detail in the paper Leichtweiss [Le 611. In this paper the Riemannian geometry of the Grassmann manifold of subspaces of an Euclidean space was mainly investigated. The quasiaf lne spaces were introduced by Rosenfeld [Ro 59] (see also Rosenfeld [Ro 96], §5.2.2). The Segre-affine spaces were first introduced in Dobromyslov [Dob 88] and were studied in detail in Rosenfeld, Kostrikina, Stepanova, and Yuchtina [RKSYu 90]. The stereographic projection of the Grassmann manifold G(1, 3) was considered in

the book Semple and Roth [SR 85), and for the general Grassmann manifold G(m, n), it was considered in the paper Semple [Sem 321.

266

6. GEOMETRY OF THE GRASSMANN MANIFOLD

The stereographic projection, the m-quasiaffine space and the Segre-affine space were studied in Dobromyslov [Dob 88].

Chapter 7

Manifolds Endowed with Almost Grassmann Structures 7.1 1.

Almost Grassmann Structures on a Differentiable Manifold

As we saw in Chapter 6, the Grassmannian G(m, n) of m-dimensional

subspaces of a projective space P^ admits a bijective mapping on the algebraic variety 1(m, n) of dimension p = (m+1)(n-m) belonging to a projective space

P", where N = (n) - 1. This mapping was called the Grassmann mapping. Under this mapping, to the family of m-dimensional subspaces of the space P"

intersecting a fixed subspace Pm = x in a subspace of dimension m - 1, on the variety 1 = f2(m, n), there corresponds a cone with vertex at the point x. (Note that we denoted the subspace P" and the corresponding point on fl (m, n) by the letter x not p as in Chapter 6.) This cone is a Segre cone, and it carries two families of plane generators f and 0 of dimensions p = m + 1 and q = n - m, respectively. We denote this cone by SC., (p, q). It is located in the tangent subspace T,, (n) and is the intersection T. (n) fl f2. Now we define the notion of an almost Grassmann structure on an arbitrary differentiable manifold M of dimension pq.

Definition 7.1.1 Let M be a differentiable manifold of dimension pq, and let SC(p, q) be a differentiable field of Segre cones with the base M such that SC,, (p, q) C Tt(M), x E M. The pair (M,SC(p,q)) is said to be an almost Grassmann structure and is denoted by AG(p - 1, p + q - 1). The manifold M endowed with such a structure is said to be an almost Grassmann manifold.

267

268

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

It follows from our previous considerations that the algebraic variety SZ(m, n),

onto which the Grassmannian G(m, n) is mapped and along with it the Grass-

mannian G(m, n) itself are endowed with an almost Grassmann structure AG(m, n), since on the variety 11(m, n), the field SC(p, q), where p = m+ 1 and q = n - m, a field of Segre cones is defined naturally. As was the case for Grassmann structures, the almost Grassmann structure

AG(p - 1, p + q - 1) is equivalent to the structure AG(q - 1, p + q - 1), since both of these structures are generated on the manifold M by a differentiable field of Segre cones SC. (p, q). The structural group of the almost Grassmann structure is a subgroup of the general linear group GL(pq) of transformations of the space T= (M), which leave the cone SC: (p, q) C T=(M) invariant. We denote this group by G = GL(p, q).

To clarify the structure of this group, in the tangent space T=(M) we consider a family of frames {e° }, a= 1,...,p; i =p+ 1, ... , p + q, such that for any fixed i, the vectors e; belong to a p-dimensional generator of the Segre cone SC =(p, q), and for any fixed a, the vectors e; belong to a q-dimensional generator q of SC= (p, q). In such a frame the equations of the cone SC= (p, q) can be written as follows:

za=tas',

a=1,...,p; i=p+1,...,p+q,

(7.1.1)

where zQ are the coordinates of a vector z = z ea belonging to the space T,(M), and to and s' are parameters on which a vector z E SCA(M) depends. The family of frames {e°} attached to the cone SC= (p, q) C T=(M) admits a transformation of the form 'e° = AapA;ea,

(7.1.2)

where (Aa) and (A') are nonsingular square matrices of orders p and q, respectively. These matrices are not defined uniquely since they admit a multiplication by reciprocal scalars. However, they can be made unique by restricting

to unimodular matrices (Ap) or (A;): det(A') = 1 or det(A;) = 1. Thus the structural group of the almost Grassmann structure defined by equations (7.1.2) can be represented in the form

G = SL(p) x GL(q)

GL(p) x SL(q),

(7.1.3)

where SL(p) and SL(q) are special linear groups of dimension p and q, respectively. Such representation has been used by T. Hangan [Han 66, 68, 80], V. V. Goldberg [Go 75a) (see also the book [Go 88], Ch. 2), and Yu. I. Mikhailov [Mi 78]. Unlike this approach, we will assume that both matrices (Ap) and (A;) are unimodular though the right-hand side of equation (7.1.2) admits a multiplication by a scalar factor. As a result we obtain the more symmetric representation of the group G:

G = SL(p) x SL(q) x H,

(7.1.4)

7.1

Almost Grassmann Structures on a Differentiable Manifold

269

where H = R' ® Id is the group of homotheties of the T=(M), and R' is the multiplicative group of real numbers. It follows from condition (7.1.1) that the p-dimensional plane generators of the Segre cone SC1 , (p, q) are determined by values of the parameters s' and

that tQ are coordinates of points of a generator . But a plane generator is not changed if we multiply the parameters s' by the same number. Thus the family of plane generators l; depends on q - 1 parameters. Similarly, q-dimensional plane generators rl of the Segre cone SC= (p, q) are

determined by values of the parameters tQ, and s' are coordinates of points of a generator q. But a plane generator rl is not changed if we multiply the parameters tQ by the same number. Thus the family of plane generators r) depends on p - 1 parameters. The p-dimensional subspaces l: form a fiber bundle on the manifold M. The base of this bundle is the manifold M, and its fiber attached to a point x E M is the set of all p-dimensional plane generators f of the Segre cone SC= (p, q). The dimension of a fiber is q - 1, and it is parameterized by means of a projective space Pa, dim P. = q - 1. We will denote this fiber bundle of p-subspaces by

Ea = (M, Pa)

In a similar manner q-dimensional plane generators r) of the Segre cone

SCy (p, q) form on M the fiber bundle EE = (M, Pp) with the base M and fibers

of dimension p - I = dim P. The fibers are q-dimensional plane generators r/ of the Segre cone SC,(p,q). Consider the manifold M. = M x P. of dimension pq + q - 1. The fiber bundle E0 induces on MQ the distribution Da of plane elements e of dimension p (see Figure 7.1.1). In a similar manner, on the manifold Mp = M x Pp, the fiber bundle Ep induces the distribution Op of plane elements q, of dimension q (see Figure 7.1.2).

Figure 7.1.1

Figure 7.1.2

270

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

Definition 7.1.2 An almost Grassmann structure AG(p-1,p+q-1) Is said to be a-semiintegrable if the distribution A. is integrable on this structure. Similarly an almost Grassmann structure AG(p - 1, p + q - 1) is said to be p-semiintegrable if the distribution Op is integrable on this structure. A structure AG(p-1, p+q-1) is called integrable if it is both a- and /3-semiintegrable.

Integral manifolds V of the distribution A, of an a-semiintegrable almost Grassmann structure are of dimension p. They are projected on the original manifold M in the form of a submanifold V, of the same dimension p, which, at any of its points, is tangent to the p-subspace {o of the fiber bundle E0. Through each point x E M, there passes a (q - 1)-parameter family of submanifolds Va. Similarly integral manifolds f/0 of the distribution Op of a p-semiint.egrable

almost Grassmann structure are of dimension q. They are projected on the original manifold M in the form of a subtanifold Vi3 of the same dimension q, which, at any of its points, is tangent to the q-subspace tlp of the fiber bundle E0. Through each point x E M, there passes a (p - 1)-parameter family of submanifolds Vp.

If an almost Grassmann structure on Af is integrable, then through each point x E M, there pass a (q - 1)-parameter family of submanifolds 1". and a (p - 1)-parameter family of submanifolds Vp which were described above. The Grassmann manifold G(m, n) is an integrable almost Grassmann structure AG(m,n), since it admits a bijective mapping onto the manifold 0(m, n) of dimension pq, p = m + 1, q = n - m, through every point x of which there pass a (q - 1)-parameter family of p-dimensional plane generators (which are and a (p - 1)-parameter family of q-dimensional plane the submanifolds generators (which are the submanifolds Vp). In the projective space P", to submanifolds V. there corresponds a family of m-dimensional subspaces belonging to a subspace of dimension m + 1, and to submanifolds Vp there corresponds a family of rn-dimensional subspaces passing through a subspace of dimension

m - 1. 2. Let us consider some examples. First, we consider a pseudoconformal CO(2, 2)-structure on a four-dimensional manifold Af (see Section 5.1). The isotropic cones C. of this structure carry two families of plane generators. Therefore a pseudoconformal Hence these cones are Segre cones CO(2, 2)-structure is equivalent to an almost Grassmann structure AG(1, 3). If we complexify the four-dimensional tangent subspace T=(M) and consider Segre cones with complex generators, then conformal C0(1, 3)- and CO(4, 0)structures can also be considered as complex almost Grassmann structures of the same type AG(1, 3). However, in this book we will consider only real almost Grassmann structures. Almost Grassmann structures arise also in the study of multidimensional webs (see Akivis and Shelekhov (AS 92J and Goldberg [G 88]). We consider first a three-web W(3, 2, q) formed on a manifold AP 9 of dimension 2q by three

foliations A o = 1, 2, 3, of codimension q that are in general position.

7.1

Almost Grassmann Structures on a Differentiable Manifold

271

Through any point x E M29, there pass three leaves F, belonging to the foliations a,. In the tangent subspace T= (M2q), we consider three subspaces T. (.F.,) that are tangent to F, at the point x. If we take the projectivization of this configuration with center at the point x, then we obtain a projective space P2q-1 of dimension 2q - 1 containing three subspaces of dimension q - 1 that are in general position. As we saw in Subsection 6.1.6, these three subspaces determine a Segre variety S(l,q - 1), and the latter variety is the directrix for a Segre cone SC.(2,q) C T.(M2q). Thus on M2q a field of Segre cones arises, and this field determines an almost Grassmann structure on M2q. However, the structural group of the three-web in question is smaller than that of the induced almost Grassmann structure, since transformations of this group must keep invariant the subspaces TT(F,). Thus the structural group of the three-web is the group GL(q). In the same manner we can prove that an (p+ 1)-web W (p+ 1, p, q), formed on a manifold M of dimension pq by p + 1 foliations a or = 1, ... 'P+ 1, of codimension q which are in general position, generates an almost Grassmann structure on M. The structural group of the web W (p + 1, p, q) is the same group GL(q), and this group does not depend on p. 3. Let us reduce the structure equations of the Grassmannian G(m, n) that have been already considered in Chapter 6 (see Section 6.5) to a form convenient for a further generalization. As earlier, we denote the points of a moving frame of the space P" by AE and write the equations of infinitesimal displacement of this frame in the form dAf = 0"A,,,

0,

. .

. ,nn..

(7.1.5)

Since the fundamental group of the space P" is locally isomorphic to the group

SL(n + 1), the forms 0 are connected by the relation tE = 0.

(7.1.6)

As was indicated in Section 6.3, the structure equations of the space P" have the form

dln=9 A9.

(7.1.7)

By (7.1.7), the exterior differential of the left-hand side of equations (7.1.6) is identically equal to 0, and hence this equation is completely integrable.

Let a subspace P' = x be an element of the Grassmannian G(m, n). We place the points Ao, A1, ... , A,,, of the frame into this subspace. Since by (7.1.5), we have

dA,,=8 A0+0'Ai,

(7.1.8)

where here and in what follows in this subsection, a,# = 0, ... , m and i = m + 1, ... , n, the 1-forms 9 , are basis forms of G(m, n). These forms are linearly independent, and their number is equal to p = (m + 1)(n - m) = p q, where p = m + 1, q = n - m; that is, it equals the dimension of the Grassmannian G(m, n). We will assume that the integers p and q satisfy the inequalities

272

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

p > 2 and q > 2, since for p = 1, we have m = 0, and the Grassmannian G(0, n) is the projective space P", and for q = 1, we have m = n - 1, and the Grassmannian G(n - 1, n) is isomorphic to the dual projective space (P")'. Let us rename the basis forms by setting 90 = wi and find their exterior differentials: (7.1.9)

dwp = Ba n wp +WJ- A Off.

Define the trace-free forms

wa = 6a - pda97, wj = B

- Qk

(7.1.10)

satisfying the conditions

wa = 0, w; = 0.

(7.1.11)

Excluding the forms 9 and 9 from equations (7.1.9), we find that

dwawanwp+wfanw;+wnwa

(7.1.12)

where

w = pe7 - qek. But since, by (7.1.6), we have

ek = -e7

y,

A:

then the expression for the 1-form w can be written in the form

(1+1)e;. P

(7.1.13)

4

Taking the exterior derivatives of equations (7.1.10) and eliminating the

forms 9 and t from the equations obtained, we find that dwa =wQnwA+wkn (aa k - I 0a8k p

(7.1.14)

dw = w; A wk + (ake; - look) n wQ. Finally, taking the exterior derivative of equation (7.1.13), we find that

dw=

(1+1)waA9°.

p

(7.1.15)

q

If we set

p+q)Ba,

(7.1.16)

then equation (7.2.15) takes the form

dw=w°nwQ,

(7.1.17)

7.1

Almost Grassmann Structures on a Differentiable Manifold

273

and equations (7.1.14) become dwa = wa A wA +

wjkA wk +

p

+ q (ba wk A ,, _Y - pw Q A wk) , q

p+q

Awry

- qw

(7.1.18)

wy) .

Taking the exterior derivatives of equations (7.1.16) and applying equations (7.1.7) and previous relations between the forms BE and wf, we find that

Awl +Awp°+w°Aw.

(7.1.19)

Finally, exterior differentiation of equations (7.1.19) leads to identities. Thus the structure equations of the Grassmannian G(m, n) take the form (7.1.12), (7.1.17), (7.1.18) and (7.1.19). This system of differential equations is closed in the sense that its further exterior differentiation leads to identities.

If we fix a subspace x = PI C P" (an element of the Grassmannian G(m, n)), then we obtain wQ = 0, and equations (7.1.17) and (7.1.18) become d7r13 = nQ A rrA, drr = n A Irk, d7r = 0,

(7.1.20)

where, as in previous chapters, it = w(b), as = w13 (b), it = w (b), and b is the operator of differentiation with respect to the fiber parameters of the frame bundle associated with the Grassmannian G(m, n) (see Subsection 6.3.1). Moreover the forms xa and 7r satisfy equations similar to equations (7.1.11), so these forms are trace-free. The forms 7rO are invariant forms of the group SL(p) which is locally isomorphic to the group of projective transformations

of the subspace P"'. The forms ar are invariant forms of the group SL(q) which is locally isomorphic to the group of projective transformations of the bundle of (m + 1)-dimensional subspaces of the space P" containing P"`. We

will denote this bundle by P"/P'". The form it is an invariant form of the group H = R' ® Id of homotheties of the space P" with center at P'". The direct product of these three groups is the structural group G of the Grassmann manifold G(m, n):

G = SL(p) x SL(q) x H.

(7.1.21)

Finally, the forms ir° = w° (b), which by (7.1.19) satisfy the structure equations dir° = rr, A rra + 7rR A ir; + a° Air, (7.1.22) are also fiber forms on the Grassmannian G(m, n), but unlike the forms Ira, ir; , and it, they determine admissible transformations of second-order frames associated with the Grassmannian G(m, n).

It follows from Subsection 6.6.3 that a projective space P", in which an m-dimensional subspace PI is fixed, is an (n-m-1)-quasiaffine space An_m_1 Its dimension coincides with the dimension of the Grassmannian G(n-m-1, n), and this dimension is the same as the dimension of the Grassmannian G(m, n):

274

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

p = (m + 1)(n - m). The forms a° determine the parallel translation of the element P"-m-1 = Am+1 A ... A An of the space An_,,,_,. All together, the forms ire, 1T , it, and n°, satisfying the structure equations (7.1.20) and (7.1.22),

are invariant forms of the fundamental group G' of the space An-m-1 which is the semidirect product of the groups G and T(pq):

G' = G a T(pq),

(7.1.23)

where T(pq) is the group of parallel translations of the space The group G' coincides with the stationary subgroup H. of the element x = P" of the Grassmannian G(m, n). Ann_

7.2

Structure Equations and Torsion Tensor of an Almost Grassmann Manifold

1. Consider a differentiable manifold M of dimension pq endowed with an almost Grassmann structure AG(p-1, p+q -1). Suppose that x E M, T.(M) is the tangent space of the manifold M at the point x and that {e°) is an adapted frame of the structure AG(p - 1,p + q - 1). The decomposition of a vector z E Tt(M) with respect to this basis can be written in the form z = wa(z)e°,

where w, are 1-forms making up the co-frame in the space T=(M). If z = dx is the differential of a point x E M, then the forms wa(dx) are differential forms defined on a first-order frame bundle associated with the almost Grassmann structure. These forms constitute a completely integrable system of forms. As a result we have (7.2.1)

dw' = w'p A w'Q .

The forms wa are called also the basis forms of the manifold M. As earlier, we set rr'a°. = w'', (b), where b is the operator of differentiation with respect to the fiber parameters of the frame bundle. These forms determine an infinitesimal transformation of the adapted frames: be'? = it ea.

(7.2.2)

On the other hand, the admissible transformations of adapted frames can be written as closed form equations (7.1.2). Solving equations (7.1.2), we obtain e? = A°A 'ep

(7.2.3)

where (Ap) and (Ai) are the inverse matrices of the matrices (Ap) and (A'), respectively:

Ay AF = ApA° = 6, AkA = AJAk = 5

.

(7.2.4)

Structure Equations of an Almost Grossmann Manifold

7.2

275

It follows from (7.2.4) that

Ap . 6Ay = -AI bAp, A; bA1 = -A3 dAk,

(7.2.5)

Suppose now that (x, 'e°) is a fixed frame, d('e;) = 0. Then, differentiating (7.2.3) for a fixed x E M and using (7.2.4) and (7.2.5), we obtain Sea = (dp°ir; - d, rp )eA,

(7.2.6)

1r = Ak dA, , ira = Aa bAp.

(7.2.7)

where

Comparing formulas (7.2.2) and (7.2.6), we find that = bairi - b; lrpa.

7r'

(7.2.8)

In these formulas the forms lri are invariant forms of the group GL(q), the forms ira are invariant forms of the group GL(p), and the forms ap' are invariant forms of the structural group G of the almost Grassmann structure AG(p - 1,p + q - 1). If a point x E M is variable, then from equations (7.2.8) we find that wAi = by°

- b w+

uQ kWy0

(7.2.9)

,

where uQik are certain functions defined on the first order frame bundle. Substituting for WQa in (7.2.1) their values taken from (7.2.9), we obtain duly = Wa Awp +wa AWE +u'a kwp Au)

(7.2.10)

where ua k denotes the result of alternation of the quantities ii' O" occurring in

(7.2.9) with respect to the pairs of indices (Q) and (k): u` k = -u«' . If we set

lbiWk, aQWry, W! =WT + p q then it is easy to see that wQ = 0 and wk = 0, and the above structure equations take the form wa ==Wa +

dwQ = w A

V,' A wa + W A wa' + uQ kwp A Wy,k,

where w = I_W7 - QWk. If we suppress -, then the structure equations take the form: du,a =Wa AWj +W , AWR +W AWc,

c

kWp AWk

(7.2.11)

where i7A io-V TEajk = -uakf

(7.2.12)

276

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and

W7 = 0, Wk = 0.

(7.2.13)

Conditions (7.2.13) mean that the subgroups GL(p) and GL(q) of the structural group G of the almost Grassmann structure AG(p-1, p+q- 1) are reduced to the groups SL(p) and SL(p), respectively, and that the group G itself is represented in the form (7.1.4). As for the Grassmannian G(p - 1, p+ q - 1) (see Subsection 7.1.3), for the almost Grassmann manifold the forms wa, w; , and w are fiber forms defined on the second-order frame bundle associated with the

almost Grassmann manifold AG(p - 1,p+ q -1). The structure equations (7.2.11) differ from the structure equations (7.1.9) of the Grassmannian G(p - 1, p + q - 1) only by the last term. 2. We obtain the remaining structure equations of the almost Grassmann manifold M by exterior differentiation of (7.2.11). This gives

nflAwp- Q' A wQ+(Vu'Q k+u'Q."W)AwpAwk

(7.2.14)

+dw Aw, + 2uaic-fmkum,Rwfs A WQ Awy = 0, where Q0 = dwQ - wa A wA, fl' = dw' - wk A wk, VuQ,k = duopjk - u6 kWQ

- tiafkwj - uo'Iwk +

uIQ k-f

wf + uokc

i

+ ua kwe .

To solve equations (7.2.14), we represent the forms and dw from the left-hand side of equations (7.2.14) as a sum of terms containing the basis forms and the terms not containing these forms:

1l =w7Awak+4i«, fly=wokAwk+dw=w°AWQ+4',

(7.2.15)

where 41,a, I , 4 and wak,w4k are certain 2- and 1-forms not expressed in terms of the basis forms wi, only. By (7.2.13), we have fly = 0 and ftk = 0, which implies that (7.2.16)

4K7 = 0, 4kk = 0. and

wak=0,

;k=0

(7.2.17)

Substituting (7.2.15) into equations (7.2.14), we obtain 16 A 4 A wry (b 4ip - 6131'. + 606' fl A w! + 2uaml0l-OU

+(f5(,Wlolk1 +

6Qry61jlw,) + Vua k +

Awp AWy = 0. (7.2.18)

7.2

Structure Equations of an Almost Grassmann Manifold

277

The first term in the left-hand side of (7.2.18) does not have similar terms among other terms of this side. Thus this term vanishes. But since the first factor of this term does not contain the basis forms, this factor itself vanishes:

+b 54' = 0.

(7.2.19)

Contracting (7.2.19) with respect to the indices a and Q, applying (7.2.16), and dividing by p, we find that

-4i +bj4' = 0. Contracting this equation with respect to the indices i and j, we obtain 4' = 0, and consequently 1i = 0. Finally, by (7.2.19), we find that 4 = 0. Now equation (7.2.18) contains only the last two terms. It follows that the 1-form which is multiplied by wR A wk is expressed only in terms of the basis forms. Therefore, if the principal parameters are fixed (i.e., if wi = 0), then we obtain R7 r ak

- b'

k

7R + *5ait,k- ba77r'Rkj

aj

u' + u' aQjk7 + ba7bk' 7rp - 60b a ` ak7 + 2 (V 6 k aR7j

it) = 0. (7.2.20)

It follows from equation (7.2.20) that the quantities u'a k form a geometric object that is defined in a second-order differential neighborhood of the almost Grassmann structure AG(p - 1, p + q - 1). Consider the quantities iR7

R7

ia7

i7

(7.2.21)

link - uaik, ujk = U01 jk.

If we contract equations (7.2.20) with respect to the indices i and j, then after some calculations we find that uR7 + uR7ir = _ 2 r X07 q( ok 6 ak ak

by it

- aitk) - ,rak - a ( 5

-YO

ki

- aRk) ]

'

( 7.2.22)

Similarly, contracting equations (7.2.20) with respect to the indices a and fl, we obtain VJujki-f

2[p(xjik-Miak)-

i7

k j- bk (Ira7aj

- sj7)]

(7.2.23)

Formulas (7.2.22) and (7.2.23) show that each of the quantities uQk and u"k form a geometric object that is defined in a second-order differential neighborhood of the almost Grassmann structure AG(p - 1,p+ q - 1). Let us prove that if we make a specialization of second-order frames, then we can reduce these geometric objects to 0. In our proof we will apply the same method that we used in constructing the tensor of conformal curvature in Chapter 4 (see Subsection 4.1.3). We will prove this for the geometric object uo". To this end, we must show that the 1-forms in the right-hand sides of equations (7.2.22) are linearly

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

278

independent. First, we note that the forms 7rak are linearly independent in the set of second-order frames. Let us equate to 0 the right-hand sides of equations (7.2.22): -Yo

0

9(7rok - balk) - yak - 6-V (71j, - k)

(7.2.24)

If we contract equations (7.2.24) first with respect to the indices a and 0 and second with respect to the indices a and -y, we arrive at the system pq)ir

1ak + 7rk;

(7.2.25)

gook - PAki = (q - p)ir If we solve this system, we find the quantities 7r"ak and irki:

rya -- 7rak

q(p2 - 1)

P(q2 - 1) " Irk.

,. = p+q k+ ski - - p+q If

(7.2.26)

Substituting these values of 7rk7 into equations (7.2.24), after some calculations, we reduce the equations obtained to the following form: (7.2.27)

-yo = 0, 9Wak - yak

where

irek

- 7rak - p

q (6.07rk

- p8,prk )

.

+ Interchanging in (7.2.27) the indices 0 and y, we obtain Ary

ryR

- lrak + gook -

(7.2.28)

O.

Since the determinant of the system of equations (7.2.27)-(7.2.28) is equal to q2 - 1 0 0, the system has only the trivial solution. But the forms *Qk as the forms 7rak are linearly independent. Thus the

forms q;rak - a.k are linearly independent too. But, up to the factor -, a the latter forms coincide with the right-hand sides of equations (7.2.22).

Hence the geometric object uek = u'Q k can be reduced to 0. Similarly the geometric object ufk = can be reduced to 0. This operation leads to a reduction of the set of second order frames of the almost Grassmann structure AG(p - 1, p + q - 1). Before this reduction, the set of second-order frames depended on pq(p2 + q2) parameters equal to the number of linearly independent forms among the forms 7rak and 7r"k. After the reduction, the forms WCkk3" and vanish, and the forms 7rak and 7r,k are expressed in terms of the 1-forms irk: 7rak - p

q (b°7rk

+

- pbaak)

,

7r;k - p

q

+

(bj7rk

- gbk7r,7).

(7.2.29)

7.2

Structure Equations of an Almost Grassmann Manifold

279

Since there are pq forms Irk , and they are linearly independent, the reduced family of second-order frames depends on pq parameters. The 1-forms Irk define

admissible transformations of frames in this reduced family of second-order frames.

Denote by a.k the quantities ua k after the specialization indicated above. Then the quantities a'Q k satisfy the conditions iory

iRry

aajk = 0, aaik = 0

(7.2.30)

and

as k = -

(7.2.31)

Qk,.

The last relations follow from conditions (7.2.12). Substituting expressions (7.2.29) into equations (7.2.20), we find that

a, klr = 0.

V6

(7.2.32)

This implies the following theorem:

Theorem 7.2.1 The quantities a`o k, defined in a second-order neighborhood by the reduction of second-order frames indicated above, form a relative tensor of weight -1 and satisfy conditions (7.2.30) and (7.2.31).

Definition 7.2.2 The tensor {aQ } is said to be the first structure tensor, or the torsion tensor, of an almost Grassmann manifold AG(p - 1, p + q - 1). After the specialization of second-order frames has been made, the first structure equations (7.2.11) become

dwa=wQnwj' +wQAwp+wAw.' +aakwRnw1

(7.2.33)

3. We will now find the expression for the tensor a'Q k in terms of the quantities

uQ k occurring in equations (7.2.11). We assume that the specialization of second-order frames indicated above has not been made and that the quantities uQ k satisfy equations (7.2.20), which we write in the form iRry iRry 06uojk + uojk7r =

1

2

i

i

iR rya Qkl - tSi7rak + koj ry

Rry

R

iry

(7.2.34)

+a og7rk -

We will eliminate the fiber forms irak, and Irk from equations (7.2.34). To this end, we construct the following three objects: iRry

_-

yolk iRry _

2

,(

q2 - 16 i 2

k

(aryl

IoIkJ

I7R) 1 + ulolkll,

Ikil(((7.2.35)

b(R ( ulilryl + ulilryll

yOjk - -p2 -1 o zQ

.

q likkri

(p2 - 1)(q2 - 1) L(pq - 1) \aljalalullIkl + blkoIQlujliryllJ +(p - q) (a[i Jalukit + a(kbJai 111ji 9] 1

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

280

where the quantities uak and u,k are defined by formulas (7.2.21). A straightforward calculation with help of equations (7.2.22) and (7.2.23) gives the following differential equations for the objects x'a k, y;k, and zQ k x'RY vdxi0Y Lj ajk + ajk ir = 2 [6

a k) 6t-( irYo kja - 6Yxo)] a3

607rY

ak

66a>
Vdya

k +ypkir _

±2(921- 1) l60(7r k - a;ak) - 6a(rrk - 6k1rj 1

QdzioY + zil3Yir = ajk ajk

i

Y Rd - 1) l p(6j6a7dk -

2(p2 - 1)

Ir

i

Y pd

t

0 Yd

t

- 6j6ankl]

-yd

Rd adj - 6j6aa6k]

6' 6'Y'

i p Yd + 6k6aadj Y Rd dkbairdj)

X Q Yd

1 [q(6'6'3 1(q21 _) k oirlYp- 6i6Yalo) j a ki + MR k arlpjl - 6i6aal' j a kl

2

(7.2.36)

If we set

aipY

,uiOY + xiAY + t9Y + iRY ajk yajk ajk

ajk = ajk

(7.2.37)

then by (7.2.34) and (7.2.36), it is easy to check that

Vd'aa k+a,,jkir=0. This means that the quantities ask form a relative tensor of weight -1. Using (7.2.12), it is easy to verify that the tensor ask satisfies the conditions similar to conditions (7.2.30) and (7.2.31). We will prove now the following proposition: Proposition 7.2.3 The relative tensor a = {aapk} defined by formulas (7.2.37) and (7.2.35) and satisfying the conditions similar to conditions (7.2.30) and (7.2.31) coincides with the tensor a = {a'a as = a. k}:

Proof. Let us assume that from the beginning we have the tensor a = {a k } in place of the object u °k and that tensor satisfies conditions

(7.2.30) and (7.2.31); in other words, we have equations (7.2.30)-(7.2.33) and (7.2.29). Then, applying (7.2.30), we find from (7.2.35) that xi k = yQ = k za k = 0, and consequently from formula (7.2.37) we find that a« k = aQ k. By Proposition 7.2.3, if we substitute for xQpk, yQFk and za k in equations (7.2.37) their values (7.2.35), we find that i1Y aajk

=

uipY

ajk

2 610 ( 14-11 + ulil") + UN01 - q22- 1 6i (uIAYI q 101k) )-1k) - p2 - 1 a Pulikl lkil 2

(p2 - 1)(q2 - 1) l(pq -

1)(6U61QIuk +6Ik6Ia1 ll)

+(q - P) (6(i6jai kl + 6lk61-1"il )] (7.2.38)

The Complete Structure Object of an Almost Grassmann Manifold

7.3

281

where the alternation is carried out with respect to the pairs of indices (p), (k) (! " (ary ory --'Y - (y lory - fry' ry ry rya

or (k), (j), and uk = uA - ualk - uok+ Uk - uk1 - v'okl - -uolk - -ubk-

The expression (7.2.38) for the tensor ai k was obtained by Goldberg [Go 751 (see also [Go 88J, §2.2).

7.3

The Complete Structure Object of an Almost Grassmann Manifold

1. In this section we will construct the second and the third structural objects of the almost Grassmann structure. From equations (7.2.29) it follows that, after the reduction of the secondorder frame bundle made in Section 7.2, we have the following equations on the manifold AG(p - 1, p + q - 1): pry6wl6+ = p +q q (a°a Wk - }tow ) + wakl k 6 ry i 7 iry _ p w,jk - p+q (ai)w k g bkifj )+ w,jklw6 Wa

k

-

.

(7.3.1)

1

Substituting these forms into system (7.2.15), we obtain (&0

YAk- pwa A wk)+wakiwkA W6,l

W'

dWl - w A Wk=

Awy)+wIkiwryAwa,

(7.3.2)

dW=w°AwQ, where wak15=,Wa[kijl an d w'jkI =

The latter quantities satisfy the relations

i6ry iry6 0-Y6 136-Y v)Okl = -walk + WJkl - -wjlk

and also the relations

W ki =

0, wild = 0,

(7.3.3)

(7.3.4)

which follow from equations (7.3.1) and (7.2.17). Exterior differentiation of equations (7.3.2) gives the following exterior cubic equations:

kl + 2wakl W) A Wry A W6 - 2waki a'6mnw A wm A WS (Vwao'j

+p

(7.3.5)

q

(Ilk +W AWk) AW'

(Owjkl + P4

p+9

aokl6wm A wry A Wit] = 0,

AWry AW6 - 2wjkladmnwry A W- A WC (7.3.6)

(SZf + W A Wj) A w ai"t wJ A wl6 A WE"J = O,

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

282

and

-aak w°AwpAwy=0,

(fl°+w At )Ac

(7.3.7)

where

n° =d4 - w; A w,°r - w, A wr , = dwp,6 - wp7awe - W 0-16 ld1k wpryV, + w`76wp + wpeaw7 + wp7ew6 vwp7a akj a akj e+ akl ail k - akj 1 akj aki Chi a wi76 wir6we + wi7c r wm - wi76 Vwi7a 7k! 6 kmwm 1 + wM-16wi ,7m1 k jk! m + ,7kl 7 mklWm j - wi76 jk1 - dwi76 jkl ,

-

From equations (7.3.5)-(7.3.7) it follows that the 2-form fl° + w A w; can be expressed as follows:

are defined on the fourth-order frame bundle associated with The forms the almost Grassmann structure AG(p -1, p + q - 1). Substituting expressions (7.3.8) into (7.3.5)-(7.3.7), we obtain p7s

wakl

q

-2w [°w;-1a k!

(aawkl + aklawm), A wy A wa

+2wakl U j_

(7.3.9)

r. oklAwE"Aw7Awa=0, w + P4 (bkw + aChwi ), A w7 A wa

p+q

(7.3.10)

-2w",,° aki A wm A wry A w6 = 0, and

(wp + a7 awk) A w, A w'p = 0.

(7.3.11)

It follows from (7.3.9)-(7.3.11) that for w, = 0, we have 06weki + 2waki 7r

1 + aklair ' = 0,

- - --

Vswik6 + 2w"ki 7r +

pq (bik7r(Y61 + a'7airE) = 0, ekl (

p+q

101

(7.3.12) ( 7.3.13 )

and 7r(k111 + aekl67rm

- 0,

(7.3.14)

where the alternation is carried over the pairs of indices (k) and (i) and as usual Irk = wk(a), 7rAk = wpk (6), and 7r = w(d).

From (7.3.14) it follows that the form Irk, can be written as 'Yd

_16

7rk! = rk! - a k! 67rm,

(7.3.15)

The Complete Structure Object of an Almost Grassmann Manifold 283

7.3

where w y6

kl -

h

TIk

(7.3.16)

The 1-forms Wk determine admissible transformations of third order frames associated with the almost Grassmann structure AG(p - 1, p + q - 1). If we apply equation (7.3.15), we can write equations (7.3.12) and (7.3.13) in the form: V Jwakt + 2w0__'6-,r

w06

- 2(p + q) e

a elk

a eki

+2(P+pq

--Y -

q) lbk-i,J

+(26ma'-" eki

(7.3.17)

6yamp6 + b6 amo` ) Ire

akl

OJw,ki +2w"ki7r

- 66 rlky

-

71

m =- 0 J

blr7k

(7.3.18) 1

64 amry6 + t6"a"Jy ejk) 7rc m 1 = 0. k cjl

2. If we contract equation (7.3.17) with respect to the indices a and ry, equation (7.3.18) with respect to the indices i and k and change the notation of some indices, we obtain

akl + (a ki - qa kl )remI = 0. (7.3.20)

VJwkit + 2wkit + 2(p + q)

Using the quantities wnk1 and wkit, we now construct the following new object: yJ

yaJ

i-y6

Jay

y

i5lik

(7.3.21) wkl = waki - wkit + walk - w By means of (7.3.19), (7.3.20), and (7.3.16), it is easy to prove that the quantities wkt defined by (7.3.21) satisfy the following differential equations:

V6wkt + 2wk61r = p

q [(p + q)Frkt - 27rki ] .

(7.3.22)

Formulas (7.3.22) show that the quantities wkt form a geometric object which is defined in a third-order differential neighborhood of the almost Grassmann structure AG(p - 1, p + q - 1). Let us prove that if we make a specialization of fourth order frames, then we can reduce the geometric object wkt defined by (7.3.21) to 0. In our proof we will again apply the method used in Subsections 4.1.3 and 7.2.2. We must show that the 1-forms on the right-hand sides of equations (7.3.22) are linearly independent. First, we note that by (7.3.16) and (7.3.21), both the

components wkt and the forms ikt are symmetric with respect to the pairs of indices (k) and (,), and there are Zpq(pq + 1) linearly independent forms

284

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

among the 1-forms irki . Let us equate to 0 the right-hand sides of equations (7.3.22): (7.3.23) (p + q)irki - 2ak = 0.

Interchanging the indices -y and 6 in (7.3.23), we obtain (7.3.24)

- 2irki + (p + q)3ki = 0.

Since the determinant of the system of equations (7.3.23)-(7.3.24) is equal to (p + q)2 - 4 76 0, the system has only the trivial solution.

There are 2pq(pq + 1) linearly independent forms among the 1-forms (p + q)ifkl - 2Tk1. But, up to the factor --M, the latter forms coincide with the right-hand sides of equations (7.3.22). Hence the geometric object wki can be reduced to 0. This operation leads to a reduction of the set of third-order frames of the almost Grassmann structure AG(p- 1, p+q- 1). Before this reduction, the set of third-order frames depended on 2pq(pq + 1) parameters equal to the number of linearly independent forms among the forms F`6. After the reduction, the forms if vanish,

ak6 = 0,

(7.3.25)

and the forms irkj are expressed in terms of the 1-forms ak, m76 e klyd = -aekl Ire..

(7.3.26)

But as we saw earlier, the forms irk determine admissible transformations of second-order frames. This means that the group of admissible transformations of third-order frames coincides with the group of admissible transformations of second-order frames. This implies the following result:

Theorem 7.3.1 The almost Grossmann manifold AG(p - 1, p + q - 1) is a G-structure of finite type two.

This result is analogous to the result for the conformal CO(p, q)-structures proved in Subsection 4.1.3. Denote the values of the quantities wakl and w,k in the reduced fourthorder frames by bald and respectively. Then the quantities bald and L"Y' satisfy the differential equations obtained from equations (7.3.17) and (7.3.18) by means of (7.3.25): pq V6bv,6 + 2bpry6ir . Okl Okl Okl 2(p + q) (260a'&

- 6'amf6 a eki + a6amf')ir a elk J) M = 0

(7.3.27)

- 6` amy6 ejl + 6`I ai6y) ejk am = 0

( 7. 3 . 28 )

and pq V 6 b"6 j6 jkl + 2b''6 ) aekl jkl a + 2(p + q) (26"'

k

e

.

The Complete Structure Object of an Almost Grassmann Manifold 285

7.3

They also satisfy the conditions (cf. (7.3.21)) baki and

- bkil + balk - beak - 0

jkl -akl -- -bpd' alk, bi7b

(7.3.29)

bQtia

(7.3.30)

,ilk,

and the relations (7.3.31)

baki = 0, baki = 0.

The relations (7.3.30) and (7.3.31) follow from condition (7.3.3) and (7.3.4). Equations (7.3.26) show that the 1-forms wkf occurring in equations (7.3.8) are expressed in terms of the forms wk and the basis forms wry: aQ _ ^a07 k Wij - -awika0 Wk If+Cijk W7.

(7.3.32)

This means that the fiber forms w°Q associated with the fourth-order frame bundle are expressed in terms of the fiber forms wk, defined on the third-order frame bundle, and the basis forms wry. 3. Substituting for the forms in equations (7.3.8) their values (7.3.32), we find that dw° - WQ AWaQ -41l AW°)+wAWa =CopYWk /WW1Q

where c,k7 =

i

akaOWk A wd - 7ij Q'Y

,

(7.3.33)

ik; 1, and the alternation is carried over the vertical pairs of

indices.

If we substitute for the forms w,f in equations (7.3.11) their values (7.3.32),

we find that the quantities k' must satisfy the following condition: o071

ijkl

= 0.

(7.3.34)

Equations (7.3.33) together with equations (7.2.33) and (7.3.2) make up the complete system of the structure equations of the almost Grassmann structure AG(p - 1, p+ q - 1): dwa - w , A u - wp n wp - co A w = aQ kwp n wy, dWa -wQ Awp =

dw --j Awk =

p+q(buwk

'Y

Awk -PWk a Awk) +bakiwk Awl

p+q(dfwk Awry -qwj' Awy) +bfkwk Awd,

dW=w°AwQ, dw2 - wa A wQ - wi A w + w A w° = ca,k7Wry A wp - ayi Awk A wQ. (7.3.35)

We have proved above that a structure AG(p - 1, p + q - 1) is a G-structure of finite type two. The invariant forms of this structure are divided into three

286

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and {w°} which are defined in the frame bundles of groups: {wa}, first, second, and third order, respectively. The forms wQ define a displacement

of a point x along the manifold M on which the almost Grassmann structure is defined. By (7.3.35), for wa = 0 these forms satisfy the equations d7ra = xo A ae, &4 =

and the forms trp and

j WI

wk,

(7.3.36)

dir = 0,

(7.3.37)

dir°=apAwp+rr;Aar,

(7.3.38)

satisfy the conditions (7.2.13), ?rs = 0, Trj = 0.

In view of (7.3.37), the form it is an invariant form of the group H of and it are homotheties of the tangent space TT(M), and the forms invariant forms of the group G L SL(p) x SL(q) x H, whose transformations leave the cone SC=(p,q) C T=(M) invariant. The group G is the structural group of the almost Grassmann structure AG(p - 1,p + q - 1). Equations (7.3.36), (7.3.37), and (7.3.38) prove that the forms and ir° are invariant forms of the group G' which is obtained as a differential prolongation of the group G. The group G' is isomorphic to the group GxT(pq) whose subgroup T(pq) is defined by the invariant forms Ire. To describe the group G' geometrically, we compactify the tangent subspace T=(M) by enlarging it by the point at infinity and the Segre cone SC". (p, q) with its vertex at this point. Then the manifold TZ(M)f1SCOJp, q) is equivalent to the algebraic variety f2(p-1, p+q-1). Since the point x at which the variety Il(p - 1, p + q - 1) is tangent to the manifold M is fixed, the geometry defined

by the group G' on l(p - 1,p + q - 1) is equivalent to that of the flat Segreaffine space SAP of dimension p = pq, on which the variety ft(p - 1, p + q - 1) is projected by means of a stereographic projection from the point x (see Subsection 6.6.3). The group G' is the group of motions of this space, its subgroup G is the isotropy group of this space, and the subgroup T(pq) is the subgroup of parallel translations. 4. After the first reduction of second-order frames associated with the almost Grassmann structure AG(p - 1, p + q - 1), we introduced the first structure tensor (the torsion tensor) a = {a fl } of AG(p - 1, p + q - 1). Let us set k

b' _ {b6jk,b2 = {b;ak} and b = (b',b2).

Equations (7.3.27) and (7.3.28) show that the quantities (a, b1) and (a, b2) form linear homogeneous objects. They represent two subobjects of the second structure object (a, b) of the almost Grassmann structure AG(p - 1, p + q - 1). Taking the exterior derivatives of the last equation of (7.3.35), we arrive at the following exterior cubic equation:

7.3

r

The Complete Structure Object of an Almost Grassmann Manifold 287

aR7

00"

a7R ° s7R a +3cijk w - b°kj wi + bikj ws

+(2c°c°as7Rwm tma °kj c

(saR m°ry

a, ,3 acak +

ma° s7R

c

acis a°kj )wm

)wm1 A wk Awl - 0 - CsjkR7aaac aim e 7

° - c7+°7° ilk +C ilk

Cao'tiws _ R7ws where OcaRti iik - dcap'' ijk - Rik i isk j ids k For wQ = 0, it follows from equation (7.3.39) that

V6c007+3C°R'7r-b°70,n°+b'tif,ra+(asaparn°ry+ama°a'yR cis ikj s °tj csk ilk iik a, kj s

°

(7.3.39)

°

+C°R°wy

iik

0. (7.3.40)

Let us set c = (cajk7). Equations (7.3.40), (7.3.27), (7.3.28), and (7.2.32) prove that S = (a, b, c) form a linear homogeneous object, which is called the third structure object of the almost Grassmann structure AG(p - 1,p + q - 1). It is defined in a fourth-order differential neighborhood of AG(p - 1, p + q - 1).

As we proved earlier, its subobject a is a relative tensor (the torsion tensor) defined in a second-order differential neighborhood of AG(p - 1, p + q - 1), and the subobjects (a, b'), (a, b2), and (a, b) are defined in a third-order differential neighborhood of AG(p - 1, p + q - 1). The third structural object S = (a, b, c) is the complete geometric object of

the almost Grassmann structure AG(p - I,p + q - 1), since if we prolong the structure equations (7.3.35) of AG(p - 1, p + q - 1), all newly arising objects are expressed in terms of the components of the object S and their Pfaffian derivatives. This follows from Theorem 7.3.1. 5. Now we will find new closed form equations and differential equations

that the components of b' and b2 satisfy. First, note that since the object a is a relative tensor of weight -1, its components satisfy the equations iRti iRY aajk + aajkw =

'R76

t

(7.3.41)

aajktw6.

This equation is equivalent to the equation (7.2.32). The quantities as ki occurring in equations (7.3.41) are Pfaffian derivatives of the components of the tensor a with respect to the basis forms wt6. With respect to the indices i,0171 a, j, k, they satisfy the same relations (7.2.30)-(7.2.31) as the tensor a. Substituting for Vaa k + a' kw and for the forms f1a,11 , and dw in equations (7.2.14) their values from expansions (7.3.41) and (7.3.35) and equating to zero the coefficients of wt6 A rr'R A wy, we obtain the following equations: a ['jblal

I] - 6a 1001

X61

+ °oljktj + 2aaljllma Ik

= 0.

(7.3.42)

As earlier, in this formula, the alternation is carried over the pairs of indices (k', and (r). Equation (7.3.42) can be written in the form a(iblalkil - as100 [jl ill

_ Aakl I

(7.3.43)

288

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

where the quantities A'Q ki are skew-symmetric with respect to the last three pairs of indices and are expressed in terms of the components of the tensor a and their Pfaffian derivatives.

We will now prove that for p > 2 and q > 2, the components of bl and b2 are expressed in terms of the components of the tensor a and their Pfafan derivatives. In fact the components of bl and b2 satisfy equations (7.3.43) which are a nonhomogeneous system of linear equations with respect to the quantities bQk6n and b;jk . Consider the homogeneous system corresponding to

this nonhomogeneous system; that is, set a' tk = 0 in this nonhomogeneous system. This gives b(jb1a i

1 - blablit-] = 0

or iR6 @ i6e i e08 d a0 i Rde + bli 6e0 bjbaim bamj + bmbajl - babjlm - bablmj - ba abmjl = 0.

Contracting the latter equations with respect to the indices i and 1, i and j, and i and m and applying conditions (7.3.31), we obtain be 6ep e"' -_0 gbamj + ajm + bajm

bamj + q ajm + damj = 0,

(7.3.44)

b6`p amj = 0. ajm + ba6E aim + qb`#6

If we symmetrize equations (7.3.44) with respect to the indices j and m, we obtain a homogeneous system of equations with respect to bQ6am1, ba(im), and b`a im). The determinant of the matrix of coefficients of this system is equal to

= (q - 1)2(q + 2),

and it does not vanish if q > 2. Hence this system has only the trivial solution: b6ep a(mj) = 0. In the same manner, if we alternate equations (7.3.44) with respect to the indices j and m, we obtain a homogeneous system of equations with respect bpd' to b6`# and bep6 . The determinant of the matrix of coefficients of slim)' a1jm1' CO-) this system is equal to -q

1

1

q

1

1

1

-q

1

= (q + 1)2(q - 2),

and it does not vanish if q > 2. Hence this system has only the trivial solution: b4 ml = 0. As a result the homogeneous system in question has only the trivial solution: bajp,,, = 0 provided that q > 2; thus the original nonhomogeneous

The Complete Structure Object of an Almost Grassmann Manifold 289

7.3

system has a unique solution expressing the quantities bolo, in terms of the components a'Q k of the tensor a and their Pfaffian derivatives. In a similar manner we can prove that if p > 2, then the quantities are expressed in terms of the components ask of the tensor a and their Pfafflan derivatives. Note that the condition q > 2 is required only for finding of beo ) = b`('") and the condition p > 2 for finding of bj(k (see Lemma 7.4.1, p. 292).

Now we can see that the tensor a satisfies certain differential equations. These equations can be obtained if we substitute for the components of bl and b2 in equations (7.3.43) their values found in the way indicated above. The conditions obtained in this manner are analogues of the Bianchi equations in the theory of spaces with affine connection. 8. Next we will find new closed form equations and differential equations in equathat the components of c satisfy. If we substitute for the 1-forms tions (7.3.9) and (7.3.10) their values taken from (7.3.32) and apply (7.3.34), we arrive at the following exterior cubic equations: Abo-Y6 A wl6 akl A Wk Y

-

she)w,- /1 wY p pq + q6' aa°me°Y6 ski +2b O-Yu aks aaim

k

1

/1 w6

0 (7.3.45)

_

2b sm oki J )we"' / wk / w6 = 0,

ob kt n wk /1 wa + ( p +

(7.3.46)

where Ob#-ry6 aki = VbQy6 aki + 2b/iY6 akl w+

ObiY6 ,lkl + jkl = Obi" Jkl+ 2b'ry6w

p-Y +q(s

R m sY6) we mA6 -6e6a aaklwml

6a6kaCal

(&o mai

ej

+ ppq

oM

amo61we c55' o k ejl 11 m

It follows from equations (7.3.45) and (7.3.46) that

Qb"' = bY6e wm QbJYd - bf7de m jklm e , 1M akl aklmwe

(7.3.47)

where bjtktm and bakim are the Pfaffian derivatives of 6akl and bake respectively. Substituting (7.3.47) into equations (7.3.45) and (7.3.46), we find the following differential equations for the components of b: bplY6e1

a(klm]

_

P9 6(ecIAh6)

p + q a lmkil

- 2bp(Yleasl6e) a(kle olim) -- 0 (7.3.48)

6 (klml

p + g6lmcljl M) +

0.

Equations (7.3.48) can be written in the form 6[f Clmkll]

-

BO-The

(7.3.49)

290

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and

(7.3.50)

b1-clilk) = Bjki-I

where the quantities Bakem and Bjkl,`,+ are skew-symmetric with respect to the last three pairs of indices and are expressed in terms of the components of the subobjects (a, b') and (a, b2), respectively, and their Pfaffian derivatives. We will now prove that if p > 2, then the components of c are expressed in terms of the components of the subobject (a, b') and their Pfafan derivatives,

and that if q > 2, then the components of c are expressed in terms of the components of the subobject (a, b2) and their Pfaffian derivatives. We will prove only the first part of this statement. The proof of the second part is similar. The components of c satisfy equations (7.3.49) that are a nonhomogeneous system of linear equations with respect to c . Consider the homogeneous

system corresponding to this nonhomogeneous system; that is, set a = b' = 0 in this nonhomogeneous system. This gives a;ckld + batik + ba

j

kf = 0.

Contracting this equation with respect to the indices a and e, a, and S, and a and ry, we obtain

k jkl + Ije"Y

+ klj =

pcas7

76 _

-a-Y6

jkl +

0,

Ijk + klf - 0+

fOZ, + co" + l _c'klj - 0. jkl

(7.3.51)

If we symmetrize and alternate equations (7.3.51) with respect to the indices

'y and d, we obtain two homogeneous systems of equations with respect to 6) andkl ai with different order of lower indices (cf. Subsection 7.3.5). Cak1 '1 he determinants of the matrices of coefficients of these systems are equal to (p - 1)2(p + 2) and (p + 1)2(p - 2), respectively. They do not vanish if p > 2. Hence these systems have only the trivial solution. As a result the homogeneous system in question has only the trivial solution c0,216 = 0 provided that p > 2; thus the original nonhomogeneous system has a unique solution expressing the components of c in terms of the components of the subobject (a, b') and their Pfaffian derivatives. Now we can see that the object (a, b) satisfies certain differential equations.

These equations can be obtained if we substitute for the components of c in equations (7.3.49) and (7.3.50) their values found in the way indicated above. The conditions obtained are other analogues of the Bianchi equations in the theory of spaces with affine connection. 7. An almost Grassmann structure AG(p - 1, p + q - 1) is said to be locally Grassmann (or locally flat) if it is locally equivalent to a Grassmann structure. This means that a locally flat almost Grassmann structure AG(p - 1, p + q - 1) admits a mapping onto an open domain of the algebraic variety fl(m, n) of a

7.3

The Complete Structure Object of an Almost Grassmann Manifold 291

projective space PI, where N = (m+i) - 1, m = p - 1, n = p + q - 1, under which the Segre cones of the structure AG(p - 1, p + q - 1) correspond to the asymptotic cones of the variety 1(m, n). From the equivalence theorem of E. Cartan (see Cartan [Ca 08) or Gard-

ner (Gar 891), it follows that in order for an almost Grassmann structure AG(p -1, p + q - 1) to be locally Grassmann, it is necessary and sufficient that its structure equations have the form (7.1.9), (7.1.14), (7.1.15), and (7.1.17).

Comparing these equations with equations (7.3.35), we see that an almost Grassmann structure AG(p - 1, p + q - 1) is locally Grassmann if and only if its complete structure object S = (a, b, c) vanishes. However, we established in this section that if p > 2 and q > 2, the components of b are expressed in terms of the components of the tensor a and their Pfaffian derivatives, and the components of c are expressed in terms of the components of the subobject (a, b) and their Pfaffian derivatives. Moreover it follows from our considerations that the vanishing of the tensor a on a manifold M carrying an almost Grassmann structure implies the vanishing of the components of b and c. Thus we have proved the following result:

Theorem 7.3.2 For p > 2 and q > 2, an almost Grassmann structure AG(p - l,p + q - 1) is locally Grassmann if and only if its first structure tensor a vanishes. 8. We will now write the structure equations (7.3.35) in index-free notation. To this end, we denote the matrix 1-form (w.), defined in a first-order frame bundle, by w and write equation (7.2.33) in the form

dw=r. Aw-w A8 -90Aw+fl,

(7.3.52)

where 0. = (90*) and Bp = (9) are the matrix 1-forms defined in a second-order frame bundle for which

tr 0o = 0, tr Bp = 0; by the letter k we denote the scalar form w occurring in equation (7.2.33) and also defined in a second-order frame bundle. Note that in the exterior products of 1-forms, occurring in equations (7.3.52) and in further structure equations of this subsection, multiplication is performed according to the regular rules of matrix multiplication-row by column (see Subsection 4.1.6). The 2-form H = (f)a) is the torsion form with the components (7.3.53)

292

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

The remaining structure equations (7.3.35) can be written in the form dea + 9a, A 8a =

[-Ia tr (co A w) + pcp A w] + 9a,

p+ q p d00 + Oa A 90 = P q [-Io tr (cp A w) + qw A,p]

+00, (7.3.54)

dx=tr(,pAw), dip + ea A ,p + W A Bo +K AV = - (ap) A w + +,

where V = (w°) is a matrix 1-form defined in a third-order frame bundle; In = (ba) and to = (b;) are the unit tensors of orders p and q, respectively; and 2-forms 9a = (9p), 90 = (9'), and 4' = ($i) are the curvature 2-forms of the AG(p - 1, p + q - 1)-structure whose components are 9a = baryfwk Awl

7.4

9' = b'`kl6wk Awl

4'° = ca7dwl A wk

(7.3.55)

Manifolds Endowed with Semiintegrable Almost Grassmann Structures

1. In this section we will establish geometric conditions for an almost Grassmann structure AG(p - 1, p + q - 1) defined on a manifold M to be semiintegrable. The conditions will be expressed in terms of the structure object S of the almost Grassmann structure AG(p - 1, p + q - 1) and its subobjects Sa and So which will be defined in this section. In what follows, we will often encounter quantities satisfying the conditions similar to conditions (7.2.12) for the quantities uapk. For calculations with quantities of this kind, the following lemma is very useful:

Lemma 7.4.1 If a system of quantities T:'.vo is skew-symmetric with respect to the pairs of indices (a.) and (0), namely satisfies the conditions :..00 = -T :'. °,

(7.4.1)

then the following identities hold:

j

T...]ao] = T....a0

T ..

l

T..1

T...(a0) ..*J

= T...no lif]+

-T'jao) = T..(ll )$ T..li l = -T..° On = T...lijl .

] = 0,

(7.4.2)

T.. gyp) = 0.

In these relations the symmetrization and the alternation are carried separately over the lower indices and the upper indices. In addition the following decompositions take place: .. no =T...iao) +T...no

T".: ro

=

T..I;aal

+T...no.

(7.4.3)

Manifolds with Semiintegrable Almost Grassmann Structures

7.4

293

Proof. All these identities can be proved by direct calculation with help of (7.4.1).

In addition, in the proof of the main theorem, we will use the following lemma:

Lemma 7.4.2 The condition Tlikl7l

-0

(7.4.4)

where the alternation is carried over three vertical pairs of indices, implies the condition (7.4.5) T(ijk) l - 0 where the alternation and symmetrization are carried separately over the upper triple and the lower triple of indices. and collect Proof. To prove this, one should write down 36 terms of T(la°.p71 k) from them 6 groups of 6 terms to each of which the hypothesis (7.4.4) can be

applied. 0 Next we will prove the following important result on the decomposition of the torsion tensor of an almost Grassmann structure AG(p - I, p + q - 1):

Theorem 7.4.3 The torsion tensor a =

} of the almost Grassmann struck ture AG(p - 1, p + q - 1) decomposes into two subtensors: aiO'Y

a=as-f ao, where

(7.4.6)

i(o7) }. io7 na = {aCr(jk)}, ao = {aOjk

Proof. Since the tensor ak is skew-symmetric with respect to the pairs of indices (o) and (k), then, by Lemma 7.4.1, the decomposition (7.4.6) is equivalent to the obvious decomposition io7 io7 07 aajk - aa(jk) + aaijkl'

U Note that by Lemma 7.4.1, the subtensors a,, and ao can be also represented in the form

a,, = (a`a,,)), ao = {aiA7 } Q[jkl

Note also that like the tensor a, its subtensors aQ and ao are skew-symmetric with respect to the pairs of indices (Q) and (k): iW7 i7d ' iA7 a«(jk) = -aa(kj)' a.(jk) = -aO(kj),

294

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and they are also trace-free, since it follows from (7.2.30) that

ao(jk) - 0, a«k - 0+ aelk = 0, aaukl = 0.

(7.4.7)

Theorem 7.4.4 If p = 2, then as = 0, and if q = 2, then ap = 0. Proof. Suppose that p = 2. Then a,#, ry = 1, 2. Since, by definition and Lemma 7.4.1, the tensor as is skew-symmetric with respect to the indices A and ry, we have aill

a(lk)

ai22 = a7k) = 0.

But the first condition of (7.4.7) gives i2l i12 i22 al(ik) + a2 ()k) - 0, al (1k) + a2(Jk) = 0.

It follows from these relations that a2(fk) = aI(ik) = 0+

that is all components of the tensor as vanish. For the case q = 2, the proof is similar. 0 2. Now we will prove the following necessary and sufficient conditions for an almost Grassmann structure AG(p - 1,p + q - 1) to be a-semiintegrable or p-semiintegrable.

Theorem 7.4.5 (i) If p > 2 and q > 2, then for an almost Grassmann structure AG(p - 1, p + q - 1) to be a-semiintegrable, it is necessary and sufficient that the following condition holds: as = 0.

(ii) If p > 2 and q >

2,

then for an almost Grassmann structure

AG(p - 1, p + q - 1) to be A-semiintegrable, it is necessary and sufficient that the following condition holds: ap = 0.

Proof. We will prove part (i) of theorem. The proof of part (ii) is similar. Suppose that 0a, a = 1, . . . , p, are basis forms of the integral submanifolds Va, dim Va = p, of the distribution Aa appearing in Definition 7.1.2. Then w4 = s`ea,

a = 1,...,p; i = p+ 1,...,p+q.

(7.4.8)

For the structure AG(p - 1, p + q - 1) to be a-semiintegrable, it is necessary and sufficient that system (7.4.8) be completely integrable. Taking the exterior derivatives of equations (7.4.8) by means of structure equations (7.2.11), we find that (ds' + sjw - siw) A 0a + si(dOa - wQ A 00) = a`o ks3sk0s A 0.y.

(7.4.9)

7.4

Manifolds with Semiintegrable Almost Grassmann Structures

295

It follows from these equations that

dOa - won 8p =

(7.4.10)

A 80,

where vg is an 1-form that is not expressed in terms of the basis forms O. For brevity, we set gyp' = ds' + s'wj - s'w. (7.4.11) Then the exterior quadratic equation (7.4.9) takes the form (6arP' + 8V O.) A 8Q

(7.4.12)

= a'a ksisk8p A 8y.

From (7.4.12) it follows that for 00, = 0, the 1-form

s'W« vanishes:

69(p'(6) + s'cpa(6) = 0.

(7.4.13)

Contracting equation (7.4.13) with respect to the indices a and Q, we find that gyp' = -s' 'P(6),

= 6!W(6),

(7.4.14)

where we set w(6) = It follows from (7.4.14) that on the subvariety V0, the 1-forms W' and WO can be written as follows: 'P' _ -s'W + SiO00, woo = 6aW + 80."0-,.

(7.4.15)

Substituting these expressions into equations (7.4.10) and (7.4.11), we find that d8,, - w« A 8Q = P A Ba + say 8y A Bp

(7.4.16)

where say = AC OP) and

ds' + sjw - s'w = -s'W + sio0o.

(7.4.17)

Substituting (7.4.16) and (7.4.17) into equation (7.4.9), we obtain

- s'soy - 6QRslilyl = aapk lsjsk.

(7.4.18)

Contracting equation (7.4.18) with respect to the indices a and Q, we obtain

-2s'say - ps'y + 8'y = 0, from which it follows that

s'y = s'sy,

(7.4.19)

where we set sy = -P2 sa'r. Substituting (7.4.19) into (7.4.18), we find that s'(6yso - dasy - 2,,O-f) =

2a'lRklsisk.

(7.4.20)

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

296

It follows that basa

- 60s'y - 2sQ' =

(7.4.21)

where so ctj = -s"A. a.? Substituting (7.4.21) into (7.4.20), we arrive at the equation = a'"07

sa(j a7 d$

(7.4.22)

a(jk),

k)

where the alternation sign in the right-hand side is dropped by Lemma 7.4.1.

Contracting (7.4.22) with respect to the indices i and j and taking into account equations (7.2.30) and (7.2.31), we obtain s a, ak = O ,

(7 4 23) .

.

from which, by (7.4.22), it follows that aia, a(jk)

= 0.

(7.4.24)

This proves that if an almost Grassmann structure AG(p - 1, p + q - 1) is a-semiintegrable, then its torsion tensor satisfies the condition (7.4.24), as = 0. Since, by Theorem 7.4.4, for p = 2 the subtensor as = 0, condition (7.4.24) is identically satisfied. Hence, while proving sufficiency of this condition for a-semiintegrability, we must assume that p > 2. Let us return to equations (7.4.16) and (7.4.17). Substitute into equation (7.4.17) the values s'a taken from (7.4.19) and set

P = V - saga.

(7.4.25)

In addition, by (7.4.23), relations (7.4.21) imply that sa, = altisal a

a

Then equations (7.4.16) and (7.4.17) take the form d9a - (wQ + 6.00) A Bp = 0

(7.4.26)

ds'+sjwj' -s'(w-W-)=0.

(7.4.27)

and

Taking the exterior derivatives of (7.4.27), we obtain the following exterior quadratic equation: s'4i +

A 9Q = 0,

(7.4.28)

where 4

= dip - (p+1)g s kwk A9 ry

p+q

Next, taking the exterior derivatives of (7.4.26), we find that

A06=0.

(7.4.29)

7.4

297

Manifolds with Semiintegruble Almost Grassmann Structures

Equation (7.4.28) shows that the 2-form 4i can be written as 4i = A- sks'O,, A 06,

(7.4.30)

where the coefficients Ak6 are symmetric with respect to the lower indices and

skew-symmetric with respect to the upper indices. Substituting this value of the form t into equations (7.4.28) and (7.4.29), we arrive at the conditions b(k,)

0

(7.4.31)

blaktj + a[a Aki 1 - 0.

(7.4.32)

and

Contracting equation (7.4.31) with respect to the indices i and j and equation (7.4.32) with respect to the indices a and Q, we find that 2(q + 2)Aki + bk1i + bk;l + bi k + b,4k; = 0

(7.4.33)

and

2(p

- 2)Aki + bOk + boik + baki + balk = 0.

(7.4.34)

Note that for p = 2 equation (7.4.32) becomes an identity, and we will not obtain equation (7.4.34). If we add equations (7.4.33) and (7.4.34) and apply condition (7.3.29), we find that Aki = 0. (7.4.35) As a result equations (7.4.31) and (7.4.32) take the form

0kl)

= 0+ 6[a ki) = 0.

(7.4.36)

By Lemma 7.4.1, conditions (7.4.36) are equivalent to the conditions b(jkt) = 0+ b[Oki 61 - 0.

(7.4.37)

It follows from equations (7.4.35) and (7.4.30) that d 4p =

(p + 1)g s k Wk n By.

p+q

(

7 4 38) .

.

Finally, taking the exterior derivatives of equations (7.4.38) and applying (7.4.26), (7.4.27), and (7.3.35), we obtain the condition [QQyI - 0. c(ijk)

(7.4.39)

These equations will not be trivial only if p > 2. But, by Lemma 7.4.2, conditions (7.4.39) follow from integrability conditions (7.3.34).

298

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

Thus the system of Pfaf flan equations (7.4.8), defining integral submanifolds of an a-semiintegrable almost Grassmann structure, together with Pfaffian equations (7.4.17) and (7.4.38) following from (7.4.8), is completely integrable if and only if conditions (7.4.24) and (7.4.37) are satisfied. But as we showed

earlier, for p > 2, conditions (7.4.37) follow from condition (7.4.24). Hence only condition (7.4.24) is necessary and sufficient for complete integrability of the system of equations (7.4.8), (7.4.17), and (7.4.38), that is, for the almost Grassmann structure to be a-semiintegrable for p > 2. This proves part (i). As we noted in the beginning, the proof of part (ii) is similar. We note only that the equations of integral submanifolds Vo, dim Vp = q, of the distribution G10 appearing in Definition 7.1.2 can be written in the form

a=1,...,P; i=p+I....,P+q,

wQ=s09$,

where the 1-forms 9' are linearly independent on the submanifold V0. We introduce the following notations: ba

=

{b{ikt)}+ ba = 1 okl l}, Ca =

6p = 1

[jkll},

ICI- j01 11

b0 = {baki6)}, co =

It follows from our previous considerations that

1. for p = 2 we have ba, = O and ca = 0;

2. for q = 2 we have by = 0 and co = 0;

3. forp > 2 we have ca = 0; and

4. forq>2wehave c0=0. The last two results follow from conditions (7.3.34) and Lemma 7.4.2. These

results combined with equations (7.3.27) and (7.3.28) imply that the tensors a,, ao and the quantities b,,bQ,ba,by form the following geometric objects: (aa, bI), a

z (aa, ba), S. = (/aa, bia, b2a),

(a0, bb),

(ao,bo), So = (ao,bp,b2),

which are subobjects of the second structural object and the complete structural object of the almost Grassmann structure. From the proof of Theorem 7.4.5 it follows that for p > 2 the condition as = 0 implies the conditions b l = bQ = 0. Similarly for q > 2 the condition ao = 0 implies the conditions bQ = bQ = 0. Now we consider the cases p = 2 and q = 2. For definiteness we take the case p = 2. As we have already seen, for p = 2, the tensor as as well as the quantities bQ and ca vanish (aa = bQ = ca = 0), and the object bQ becomes

7.4

Manifolds with Semiintegrable Almost Grassmann Structures

299

a tensor. Thus the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(1, q + 1) to be a-semiintegrable. Hence we have proved the following result:

Theorem 7.4.6 (i) If p = 2, then the structure subobject Sa consists only of the tensor b'', and the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(1, q + 1) to be a-semiintegrable.

(ii) If q = 2, then the structure subobject So consists only of the tensor bQ, and the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(p-1, p+ 1) (which is equivalent to the structure AG(l,p+ 1)) to be p-semiintegrable.

(iii) If p = q = 2, then the complete structural object S consists only of the tensors b1, and b2,, and the vanishing of one of these tensors is necessary

and sufficient for the almost Grossmann structure AG(1,3) to be a- or (3-semiintegrable, respectively.

We will make two more remarks:

1. The tensors ba and bQ are defined in a third-order differential neighborhood of the almost Grassmann structure.

2. For p = q = 2, as was indicated earlier (see Subsection 7.1.2), the almost Grassmann structure AG(1,3) is equivalent to the conformal CO(2, 2)-structure. Thus by results of Subsection 5.1.3, we have the following decomposition of its complete structural object: S = b,4-b2 . This matches the splitting of the tensor of conformal curvature of the CO(2, 2)-structure: C = C. 4-Co. 3. Now we can compare the differential geometry of conformal and pseudoconformal structures with that of almost Grassmann structures. A conformal or pseudoconformal structure CO(p, q) is defined on a differentiable manifold M of dimension n = p + q by a differentiable field of second-order cones Cz(p, q) of signature (p, q) lying in the tangent space T,, (M). A cone Cx(p,q) is invariant under transformations of the group G °_w SO(p,q) x H, where SO(p,q) is the pseudoorthogonal group of signature (p, q) and H is the group of homotheties.

When we derive the structure equations (4.1.31)-(4.1.35) of a conformal (SO (p, q) x H) a T(n) structure, we prolong the group G to the group G' which is the group of motions in the compactified space T2(M) enlarged to an n-dimensional quadric Qy of index q. This quadric can be embedded into a projective space P"+l of dimension n + 1 and is determined in it by a homogeneous equation of second order whose left-hand side is a quadratic form of signature (p + 1, q + 1). Since a point x E Q, at which Qx is tangent to M, is fixed, the geometry of Qy is equivalent to that of a pseudo-Euclidean space RQ .

The group G' is isomorphic to the group of motions of this space, and T(n) is its group of translations.

300

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

The first structure tensor appearing in the structure equations (4.1.34) is the tensor of conformal curvature which is determined in a third order differential neighborhood. In general, if n > 4, this tensor does not vanish. The vanishing of this tensor leads to a local conformally flat structure.

In the case of a four-dimensional conformal structure CO(p, q), where p + q = 4, the tensor of conformal curvature splits into two subtensors which are the curvature tensors of two fiber bundles Ea and E0 associated with this conformal structure. The vanishing of one of these subtensors leads to a semiintegrable conformal structure. An almost Grassmann structure AG(p - l, p + q - 1) is defined on a differentiable manifold M of dimension n = pq by a differentiable field of algebraic Segre cones SC,. (p, q) C T., (M) whose projectivizations are the Segre varieties S(p - 1, q - 1). Each of these cones carries two families of plane generators of dimensions p and q that form two fiber bundles Ea, and Ep on the manifold M. A cone SC,(p, q) is invariant under transformations of the group G °-' SL(p) x SL(q) x H where SL(p) and SL(q) are the special linear groups of orders p and q, respectively. When we derive structure equations (7.3.35) of an almost Grassmann structure, we prolong the group G to the group G' °-r (SL(p) x SL(q) x H) of T(pq). The group G' is the group of motions in the compactified space T=(M) which is obtained by joining to the space T=(M) the Segre cone SC,,. (M) with vertex at the point at infinity of the space T,T(M). The compactified space Ts(M) is equivalent to the algebraic variety fl(p-1, p+q-1) with a fixed point x at which this space is "glued" to the almost Grassmann manifold AG(p - 1, p + q - 1). The variety l(p - 1, p + q - 1) itself is the image of the Grassmannian in the projective space PN, where N = (Dq°) - 1. The variety 0 (p-1, p+q-1) with a fixed point x is equivalent to the Segreaffine space SAD" (see Subsection 6.6.3) of dimension pq. The latter space is a stereographic projection of the variety fl(p - 1,p+ q - 1) from the point x onto a flat space of dimension pq. The group G' is the group of motions of the space SAD", and the group T(pq) is the group of translations of this space. Unlike the CO(p, q)-structure, the first structure tensor (the torsion tensor) of the almost Grassmann structure AG(p - l,p + q - 1) is determined in its differential neighborhood of second order. If p > 2 and q > 2, then just like for CO(2, 2)-structure, this tensor splits into two subtensors that are the first

structure tensors of two fiber bundles E, and E. The vanishing of one of these subtensors leads to a semiintegrable almost Grassmann structure.

On the other hand, if p = 2 or q = 2, then the corresponding torsion tensor vanishes, and the condition of semiintegrability of an almost Grassmann structure will be connected with the vanishing of the second structure tensorthe curvature tensor of the corresponding fiber bundle. Finally, if p = q = 2 (note that this is the only positive integer solution to the equation p + q = pq), then the almost Grassmann structure AG(1, 3) becomes

7.5

Multidimensional (p + 1)-Webs and Almost Grassmann Structures

301

Table 7.4.1

#

Property

CO(p, q)

1.

dimM

n=p+q

2.

Invariant construction inT=(M)

2nd-order cone

Segre cone

C.(p,q)

SC.(p,q)

Order of

s = 1

s = 1

3.

AG(m,n)

p=m+l, q=n-m

G-structure 4.

Structure group

5.

Prolonged structure group

6.

Type of

SO(p, q) x H G'

G x T(p + q)

G'-° SL(p) x SL(q) x H G' '-5 G x T (p q)

t=2

t=2

Torsion-free

With torsion

(b, c)

(a, b', V, c)

G-structure 7.

Existence of

torsion 8.

Complete

structure object 9.

Local space

(q).

(G(m,n))=

10.

Locally flat

Cq

G(m, n)

structure

11.

Existence of isotropic bundles

`dp & q : p = q = 2: E,, (M, SL(2)) and & (M, SL (p)) and Ea(M, SL(q)) E,(M, SL(2))

12.

p+q=pq

CO(2,2)

AG(1, 3)

p=q=2 the CO(2, 2)-structure. Its torsion tensor vanishes, and the role of its curvature tensor was described earlier (see Sections 5.1 and 5.4). The preceding comparison of the conformal CO(p, q)-structures and almost Grassmann AG(m, n)-structures is summarized in Table 7.4.1.

7.5

Multidimensional (p + 1)-Webs and Almost Grassmann Structures Associated with Them

1. We will define the notion of a d-web of codimension q given on a differentiable manifold of dimension pq.

Definition 7.5.1 Let M be a C'-manifold of dimension pq, p > 2, q > 1, s > 3. We say that a d-web W (d, p, q) of codimension q is given in M if

302

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

1. d foliations A, a = 1,... , d, of codimension q are given in M; and 2. d leaves (of the foliations A,) passing through a point x E M are in general position; namely any p of the d tangent subspaces to the leaves at the point x have in common only the point x. There exists a neighborhood of each point x of the web W (d, p, q), t where the foliations A, are fibrations. Therefore, from a local point of view, a d-web

can be considered as formed by d fibrations. We denote the bases of these fibrations by X,. Example 7.5.2 Consider in an affine space AP4 of dimension pq d families of parallel (p - 1)q-dimensional planes that are in general position. They form a d-web called a parallel d-web.

Example 7.5.3 Let X o = 1, ... , d, be d smooth submanifolds of dimension q in a projective space pP+q-1, and let L be an (p - 1)-plane that intersects each X, at the points x,. A d-web arises in a neighborhood of the (p-1)-plane L on the Grassmannian G(p - 1,p + q - 1) of all (p - 1)-planes of the space pp+q-t (dimG(p- l,p+q-1) = pq (see Section 6.1)). The leaves of this web are bundles of (p - 1)-planes with vertices located on the submanifolds X,. Such a web is called the Grassmann d-web and is denoted by GW (d, p, q). Our definition of the Grassmann d-web is essentially of local nature, since, only for (p - 1)-planes sufficiently close to x, can we assert that they intersect each of

the submanifolds X, only at one point, as it was for the (p - 1)-plane L.

Example 7.5.4 A Grassmann d-web is called algebraic and is denoted by AW (d, p, q) if the submanifolds X, defining it belong to the same algebraic q-dimensional submanifold Vd of degree d. There are many special cases of algebraic d-webs. They are characterized by the fact that the submanifold Vd is decomposed into two or more submanifolds Vdk, 0 < dk < d, Ek dk = d, and each X, belongs to one of V9. For example, all submanifolds X, can be q-planes. In this case the submanifold Vd is decomposed into those q-planes.

Definition 7.5.5 Let M and M be two manifolds of the same dimension pq. Two webs W (d, p, q) and W (d, p, q) defined in M and M are said to be equivalent if there exists a local diffeomorphism w: M -4 M that transfers the foliations of the first web W (d, p, q) into the foliations of the second web W (d, p, q).

In particular, d-webs equivalent to the parallel, Grassmann, and algebraic d-webs, which were considered above, are called parallelizable, Grassmannizable, and algebraizable, respectively. Thus a parallelizable web W (d, p, q) is equivalent to a web consisting of d families of parallel planes of codimension q. If d < p, a web W (d, p, q) is always For brevity, we will use these words instead of the words "each point x of a manifold M carrying a d-web W".

7.5

Multidimensional (p + 1)-Webs and Almost Grassmann Structures 303

parallelizable. Because of this we will assume that d > p + 1. In Sections 7.6 and 7.8 we will study Grassmann and Grassmannizable d-webs in more detail. 2. Consider a (p + 1.)-web W (p + 1, p, q) defined on a manifold M. The foliations .10, a = 0,1, ... , p, forming this web can be defined by the following completely integrable systems of Pfaffian equations: W4 = 0, c

o = O,l,...,p; i = p + 1,...,p + q.

(7.5.1)

Since the number of 1-forms on the left-hand sides of equations (7.5.1) is (p+l)q

and dim M = pq, the forms w{ are connected by linear equations. It can be proved (see Goldberg [Go 73, 74a] or Goldberg [Go 88], §1.2) that these equations can be reduced to the following form:

w'+w'+...+w'=0. 0 p

(7.5.2)

1

Relations (7.5.2) remain invariant under the transformations

AJw', det(AJ) # 0,

(7.5.3)

forming the group G = GL(q)-the structure group of a web W (p + 1, p, q) (see Subsection 7.1.2). By conditions (7.5.2), the structure equations of a web W (p + 1, p, q) can be reduced to the form

dw'=wiA ,'+E 003k, a R#a

p

o

o

where a,,0 = 1, ... ,p, and a'.k is the torsion tensor of the web satisfying the conditions p,k

Rakj, E a ik

(7.5.5)

0

CO

(see Goldberg [Go 73, 74a] or Goldberg [Go 881, §2.1). In addition, we suppose that 0. The forms ww satisfy the structure equations

as k =

dwf = w A wk +

b

klwk A wl,

(7.5.6)

Ck,

where b

ki

is the curvature tensor of the web, and define an affine connection

t on the manifold MPQ (see Goldberg [Go 88], §1.3). The tensors a

and

no

bp';ki are the torsion tensor and the curvature tensor of this connection.

a

Equations (7.5.6) are differential prolongations of equations (7.5.4). In addition, as another result of exterior differentiation of equations (7.5.4), we obtain the Pfaffian equations p

VClip a ka ikt +apya ,,k noa i + a ma a kl)w1, ya 7_1

R7

'Y

a

(7.5.7)

304

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and the closed form relations (7.5.8)

(7.5.9)

p(ikll = 0,

b,';k+2bbkll=0.

a«

ao

(7.5.10)

Relation (7.5.8) shows that if p > 2, then the curvature tensor of a web W (p + 1, p, q) is completely determined by the covariant derivatives of the torsion tensor of this web. This implies the following result:

Theorem 7.5.6 For a web W (p + 1, p, q), p > 2, to be parallelizable, it is necessary and sufficient that its torsion tensor vanishes, ask = 0. Proof. By (7.5.7) and (7.5.8), the condition «a k = 0 implies that b '

= 0.

As a result the structure equations of the web under consideration take the

form dw' 0

0

dw' =wh Awk,

o =0,1,...,p; i,j,k=p+1,...,p+q.

But these equations determine a (p + 1)-web in an afiine space APQ of dimension pq formed by foliations of parallel planes of codimension q (see Goldberg [Go 73, 74a] or 88), §1.5). The converse follows immediately from the previous equations. Note that for a multidimensional three-web W(3, 2, q) the symmetric part of the curvature tensor cannot be expressed in terms of the covariant derivatives of the torsion tensor (see equations (1.31) and (1.33) in Akivis and Shelekhov [AS 92]). The parallelizability condition for webs W(3, 2, q) is expressed in

terms of both the torsion and curvature tensors (see Akivis and Shelekhov [AS 92), §1.5).

3. We will now show that an almost Grassmann structure AG(p-1, p+q-1) is associated with each (p + 1)-web W (p + 1, p, q).

Let T,,(M) be the tangent space to M at the point z. The co-basis forms p; i = p + 1, ... , p + q, of the (p + 1)-web introduced above can be taken as coordinates in the space TT(M). Then the equations of the subspaces T, of this space that are tangent to the leaves of the web passing through the point x can be written in the form (7.5.1). By virtue of (7.5.2), we can see that the relations

hold. Equations (7.5.1) and (7.5.2) are invariant under transformations of the group GL(q) of the web W(p+ 1,p,q).

Grassmann (p+ 1) -Webs

7.6

305

Let (r, K, 01, ... , o _ 1) be a permutation of the indices (0, 1, ... , p). In T=(M), we consider the intersection of the subspaces T,,, k = 1, . . . , p - 1. Denote this intersection by TT,. Its dimension is q, and it is defined by the equations w' = 0. The number of such subspaces is (PZ1) = P(P'). z If p = 2, o. this number is equal to 3, and the subspaces TTK coincide with the subspaces T. tangent to the leaves of the web passing through the point x. In the space T=(M) there exists a unique Segre cone SC=(p,q) containing all subspaces TT,,. This cone can be defined by parametric equations (7.1.1) where zo = w'. By (7.5.2), it follows from these equations that Q

W` _ -o'rb' o

where rro = - EP.=1 j7,,. The subspaces TTK belonging to the Segre cone can be given on this cone by the equations q,. = 0, where the indices ok take the values indicated above. These subspaces belong to the family of the q-dimensional plane generators q4 of the Segre cone SC, (p, q). Since the family of Segre cones SC= (p, q) given in the tangent spaces T. (M)

defines an almost Grassmann structure AG(p -1, p+ q - 1) in the manifold M, the following theorem holds:

Theorem 7.5.7 An almost Grassmann structure AG(p-1, p+q-1) is invariantly connected with an (p + 1) -web W (p + 1, p, q) given on a smooth manifold

M of dimension pq. The structure group of this web is a normal subgroup of the structure group of the almost Grassmann structure.

Note that the last statement of Theorem 7.5.7 follows from the fact that the structural group of a (p + 1)-web is the group GL(q) and the structure group of the almost Grassmann structure is either of two following isomorphic groups: SL(p) x GL(q) SL(p) x SL(q) x H. The q-dimensional plane generators r)q of the Segre cones SC, (p, q) associated with a web W (p + 1, p, q) are called its isoclinic subspaces. In addition to them, the Segre cones SC., (p, q) carry the p-dimensional plane generators cP. They are called the transversal subspaces of this web.

7.6

Grassmann (p + 1)-Webs

1. In Section 7.5 we constructed an important example of a web W (p+ 1, p, q)-

the so-called Grassmann (p+1)-webs in the Grassmannian G(p-1,p+q-1) of (p-1)-planes of a projective space PP+q-1 of dimension p+q-1 (see Example 7.5.3). In this section we will study Grassmann (p + 1)-webs in more detail. First, note that dim G(p -1, p + q - 1) = pq. Next, consider a submanifold in G(p - 1, p+ q - 1) formed by (p - 1)-planes passing through a fixed point

306

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

z E PP+q-I. We will call this manifold a bundle of (p-1)-planes and denote it by S. It is easy to see that the bundle S. is isomorphic to the Grassmannian G(p - 2, p + q - 2) and that dim S= _ (p - 1)q and codim S., = q. Using bundles of (p - 1)-planes, we can construct foliations and webs of codimension q in the Grassmannian G(p - 1, p + q - 1). In fact, in the space pP+q-1, let us consider a smooth manifold X of dimension q and the set of the bundles S,T with vertices belonging to X: x E X. If we exclude the (p - 1)planes tangent to the manifold X and the (p-1)-planes intersecting X at more than one point from each of the bundles St, the remaining parts Sx of S. form a foliation in an open domain D of the Grassmannian G(p - 1, p + q - 1). In the space PP+q-1, we further consider submanifolds X a = 0,1, ... , p, of dimension q in general position. Each of these submanifolds generates a

foliation in an open domain D. C G(p - 1, p + q - 1) described above. All foliations constructed in this manner generate a (p + 1)-web of codimension q in the domain D = ns=o D. In Section 7.5 the (p + 1)-webs described above were called Grassmann (p + 1)-webs and were denoted by GW(p + 1,p, q). Next, denote by L a moving (p - 1)-plane of a Grassmann (p + 1)-web and

by A, the points of intersection of L and the submanifolds X,. Since the submanifolds X. are in general position, p of those points, for example, the points A1,.. . , AP, can be taken as the vertices of a projective frame of the (p - 1)-plane L. We also take the vertex A0 as the unit point of this frame. Thus A0 = Al + A2 +... + AP. Let us take the points A,, i = p + I.... ,p+ q, that supplement the points Aa, a = 1,. .. , p, to a complete frame of the space PP+q-1 As usual, the equations of infinitesimal displacement of this moving frame can be written in the form dAE = wf An,

,n = 1,...,p,p + I,...,p + q,

(7.6.1)

and the structure equations which the forms wf satisfy, in the form: d w E = wf A t,

f,i,C = 1,...,p,p + 1,...,p + q.

(7.6.2)

Since the (p-1)-plane L is not tangent to any of the submanifolds X,,, the 1-forms w,, can be taken as co-basis forms on these submanifolds. Since the points Aa are fixed when the subspace L is fixed, we must have wa = Aoiwa,

0 # a; a,Q = 1,...,p; i = p+ 1,...,p+q,

(7.6.3)

and

dAa = w°A,, +wa 1 Ai +

\

AaiAp f .

(7.6.4)

p#a

Here and in what follows, the summation is carried over the indices i, j, k according to the usual rule, while the summation is carried over the indices a, j3, ry only if there is the summation sign.

7.6

Grassmann (p + 1)-Webs

307

Let us locate the points A; in the space TA,, tangent to the manifold Xe generated by the point A0. Then we have dAo = wAo + Ai

wQ

(7.6.5)

where (7.6.6) Q

We define

wQ=-wo.

(7.6.7)

Since the point AO generates a q-dimensional manifold X0, the forms wo are linearly independent. In the frame that we have constructed the equations

wa =0

w, = O, ... , wyi= 0,

determine p + 1 foliations in the Grassmannian G(p - 1, p + q - 1), and these foliations form a Grassmann (p+ 1)-web GW(p+ 1,p, q). The forms w. are the co-basis forms of this web, and equations (7.6.3) and (7.6.7) are its fundamental equations. 2. Let us find the torsion and curvature tensors of a Grassmann web GW (p + 1, p, q). To this end, we first prolong equations (7.6.3); that is, we take the exterior derivatives of these equations and apply the Cartan lemma to the exterior quadratic equations obtained as the result of exterior differentiation. As a result we obtain

Oxa; + \n;Aojw0 + E (Aai

- Aoi)(A'3 - )'3 )w,, + wR = Aa;jwJo+

(7.6.8)

7¢a,Q

where a,$ and -y are distinct and Vij = AQji. In equations (7.6.8) we used the notations

Vap;=dAp;-aaj9;, 9 =w -b; w,

(7.6.9)

and the forms wo are determined by formula (7.6.7). Exterior differentiation of relation (7.6.6) leads to the equation

Awo=0. The solution of this equation can be represented in the form w° = w° +

.?

wo

where

P,=Pji,

EP,a.=0, a

(7.6.10)

308

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

and

(7.6.11)

P

are fiber forms, so they are not linear combinations of the co-basis forms wQ.

Let us fix the (p - 1)-plane L of the web under consideration, that is, set w`o = 0. Then equations (7.6.8) take the form

VsAoi + np = 0,

(7.6.12)

where as usual b denotes the differentiation symbol with respect to the fiber parameters and aR = wo(b). We define the quantities

A,=

1

R

p(P-.00 "

(7.6.13)

It follows from equations (7.6.12) that when a point x is fixed, the quantities Ai satisfy the equations

Daai+rro=0. On the other hand, equations (7.6.1) and (7.6.11) imply that

bAi = iA; + r°Ao. The last two relations lead to the equations: b(Ai + AiAo) = 7r, (Ai + A,Ao),

which show that the plane L spanned by the points A, + A,A0 is invariant. Let us locate the points A, in L. Then the quantities A, vanish, and (7.6.13) gives

E 1oi = 0.

(7.6.14)

o#p

The forms ir? become zero, and the forms w° become linear combinations of the co-basis forms wi : gAwA.

(7.6.15)

A

Equations (7.6.15), (7.6.10), and (7.6.7) imply that (qQ

- p,)wa.

(7.6.16)

We can now find the torsion and curvature tensors of the web GW(p + 1, p, q). Applying exterior differentiation to its co-basis forms wa, we get dwQ=w1Awj' +w.0 Awa.

7.7 Transversally Geodesic and Isoclinic (p + 1) -Webs

309

Next from equation (7.6.6) we obtain

Wa =w- wp. 00a

Substituting these expressions into the previous formulas and applying relations (7.6.3), we obtain

dw,, =wi A0 +

Aw,a,

kAO

(7.6.17)

p#a

where the 1-form B is determined by formula (7.6.9). By (7.6.14), the expressions bkAO + 6IAak satisfy equations (7.5.5). Therefore equations (7.6.17) are the structure equations of a Grassmann web GW(p + 1, p, q), and the torsion tensor of this web has the form (7.6.18)

Qik = bkA«j + 6 apk.

Since for p > 2 the torsion tensor completely defines the geometry of a web W (p + 1, p, q) (see Section 7.5), we arrive at the following result:

Theorem 7.6.1 For a web W (p + 1, p, q), p > 2, to be Grassmannizable, that is, to be equivalent to a Grassmann web GW (p + 1, p, q), it is necessary and sufficient that its torsion tensor has the form (7.6.18). The forms B determined by equations (7.6.9) define an affine connection 1' on a web GW (p + 1, p, q). It follows from the previous equations that

dw, - w n wk

w n w'a,

dw = -wok A w°t.

a

By virtue of formulas (7.6.9), (7.6.15) and (7.6.16), we find from this equation

that

d9i - ej kA eki =

is

i s

(btgjk + bjglk

i s wak A w0. 1 bkPjl)

04 This gives the following expression for the curvature tensor of the Grassmann (p + 1)-web under consideration:

b'jki = 2

7.7

bjgik + bkipp) -

2

(6'

'3, +

5 pQk).

(7.6.19)

Transversally Geodesic and Isoclinic (p + 1)-Webs

1. A web W (p+ 1, p, q) is called transversally geodesic if the almost Grassmann structure AG(p - 1, p + q - 1) associated with this web is a-semiintegrable. A

310

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

web W (p + 1, p, q) is called isoclinic if the structure AG(p - 1, p + q - 1) is p-semiintegrable. We first consider transversally geodesic (p + 1)-webs W and find analytic conditions characterizing them. The semiintegrability of the almost Grassmann structure associated with a web means the existence of a family of subvarieties

VP on the manifold M that are tangent to the p-planes c". The equations of these subvarieties have the form w' = '9a,

a = 1,...,p; i = p + 1,...,p + q,

(7.7.1)

Cr

where 9a are 1-forms independent on VP, and ' are the coordinates of a vector

determining the location of the transversal subspace of the web. By means of formulas (7.5.4), exterior differentiation of equations (7.7.1) leads to the following exterior quadratic equations:

a' 8,) A 0a =

(7.7.2)

p#a where we used the notations

e'w, ,

V'=

a"= aap.kf'k. ap

(7.7.3)

The quantities a' satisfy the equations aA

a = Ra a', > a'aA= 0.

a13

(7.7.4)

a.A

If we add up equations (7.7.2) written for all a = 1,. .. ,p, and use conditions (7.7.4), we find that

V ' A (o) = -'d (> 90)

.

(7.7.5)

01

Ck

Equations (7.7.5) show that

d(1: where 0 is an 1-form. Substituting the last expression into equations (7.7.5), we obtain the equation

(Vt' - e'9) A

1>

9a) = o.

By Cartan's lemma, we find that V{' = {`9 +

0a.

a

(7.7.6)

7.7 Transversally Geodesic and Isoclinic (p + 1)- Webs

311

On a submanifold VP, the foliations of a web W (p + 1, p, q) cut out a (p + 1)-web W (p + 1, p, 1) of codimension 1. The forms 0. are the basis forms of this web, and the leaves of its (p - 1)-dimensional foliations are determined by the following systems of Pfaffian equations: 0« = 0,

0a = 0.

By (7.5.4), the structure equations of a web W(p+ 1,p,1) have the form

a0aA00,

(7.7.7)

096000

where

Ea=0. -'000

Substituting expressions (7.7.6) and (7.7.7) into equations (7.7.2), we find that

w=0and

a'+a0a'=f'a. a0

Summing up all these equations in a and 3, we obtain a' = 0, a0' = f' R.

(7.7.8)

By (7.7.8), equations (7.7.6) and (7.7.3) take the form Vf' = f'0,

(7.7.9)

and

aaaXfk = a0f'.

(7.7.10)

The following theorem gives the geometric meaning of relation (7.7.9):

Theorem 7.7.1 The subvarieties VP defined on M by equations (7.7.1) are totally geodesic in the connection 1' induced by a web W (p + 1, p, q).

Proof. Denote by lei) the frame that is dual to the co-frame 10) Q consisting of the co-basis forms of the web W (p + 1, p, q), and consider the vectors fa = f'e1. By (7.7.1), these vectors are tangent to VP. By (7.7.9), they, as a well as all vectors of the form cafes where ca are constants, can be parallel translated along VP. Therefore, VP is a totally geodesic submanifold. It follows that the submanifolds VP cut out the leaves of the (p + 1)-web (which are themselves totally geodesic submanifolds on M) along geodesic lines.

This is the reason that the submanifolds VP are called transversally geodesic submanifolds of a web W (p + 1, p, q), and the web itself is called transversally geodesic.

312

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

Next, we will study equations (7.7.10). First of all, note that if p = 2, the left-hand side of (7.7.10) is identically zero, since by (7.5.5) the torsion tensor a' -k of a web W(3, 2, q) is skew-symmetric in the indices j and k. We can see 123

from relations (7.7.3) and (7.7.8) that in this case la' = 0 and equation (7.7.10)

becomes an identity. Therefore, if p = 2, the property of a three-web to be transversally geodesic can be expressed in terms of the curvature tensor (see Akivis and Shelekhov [AS 92], §3.1).

Suppose further that p > 2. Since equation (7.7.10) must be identically satisfied with respect to C', the expression p on its right-hand side is linear in {'; namely a = a kek.

ao

(7.7.11)

no

hold. Substituting (7.7.11) into (7.7.10), we obtain the equation

bk

( .03 k - .1Q)(3e

0,

pk).

(7.7.12)

Theorem 7.7.2 For a web W (p + 1, p, q), p > 2, to be transversally geodesic, it is necessary and sufficient that the symmetric part of its torsion tensor has the form (7.7.12). Proof. The necessity of the condition of Theorem 7.7.1 was proved above. Its sufficiency follows from the fact that by (7.7.12), the system of Pfaffian equations (7.7.1) and (7.7.9) defining the transversally geodesic submanifolds of a web W (p + 1, p, q) is completely integrable. 0 Since the almost Grassmann structure associated with a transversally geodesic (p + 1)-web is a-semi integrable, the condition (7.7.12) of transversal geodesicity we have obtained is equivalent to the condition aQ = 0 of semiintegrability of this structure. It follows that the vanishing of the tensor as is equivalent to the condition (7.7.12), and conversely. In view of this, the tensor as must be expressed in terms of the tensor ai(ik), and conversely. This was proved analytically in Goldberg [Go 75a] (see also Goldberg [Go 88], §2.4). To this end, the following formula derived in Goldberg [Go 75a] was used: aQ(,kl = 2 a (jk)

- ,.

2

,

6(i (a k)i + a irlk) J ' p

E (a 1.)r + a.rizIk)) + (q + 1)(p - 1) &(j6*y

-

6 tjk) 6#Y

(a not summed). (7.7.13)

2. We now consider isoclinic webs W (p + 1, p, q). The almost Grassmann structure associated with such a web must be a-semiintegrable. Therefore

7.7 Transversally Geodesic and Isoclinic (p + 1) -Webs

313

on the manifold M, there exists a family of submanifolds Va tangent to the isoclinic subspaces rlo. The equations of these submanifolds can be written in the form

a=1,...,p; i=p+1,...,p+q,

w'=r1a9', a

(7.7.14)

where the 9i are 1-forms, which are linearly independent on VQ, and qa are parameters determining the location of the isoclinic subspace of the web.

Theorem 7.7.3 For a web W (p + 1, p, q), p > 3, q > 2, to be isoclinic, it is necessary and sufficient that the skew-symmetric part of its torsion tensor has the form: p[jkl = b [i6kl. (7.7.15) 00

The proof of this theorem is similar to that of Theorem 7.7.2. Note that the condition of Theorem 7.7.2 is equivalent to the condition for an almost Grassmann structure associated with a web W (p + 1, p, q) to be /3-semiintegrable. As was proved in Section 7.4, this condition has the form ao = 0. It follows that the tensor ao must be expressed in terms of the tensor aa and, conversely.

These expressions were also found in Goldberg [Go 75a] (see also Goldberg [Go 88], §2.4):

aa(jkl =

2

2

[jk] +/

q

2 1

a(j a7l1Ikl

2

(a

blj

\4 - 1)(p + 1)

p+

d#7

1

6#y

a7(jkl

(7.7.16)

(a not summed).

a 67

d7

The following important theorem follows from Theorems 7.7.2 and 7.7.3;

Theorem 7.7.4 A web W (p + 1, p, q), p > 2, q > 2, is Grassmannizable if and only if it is both isoclinic and transversally geodesic.

Proof. Suppose that a web W (p + 1, p, q) is Grassmannizable. Then it is equivalent to a Grassmann web GW (p + 1, p, q). As was proved in Section 7.6 (see formula (7.6.18)), the torsion tensor of a Grassmann web has the form ajjk = bkaaj

a

ki

This implies the relations

p[jkl -

Qfi -

Xolj)akl'

aR(jk) = (\a(j +'\R(j)ak);

clearly the torsion tensor of the web under consideration satisfies the condition of Theorems 7.7.2 and 7.7.3. Thus a Grassmannizable web is both isoclinic an transversally geodesic.

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

314

Conversely, suppose that a web W (p+ 1, p, q) is both isoclinic and transver-

sally geodesic. Then relations (7.7.12) and (7.7.15) hold on this web. So we have

Qaik = as

+ 0ljkl = aa(jbk) + b 1,j 00

l = (a0j + pj)bk + ( k - k)bj a

Thus the torsion tensor of this web has the form (7.6.18), and therefore the web is Grassmannizable. 0 Hence relations (7.7.12) and (7.7.15) are analytic conditions of the Grassmannizability of a (p + I)-web W (p + 1, p, q). These conditions are conditions for an almost Grassmann structure associated with an (p+ 1)-web to be locally integrable, and hence, for p > 2 and q > 2, they equivalent to the vanishing of the torsion tensor a = of the almost Grassmann structure. If p = 2, then a web becomes a three-web, and its condition of Grassmannizability is expressed not only in terms of the torsion tensor but also in terms of the curvature tensor of the three-web. Clearly this condition is connected with a differential neighborhood of third order (see Akivis and Shelekhov [AS 921, §3.4).

A similar situation occurs for q = 2. If p = q = 2, then we have a three-web on a four-dimensional manifold M. This web induces on M a CO(2, 2) -structure which is torsion-free, and all main properties of this structure are expressed in terms of its tensor of conformal curvature.

7.8

Grassmannizable d-Webs

1. In Sections 7.5-7.7 we studied the webs W (p + 1, p, q) formed by p + I foliations of codimension q on a manifold M of dimension pq. In this section

we consider the webs W (d, p, q), d > p + 1 on a manifold M of the same dimension pq.

As in Section 7.5 we define the foliations as and A a = p+q+ 1,. .., q+d, forming the web W (d, p, q) on the manifold M by the following completely integrable systems of equations:

WI=0, W'=0, P O

(7.8.1)

where i = p + 1 , . . . , p + q; a = 1 , . . . , p ; o = p + q + 1,...,q + d. Since the foliations as and a, are in general position, each of the subsystems of system (7.8.1) corresponding to p values of indexes a and a is linearly independent.

We take the forms w', or = 1,. .. , p, as co-basis forms of the manifold M. a Then other forms of system (7.8.1) are their linear combinations:

a=p+q+1,...,q+d,

315

Grassmannizable d- Webs

7.8

where all matrices (A ) are nonsingular. By a change of the co-bases in the foliations Aa and A we can reduce the last equations to the form go.?

P+q+1

"P-1Jp-1 +A'wJ+...+ . a2J 2

a1J I

a

(7.8.2)

p

2

1

J.

(7.8-3) P

where a = p + q + 2, ... , q + d. Now all the forms w' and w' admit only the a o concordant transformations of the form 'w' =

Ja

a

Jo

o

These transformations form the structural group GL(q) of the web W (d, p, q), and the matrices (A ) become tensors with respect to these transformations. as We denote these tensors by A and note that Af = b). as ap Besides being nonsingular, the tensors A satisfy an additional condition: as their differences A - A as well as some other of their combinations must be as ap nonsingular. These conditions follow from the fact that the foliations Af are in general position.

The structure equations of a web W(d,p,q) consist of equations (7.5.4) (which are the conditions of integrability of the systems of equations w' = 0, a 0), and the conditions of integrability of the systems a = 1, ... , p, and w p+q+1

of equations w' = 0, a = p + q + 2,...,q + d, which contain the differentials of a

the tensors A . We will not write the general form of all these equations. as 2. Let us consider a four-web W (4, 2, q) in more detail. For this web, equations (7.8.2) and (7.8.3) take the form

-w' = w' + w', -w' = 3

1

2

4

J1

w' 2

where the tensors A' and b - M must be nonsingular. The tensor M is called the basis afnor of the web W (4, 2, q) and is denoted by A. Let us clarify its geometric meaning.

Let x be an arbitrary point of a manifold M of dimension 2q carrying a web W (4, 2, q), and let T=(M) be the tangent space to M at this point. Denote

by T a = 1,2,3,4, the subspaces of TT(M) tangent to the leaves 1, of the web W passing through the point x. Under the projectivization of the tangent space TT(M) with center at the point x, the projectivizations of the subspaces T. are subspaces P, C P2q-1. Consider in the space T. (M) the frame {e;, e;} which is dual to the co-frame 1 2 {w',w'} and such that any vector l; E Ts(M) can be written in the form 1

2

t = w'(t)ej - w(t)e+. 2 2 1

1

316

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

Then e; E T1, e; E T2, + e; E T3, and e; E T4, and each of these e, systems of vectors form a basis in the corresponding subspace T,. The triples of subspaces (TI, T2, T3 ) and {T1, T2, T4 } define in the space T,, (M) two systems of transversal bivectors, and the triples {PI, P2, P3} and

{PI,P2iP4} define in the space p2,7-1 two Segre varieties, S(1, q - 1) and S(1, q - 1), which are the projectivizations of these systems of bivectors. Let

1 = E'e; be an arbitrary vector from T1. The bivector H = £I A 6, where 1:2 = ej, passes through ti, and this bivector H is transversal to the first Iel, triple of subspaces. Similarly the bivector H = E .j 2, where SI = passes through t;2 = ty'e;, and this bivector H is transversal to the second 2

triple of subspaces. This defines the linear transformation A: T1 -+ TI which can be written in the form £' = A'tyj (see Figure 6.1.2, p. 228). Thus, as was proved in Subsection 6.1.5, the operator A is the cross-ratio of the quadruple of subspaces P1, P2, P3, and P4 which also can be considered as the cross-ratio of four subspaces T1,T2iT3, and T4. Let rl be the eigenvector of the operator A: T1 -4 T, corresponding to an eigenvalue A. Since it = Ail, the transversal bivectors H and k, defined by the eigenvector rl, belong to the common transversal subspace of the quadruple of subspaces T. We arrive at the following theorem:

Theorem 7.8.1 At each point x E M, the basis affinor A =

of a fourweb W (4, 2, q) is the cross-ratio of four subspaces T1, T2, T3 and T4 which are tangent to the leaves of the web passing through the point x: A = (T1, T2; 2'3, T4 ). To the eigenvectors of the operator A, there corresponds the common transversal subspace of the quadruple of subspaces T.

3. Now we return to the study of the general webs W (d, p, q). Each subsystem of foliations Ar, , ... Arp+, , where r = {a, o) is the combined index taking the d values, 1 , ... , p, p+ q + 1, ... , q + d, forms a (p+ 1)-subweb on the manifold M. We denote this subweb by ITI, ... , Ty+I ]. The total number In the tangent space TT(M) each of of such subwebs is (p+1) = e these subwebs determines (see Section 7.5) the Segre cone SCz(p, q) and consequently the almost Grassmann structure AG(p - 1, p + q - 1) in the manifold M. Thus a system of almost Grassmann structures arises in the manifold Al. However, the most interesting case is indicated in the following definition (cf. Section 7.5):

Definition 7.8.2 A web W (d, p, q) is said to be almost Grassmannizable if all almost Grassmann structures defined by its (p + 1)-subwebs coincide. Theorem 7.8.1, proved in Subsection 7.8.2, implies that the web W (4, 2, q) is almost Grassmannizable if and only if its basis affinor is scalar: A = AI. In fact in this case all transversal bivectors of the subweb [1, 2,31 are also

7.8

Grassmannizable d- Webs

317

transversal bivectors of the subweb [1, 2, 4], and consequently for all of its other three-subwebs, [1, 3,41 and 12,3,4]. But the transversal bivectors constitute one of the families of the plane generators of the Segre cones SC(2, q) associated with the web W. Therefore, if A = AI, then the Segre cones defined by different subwebs of the web W(4, 2, q) coincide. The converse is obvious. In the general case we have the following result:

Theorem 7.8.3 For a web W (d, p, q) to be almost Grassmannizable, it is necessary and sufficient that all its basis affinors A be scalar, that is, proportional as

to the identity afnor I =

Proof. The almost Grassmann structures, determined on the manifold M by two (p + 1)-subwebs of a web W (d, p, q), coincide if and only if at each point

x E M the Segre cones located in the tangent space T=(M) and determined by the tangent subspaces to the leaves of these subwebs coincide. Consider the subwebs [1, ... , p, p + 1] and [1, ... , p, a] on M. As shown in Section 7.5, the Segre cone determined in T=(M) by the first subweb can be given by the equations (7.7.1). In a similar way we can show that the Segre cone determined in T=(M) by the second subweb can be given by the equations

zo=?.(asafli).

(7.8.4)

A; = ao a dj

(7.8.5)

If ao

then equations (7.8.4) take the form

zi = 111X an

(7.8.6)

and determine the same Segre cone as equations (7.7.1). Conversely, if equations (7.8.4) define the same Segre cone as equations (7.7.1), the tensors ao

have the form (7.8.5); that is, they are proportional to the identity tensor. It follows from Theorem 7.8.3 that for an almost Grassmannizable web W (d, p, q), equations (7.8.3) take the form

-w`=\w'+...+ A all a,p-1P-I

+w' P

a

Since the foliations ar are in general position, in the matrix I

...

1

1

A

...

A

1

P+q+2,1

p+q+2,p-l

..........................

A

q+d,l

...

A

q+d,p-l

1

I

(7.8.7)

318

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

composed of the coefficients on the right-hand sides of equations (7.8.2) and (7.8.7), all the minors of any order are different from zero. Denote an almost Grassmannizable web W (d, p, q) by AGW (d, p, q) and consider the almost Grassmann structure AG(p - 1, p + q - 1) associated with this web. If this structure is a-semiintegrable, then the web AGW (d, p, q) is called transversally geodesic (cf. Section 7.7). If the almost Grassmann structure AG(p - I, p + q - 1) is 0-semiintegrable, then the web AGW (d, p, q) is called isoclinic. A web AGW (d, p, q) is called Grassmannizable if it is equivalent to a Grass-

mann web GW (d, p, q) formed on the Grassmannian G(p - 1, p + q - 1) by d foliations AE whose structure has been described in Section 7.6. It follows from Theorem 7.7.4 that a web AGW (d, p, q) is Grassmannizable if and only if it is both isoclinic and transversally geodesic. However, these conditions of Grassmannizability can be weakened, since the following theorem holds:

Theorem 7.8.4 If d > p + 2 and q > 3, an almost Grassmannizable web AGW(d, p, q) is isoclinic.

Proof. We write the system of Pfaffian forms defining the foliation Ap+q+2 on the web AGW (d, p, q) in the form (7.8.8)

p+q+2

pP

II

In equation (7.8.8) we omitted the index p + q + 2 in the coefficients

A

p+q+2,a

and assumed that A is not necessarily equal to one. By the Fobenius theorem, P

the condition of complete integrability of system (7.8.8) can be written in the form

d w

w J A O'. 3 p+q+2

p+q+2

(7.8.9)

By virtue of formulas (7.5.4), exterior differentiation of (7.8.8) leads to the exterior quadratic equations

-d w '_- w jAw'+EdAAw'+EAa""'{{,twiAWk. p+q+2 j a a a

p+q+2

a

ZOO. (f'

(7.8.10)

Q

In these equations the coefficients A are relative invariants. This implies that a

dA = AA9+E Afp . Q

Substituting these expansions into (7.8.10), we obtain

-d w

p+q+2

w

p+q+2

a,Q

Aw k. A dk+Aa'k aaQ )wj a Q

a82

Notes

319

From condition (7.8.9) it follows that the second term on the right-hand side of the last equation must have the form - w A o`, where a` _ E µ'k wk. P+q+2

>

>

a a3a

Equating these two expressions and applying (7.8.8), we get

(A jbk + A a'k)wj A wk = E a,0aj3 aaft a 0 ap ap

a

A wk. R

Comparing the alternated coefficients and applying relations (7.5.5), we arrive at the equations

(a -

Rkb - jbk' + 'jk - pUkj.

aa

(7.8.11)

Setting a: = A in (7.8.11), we find that ki

aQU6k) a

By virtue of these equations, the alternation of relations (7.8.11) with respect to the indices j and k gives

ljbkl,

(7.8.12)

where we used the notation 1

AQ k

a Ak

From relations (7.8.12) and Theorem 7.7.3 it follows that if q > 3, then the (p + 1)-subweb [1,. .. , p, p + 1) of the web AGW (d, p, q) is isoclinic. This immediately implies the isoclinicity of the web AGW (d, p, q). Theorems 7.7.4 and 7.8.4 give another result:

Theorem 7.8.5 If d > p + 2 and q > 3 and a web W (d, p, q) is almost Grassmannizable and transversally geodesic, then it is Grassmannizable.

NOTES 7.1. Almost Grassmann manifolds were introduced in Hangan [Han 66) as a generalization of the Grassmannian G(m, n). Hangan [Han 66, 681 and T. Ishihara [I 721 studied mostly some special almost Grassmann manifolds, especially locally Grassmann manifolds. A. B. Goncharov [Gon 871 considered the almost Grassmann manifolds as generalized conformal structures. R. J. Baston [Bas 91a1 constructed a theory of a general class of structures, called almost Hermitian symmetric (AHS) structures,

320

7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES

which include conformal, projective, almost Grassmann, and quaternionic structures and for which the construction of the Cartan normal connection is possible. He constructed a tensor invariant for them and proved that its vanishing is equivalent to the structure being locally that of a Hermitian symmetric space. Subsequently Baston [Bas 91b) computed an algebra of differential invariants of the AHS structures. In Goncharov [Con 87] the AHS structures were studied from the point of view of cone structures (see Baston [Bas 91a, b] and Concharov [Gon 87] for further references on generalized conformal structures and their invariants). The local twistor theory of almost Grassmann structures was constructed in the recent paper by Bailey and Eastwood [BE 91] where the almost Grassmann structures were called paraconformal structures. In the another recent paper, Dhooghe [Dh 94] (see also Dhooghe (Dh 93]) considered the almost Grassmann structures (he called them Grassmannian structures) as subbundles of the second-order frame bundle and constructed a canonical normal connection for these structures. As we noted in Subsection 7.2.2, a pseudoconformal CO(2, 2)-structure is equivalent to an almost Grassmann structure AG(1,3). Since, as we saw in Chapter 5, fourdimensional conformal structures play an important role in general relativity, this provides a physical justification for studying the general almost Grassmann structures AG(m, n). In our exposition we defined the almost Grassmann structures geometrically following Akivis [A 80, 82a] (see also the paper Mikhailov (Mi 78] and the books Akivis and Shelekhov [AS 92], §8.3, and Goldberg [Go 88], §§2.1 and 2.2). 7.2-7.4. In Goldberg [Go 75a] (see also Goldberg (Go 881, §2.2, Eq. (2.2.38)) the expression (7.2.38) of the torsion tensor a'o k in a general (not specialized) frame was constructed for the first time. Using another method, Hangan [Han 80] deduced this expression again. The theorem similar to Theorem 7.3.1 was proved in Hangan [Han 80] in terms of Lie algebras. Mikhailov (Mi 72, 74, 77, 81) considered almost Grassmann structures and found their realizations in the frame of theory of two-webs.

Our structure equations (7.3.35) are very close to the structure equations in Dhooghe [Dh 94].

7.5. The basic equations of the theory of (p + 1)-webs W (p + 1, p, q) as well as the connection r were obtained in Goldberg [Go 73, 74a) (see also the book Goldberg (Go 88], Chapter 1). Theorem 7.5.6 can also be found in these papers and the book. 7.6. Grassmann webs GW (p + 1, p, q) for p = 2 were considered in Akivis [A 731, for p = 3 in Akivis and Goldberg [AG 74], and for any p in Goldberg [Go 75b]. For examples of Grassmann (and algebraic) webs GW (4, 2, q), see Goldberg [Go 82b]. 7.7. Transversally geodesic webs W (p + 1, p, q) were introduced in Goldberg [Go 73, 74a], and isoclinic webs W (p + 1, p, q) were introduced in Goldberg [Go 74b]. In connection with the theory of almost Grassmann structures, these webs were considered in Goldberg (Go 75a) and in Akivis [A 80, 82a]. The Grassmannizability problem was solved for webs W(3, 2, q) in Akivis [A 74] and for webs W (p + 1, p, q) (Theorem 7.7.5) in Akivis (A 80, 82a] and Goldberg [Go 82a].

7.8. The theory of webs W(4, 2, q) was constructed in Goldberg [Go 77, 80]. The geometric meaning of the basis affinor a' (Theorem 7.8.1) was also established there. A geometric definition of almost Grassmannizability for webs W (d, p, q) in the

case d > p + I was introduced in Akivis [A 83b]. But actually this kind of webs

Notes

321

was considered by Akivis (A 81) who gave the analytical characterization of these webs. Analytically a definition of almost Grassmannizable webs was given in Goldberg

[Go 84) (in this paper they were called scalar webs). Theorem 7.8.4 supplements Theorem 8.1.10 on almost Grassmann webs AGW (d, 2, q) from Goldberg [Go 881. It has appeared that the almost Grassmannizable webs are related to webs W (d, p, q) of maximum q-rank. S. S. Chern and P. A. Griffiths [CG 78] proved a geometric theorem which, in terms of almost Grassmannizable webs, can be formulated as follows: for d > 2p + 1 and p > 3, a web W (d, p, 2) of maximum 2-rank is almost Grassmannizable. J. B. Lit-

tle [Lit 89] extended this result to webs W (d, p, q). He proved that if q > 2 and d > q(p - 1) + 2, then every web W (d, p, q) of maximum q-rank is almost Grassmannizable. The last result (partially) gives an affirmative answer to a problem posed by V. V. Goldberg whether every web W (d, p, q) of maximum q-rank is almost Grassmannizable. It follows from Little's result mentioned above that a web W (d, 2, q) of maximum q-rank is almost Grassmannizable if q > 2 and d > q+2. Since the last two inequalities imply that d > 4, the case d = 4 should be studied separately. This case was considered earlier by V. V. Goldberg [Go 85) (see also Goldberg [Go 88), §8.3) who showed that if q > 2 and a web W(4, 2, q) admits at least one abelian equation, then the web is almost Grassmannizable. Recently V. V. Goldberg [Go 92) gave a description of almost Grassmannizable 6-webs AGW (6, 3, 2) of maximum 2-rank.

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Yano, K., and Y. Mutb, Sur la determination dune connexion conforme, Proc. Physico-Math. Soc. Japan 20 (1938), 267-279. (Zbl 18, 376.)

(YM 391

Yano, K., and Y. Muto, A projective treatment of a conformally connected manifold, Proc. Physico-Math. Soc. Japan 21 (1939), 270-286. (ZbI 21, 427.)

(YM 41a]

Yano, K., and Y. Muto, Sur la thdorie des hypersurfaces dans un espace a connexion conforme, Japan J. Math. 17 (1941), 229-288. (MR 7, 331; Zbl 60, 389.)

[YM 41b]

Yano, K., and Y. Muto, Sur la thdorie des espaces d connexion conforme normale et la gComdtrie conforme des espaces de Riemann, J. Fac. Sci. Imp. Univ. Tokyo Sect. 14 (1941), 117-169. (MR 3, 192; Zbl 63, A1045.)

[YM 42a]

Yano, K., and Y. Muto, Sur le thdorbme fondamental dons la geomdtrie conforme des soul-espaces riemanniens, Proc. PhysicoMath. Soc. Japan 24 (1942), 437-449. (MR 7, 332; Zbl 60, 389.)

[YM 42b]

Yano, K., and Y. Muto, On the fundamental theorem of conformal geometry, Tensor 5 (1942), 51-59 (Japanese). (MR 9, 203; Zbl 63, A 1045.)

[Z 78]

Zhogova, T. B., On the focal three-web of a two-parameter family of two-dimensional planes in P5, The Geometry of Imbedded Manifolds, Moskov. Gos. Ped. Inst., Moscow, 1978, 40-46 (Russian). (MR 82a:53010; Zbl 444:53009.)

(Z 79]

Zhogova, T. B., On a class of two-parameter families of twodimensional planes in Ps with a hexagonal focal three-web, The Geometry of Imbedded Manifolds, Moscow, 1979, 44-50 (Russian). (MR 82f:53026; Zbl 484:53004.)

(Zi 02]

Zindler, K., Liniengeometrie mit Anwendungen, G. J. Goschen, Leipzig, 1. Band, 1902, viii+380 pp. (Jbuch 33, 682-683); 2. Band, 1906, vii+248 pp. (Jbuch 37, 673-674.)

Symbols Frequently Used The list below contains many of the symbols whose meaning is usually fixed throughout the book.

I. Groups

G': G2(n):

structure group of CO(p, q)-structure and AG(p - 1, p + q - 1)-structure, 120, 166, 268, 273, 275,286,299-301 prolonged group of G, 128, 166, 274, 286, 299-301 group of admissible transformations of second-order

GL(n):

frames, 121 n-dimensional general linear group, 34, 120, 166, 254

G:

SU(n): T(n):

structural group of AG(p - 1, p + q - 1), 268 group of homotheties, 34, 120, 166, 254, 268 stationary subgroup (isotropy group) of a point x E V, 2, 34, 41, 44, 75, 80 n-dimensional orthogonal group, 2, 175 n-dimensional pseudoorthogonal group of signature (p, q), n = p + q, 7, 17, 120 group of conformal transformations of C", 7, 75 group of conformal transformations of Ca, 17, 127 fundamental group of projective transformations of the space P"+1, 150 multiplicative group of reals, 269 special n-dimensional linear group, 151, 166, 268 special n-dimensional orthogonal group, 120, 172 Lorentz group, 169 special n-dimensional pseudoorthogonal group of signature (p, q), n = p + q, 6, 7, 17, 120, 299 special n-dimensional unitary group, 172 n-dimensional group of parallel translations, 2, 34,

Z2:

127, 166, 264, 274 cyclic group of second order, 7

GL(p, q): H:

H, H= (V): O(n): O(p, q):

PO(n + 2, 1): PO(n + 2, q + 1): PSL(n + 2):

R': SL(n): SO(n): SO(1,3): SO(p,q):

355

SYMBOLS FREQUENTLY USED

356

II. Manifolds, Submanifolds, Spaces and Structures

AG(p-1,p+q- 1): A":

A: m AGW(d,p, q):

almost Grassmann structure on Ma9, 267 affine space of dimension n, 263, 302 m-quasiaffine space of dimension n, 263, 273 almost Grassmannizable d-web of codimension q on MDO: 318

algebraic d-web of codimension q on MDQ: 302 field of complex numbers, 170

AW(d,p,q): C:

C".:

isotropic cone with vertex at oo, 16 conformal space of dimension n, 3, 89 n-dimensional pseudoconformal space of index q,

C': C"v (C")Z and

(C,").

:

CO(p, q):

CPa and CPg: 'ipk). CT=(M): C=:

Cr"":

0(a) and o(p): O(k)(a) and E« and E0: G(m, n): G" (m, n): G(1, 3): GW (d, p, q):

H": G:

AI:

f2(m,n): fl (1, 3):

PN: (PN)*: papa,... a-: p`j: Q": QQ

R:

R":

yk)(f3)

16, 31, 100, 127, 141, 169, 221, 264 local conformal and pseudoconformal space at point x, 126, 132 pseudoconformal structure of signature (p, q), 120, 163 complex projective lines, 171 asymptotic cone of order k of fl(m, n) at p, 240 complexified tangent space at x E M, 170 isotropic cone with vertex at x, 6, 13, 100, 141, 164 osculating (m + ml)-sphere of VI, 85 distributions of two-dimensional elements on EQ and E0, 186 plane generators of Cpk), 241-243 isotropic fiber bundles and fiber bundles of plane generators of Segre cones: 165, 269 Grassmannian of m-dimensional subspaces in P", 221, 267 normalized Grassmann manifold, 253

Pliicker manifold of straight lines in P3, 19, 108, 221 Grassmann d-web of codimension q on M": 302, 306 n-dimensional hyperbolic (Lobachevsky) space, 66 Lie hyperquadric, 25 differentiable manifold, 119 image of the Grassmannian G(m, n), 223 Pliicker hyperquadric, 20 projective space of dimension N, 3, 7 projective space dual to pN, 224 Grassmann coordinates of P"' C P", 222 Pliicker coordinates of a straight line in p3, 20, 232 hyperquadric in Pn+1, 6, 66, 87 hyperquadric of index q in Pn+i, 16, 131, 141 field of real numbers, 19, 230 vector space of all ordered n-tuples of real numbers,

86,198,224

SYMBOLS FREQUENTLY USED

357

RPo and RP0: RP2: Rk(X):

n-dimensional Euclidean space, 1, 34, 66, 115, 264 n-dimensional pseudo-Euclidean space of index q, 14, 127, 166, 264 real projective lines, 165 real projective plane, 193 bundle of frames of order k over X, 31, 40, 44, 49, 75,

Sk:

120, 141, 145 k-dimensional elliptic space or k-dimensional sphere

R": R' n:

S"'1: SQ :

SA': SC. (p, q):

in C", 17, 66, 72, 145 hypersphere in C", 72 n-dimensional pseudoelliptic space of signature q, 262 Segre-affine space of dimension p, 264, 300 Segre cone of G(m, n) or AG(m, n) with vertex at x, 267

S(k, 1):

T=(M): T.kl (M): U°: V'": V'": VQ

V"-1: W (d, p, q):

n-dimensional Segre variety (Segrean), n = k + 1, 225 tangent space of M at x, 100, 120, 126, 268 osculating subspace of order k of M at x, 211, 239 normalized domain of the Grassmannian, 253 submanifold of dimension in, 73 tangentially degenerate submanifold of dimension m and rank r, 68, 108 n-dimensional Riemannian manifold of signature q, 140 hypersurface in a n-dimensional space, 31 d-web of codimension q on MDQ, 195, 270, 301

III. Tensors and Geometric Objects alk: aq or b..: a

a= {a' k}: (a, b): B;,,k:

b = (CJkg): CQ and CO:

filzl: 4iZi g:

torsion tensor of a web W (d, n, q), 303

second fundamental tensor of V"-1 C P", 38, 152, 208 torsion tensor of AG(p - 1, p + q - 1), 279 second structure tensor of AG(p - 1, p + q - 1): 286 Darboux tensor of a hypersurface, 152, 209 curvature tensor of a web W(d, n, q), 303 tensor of conformal curvature (Weyl tensor), 125, 142 subtensors of the tensor of conformal curvature of CO(2, 2)-structure, 168

second fundamental form of V"` in C" (or P"), 37 second fundamental forms of V'" in C", 78 fundamental form of CQ , CO(p, q)-structure and AG(p - 1, p + q - 1)-structure, and also first fundamental form of V'" C C" and V'" C CQ, 13, 18, 37, 74, 100, 119, 141, 148, 163

9;j: K(!; A ri):

fundamental (metric) tensor, 9, 73 conformal sectional curvature, 183

SYMBOLS FREQUENTLY USED

358

scalar curvature, 52, 133, 190 Ricci tensor, 51, 133, 190 torsion tensor of an affine connection, 147 curvature tensor of affine connection or Riemannian manifold, 51, 130, 133, 147

R R;i : R2 k:

Rj'kI: S = (a,6,c):

third (complete) structure tensor of AG(p - I, p + q - 1), 287

S and S0: Tjk:

structure subobjects of the structure object S of AG(p - I, p + q - 1), 298 deformation tensor, 47, 137

IV. Other Symbols d: 6:

6ij,6;i:

I'°: ry:

Id:

a: V: Va: v:

PT: a': ®: ®:

x: A:

oo:

(, ):

exterior differential, 11 symbol of differentiation with respect to fiber parameters, 122 Kronecker symbol, 11, 54, 91 affine connection on G°(m,n), 258 affine connection, 136, 147, 196 identity operator, 115, 179, 231 semidirect product, 2, 41, 128, 173, 264, 274, 286 operator of covariant differentiation, 76, 123 operator of covariant differentiation with respect to fiber parameters, 35, 75 normalizing mapping, 253 projectivization of T, 23, 221, 232 symbol of local isomorphism of groups, 2, 299 direct sum, 180 tensor product, 170 direct (Cartesian) product, 44, 166, 216, 225, 264, 273 symbol of equivalence of matrices or structures or spaces, 17, 256 exterior multiplication, 11 point at infinity, 3, 16 scalar product, 1, 32, 105, 126, 261

Author Index Adati, T., 135, 187, 323

Darboux, G., ix, 28, 70, 72, 331 Decuyper, M., 333 Delens, P. C., 70, 331 Demoulin, A., 28, 70, 72, 331 Deszcz, R., 115, 331 Dhooghe, P. F., 320, 331 Dieudonne, J., 1, 19, 332 Dobromyslov, V. A., 263, 265, 266, 332 Dubrovin, B. A., 28, 130, 134, 332 Dupin, C., 72, 332

Akivis, M. A., x, xiii, 6,23, 24, 33, 38, 67, 68, 71, 72, 87, 90, 97, 99, 107, 110, 111, 116, 117, 152, 153, 156, 161, 195, 196, 213, 218,

219, 221, 239, 244, 249, 265, 270, 304, 312, 314, 320, 321,

323-325

Alekseevskii, D. V., 2, 218, 325, 326 Atiyah, M. F., 163, 180, 217, 218, 326

Backes, F., 28, 72, 326 Bailey, T. N., 320, 326 Garner, M., 115, 326 Barrett, J. W., 120, 163, 218, 326 Baston, R. J., 319, 320, 326, 327 Berwald, L., ix, 70, 327 Blair, D. F., 115, 327

Eastwood, M. G., 320, 326 Einstein, A., 15, 219, 332, 343 Eisenhart, L. P., 70, 129, 130, 134, 136, 150, 160, 161, 177, 332

Euler, L., 28, 332 Fialkow, A., 115, 116, 333 Finikov, S. P., 112, 113, 117, 265, 333 Finzi, A., 71, 161, 333 Fomenko, A. T., 28, 130, 134, 332 Fubini, G., ix, 70, 72, 160, 333 Fuhrman, A., 265, 334

Blaschke, W., ix, 28, 29, 63, 70, 72, 94,

115, 327

Bol, G., 117, 195, 327, 333 Bompiani, E., 111, 117, 327 Bryant, R. L., xiii, 33, 115, 124, 327 Bubyakin, I. V., 251, 265, 328

Buchner, K., 116, 346 Burali-Forti, C., 52, 328 Bushmanova, G. V., 115, 328

Gantmacher, F. R., 63, 334 Gardner, R. B., xiii, 14, 33, 124, 131, 291, 327, 334

Carmo, do M., 71, 72, 115, 162, 328 Cartan, It., ix, 14, 24, 28, 72, 98, 128, 150, 160, 161, 219, 291, 328, 329 Cartan, H., 35, 329 Cayley, A., 28, 329 tech, E., ix, 72, 333 Cecil, T. E., 29, 72, 330 Chandrasekhar, S., 140, 170, 189, 191,

Geidelman, R. M., x, 72, 244, 265, 334 Gelfand, 1. M., 250, 334 Gheysens, L., 116, 334, 335 Gibbons, G. W., 120, 163, 218, 326 Gindikin, S. C., 163, 217, 335

Goldberg, V. V., xiii, 6, 23, 33, 38, 67, 68, 72, 90, 94, 97, 99, 107, 110,

117, 152, 153, 156, 187, 198202, 213, 219, 239, 249, 265, 268, 270, 281, 303, 304, 312,

202, 206, 207, 218, 219, 330 Chebysheva, B. P., 265, 325

Chen, B. Y., 70, 72, 115, 116, 330, 353,

313, 320, 321, 325, 335, 336

Goldsmith, H. 1.., xiii, 33, 124, 327 Goncharov, A. B., 319, 320, 336

354

Chern, S. S., xiii, 33, 70, 72, 124, 321, 327,

Graev, M. 1., 218, 250, 325, 326, 334 Grassmann, H., 28, 29, 336

330, 331

Chupakhin, N. P., 244, 344 Cicco, de J., 115, 339

Grifiths, Ph. A., xiii, 33, 72, 124, 161, 239, 265, 321, 327, 331, 336 Grincevicius, K. 1., 244

Dajczer, M., 71, 72, 115, 162, 328 359

AUTHOR INDEX

360 Haantjes, J., 28, 71, 115, 160, 337, 348 Haimovici, A., 115, 337

Hangan, T., 268, 319, 320, 337 Harris, J., 161, 224, 239, 265, 336, 337 Hitchin, N. L., 163, 180, 217, 218, 326 Hlavaty, V., ix, 117, 161, 337 Hodge, W. V. D., 218, 221, 223, 265, 338 Houh, C. S., 116, 338

Hsiung, S. S., 70, 338 Hua, L. K., 265, 338 Huggett, S. A., 218, 338

Matsumoto, M., 71, 72, 343 Mercuri, F., 71, 162, 328 Michor, P., V. xiii, 340 Mikhailov, Yu. 1., 268, 320, 343 Minkowski, H., 28, 332, 343 Miyaoka, R., 72, 343, 344 Miyazawa, T., 135, 187, 323 Mizin, A. G., 244, 344 Monge, G., 28, 342 Mugridge, L. R., 70, 338 Musso, E., 72, 338 MOto, Y., 71, 115, 160, 344, 354

Ishihara, S., 115, 354 Ishihara, T., 319, 338

Jensen, C., 72, 338

Newman, E. T., 170, 218, 344 Nishikawa, S., 71, 344 Nomizu, K., xiii, 46, 94, 131, 147, 258, 340

Karapetyan, S. E., 233, 236, 244, 265,

Norden, A. P., x, 115, 132, 136, 137, 149, 161, 265, 328, 344

338, 339 Kasner, E., 115, 339 Kerr, R. P., 219, 339 Klein, F., x, 7, 19, 20, 28, 29, 54, 221, 230, 339, 340

Ogiue, K., 160, 345 Ozawa, T., 72, 344

Kaplenko, A. F., 265, 338

Klekovkin, G. A., 134, 219, 340 Kobayashi, S., xiii, 46, 94, 120, 121, 128, 131, 147, 160, 258, 340

K61at, I., xiii, 340 Kiinigs, G., 117, 340 Konnov, V. V., 161, 162, 218, 219, 325, 340

Korovin, V. 1., 265, 341 Kossowski, M., 101, 341 Koetrikina, L. P., 265, 347 Kovantsov, N. 1., 113, 117, 341 Kowalski, 0., 115, 341 Krivonosov, L. N., 72, 341 Kruglyakov, L. Z., 244, 341 Kulkarni, R., 160, 341

Lagrange, R., 115, 341 Laptev, G. F., 36, 71, 77, 116, 121, 124, 147, 153, 156, 258, 341, 342

LeBrun, C., 218, 342 Leichtweiss, K., 221, 265, 342 Lichnerowicz, A., 147, 258, 342 Lie, S., 27, 342

Liebmann, H., 115, 342 Liouville, J., 2, 28, 342 Little, J. B., 321, 342 Lorentz, H. A., 332, 343 Lumiste, Yu. G., 71, 116, 342

Nordstrom, G., 219, 344 Novikov, S. P., 28, 130, 134, 332

Pedoe, D., 221, 223, 265, 338 Pendl, A., 115, 345 Penrose, R., 163, 165, 170, 189, 191, 217-219, 344, 345 Perepelkine, D., 115, 345

Perry, M. J., 120, 163, 218, 326 Petrescu, S., 115, 345 Petrov, A. Z., 191, 219, 262, 345 Pinkall, U., 72, 346 Pirani, F. A. E., 191, 219, 346 Pliicker, J., 19, 28, 29, 346 Poincark, H., 28, 346 Ponomarev, V. A., 265, 338 Pope, C. N., 120, 163, 218, 326 Rawnsley, J. H., 218, 346 Reissner, H., 219, 346 Rham, G. de, 218, 346 Ribaucour, A., ix, 70, 72 Rindler, W., 165, 189, 191, 218, 219, 345 Room, T. G., 265, 346 Rosca, R., 116, 346 Rosenfeld, B. A., x, 1, 7, 28, 115-117, 151, 226, 229, 265, 338, 346, 347 Roth, L., 265, 349 Rouxel, B., 116, 347 Rubak, P., 120, 163, 218, 326 Ryan, P., 72, 330

Maeda, J., 115, 343 Maeda, Y., 71, 344

Saksteder, R., 72, 348

Manin, Yu., 1. 217, 218, 343

Sasaki, S., ix, x, 115, 160, 161, 348

AUTHOR INDEX Sasaki, T., 162, 348 Scheffers, G., 27, 342

Schiemankgk, C., 49, 52, 115, 348 Schild, A., 219, 339 Schouten, J. A., ix, x, 28, 52, 160, 161,

348

Schubarth, E., 115, 348 Schwarzschild, K., 219, 349 Segre, C., 28, 349 Semple, J. G., 265, 349 Seven, F., 224, 265, 349 Shcherbakov, N. R., 244, 344 Shcherbakov, R. N., 244, 349 Shelekhov, A. M., 195, 196, 270, 304, 312, 314, 320, 325 Singer, 1., 163, 180, 217, 218, 326

SiovSk, J., xiii, 340 Solodovnikov, A. S., 2, 326 Stepanova, G. B., 265, 347

Sternberg, S., xiii, 125, 160, 72, 349 Struik, D. J., 28, 52, 160, 161, 348 Sulanke, R., 49, 52, 71, 72, 115, 348-350

Takasu, T., ix, 70, 71, 115, 350 Thomas, J. M., ix, 160, 350 Thomas, T. Y., ix, 160, 350 Thomsen, G., ix, 70, 71, 115, 350 Thorbergsson, G., 72, 346, 350 Tikhonov, V. A., 72, 115, 350, 351 Timoshenko, T. A., 116, 347

361

Tod, K. P., 218, 338 Tresse, A., 70, 350

Vasilyev, A. M., 24, 351 Vedernikov, V. I., x, 72, 115, 116, 161, 351 Verbitsky, L. L., x, 115, 116, 160, 161, 351 Verheyen, P., 115, 116, 334, 335, 351 Verstraelen, L., 115, 116, 334, 335, 351 Vessiot, E., ix, 70, 351, 352

Vinberg, E. B., 2, 326 Vlasov, A. K., 233, 265, 352 Voss, A., ix, 70, 352 Vranceanu, G., 160, 352 Wells, R. 0., Jr., 177, 218, 352 Weyl, H., ix, 160, 161, 332, 343, 352 Wilczynski, E. J., ix

Wolf, J. A., 14, 352 Wong, Y. C., 71, 221, 352 Woude, W. van der, 115, 353 Yano, K., ix, 28, 72, 115, 160, 330, 353, 354

Yuchtina, T. 1., 265, 347

Zamakhovskii, M. P., 116, 347 Zayatuev, B. V., 219, 325 Zhogova, T. B., 245, 265, 354 Zindler, K., 117, 354

Subject Index variety 11(1,3), 109-114, 117, 232 variety 11(1,4), 232, 233

Abelian equation, 321 Absolute, 18, 66, 127, 264

variety 11(2,4), 235, 236

parallelism, 148 tensor, 36, 40

variety n(2,5), 245, 250

Adapted frames, 164, 165, 188, 218, 274

Algebraizable webs, 302 Almost Grassmann manifold, 319

Admissible transformation of

adapted frames, 135, 188, 214, 274 frames of first order, 148 m-pair, 254

Almost Grassmann structure, ix-xii, 221, 267-300, 320 associated with web, 304, 305, 312,

316, 318

integrable, 270 semiintegrable, xii, 270, 292, 300 structure equations of, xii, 274-276, 285, 300, 320

second order, 121, 273, 284

third order, 125, 283, 284

reduced family of second-order frames, 279 Affine connection, 132, 133, 137, 141,

structure group of, 268

torsion tensor of, 274, 279, 287, 293 subtensors of, 293

147-150, 252, 264

associated with web, 196,309, 311 curvature tensor of, 136, 147, 150 connection forms of, 136, 147 P-, 258-264

Almost Grassmannizable web, 316-320, 321 Almost liermitian symmetric structure, 319

o-plane(s), 164, 166, 171, 184, 185, 196, 215, 218

on normalized submanifold, 115,

o-semifiat conformal structure, 183 a-semiintegrable almost Grassmann structure, 270, 294-299, 309,

141

torsion-free, 115, 148, 258 torsion tensor of, 147 coordinate system, ix differential geometry, ix

318

a-semiintegrable CO(2,2)-structure, 186 o-semirecurrent CO(2, 2)-structure, 187

space, 117, 150, 161, 263, 264, 302,

a-submani fold (s), 215

304

a-subspace(s), 242, 243

transformation(s), 147

Alternation, 40, 275, 285, 287, 292, 293,

Weyl connection, 133, 149, 150, 205

319

Affinor, 62

Anti-involutive operator, 180 Anti-self-conjugate subspaces, 182 Anti-self-dual

Burali-Forti, 52, 113 symmetric, 52, 107 AG(m, n)-structure, 267, 301 AG(1,3)-structure, 270, 300, 301 Algebraic d-web, 302 geometry, 224, 225 of Grassmannian, 265

asymptotic CO(2, 2)-structure, 215 CO(1,3)-structure, 218 CO(2, 2)-structure, 197, 199 part of Weyl tensor, 217 structure, 183, 185, 186 subspace, 180, 182, 183, 218

variety A(m,n), 20-23, 28, 223-225, 238-244,267-271,286,290,291,

Apolarity, 51, 78, 85, 92 Apolar tensors, 38, 39, 58, 61, 66 Asymptotic

300

cone on, 225 manifold, 240

cone(s),

363

SUBJECT INDEX

364

of algebraic variety fl(m, n), 240-246, 250, 291 of CO(2, 2)-structure, 208, 209, 214 of Darboux hyperquadric, 104, 106 filtration of, 240 of hyperquadric, 17-18 of hypersurface, 208, 209 of order k, 240, 242, 243

of tangent space, 178 0-plane(s), 164, 166, 171, 184, 185, 196, 215, 218

l-semiflat conformal structure, 183 fl-semiintegrable almost Grassmann structure, 270, 294, 299, 310, 318

#-semi integrable CO(2, 2)-structure, 186, 194, 202

of second order, 240, 241, 243, 245, 250

of submanifold, 106 of third order, 241, 242, 245-247, 250

CO(2,2)-structure, 208, 209, 210 semiflat, 215, 216 conformal connection, 157 conformal structure, 150, 153, 161 flat, 155, 158, 161 direction(s) of second order, 250

of submanifold, 245, 246, 250 of variety f1, 239, 240, 244 form of second order, 152 line(s) of order k, 244 of submanifold, 250 of surface, 110, 111 of third order, 246 of variety fl, 244 Axially symmetric metric, 202, 204, 206, 207

fl-semirecurrent CO(2,2)-structure, 187

$-submanifold, 215 fl-subspace, 243

Bianchi equations, 289, 290 Bijective correspondence, 171 Bijective mapping, 245, 267, 270 Biquadratic algebraic submanifold, 155 Bisecant variety, 241 Bivector(s), 178, 183, 259 indices, 179, 181 isotropic, 184 space, 178, 180

Bundle of central hyperspheres, 85 frame, 11, 176 invariant, of 2nd fundamental forms, 78, 90 of isotropic frames, 204 of m-subspaces, 243, 273, 302, 306 of normal hyperspheres, 82

of second fundamental forms, 79 of straight lines, 22, 114 of subspaces, 224, 231 tangent, 46, 75, 238

Base

of bundle of first-order frames, 75,

of tangent hyperspheres, 78

120

of fibration, 302 forms, 33, 238 of frame fiber bundle, 11, 75, 121 hypersurface, 151 frame fiber bundle, 11 of isotropic fiber bundle, 164, 171,

176

parameters, 122 variables, 121 Basis affinor, 315, 316, 320 Form(s), 74

of affine connection, 136 of almost Grassmann structure, 274 of frame bundle of Grassmannian, 253, 271 of Grassmannian, 238, 253, 254, 257

of hypersurface, 34, 35, 46, 48, 49 of manifold, 177 of subspace, 229

Canal hypersurface, xi, 57, 60, 61 submanifold, 145 surface, ix, 70, 72 Canonical

form, 262 frame, 70 normal connection, 320

Cartan's lemma, 33 normal connection, 128, 320 number, 99 test, 99, 250 variety, 98, 213, 215

Cartesian coordinates, 127, 159 coordinate system, ix, 3, 159 Center of bundle of subspaces, 232, 243 hypersphere, 2, 4 inversion, 3

SUBJECT INDEX pencil, 251, 252 projectivization, 225 Central rn-sphere, 78, 79, 145, 1.16 tangent complex, 112 tangent hypersphere, 40, 41, 45, 47, 61, 66, 78, 79, 81, 84

Character, 99 Characteristic(s), 57, 155, 157 cone, 79 equation, 54 polynomial, 181 straight line of subspace, 247-252

straight line of torse, 247, 248 subspace(s), 251 Classical differential geometry, ix, 70 Closed

form equation(s), 71, 198, 200, 201, 215, 287

365

linear, 21, 107, 109-112 plane, 189, 231 projective line, 171 representation, 173, 218 Riemannian manifold, 218 of straight lines, 108-113, 117 of general type, 109, 111 linear, 21-23, 28, 109 special, 109-111

of subspaces, 244 of tangents, 117 transformation, 170 of 2-subspaces, 245, 250-252, 265

Complexification, xi, 217, 270 Complexified space, 180

Complexified tangent space, 170-172 Complex projective line(s), 171, 172 Cone(s) asymptotic, see Asymptotic cone

system, 99, 273 Co-basis, 35

of asymptotic directions of variety

forms, 304, 306 Co-frame, 120, 274 Commuting of lnors, 62 Compact differentiable manifold, 3 pseudoconformal space, 16 Compactification, 3, 16, 64, 126, 127, 204,

characteristic, 79

286

Compactified tangent space, 127, 300 Complementary subspaces, 223, 226, 242, 245

Complete fundamental object, 71, 77, 87 invariant normalization, 82 structure object, 281, 287, 291, 298, 299, 301

Completely integrable system, 11, 56, 60, 64, 71, 148, 186, 271, 274, 294, 296, 312

isotropic submanifolds, xii, 183, 185, 186, 194, 211-215

Complex, 28, 29 conjugate curvature tensors, 174-176 directions, 188

eigenvalues, 230 fiber bundles, 171 forms, 170 generators, 171

subspaces, 180, 182 transversals, 230

coordinates, 2 form(s), 173, 174 generator(s), 270

fl, 240 degenerate, 54

field of, x imaginary, 6 isotropic, see Isotropic cone of second order, x, 53, 79, 80, 120, 141, 195, 243, 299, 301

plane generator of, 195 structure, 320 Conformal connection, 161, 204 correspondence, 64, 65 deformation, 116, 161 differential geometry, ix, x of submanifolds, x, 71, 115 differential invariant, ix, 70 flatness, 209 geometry, ix, x, 70-72, 115, 121 invariance, 160 mapping, 131, 145, 161 model of H", 66 model of S", 66 moving frame, 8, 9, 66, 87 of CO(p, q)-structure, 126 of hypersurface, 31 of submanifold, 73, 90 rigidity, 46, 72 sectional curvature, 184

semiflatness, 209 space, ix-xi, 3, 8, 25, 28, 31, 33, 45, 87, 116, 141, 144, 150, 161, 221, 264 of Lorentzian signature, xi projective interpretation of, 28

proper, 16

SUBJECT INDEX

366

structure, ix, x, xii, 37, 119, 141,

144, 148, 150, 153, 160, 161,

202, 208, 209, 217, 299, 320

of third class, 236

of 2-subspaces, 245-247

Conjugate

conformally flat, xii, 175 curvature forms of, 126

bundles, 164

CO(p,q), 120, 125, 126, 132, 141,

net, 67, 213, 249 points, 69

142, 146, 284, 301

on four-dimensional manifold, 175 on hypersurface, 153, 161 realization of, xii on submanifold, 90 ultrahyperbolic, 120, 163 theory of spheres, x

transformation(s), 2, 3, 6, 8, 14, 28, 46, 49, 52, 70, 85, 87, 119, 136, 140, 160

group of, 3, 6, 28 of pseudo- Riemannian metric, 136

of Riemannian metric, 136, 137, 140

Conformally connected space, ix

equivalent hypersurfaces, 46, 49 equivalent Riemannian metrics, 132, 189

Euclidean space, 161

flat conformal structure, xii, 175, 300 CO(1,3)-structure, 175, 218 CO(2,2)-structure, 169, 185, 194, 209, 215, 218

hypersurface, 217 submanifold(s), 144 invariant form, ix geodesics, 161 metric, 116 operator, 179, 218 properties, 161 tensor, 116 Lorentzian structure, 120 recurrent structure, 135, 136 semiflat CO(1,3)-structure, 175 semiflat CO(2, 2)-structure, 209 symmetric invariant connection, 136 Congruence(s), 28, 29 of circles, ix

normal, 64 of hyperapheres, 72 of isotropic geodesics, 189, 191-193 pair of, 117 of pairs of points, 115 quadratic, 117

of spheres, 72 of straight lines, 108, 109, 113, 117 of subspaces, 244

directions, 67

subspace(s), 260

Connected

component of identity, 10 conjugate, domain, 86, 87 hypersurface, 45 submanifold, 73 Connection, affine, see Affine connection Connection forms of

affine connection, 136, 147 Riemannian connection, 136-140

Connection r,, 258-264 curvature tensor of, 258 Contact hypersphere, 54, 56 Coordinate(s) of bivector, 184, 259 Cartesian, 127, 259

Grassmann, 222, 223, 232, 234, 235 homogeneous, see Homogeneous coordinates of linear subspace, 28 nonhomogeneous, see Nonhomogeneous coordinates PlOcker, 20

projective of point, 227 simplex, 224

of straight line, 28 tangential, 236 transformation, xi Correlation of projective space, 224 Covariant derivative, 50, 148 of curvature tensor, 187 differential, 259

differentiation, 123, 136, 148, 260 Covariantly constant fundamental tensor, 260, 262 tensor, 135 Covector, 133 form, 131 CO(1,3)-structure, 163, 169-172, 174-176, 180, 183, 188-193, 202-204, 217-219, 270 CO(2, 2)-structure, 163-169, 172, 175, 176, 180,194,195,217-219,270,301 CO(4,0)-structure, 163, 172, 175, 176, 180, 183, 189, 190, 217-219, 270 Cross-ratio of four points, 229

SUBJECT INDEX of four subspaces, 265, 316

of quadruple of plane generators, 231

of two m-pairs, 228-231, 256, 257, 260, 265

of two points and two hyperplanes, 229 Cross-section of bundle of frames, 204 Cubic cone of directions, 245 curve, 246, 247 Darboux form, 153, 212, 214 equation, 236

hypersurface, 250 submanifold, 250 Curvature form(s) of almost Grassmann structure, 292

connection r,, 258 CO(p,q)-structure, 132 isotropic fiber bundle, 169, 182, 183 Weyl connection, 51 Curvature lines, 52, 53, 62-64, 70, 89, 92 isothermic, ix, 70

spherical, ix, 70 Curvature object of CO(p, q)-structure, 126 Curvature tensor, 129 of affine connection, 51, 133, 136, 147, 303

of almost Grassmann structure, 300 of connection C", 258, 259, 262 of CO(1,3)-structure, 205 of CO(2,2)-structure, 301

of empty space, 190

of fiber bundles E. and E8, 168, 174, 183, 191-193, 300 of four-dimensional Riemannian manifold, 190 of Grassmann (p + 1)-web, 307, 309 of isotropic bundle, 168, 174, 183 of (p + 1)-web, 303, 304

of Riemannian manifold, 259 of three-web, 196

367

tensor, 152, 154, 157, 209 Deformation conformal, 116, 161 of submanifolds, 72 tensor, 47, 137 Degenerate cone, 54 congruence, 114

of first kind, 114 of second kind, 114 form, 102

hypersphere, 64 inversion, 66 linear congruence, 22 m-pair, 227 null-pair, 193 Degree of Segre variety, 225, 245 variety fl(1,4), 232 variety 11(2, 5), 245 Derivational equations, 70, 71, 116 Determinantal variety, 240, 242, 265 dimension of, 242 plane generators of, 242

Developable ruled surface, 109

submanifold, 247, see also Torse surface(s), 24, 68, 69, 108, 1 t 1-113, 213, 215, 250 Differentiable correspondence, 253

field of Segre cones, 267, 268, 300 manifold, x, 20, 27, 31, 35, 119, 222-224, 267, 274 compact, 3

structure equations of, 121 mapping(s), 20 submanifold, 253 Differential covariant, 259

of Weyl connection, 51, 133, 150 Curve, 8, 13, 42, 43, 70, 161 cubic, 246, 247 integral, 188, 189 isotropic, 24, 102, 109 Curvilinear coordinates, 86, 119, 120 Curvilinear two-web, 94 Cuspidal edge, 112 Cyclic group, 7

equation(s) of distribution, 202

Darboux form, 212, 214 hyperquadric, 16, 26, 104, 105 asymptotic cone of, 106

form, xiii, 35, 120, 125 exterior, x, 71 -geometric structure, 128 geometry, ix, xiii affine, ix classical, ix

mapping, 6, 10, 13, 17, 18, 24-26, 28, 66, 67, 87-90, 108, 117

geodesics, 138, 139 geometric object, 36 hypersurface, 46 normalizing mapping, 253 relative invariant, 50, 82 submanifold, 210 tensor, 39, 77, 125, 130, 168 three-web, 195

SUBJECT INDEX

368

conformal, ix of Grassmannians, 221, 236, 265 of m-spheres, 115

proper, 253

simply connected, 86, 87 Double congruence of isotropic geodesics,

projective, ix

of submanifold, 77 of submanifold of spheres, ix invariant, 70 operator V, 123 prolongation, 128, 286, 303 Differentiation with respect to fiber parameters, 122, 176 Dilation, 160 Dimension of asymptotic cone of fl, 241 determinantal variety, 242 Grassmannian, 221, 238, 252, 271

kth osculating subspace to fl(m,n), 239 plane generator of cone, 242 plane generator of Segre variety, 264

191-193

family of isotropic lines, 102 line, 229 principal directions, 219

principal distribution, 189, 191-193, 206 quadric, 155, 157 root(s), 189, 191, 192

Dual

frame, 237, 311, 315 projective space, 224, 235, 236, 253, 272

space, 224, 235, 236 Dupin's cyclide, 58, 72 Dupin's submanifold, 72 d-web, 301, 314, 316

quasiaffine space, 273 Segre cone, 240

Eigendirection(s), 54, 108, 181

Segre variety, 225

Eigensubspace, 180-183

tangent subspace to 0(m, n), 238,

Eigenvalue(s), 54, 91, 181, 229-231

245, 246 variety fl(1,4), 232

Direction(s)

Eigenvector of operator, 316

Einstein equation, 190, 206, 219 Einstein space, 163, 190, 218

asymptotic, see Asymptotic direction(s)

of type D,

conjugate, 67 focal, 246, 247

of type N, 192, 218 of type 0, 218

isotropic, see Isotropic direction(s)

of type 1, 191, 218 of type 11, 191, 192, 218 of type 111, 192, 218

optical, 219 principal, see Principal direction(s) Direct product of projective spaces, 225, 264,

Directrices of linear congruence, 22 Directrix of Segre cone, 232, 271 Discriminant of quadratic form, 176 Discriminant tensor, 177, 180 Distribution(s), 94, 251 O(Q) and A (O), 194, 197

double principal, 189, 191-193, 206 holonomic, 94 horizontal, 147 integrable, 198, 200-202 involutive, 94, 95 of plane elements, 269, 298

of two-dimensional elements, 186

Domain connected, 86, 87 external, 102 homogeneous, 262 internal, 102

192, 218

Elation, 160 Elliptic congruence, 113 hypersurface, 208 linear congruence, 22 point, 111 space, 66 submanifold, 111 Embedding, 225, 264 Empty space, 190 curvature tensor of, 190 Energy-momentum tensor, 190 Envelope of family of hyperquadrics, 155-159 of hyperspheres, 57, 58, 60 of spheres, 71, 98, 116, 145 Envelope of 0-focal family, 244, 246 Equation(s), of asymptotic cone, 241, 245 characteristic, 54

normalized, 253, 254, 257, 260-264 open, 253, 260

of cone C:, 195, 196 of embedding, 225

of principal directions, 53

of geodesics, 138, 140, 189

SUBJECT INDEX

369

of hyperquadric, 24, 261 Maurer-Cartan, 151 of Segre cone, 268 of Segre variety, 230

of Grassmannian, 273 of normalized Grassmannian, 254, 257

of second order frame bundle, 276 of frame fiber bundle, I1

of Vlasov congruence, 236 Equiaffine Weyl connection, 134 Ricci tensor of, 134

Equivalent webs, 302 Erlanger program, 28 Euclidean geometry, 72 metric, 150 plane, 28 space, xi, 1-3, 5, 7, 8, 15, 28, 33, 66, 70, 115, 117, 264, 265 multidimensional, 70, 115, 161 three-dimensional, 28, 70, 117

of isotropic fiber bundle, 164, 171 parameters, 77, 80, 143, 176, 204, 274, 308

variables, 121 Fibering(s), 237

Fibration(s), 302

isotropic, 218 twistor, 218

Field of asymptotic directions, 250 of cones, x

of second order, 120, 299 of geometric objects, 71

Existence of geodesics, 138

totally focal congruence, 250

Exterior derivative, 272, 273 differential forms method, x, 71 differentiation, 45, 50, 64, 75-77, 82, 99,121,143,151,153,198,257, 281, 303, 307, 318

of Segre cones, 271, 300

tensor, xiii theory, xi vector, xiii, 148 Filtration of asymptotic cones, 240 Finite type C-structure, 125, 128

First fundamental tensor of hypersurface,

product, 19, 177

46, 51

quadratic form(s), 169, 178 External domain, 102

integral(s), 12 order frames, 59, 66 bundle of, 120, 121, 126, 128, 131 prolongation of group, 173 structure tensor of almost Grassmann

Family of a-planes, 166 #-planes, 166

frames, 14, 41, 44, 268 of Segre cone, 268 hyperquadrics, 155-159 hyperspheres, 26, 45 hypereurfaces, 57, 58,

plane generators, 168, 224, 242, 267 planes, 23

structure, 279, 286, 291, 300

structure tensors of bundles E. and

Ep, 306 Five-dimensional projective space, 20, 169, 216, 217 submanifold, 2,50 Five-parameter family of 2-subspaces, 250

Flat affine connection 1", 264 asymptotic conformal structure, 155,

point frames, 253, 263 projective frames, 150, 236, 237, 260

158, 161, 162

spheres, 71

CO(2, 2)-structure, 186, 210, 216, 217

straight lines, 22, 244 submanifolds, 240, 241 tangential frames, 237 torses, 250

metric conformal structure, 162 Flatness, 161

2-subspaces, 244-252

Focal

Fiber bundle(s), 243 of normalized Grassmannian, 254, 257

of plane generators of Segre cone, 269, 300 form(s), 121, 279, 308

of fourth order frame bundle, 285

isotropic distribution, 208

direction(s), 246, 247 family, 68-70 of m-planes, 265 of 2-subspaces, 251 point(s), 246 submanifold, 108 Focus

of generator, 68

370

SUBJECT INDEX

of straight line, 68 Foliation(s), 94, 188, 195-198, 202, 270, 271, 302, 304, 306, 307, 310, 314

isotropic, 186-188, 202 leaf of, 271

one-parameter, 94 Form(s) cubic Darboux, 153, 212, 214 fiber, see Fiber form(s) horizontal, 11, 238 linear, 213 principal, 135 third fundamental, 212, 213 Four-dimensional cone, 250 conformal structure, 163, 218, 219,

320

CO(1,3)-structure, 163, 174-176, 169-173 CO(2, 2)-structure, 163, 164, 166, 175, 176, 270, 314 CO(4, 0)-structure, 163, 175, 176 hypersurface,208-217 manifold, 195, 217, 232, 272, 314 projective space, 250

pseudoconformal structure, xi, xii quadric, 20

Riemannian manifold, xi, 140, 177, 190

curvature tensor of, 190 three-web, 194, 195 Fourth order geometric object, 37, 39

Four-web, 197, 219, 315, 320 of maximum rank, 219 Frame fiber bundle, 11, 176

of almost Grassmann structure of first order, 274, 286, 291 of fourth order, 282 of second order, 286, 291, 320 of third order, 285, 286, 292 base forms of, 11 base of, 11

in C.1, 128 fiber of, 11

of Grassmannian, 253 basis forms of, 253 R1 (M), 120, 121, 126, 128, 147 R2(M), 121, 123, 128, 147, 204 R3(M), 121, 124

RI(Vm), 75, 80, 141, 145 7Z2(Vm), 80, 145 7Z1(V"-1), 31, 33, 40, 41, 151, J55

R2(V"-1), 41 7Z3(Vn-1),44,49 Frenet equations, x, 71, 116

FYobenius theorem, 12, 249, 318 Fundamental form(s), 169

of pseudo- Euclidean space, 14 of variety 11(m, n), 239, 240 group, ix of conformal space, 7 of Lie sphere geometry, 27 of projective space, 19, 24, 271 of Segre-aflne space, 264

of space An274

geometric object(s) of first order, 37 fourth order, 37 kth order, 76 second order, 37, 76, 79, 81 third order, 37, 45, 86 sequence of objects, 77 tensor, 124, 125, 133, 160

of normalized Grassmannian, 258-260 of polar normalization, 261 theorem, xi, 49

Gauss equation, 52 Generalized conformal structure, 319 Segre theorem, 97 General linear group, 34-36, 120, 121 General principal distributions, 192, 192 General relativity, ix, xi, xii, 102, 108, 140, 189, 202, 217, 218, 219, 320

Generating element, 71, 221 Generatrix of Segre cone, 264 Geodesics, 137-139, 161, 189, 191, 192, 310

Geometric object, 36, 37, 277, 278, 283

Geometry algebraic, 224, 225, 265 conformal, ix, x, 70-72, 115, 221 Euclidean, 72 of Grassmannian, 221 of hypersurface, 71 non-Euclidean, 127 projective differential, ix, 88, 97

pseudoconformal, 221 of submanifold, 77 of surface, 71 Grassmann coordinates, 222, 223, 232, 234, 235 d-web, 302-309, 320

curvature tensor of, 307, 309 torsion tensor of, 307-308 manifold, see Grassmannian mapping, 221, 223, 232, 238, 244, 250, 257

371

SUBJECT INDEX

PSL(n + 2), 151 structure, ix, x, xii, 221, 268, 290, of rotations of R,-, 127 320 Grasamannian, x, xii, 221-224, 237-244, 252, 260-265, 305-307, 319 basis forms of, 238, 253, 254, 257

G(m, m + 2), 242 G(m, n), 222, 244, 267, 268, 306

G(l,n-m+1), 242

G(1,3), 19-23, 108, 221, 242, 265 G(1,4), 232, 235

G(2,4), 235 G(2,5), 245 realization of, x rectilinear generator of, 224 of straight lines, xi submanifolds on, x, xii, 253 Grassmannizability condition, 314 Grassmannizability problem, 320 Grassmannizable webs, 302, 303, 309, 313, 314, 318, 319

Gravitational constant, 190 Gravitational radius, 206 Group, 2 of admissible transformations of second-order frames, 121, 124 of of Ine transformations , 147 of conformal transformations, 3, 6, 10-12, 14, 17 intransitive, 12 cyclic, 7

R', 269

SL(p), 151, 166, 271, 273, 300 SL(2, C), 171 SO(n), 120, 172 SO(p, q), 6, 7, 17, 27, 120, 127, 169, 172-174, 299 SU(2), 172

T(n) of parallel translations, 2, 34, 41,166,173,175,264,274,286, 299

of transformations of pseudoconformal space, 128 transitive, 2 Z2, 7, 17 G-structure, 120, 125, 218, 301 of finite type, 125, 160, 284

Harmonic function, 218 intersection, 24

normalization, xii, 255, 258, 261 Hermitian symmetric space, 320 Hodge operator, 176, 178, 181, 218 Hodge tensor, xii, 178, 218 Holomorphic geometry, 217 Holonomic distribution, 94 Holonomic net of curvature lines, 55, 56, 94-98

fundamental, ix, 7

Homogeneous

general linear, 34-36, 120, 254 G0, 166, 172

coordinates of hypersphere, 16, 25, 127 point, 6, 19, 25, 223 straight line, 20 domain, 262 geometric object, 128, 130, 152 normalization, 260, 262, 264 space, 2, 71, 77, 115, 260, 264 Homothety, 2, 3, 173 Horizontal distribution, 147 form(s), 11, 238 invariant distribution, 147 Hyperbolic congruence, 113 linear congruence, 22

Gp, 166, 172 GL(n), 120, 254 GL(q), 304 H of homotheties, 34, 37, 41, 44, 120, 126-128, 166, 169,

171-173,254,269,286,299,300 isotropy, 2, 12

of motions of compactified T:(M), 299, 300

of motions and homotheties of R", 33 Rq , 127

of motions of R", 2 of motions of Ry", 299 of motions of SAP, 286

O(n), 2, 172, 175 O(p, q), 7, 17, 120, 194 PO(n + 2,9 + 1), 17, 19, 127 PO(n + 2,1), 7, 10, 12, 75 invariants of, 75 of projective transformations, 115,

254, 273 pseudoorthogonal, 7, 27, 169, 299

point, Ill T11 led submanifold, 111

space, 7, 66 Hyperboloid of one sheet, 16 Hyperboloid of two sheets, 16 Hypercomplex of subspaces, 244 Hypercubic, 250 Hypergeometric function, xi, 221 Hyperplanar element(s), 26, 27

372

SUBJECT INDEX

Hyperplane(s), 1, 3, 4, 226, 253, 264 improper, 3 at infinity, 3, 264 polar, 21 proper, 3 tangent, see Tangent hyperplane Hyperquadric, 6-8, 10, 16, 17, 70, 87-90, 97, 153, 169, 204, 217, 260, 261, 265

equation of, 24, 261 imaginary, 260, 265 nondegenerate, 155, 158 f)(1,3), 21, 22, 109-114 submanifold of, 109 oval, 6 Plucker, 20, 223 of revolution, 159 tangent hyperplane to, 6 Hyperephere, 2, 3, 14-16, 25, 26, 42, 127, 160

center of, 2, 4 contact, 54, 56 imaginary, 4, 15, 66 improper, 4, 6 orthogonal to hypersurface, 31 orthogonal to submanifold, 143

proper, 4 real, 4, 7, 15, 66 tangent to hypersurface, 32 of zero radius, 127

Hypersurface(s), 45, 98 asymptotic cone of, 209

basis forms of, 34, 35, 46, 47, 49, 151

canal, xi, 57, 60, 61 of conformal space, xi, xii, 31-71 connected, 45 cubic, 250

invariant normalization of, xi, 71 moving frame of, 104, 105, 109, 151 normalizing object of, 40, 43 in Ps, 251, 252

plane generator of, 214-216 of projective space, xii, 72, 151, 153 real, 31 of revolution, 159

in Rn, 66 ruled, 251

of second order, 6, 20 second fundamental form of, 46, 150, 152, 210, 214 second fundamental tensor of, 38, 46, 52, 66, 152, 209, 215

simply connected, 45 smooth, 45 tangentially degenerate, 68

tangentially nondegenerate, 150, 152, 154, 208 tangent hyperplane to, 35, 66, 151, 152

of third order, 236 ultrahyperbolic, 208-210 Identity of Lie group, 10 matrix, 230, 270 operator, 115, 179, 231 Imaginary asymptotic cone, 208 cone, 6

developable surface, 111 hyperquadric, 260, 265 hypersphere, 4, 15, 66 isotropic cone, 120 quadric, 7 radius, 15

Improper hyperplane, 3 hypersphere, 4, 6 Incidence condition, 237 Incident subspaces, 224 Indeterminate net, 92 Index-free notations, 131, 291 Index notations, xiii Infinitesimal displacement of frame of conformal space, 10, 14 CO(p,q)-structure, 126 null-pair, 193 projective space, 11, 23, 151, 271,

306

pseudoconformal space, 18 Cs , 24 7Z2(Vn-1), 44, 49

IZ3(Vn-1) 44 space p3, 23 submanifold, 74, 86, 90, 106, 142 Infinitesimal displacement of adapted frame, 274 invariant frame, 83 point frame, 237 tangential frame, 237 Inflectional center, 112 Integrability condition(s), 116, 315 Integrable distribution, 198, 200-202 Integrable almost Grassmann structure, 270

Integral curve(s), 188, 189 element, general, 100 geometry, xi, 221, 250 manifold(s), 100, 255, 270 submanifold, 95, 197, 243, 251

SUBJECT INDEX surface, 186, 200 Internal domain, 102 Intrinsic geometry of normalized V'", 161 Intrinsic normalization of submanifold, 115 Invariance of isotropic geodesics, 140 Invariant(s), 70, 146 bundle of normal hyperspheres, 43 bundle of second fundamental forms, 79, 88-91

373

subgroup, 2 subspace, 230 tangent m-sphere, 78 Weyl connection, 135 Inverse

matrix, 11, 241, 274 tensor, 32, 39, 60, 77, 106, 123 Inversion, 2, 15, 65, 66 center of, 3

bundle of tangent hyperspheres, 78

Involutive

circle, 115

distribution, 95, 186, 188 operator, 180 principal distribution, 187 transformation, 115 Irreducible net, 94, 98 Isoclinic

conformal connection, 128 conformal frame, 67 conformally symmetric connection, 135

connection, 134, 135 derivative, 70 differential form, 255

a-plane, 196 d-web, 318, 319

distribution, 197, 198

(p + 1)-web, 309-311, 313, 320

family of central m-spheres, 79 family of frames, 86

subspace, 305, 313

forms of

almost Graasmann structure, 285 conformal structure on hypersurface, 153 conformal structure on submanifold, 142 general linear group, 36, 121, 254 CO(p,q)-structure, 126 G-structure, 285

isotropy group, 127 group of motions of R.,-, 127 stationary subgroup H.' (Vm), 75 stationary subgroup Hs (V n- , ), 36, 41

stationary subgroup H=(Vm), 80 stationary subgroup H=(V^-I ), 41, 44

stationary subgroup H= (V n- 1), 44

stationary subgroup of m-plane, 238

structure group of CO(2, 2)-structure, 166

frame(s), 51, 63, 67, 83

of group PO(n + 2,1), 75 horizontal distribution, 147 infinitesimal operator, 70 local parameters, 116 normalization, of hypersurface, xi, 71 of submanifold, xi, 116 of surface, 115 point(s), 75, 80, 82, 229 quadratic form, 70, 112, 134, 136 relative, 50, 81-83, 86

stationary subgroup, 81

three-web, 197, 199, 201 Isothermic curvature lines, ix, 70 hypersurface, 63 surface, 63

Isotropic a-submanifold, 195, 215 Q-submanifold, 195, 215 bivector(s), 184 bundle, 171, 301 co-frame(s), 164 cone, 6, 7, 13-16, 102, 104, 112, 113,

120, 127, 139, 141, 164, 169, 170, 171, 194, 209

of CO(2, 2)-structure, 199-201, 279 plane generator of, 164, 170, 171, 270

of pseudoconformal space, 18 curve, 24, 102, 109

distribution, 186, 208 fiber bundle(s), xii, 164, 168-172, 174176, 182-197, 200, 202,

205-208

curvature tensor of, 168, 183 fibration(s), 218 foliation(s), 186-188, 202 four-web, 187

frame(s), 164, 180, 204 frame bundle, 188 geodesic congruence, 189 geodesic(s), 138-140, 161, 189 hypersurface, 102, 104, 108, 109 net, 113 submanifold, 101, 110 tangent elements, 109 Isotropy group, 2, 12, 127

374

SUBJECT INDEX of affine space, 264 invariant forms of, 127 of space SAP, 286

Kerr metric, 206, 219 Klein interpretation, 7 Kronecker symbol, 11, 12, 54, 91

Laguerre space, ix, 70 Law of transformation of basis forms, 34, 35, 254 connection forms, 161 curvature tensor, 160 invariant, 70 quadratic form, 70 Riemannian connection, 137 tensor, 35, 148 vector, 35

Left-invariant forms, 151, 271, 301 Levi-Civita connection, 136, 259, 262 Lie

algebra, 10, 320 group, 10, 71, 72, 151 hyperquadric, 25-27 hypersphere, xi mapping, 25, 26 sphere geometry, 24-26, 29 Light cone(s), 15, 102

Light impulse, 140 Lighting surface, 108 Lightlike hypersurface, 102 Lightlike submanifold, 101 Light tetrad(s), 218 Line(s) geometry, 19 of propagation of light, 108 submanifolds, xii, 108, 111, 117 Linear complex, 21 special, 22, 23 congruence,22 -fractional function, 3 homogeneous object, 286 mapping, 46, 179, 229 operator, 178, 229 scalar, 231

pencil of subspaces, 224 span, 231, 233, 237 subspace, 28 transformation(a), 6, 316 Liouville theorem, 2, 3, 28 Lobachevsky space, 7, 66 Local conformal space, 126-128, 132 diffeomorphism, 302 projective space, 127

pseudoconformal space, 128 space, 301 twistor theory, 320 Locally flat almost Grassmann structure, xii, 290, 301

conformal structure, 144 Grassmann manifold, 319 Lorentz group, 169, 173, 174 Lorentzian signature, xi, 102, 140 Lorentzian structure, 171 Mainardi-Codazzi equations, 52 Manifold(s) algebraic, 240 with conformal structure, 119 integral, 100, 255 of null-pairs, 193 of oriented hyperplanar elements, 27

real, xi Mapping bijective, 245, 267, 270 differentiable, 20 rank of, 253, 255 Matrix coordinate of subspace, 226, 227, 255,

256, 265

Grassmann, see Grassmann mapping invariant, 260 inverse, 11, 241, 274 1-forms, 131, 291, 292 PlOcker, 20, 28, 109 symmetric, 261

Maurer-Cartan equations, 151 Maximum rank d-web, 321 Maximum rank four-web, 219 Maxwell-Einstein equations, 219 m-canal hypersurface, 60, 61 m-conjugate system, Method of exterior differential forms, x of moving frames, x, xiii tensor, x, xiii Metric, xii, 119 form, 15 Riemannian, ix, 63 of Riemannian manifold, 119 tensor, 136, 139, 140, 218, 262 Middle curvature, 66 Minimal hypersurface, 66 Minkowski space, 15, 16, 28, 102, 116 Mdbius geometry, 3 M6bius space, 3 Motions of Euclidean space, 1-3 homogeneous space, 77

SUBJECT INDEX pseudoelliptic apace, 262 Moving frame of conformal space, 8, 28 hypersurface, 104, 105, 109, 151

of null-pair, 193 projective space, 271 pseudoconformal space, 18 space p3, 23 submanifold, 73, 86, 90, 106

375

form, 152 matrix, 59, 74, 268 Nonsymmetric Ricci tensor, 134 Non-umbilical point, 81 Normal

Moving frames method, x, xi, xiii

bundle of hyperspheres, 75 circle, 43 conformal connection, 128, 132, 146 congruence of circles, 64 first, 67

m-pair(s), 115, 227, 260

focal family, 69

degenerate, 227

in general position, 228, 229 nondegenerate, 227-229, 260 m-plane, 115, 237, 265 m-quasiaffine space, 263, 266 m-sphere, 81, 82, 98, 116, 144, 145

Multidimensional web(s), x, xi, 221, 270, 301, 304

Multiple eigenvalue, 58 Multiple root, 194, 198 Multiplicative group of reals, 269, 273

hypersphere, 43, 46, 80, 87, 149 m-sphere, 82

(n - m)-sphere, 149 of submanifold, 88 Normalization(s), complete invariant, 82 of Grassmannian, 252, 257, 265 harmonic, xii, 255, , 258, 261 homogeneous, 260, 262, 264 intrinsic, 115 invariant, see Invariant,

normalization Net(s)

nondegenerate, 253

conjugate, 213, 249

of conjugate lines, 67, 90, 98, 116 holonomic, 98 of curvature lines, 55, 56, 67, 89-99, 116

holonomic, 55, 56, 94-98 indeterminate, 92 irreducible, 94, 98 totally holonomic, 94, 95, 97 of developable surfaces, 113 holonomic, 55, 56, 94-98 of isotropic lines, 102 Newman-Penrose tetrad(s), 170, 218

Nondegenerate hyperquadric, 155, 158 m-pair, 227-229, 260 normalization, 253 null-pair, 193, 194, 219 projective transformation, 229 quadratic form, 119, 176, 255 tensor, 53 Non-Euclidean geometry, 127 Nonholonomic submanifold, 161 Nonhomogeneous coordinates of point, 12

coordinates of subspace, 223 projective coordinates, 164

Nonisotropic complex, 112, 113 hypersurface, 72, 109 submanifold, 141, 150 Nonsingular

polar, 260, 262, 265 of submanifold, 146 Normalized domain of Grassmannian, 253-258, 260-264 Normalized submanifold, 72, 115, 149, 150 Normalizing condition, 74 mapping, 253 of zero rank, 263 (n - m)-sphere, 149 object 40, 43, 78, 82 subspace(s), 253-256, 263, 264 (n + 3)-spherical coordinates, 26 Null-pair, 193, 194 Object C,.,,, 125, 126, 128-131, 135, 141,

145, 146, 147 complete, 71, 77, 87 fundamental, see Fundamental geometric object normalizing, 40, 43, 78, 82 1-canal hypersurface, 61 1-canal submanifolds, 145 1-form, differential, 63, 120, 123, 125, 126, 128, 131, 132

One-parameter foliation, 94 group, 166, 173 subgroup, 34 One-to-one correspondence, 45, 46, 223 One-to-one mapping, 20

SUBJECT INDEX

376

Open domain of Grassmannian, 253, 260 Open neighborhood, 224 Operation of complex conjugacy, 189 Operator of covariant differentiation, 260

of differentiation with respect to fiber parameters, 274 V, 76 e, 178, 218 w, 229 Optical directions, 219 Orientation, 25 Oriented hyperplanar element, 26 hypersphere, 25, 26 hypersurface, 45 manifold, 176 Orthonormal frame(s), 12, 14, 75 Orthogonal frame(s), 12 group, 2 hypersphere, 73 m-hedron, 79 trajectories, 69 transformation, 2 Osculating circle, 43 hypersphere, 84, 93

sphere, 85, 87, 89, 93, 96, 97 subspace, of developable surface, 213 of isotropic a-submanifold, 211-214 of submanifold, 89, 98 of surface, 211-214 of tangentially nondegenerate

submanifold, 213 of variety fl(m,n), 238, 239 Oval hyperquadric, 6 Pair of congruences, 117 Parabolic congruence, 113 linear congruence, 22 pencil of hyperspheres, 26, 27 point, 111

submanifold, 111 Paraconformal structure(s), 320 Parallel d-web, 302

translation(s), 2, 148, 274 transport, 132, 264 vector field, 148

Parallelizability condition for three-webs, 304 Parallelizable webs, 302, 304

Parameters, fiber, 77, 143, 176, 204 principal, 135, 277 p-dimensional direction, 54 Pencil of characteristic straight lines, 251, 252 hyperplanes, 236 hyperquadrics, 157 hyperspheres, 40, 65, 95, 98 normal hyperspheres, 48 oriented hyperspheres, 26, 27 second fundamental forms, 37, 67 straight lines, 21, 22 tangent hyperspheres, 33 tangent linear complexes, 112 tangents, 117 tensors, 37

Pentaspherical coordinates, ix, 4, 28, 70 Petrov classification, 189-193, 206, 218,

219

Petrov's type(s), 191-193 Pfaffian derivative(s), 287-289 equation(s), 14, 55, 69, 186 equations, system of, 71, 74, 186 completely integrable, ii, 56, 60,

64, 71, 148, 186 in involution, 100 Planar hypersurface, 245 Plane(s), 215 field of straight lines, 22, 232 generator of asymptotic cone, 208, 241-243 cone C=, 113, 168, 171, 195, 216

hypercubic, 250 hyperquadric, 17 hyperquadric fl(1,3), 22 hypersurface, 214-216 isotropic fiber bundle, 164 Segre cone, 232, 240, 245, 267-269, 300, 305, 317 Segre variety, 225, 230, 231, 264

variety f1(m, n), 224, 225 generators, family of, 168, 224, 242

at infinity, 7 Plucker coordinates, 20 hyperquadric, 20, 223 manifold, 19 mapping, 20, 28, 109 Poincar4 space, 28 Point(s), 1, 115, 116 conjugate, 69 elliptic, 111 focal, 246 hyperbolic, 111

SUBJECT INDEX at infinity, 3, 16, 66, 126, 127, 286,

377 space, 3, 10, 117, 127, 150, 154, 155,

300

158-161,216,221,223,225,237, 238, 244, 253, 262, 263, 267, 269, 271, 272, 291, 299, 300,

invariant, 82, 229 non-umbilical, 81 parabolic, 111 singular, see Singular points of tangency, 112

302

dual, 224 infinitesimal displacement of frame of, 11, 151

umbilical, 42, 46, 81

Polar

structure, 320

bilinear form, 5

structure equations of, 151, 237 P3, 232, 234, 244 P4, 232, 233, 235, 236, 251 P5, 169, 216, 223, 232, 244, 245,

-conjugate subspaces, 67, 261 hyperplane, 21

normalization, 260, 262, 265 -normalized Grassmannian, 262 Pole of hyperplane, 7, 26 Polynomials CQ(a) and Ca(p), 184-190, 194, 214

Polyspherical coordinates, xi, 3, 4, 6, 15, 28

Principal

250

Ps, 250 P9, 232-236

transformation(s), 6, 159, 229, 260 Projectivization of asymptotic cone, 241, 243 cone, 225, 251

a-plane(s), 185 13-plane(s), 185

bivector, 259

direction(s), 53, 67, 89, 188, 191, 192, 219

of affinor, 113 domain of, 53 double, 219 of hypersurface, 53, 113

orthogonal, 53 subspace of, 58

distribution, 186, 191-193 double, 189, 191-192, 206 of general type, 191-193 triple, 192, 193 forms, 135

isotropic distribution, 186-193, 209 parameters, 135, 277 subbundle, 128 two-dimensional direction, 188 Product direct, 225, 264 exterior, 19, 177 Projection, 237 center, 216, 217 of Segre variety, 216, 217 Projective coordinates of point, 227 coordinate system, ix differential geometry, ix, 88, 97 frame, 66, 222, 260, 306 line(s), 171, 230

matrix coordinates, 226, 265 plane, 216 point frame(s), 236 realization, xiii

center of, 225 Grassmannian, 221 isotropic cone, 23 Segre cone, 232, 240 system of bivectors, 316 tangent space, 315 tangent subspace, 23 variety n(2,5), 247 Prolongation, 34, 71 Prolonged G-structure, 128, 166, 173

structure equations of 128 Prolonged structure group, 301

Proper

conformal geometry, x space, xi, 16, 19, 31, 74, 89, 100, 103, 141

structure, xi, 102, 120, 128, 153, 172, 175

domain of Grassmannian, 253 hyperplane, 3 hyperaphere, 3, 4 Riemannian metric, 136, 161 subspace, 1

Pseudoconformal geometry, x space, x, xi, 14, 16, 18, 19, 21, 27, 28, 31, 72, 100, 103, 116, 127, 128, 141, 221, 264, 265 C2', 108, 109, 111, 169, 221, 265

four-dimensional, xi structure, x-xii, 102-120, 127, 142, 217, 270, 299 Pseudocongruence of m-spheres, 116 Pseudoelliptic space, 262

SUBJECT INDEX

378

Pseudo-Euclidean space, 14, 15, 18, 28, 127, 128, 131, 264, 265, 299

four-dimensional, 28 group of motions of, 127, 128

group of motions and homotheties of, 128 R4, 166

Pseudogroup of contact transformations, 27

Pseudoorthogonal frame(s), 19, 163, 299 group, 7, 27, 169, 299 transformation, 6 Pseudoorthonormal frame(s), 169 Pseudo- Riemannian manifold, 102, 138-141, 160 Pseudo- Riemannian metric, x, 136, 194, 255, 258, 262 Pseudo-Riemannian structure 0(2, 2), 194 Pure imaginary function, 203 Pure imaginary isotropic cone, 13 (p + 1)-web, 271, 301, 305, 306, 320 of codimension one, 311 structure equations of, 271, 305 q-parameter family of submanifolds, 98 Quadratic congruence, 117

form(s), 1, 9, 13, 141, 163, 255, 257, 259, 299

relations, 223 Quadric(s), 17, 18, 22 double, 155, 157 four-dimensional, 20

imaginary, 7 real, 18 three-dimensional, 22 Quadrilateral web, 94 Quadruple of plane generators, 230

of points, 230, 231 principal distributions, 192, 193

root, 200 of subspaces, 230

Quasiaffine space, 263, 265, 266, 273

Quaternionic structure, 320 Quotient, 17, 254 Range of normalizing mapping, 253 Rank of mapping, 253, 255 quadratic form, 53, 110 system of forms, 93 system of hyperspheres, 93, 96 system of tensors, 80 tangentially degenerate submanifold, 68, 108

tensor(s), 45, 58, 80, 81, 258

Real

conformal space, 46 cross-ratio, 231 Curvature tensor, 176 eigensubspace(e), 180 eigenvalue(s), 180, 181 fiber bundle, 176 four-dimensional conformal structure, 217

generator of cone C, 188 isotropic cone, 18, 104, 120 isotropic directions, 100, 102, 103 isotropic fiber bundle, 165 plane generator, 111, 113

principal isotropic directions, 188, 191 quadric, 18 rectilinear generator(s), 6, 171 root(s), 107, 187 singular point(s), 107, 108

submanifold, 100 subspace, 230

tangent space, 171 tensor of conformal curvature, 218 transformation of coordinates, 100, 164, 169

Realization of Grassmannian, x Realization, projective, xiii Rectilinear a-generator, 250 fl-generator, 250 generator(s), 14

of asymptotic cone, 208 of cone C., 113, 141, 170 of developable surface, 213

of Grassmannian, 224 of hyperquadric, 17 of hyperquadric f1(1,3), 21, 22,

111, 117

of hypersurface UI_1, 68 Lie hyperquadric, 26 of Segre variety, 230, 231

of submanifold, 107, 108 of tangentially nondegenerate submanifold, 213 of variety 11(m, n), 224 Recurrent CO(2,2)-structure, 188 Reduced family of fourth-order frames, 284 second-order frames, 278, 279, 281 third-order frames, 283 Reduced group of admissible transformations, 124 Reduction of group of admissible transformations, 134 Reissner-Nordstr8m metric, 206, 207, 219

SUBJECT INDEX Relative conformal curvature, 184, 185, 205 invariant, 50, 81-83, 86, 318

tensor, 36-39, 130, 152, 177, 279, 280, 287

Relatively invariant form, 14, 37, 46, 119, 122, 132, 148, 150, 153 Relativity theory, 108 Representation, 71

complex, 173, 218

379 submanifold, 67, 79, 249 surface, 110 Second fundamental tensor of hypersurface, 38, 46, 153 Second order asymptotic direction(s), 250 envelope, 155, 157-159 frame(s), 41, 49, 80, 121, 123, 128, 131, 146, 204, 278, 279 tangency, 41, 42, 53, 54, 79-81, 84,

Restriction of Darboux form, 212, 214 Ribaucour congruence, 63

Second sheet of envelope, 45

Ricci

Second structure object, 281, 286, 287,

112, 156

identities, 129

tensor, 51, 133, 134, 190, 258, 262 symmetric, 258 Riemannian connection, 64, 134-140 geometry, x, 71, 129, 265 manifold, 116, 130, 137-141, 155, 161, 218, 252, 259

four-dimensional, xi, 217 metric, ix, 63, 119, 132, 134, 136, 137, 140, 161, 189, 202, 218, 255, 258, 262 structure, 134

tensor, 51, 161 Rigidity

conformal, 46, 72 problem, 72 theorem, xi, 46, 71 Rotation, 160 Ruled

hypersurface, 251 submanifold(s), xii, 108, 111, 117 surface, 24, 108, 109, 112, 113, 215 of second order, 23

Scalar

298 Sectional curvature of

normalized Grassmannian, 259 polar-normalized Grassmannian, 262, 263 Segre-afine space, 264-266, 286, 300 Segre cone

with complex generators, 270 Cp(m + 1, m), 240 Cp(2, 3), 232 dimension of, 240 directrix of, 232, 271

equations of, 268 plane generator(s) of, 232, 245, 267-269 projectivization of, 232, 240

SC=(p,q), 269, 271, 286, 291, 300,

301, 305, 316, 317 vertex of, 231, 264 Segre theorem, 97 Segre variety, 216, 217, 225, 230, 231, 240, 241, 264, 265, 271, 300, 316 degree of, 225, 245

dimension of, 225 equations of, 230 plane generator(s) of, 225, 230, 231,

curvature, 52, 133, 134, 190

264

linear operator, 231

projection of, 216, 217

1-form, 131, 291

rectilinear generator(s) of, 230, 231 S(k,l), 225, 230-232, 240, 241, 243, 245, 247, 256, 316

product of elements of conformal frame, 126 elements of projective frame, 237 hyperspheres, 5, 74 points, 8 vectors, 1, 262 web, 321

Schwarzschild metric, 206, 207, 219 Secant, 233 Second differential of point, 238 Second fundamental form(s) of completely isotropic submanifold, 212, 213

hypersurface, 46, 150, 152, 210, 214

Segrean, see Segre variety Self-conjugate curvature tensor, 176 subspace, 182 Self-dual classes of complexes, 252

CO(2,2)-structure, 194, 198, 215 part of Weyl tensor, 217 structure, 183, 185, 186 subspace, 182, 183

Semidirect product, 2, 41, 128, 264, 274

Semiflat

SUBJECT INDEX

380

asymptotic conformal structure, 162 CO(1, 3)-structure, 175 CO(2, 2)-structure, 169, 186, 209, 210, 215, 216, 218

four-dimensional structures, 183 Semifocal family, 251 Semiintegrable almost Grassmann, structure, xii, 270, 292, 300 conformal structure, 300 CO(2, 2)-structure, 198 four-dimensional structure, 183, 186 Semi-Riemannian manifold, 252, 265 Semi-Riemannian metric, 255, 265 Sequence of geometric objects, 77 Signature (p,q), 18, 100, 119 Simplex, coordinate, 224

Simply connected domain, 86, 87 hypersurface, 45 submanifold, 73 Singular point(s), 107, 108, 241, 243, 244, 246, 247, 251

vector, 19, 238 Spacelike direction, 103 Spacelike hypersurface, 102, 103 Space-time, 28, 140, 190, 202, 206, 217 Span, linear, 231, 233, 237 Special complex, 109-111 linear complex, 22, 23

linear group, 151, 166, 268, 271 orthogonal group, 120 pseudoorthogonal group, 120 relativity, 28, 102 three-web, 197

Specialization of fourth-order frames, 283 Specialization of second-order frames, 277 Spectrum of Hodge operator, 218 Sphere, 17, 71 n-dimensional, 3 Spherical coordinates, 206 Spherical curvature lines, ix, 70 Spherically symmetric body, 206, 207 Spherically symmetric solution, 219 Square of Hodge operator, 178 Stationary subgroup

straight line(s), 232

of element of Grassmannian, 274

subspace, 242

of element of normalized domain of Grassmannian, 264 Hl(Vm), 75, 81 H.(V"-1)333, 34, 41 HZ(Vm), 80 H=(V"-I), 41, 44 H=(V"-1), 44

Skew-symmetric bilinear form, 210 part of torsion tensor, 313 tensor, 129, 293, 294

Smooth curve, 92, 109, 138 family, 244 hypersurface, 45 submanifold, 73, 77, 109, 245 Space with affine connection, 115, 258, 289, 290

conformal, see Conformal space with conformal connection, 115, 160 dual, 224, 235, 236 elliptic, 66 Euclidean, see Euclidean space of exterior 2-forms, 178, 180 with group connection, 71 homogeneous, 2, 71, 77, 115, 260, 264

of m-pair, 254 of m-plane, 238 of point, 12, 33

Stationary value of sectional curvature, 259 Stereographic projection, xii of conformal space, 7, 17, 18, 264 of Grassmannian, 264-266

of variety fl(m, n), 286, 300 Structure conformal, see Conformal structure of Grassmannian, 224 pseudoconformal, x-xii, 102-120, 127, 142, 217

of space Cz, 221 Structure equations of

hyperbolic, 7, 66 Laguerre, ix, 70 projective, see Projective space pseudoconformal see Pseudoconformal apace R(m+l)(n-m), 224

almost Grassmann structure, xii,

R4, 198, 200, 201 tangent, see Tangent space

differentiable manifold, 121 d-web, 315

of twistors, 218

Grassmannian, 257, 271, 273, 276

274-276, 285, 300, 320 conformal connection, 128 conformal apace, 11, 14, 87

conformal structure, xii, 126, 131, 299

SUBJECT INDEX

Grassmann (p + 1)-web, 309 group G', 128 plane RP2, 194 projective space, 23, 151, 237, 271,

306

prolonged G-structure, 128 pseudoconformal space, 18, 128 pseudoconformal structure, xii, 126, 142

(p + 1)-web, 303, 304

Structure group of

almost Grassmann structure, 268, 286, 301, 305 CO(p,q)-structure, 301 CO(1, 3)-structure, 171 CO(2, 2)-structure, 166 d-web, 315

fiber bundles E. and E, 166 Grassmannian, 273 (p + 1)-web, 305 prolonged G-structure, 128

three-web, 271 Structure tensor of almost Grassmann

structure, xii Subfamily of

orthogonal frames, 12 orthonormal frames, 75 projective frames, 237 Subgroup, invariant, 2, 34 Submanifold(s), 71, 144, 149, 150 carrying conjugate net, 90 carrying net of curvature lines,

381

singular, 242 Subtensors C. and Cp, 168, 169, 173-176,

183, 184, 186, 187, 209, 210,

218, 300

Subtensors ao and ap, 293 Subweb,316

Summation convention, 9 Surface(s), 71 asymptotic line of, 110, 111 of light absorption, 108 second fundamental form of, 110 Symbol of covariant differentiation, 261 differentiation with respect to fiber parameters, 308 exterior multiplication, 132 Symmetric affinor, 52, 107 function, 82 linear operator, 108, 179 matrix, 261

part of curvature tensor, 304 part of torsion tensor, 312 tensor, 60, 62, 68, 137, 146, 152, 255 relative, 60, 152 Symmetrization, 76, 292, 293

Symmetry, xi, 170 Symmetry figure, 115, 116 System of circles, 70 hyperspheres, 93, 96

Pfaffian equations in involution, 100

89-99, 250

completely isotropic, xii, 183, 185, 186, 194, 211-215

of conformal space, x, xi, 115, 141 connected and simply connected, 73 on Grassmannian, x, xii, 253 integral, 95, 197, 243, 251 moving frame of, 73, 86, 90, 106 normalized, 72, 115, 149, 150

parabolic, 111 ruled, xii, 108, 111, 117

second fundamental form of, 67, 79, 249 smooth, 73, 77, 109, 245 of space with conformal connection, 115 tangent subspace to, 88, 100, 107, 141

V3 C Ps, 109 Subspace(s), 1, 221-226, 237, 253 characteristic, 251

invariant, 230

linear, 28 normalizing, 253-256, 263, 264

Tangency of second order, 41, 42, 53, 54, 79-81, 84, 112, 156 Tangent bundle of hypersurface, 46, 75 bundle of variety fl(m, n), 238 bundle of second order of variety ft(m, n), 238 cone,242

hyperplane to hyperquadric, 6 hypersphere, 18

hypersurface, 35, 66, 151, 152, 212 hyperquadric, 18 hypersphere, 13, 32, 46, 61, 73, 84,

87, 145

linear complex, 112 m-sphere, 145 space, xi, 1, 120, 177, 268, 274, 286, 305, 315 subspace to

leave of web, 315 submanifold, 88, 100, 107, 141 torse, 247, 251

SUBJECT INDEX

382

two-dimensional submanifold, 245 variety fl(m, n), 238, 267

of second order to variety fl(m, n), 239

2-plane to isotropic submanifold, 211, 212 Tangential coordinates, 236

frame, 233, 237 matrix coordinate of subspace, 226,

227, 229, 256

Tangentially degenerate hypersurface, 68 submanifold, 107, 108, 111 ruled submanifold, 111 Tensor(s), 35, 36 analysis, ix, 70 apolar, 38, 39, 58, 61, 66 C. and Cg, 168, 169, 171-176, 183, 184, 186, 199, 208, 209, 218 of conformal curvature, xii, 125, 128, 130,133-136,142,144-147,153, 154,161,162,166-168,184,187,

194, 218, 300

of CO(2,2)-structure, 209 curvature, see Curvature tensor Darboux, 152, 154, 157, 209 differential equations of, 39, 77, 125, 130, 168

field, xiii invariant, 320 inverse, 32, 36, 60, 77, 123 law of transformation, 148 nondegenerate, 53

Third fundamental form of isotropic submanifold, 212, 213 Third order asymptotic line, 244, 246 cone, 241 frame, 86, 121

hypersurface, 236 object(s), 45, 82, 86 Third structure object, 281, 287 Three-dimensional cubic submanifold, 250 projective space, 232, 234, 244 quadric, 22 submanifold, 246, 249 Three-parameter group, 166 Three-web, 195-202, 219, 245, 270, 271, 304, 311, 314

Time coordinate, 15 Timelike direction, 102 Timelike hypersurface, 102, 104 Torse, 108, 109, 247-251 Torsion form, 291, 300 Torsion-free, 301 affine connection, 115, 132, 136, 137, 147, 148, 258

CO(2, 2)-structure, 314 Torsion tensor of affine connection, 147, 303 almost Grassmann structure, 274, 279, 287, 293 subtensors of, 293 Grassmann (p+1)-web, 307-309, 313 (p + 1)-web, 303, 304

rank of, 45, 58, 80, 81, 258

three-web, 196, 199, 311 Torus, 17

relative, 36-39, 130, 152, 177, 279,

Total differential, 63, 66, 84, 128, 133,

280, 287

of relative conformal curvature, 206 Ricci, 51, 134, 190, 258, 262

second fundamental, see Second fundamental tensor skew-symmetric, 129, 293, 294 symmetric, 60, 62, 68, 137, 146, 152, 255 torsion, see Torsion tensor trace-free, 130, 131, 144, 294 (p, 4)-, 36 (0, p)-, 32, 35, 38, 39, 50, 152 (0,2)-, 30, 36, 37-39 (0,3)-, 39 (1,2)-, 47

(2,0)-,36 Theorem Frobenius, 12, 249, 318 Segre, 97 Third-class congruence, 236

134, 139, 166 Totally focal congruence, 247, 250 Totally geodesic submanifold, 311 Totally holonomic net, 94, 95, 97 Totally isotropic surface, 113 Transformation(s), of basis forms, 254 complex, 170

linear, 6 projective, 6, 159, 229, 260

Transitive group, 2

Transitive subfamily of frames, 12, 19 Transversal(s) #-plane, 196 bivector, 316, 317 subspace, 305, 316

of two m-pairs, 229-231 Transversally geodesic distribution, 198 d-web, 318, 319

SUBJECT INDEX four-web, 202 (p + 1)-web, 309-311, 320 submanifold(s), 311, 312 three-web, 198, 202

Triple principal distributions, 192, 193 Triply orthogonal system of surfaces, ix,

383

Vertical form(s), 12 Vlasov configuration, 233-236, 265 Vlasov congruence, 236

equation of, 236 Vlasov hypersurface, 236

Volume element, 177

70

Twistor, 217 fibration, 218 Two-dimensional developable surface, 213 isotropic direction, 184

Web(s) AGW (d, p, q), 318, 319

AGW(d,2,q), 321 AGW(6,3,2), 321 GW (p + 1, p, q), 307-309

of maximum rank, 219, 321 multidimensional, x, xi, 221, 270, 301,

plane generator, 22, 170, 171, 214,

215

submanifold, 245, 249

304

W(d,p,q), 301, 302, 314-321

tangent subspace of, 245

three-web, 245

W(p + 1,p,q), 271, 303-305, 309-314, 320

Two-fold hyperplane, 157 Two-web, 94

W (4, 2, q), 315, 320, 321

W(4,2,2), 187, 197, 219 W(3,2,q), 270, 271, 304, 311, 320 W(3, 2, 2), 195-202, 219, 314

Ultrahyperbolic hypersurface, 208-210 Umbilical point, 42, 46, 81

Unimodular matrix, 268 Unit tensor, 292

Weight of tensor, 36, 80, 82, 130, 177, 279, 280, 287

Weingarten formulas, 70, 161 Variety

Weyl connection, 51, 64, 132, 133, 136,

algebraic, see Algebraic variety

148-150, 161, 205

Cartan, 98, 213, 215 determinantal, 240, 242, 265

curvature tensor of, 133 geometry, 161

Vector(s)

law transformation of, 35

structure, 134

space, 19, 238

tensor, 125, 130, 161, 190, 191, 217

tangent, 35, 38 Vectorial frame, 120, 170 Vertex of cone, 54, 70, 250 Segre cone, 231, 264

Zero

matrix, 226, 256 -tensor, 92

Comprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry Conformal Differential Geometry and its Generalizations is the first and only

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Clear, focused, and expertly synthesized, Conformal Differential Geometry and Its Generalizations

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