Connections, Curvature, and Cohomology Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes

Connections, Curvature, and Cohomology Volume I1 Lie Groups, Principal Bundles, and Characteristic Classes

This is Volume 47 -11 in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: PAUL A. SMITH AND SAMUEL EILENBERG A complete list of titles in this series appears at the end of this volume.

Connections, Curvature, and Cohomology Werner Grezld, Stephen Halperin, and Rg Vanstone DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO TORONTO, CANADA

VOLUME II Lie Gronps, Principal Bundles, and Characteristic Classes

A C A D E M I C P R E S S New York and London A Subsidiary of Harcourt Brace Jovanovich, Publishers

1973

COPYRIGHT 0 1973;BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY A N Y MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR A N Y INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWl

LIBRARY OF

CONQRESS

CATALOG CARD NUMBER:79-159608

AMS (MOS) 1970 Subject Classifications: 53-00,55C20,55C25,

55F10,55F25,58A05,58AlO, 58C05.58C15 PRINTED IN THE UNITED STATES OF AMERICA

Respec+?.$ dedicated to the memory HEINZ HOPF

of

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Contents xi

Preface Introduction Contents of Volumes I and 1II

...

Xlll

xxi

Chapter 0 Algebraic and Analytic Preliminaries 1

1. Linear algebra 2. Homological algebra 3. Analysis and topology 4. Summary of volume I

i 12 15

Chapter I LieGroups 24 31 39 44 48 56

1 . Lie algebra of a Lie group 2. The exponential map 3. Representations 4. Abelian Lie groups 5. Integration on compact Lie groups Problems

Chapter I1 Subgroups and Homogeneous Spaces 1. 2. 3. 4. 5.

Lie subgroups Linear groups Homogeneous spaces The bundle structure of a homogeneous space Maximal tori Problems

63 70 11

83 81 96

Chapter I11 Transformation Groups 109 114 121

1. Action of a Lie group 2. Orbits of an action 3. Vector fields vii

Contents

viii

4. Differential forms 5. Invariant cross-sections Problems

125 131 135

Chapter IV Invariant Cohomology 1. 2. 3. 4.

Group actions Left invariant forms on a Lie group Invariant cohomology of Lie groups Cohomology of compact connected Lie groups 5. Homogeneous spaces Problems

146 154 161 166 178 184

Chapter V Bundles with Structure Group 1. 2. 3. 4. 5. 6.

Principal bundles Associated bundles Bundles and homogeneous spaces The Grassmannians The Stiefel manifolds The cohomology of the Stiefel manifolds and the classical groups Problems

193 198 205 212 219 224 229

Chapter VI Principal Connections and the Weil Homomorphism 1. 2. 3. 4. 5. 6. 7. 8.

Vector fields Differential forms Principal connections The covariant exterior derivative Curvature The Weil homomorphism Special cases Homogeneous spaces Problems

236 240 247 253 257 260 272 283 290

Chapter VII Linear Connections 1. 2. 3. 4. 5. 6. 7. 8.

Bundle-valued differential forms Examples Linear connections Curvature Parallel translation Horizontal subbundles Riemannian connections Sphere maps Problems

304 31 1 318 326 330 335 341 347 352

Contents

ix

Chapter VIII Characteristic Homomorphism for X-bundles Z-bundles Z-connections Invariant subbundles Characteristic homomorphism Examples Z-bundles with compact carrier 1. Associated principal bundles 8. Characteristic homomorphism for associated vector bundles Problems

1. 2. 3. 4. 5. 6.

312 3 80 383 388 394 399 403 41 1 41 6

Chapter IX Pontrjagin, Pfaffian, and Chern Classes 1. 2. 3. 4. 5.

The modified characteristic homomorphism for real Z-bundles Real bundles: Pontrjagin and trace classes Pseudo-Riemannian bundles: Pontrjagin classes and Pfaffian class Complex vector bundles Chern classes Problems

Chapter X The Gauss-Bonnet-Chern Theorem Problems

420 422 434 443 45 I 460 477 485

Appendix A CharacteristicCoefficients and the Pfaffian 1. Characteristic and trace coefficients 2. Inner product spaces

493 500

References

509

Bibliography Chapters I-V Chapters VI-X

51 1 515

Bibliography-Books

521

Notation Index Index

529 531

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Preface This monograph developed out of the Abendseminar of 1958-1959 at the University of Zurich. It was originally a joint enterprise of the first author and H. H. Keller, who planned a brief treatise on connections in smooth fibre bundles. Then, in 1960, the first author took a position in the United States and geographic considerations forced the cancellation of this arrangement. T h e collaboration between the first and third authors began with the former’s move to Toronto in 1962; they were joined by the second author in 1965. During this time the purpose and scope of the book grew to its present form: a three-volume study, ab initio, of the de Rham cohomology of smooth bundles. I n particular, the material in volume I has been used at the University of Toronto as the syllabus for an introductory graduate course on differentiable manifolds. During the long history of this book we have had numerous valuable suggestions from many mathematicians. We are especially grateful to the faculty and graduate students of the institutions below. Our exposition of PoincarC duality is based on the master’s thesis of C. Auderset, while particular thanks are due to D. Toledo for his frequent and helpful contributions. Our thanks also go to E. Stamm and the Academic Press reviewer for their criticisms of the manuscript, to which we paid serious attention. A. E. Fekete, who prepared the subject index, has our special gratitude. We are indebted to the institutions whose facilities were used by one or more of us during the writing. These include the Departments of Mathematics of Cornell University, Flinders University, the University of Fribourg, and the University of Toronto, as well as the Institut fur theoretische Kernphysik at Bonn and the Forschungsinstitut fur Mathematik der Eidgenossischen Technischen Hochschule, Zurich. T h e entire manuscript was typed with unstinting devotion by Frances Mitchell, to whom we express our deep gratitude. A first class job of typesetting was done by the compositors. xi

xii

Preface

A. E. Fekete and D. Johnson assisted us with the proof reading; however, any mistakes in the text are entirely our own responsibility. Finally, we would like to thank the production and editorial staff at Academic Press for their unfailing helpfulness and cooperation. Their universal patience, while we rewrote the manuscript (ad infinitum), oscillated amongst titles, and ruined production schedules, was in large measure, responsible for the completion of this work.

Werner Greub Stephen Halperin Ray Vanstone Toronto, Canada

Introduction T h e purpose of this monograph is to develop the theory of differential forms and de Rham cohomology for smooth manifolds and fibre bundles. T h e present volume deals with Lie groups and with fibre bundles whose structure group is a Lie group. I n particular, the last half of the book is devoted to a detailed exposition of the Chern-Weil theory of characteristic classes. T h e characteristic classes of a bundle are the cohomology classes in the image of a canonical homomorphism (the Weil homomorphism), h : (VE*), + H(B),

where B is the base manifold and (VE*), is the algebra of invariant symmetric multilinear functions in the Lie algebra, E, of the structure group. I n Chern-Weil theory (for the case of principal bundles, where the structure group coincides with the fibre) they are constructed as follows: First, a principal connection is introduced in the bundle. This is essentially the choice of a suitable “horizontal” subbundle of the tangent bundle of the total space, and should be regarded as an additional, geometric structure. Second, the curvature is obtained from the connection. T h e curvature is a 2-form with values in E; it measures the geometric “twist” of the bundle. Finally, if r is an invariant symmetric function in E then a closed differential form in B representing the class, h(T), is obtained by replacing each of the arguments of r by the curvature form. The resulting homomorphism, h, is independent of the choice of the connection and thus is a bundle invariant. I t is essentially the only de Rham cohomology invariant of the bundle, in the sense that the cohomology algebra of the total space is completely determined by H(G), the differential algebra (A(B),a), and the Weil homomorphism (as will be shown in volume 111). xiii

xiv

Introduction

T h e material in this volume is organized as follows (a more detailed description appears below): T h e first three chapters contain the basic results (developed ab initio) on Lie groups and transformation groups. T h e reader may then proceed immediately to article 1 of Chapter V, in which principal bundles are defined, and then to the first six articles of Chapter VI, where the Weil homomorphism is constructed. T h e rest of Chapters V and VI is devoted to a wide variety of examples and special cases. Chapter IV deals with the cohomology of Lie groups, T h e following chapters do not depend on it (except for article 8, Chapter VI), so that it may be omitted without loss of continuity. Chapters VII to X describe the characteristic classes of vector bundles -an alternate approach to the Chern-Weil theory. Much of the material in these chapters is independent of the preceding work. I t culminates in the Gauss-Bonnet-Chern theorem in Chapter X, which identifies the Euler class, as defined in volume I, with a characteristic class. This volume contains about 250 problems in which a great deal of additional material is developed. For instance, the last problem of Chapter I1 leads the reader to a simple, elementary proof that n2(G)= 0 (G, a compact Lie group), while the problems of Chapter VII constitute a classical course in differential geometry. However, as in volume I, the text is self-contained and does not rely on the problems. Although most of the material in this volume is either in the literature, or is well-established folklore, we have not attempted to associate with the theorems the names of their discoverers, except where this is already common usage. This monograph is intended for graduate students, especially those interested in global analysis or differential geometry. T h e present volume relies heavily on Chapters 0-V of volume I, and, to a much lesser extent, on the rest of that volume. Aside from these prerequisites, however, it is completely self-contained. This volume will be followed by volume I11 which deals with the cohomology of principal bundles and homogeneous spaces. Chapter 0 . In this chapter the algebra, analysis, and topology prerequisites given in volume I are reproduced. I n addition the chapter contains a summary of the definitions, notation, and principal results of that volume. Chapter I, Lie Groups. A Lie group, G, is a group which is also a smooth manifold and for which multiplication and inversion are smooth maps. T h e vector fields on G, invariant under left translation, form a Lie algebra, linearly isomorphic to the tangent space, E, at

Introduction

xv

the unit element, e. Thus E becomes a Lie algebra, the Lie algebra of G. This chapter contains the standard elementary material on Lie groups (exponential map, general representations, adjoint representation, classification of abelian Lie groups) in which the relations between a group and its Lie algebra are stressed. In the last article the invariant (Haar) integral of a smooth function on a compact Lie group is defined from the point of view of differential forms. Chapter 11. Subgroups and Homogeneous Spaces. T h e first two main results of this chapter (Theorem I, sec. 2.1, and Theorem 11, sec. 2.9) assert that if K is a closed subgroup of a Lie group, G, then (1) K , itself, is a Lie group and (2) the homogeneous space, G / K , of cosets has a natural manifold structure. Theorem I is applied in article 2 to obtain Lie group structures in various subgroups of GL(F), and to determine their Lie algebras. Finally, in article 5 , these results are applied to compact connected Lie groups. It is shown (Theorem 111, sec. 2.18) that every element of such a Lie group is in a maximal torus and that any two maximal tori are conjugate. T h e same machinery yields the WeyZ integration formula (Theorem IV, sec. 2.19) which asserts that iff is a smooth function on a compact Lie group, G, such thatf(xy) =f(yx), x, y E G, then

j-p dx I WG 1-1 =

1f(r) 7-

det (c - Ad?)

4,

where T is a maximal torus. T h e results of article 5 are rarely quoted, except in Chapter IV. Chapter 111. Transformation Groups. A smooth right action of a Lie group, G, on a manifold, M , is a smooth map, M x G + M , written (x,a) +-+ x a, such that z * ab = (2 * a ) b and z * e = x. Such actions are the subject of Chapter 111. Article 1 contains definitions and elementary results. In article 2 we consider the subsets, z G, of M as embedded homogeneous spaces and prove that a smooth map o : N + M satisfying o ( N ) C x * G determines a smooth map into the corresponding homogeneous space (Theorem I, sec. 3.7). This theorem is quoted once, in article 1, Chapter VIII. An action of G on M determines a Lie algebra homomorphism: E -+ X ( M ) ( E , the Lie algebra of G). In this way E acts on the algebra of differential forms on M via the substitution operator and the Lie derivative. This material is developed in articles 3 and 4 and is chiefly applied in Chapters IV and VI.

-

xvi

Introduction

Chapter IV. Invariant Cohomology. If a Lie group, G, acts on a manifold, M , then the subalgebra, A,(M), of differential forms on M , invariant under the action, is stable under 6. T h e main result of article 1 (Theorem I, sec. 4.3) asserts that if G is compact and connected then the natural homomorphism, H ( A , ( M ) )+ H ( M ) , is an isomorphism. In article 2 we consider the case where G acts on itself by left multiplication. Then the algebra of invariant differential forms on G is isomorphic to AE* ( E ,the Lie algebra of G) and so 6 induces an operator, 6, , in A E* (this operator is carefully studied in volume 111). T h e results of article I are applied in Theorem 111, sec. 4.10, to obtain isomorphisms, H(G)

H ( A E * , )6,

(AE*), ,

if G is compact and connected. Article 4 is devoted to a theorem of Hopf (Theorem IV, sec. 4.12) which states that the cohomology algebra of a compact connected Lie group is an exterior algebra over a graded space whose dimension coincides with the dimension of a maximal torus. Finally, in article 5 we consider a homogeneous space, G / K , and discuss the algebra of differential forms on GIK, invariant under the left action of G. This algebra is identified with a subalgebra of AE*. Thus (again by article I), if G is compact and connected, H ( G / K )coincides with the cohomology of an explicit, finite dimensional, graded differential algebra. Chapter V. Bundles with Structure Group. I n article 1 principal bundles are defined. These are fibre bundles, B = (P, 7r, B, G), (G, a Lie group) together with an action of G on P such that the orbits, z * G (x E P)coincide with the fibres, G,,, . Such a bundle, together with an action of G on a manifold, F, determines (article 2) an associated bundle, [ = ( M , p, B, F ) . If F is a vector space and G acts by linear transformations then is a vector bundle. In article 3 we consider bundles whose fibre or base is a homogeneous space, G / K , and (for example) describe the tangent bundle of G / K in terms of a certain representation of K. T h e rest of the chapter deals with the Grassmann manifolds (k-planes in a real, complex, or quaternionic n-space) and the Stiefel manifolds (k-frames in n-space) and various associated bundles. These manifolds are identified with homogeneous spaces, so that the results of article 3 can be applied. T h e chapter closes with the computation of the cohomology algebra of the Stiefel manifolds (article 6). Chapter VI. Principal Connections and the Weil Homomorphism. T h e first six articles of this chapter are devoted to the construction

Introduction

xvii

of the meil homomorphism for a principal bundle, as outlined above. It is shown (Theorems I and 11, sec. 6.19) that this homomorphism is a bundle invariant, natural with respect to bundle maps. Article 7 deals with three special cases: (1) abelian structure groups, (2) reduction of structure group, (3) connections invariant under a group action. For instance, the Euler class of a principal circle bundle is identified with a characteristic class. This permits the computation of the cohomology algebra of complex projective space. Finally, in article 8, the results of the chapter are applied to the principal bundle, (G, x , G/K, K ) , ( K , a closed subgroup of a Lie group, G). I n particular, all the G-invariant principal connections are determined. Chapter VII. Linear Connections. Let 5 = ( M , x , B, F) be a vector bundle. A p-form, @, on B with values in f is a smoothly varying family of skew-symmetric p-linear maps,

aJs: T,(B) x

-.*

x T,(B)+F,.

Articles 1 and 2 establish the basic properties of bundle-valued forms and develop an “index and argument-free” notation for operations with these forms. I n article 3 a linear connection in f is defined as an [W-linear map, V : Sec f -+ A1(B;f ) , satisfying V(ju) = Sf

A

u

+f Vu,

f~ Y ( B ) ,

u E Sec

f.

Every vector bundle, f , admits a linear connection, V, and a linear connection in 5 induces connections in the dual bundle, the associated tensor bundles, and any pullback of 5. T h e curvature of a linear connection is a 2-form in B with values in L, and is defined in article 4. Fix a linear connection, V, in the vector bundle f = ( M , n, B, F)and let a,h be a smooth path in B. Then V determines a bundle map, [W x F -+ f , which induces 4 : R -+ B and restricts to isomorphisms in the fibres (article 5). It follows (Theorem I, sec. 7.18) that the pullbacks of 4 under homotopic maps are strongly isomorphic. In article 6 it is shown that a linear connection in 4 = ( M , x , B, F) determines a horizontal subbundle of the tangent bundle, T,,,, , and those horizontal bundles which correspond to linear connections are characterized. Riemannian connections (article 7) in a Riemannian vector bundle, (5, g), are linear connections, V, which satisfy Vg = 0. Their curvatures take values in Sk, . Article 8 considers smooth maps 4 : B ---t Sn ( B , an oriented compact n-manifold). T h e degree of $ is represented as the

xviii

Introduction

integral of an n-form constructed from the curvature of a Riemannian connection in the bundle # * ( T ~ , , ) (Hopf index formula). Chapter VIII. Characteristic Homomorphism for C-bundles. A Z-bundle is a vector bundle, f , together with a finite ordered set, Zt = (al,...)urn), of cross-sections in the associated tensor bundles, subject to the following condition: There is a coordinate representation, { ( U , , &)}, for ,f such that the ui correspond under #, to constant functions, U, w zli (E @F* 0 OF). T h e subgroup, G C GL(F), of isomorphisms leaving the zli fixed is called the structure group of the Z-bundle. According to Theorem I, sec. 8.2, the condition above is equivalent to the following condition on ( f , E l ) : For each pair of points, x, y, in the base there is an isomorphism, F, % Fv , carrying ut(x) to ai(y) (i = 1, ...)m). A Z-connection in a 2-bundle is a linear connection, V, such that Vai = 0 (i = 1,..., m) (article 2). With the aid of a Z-connection we construct, in articles 3 and 4, a characteristic homomorphism, h, : (VE*)I 4H(B),

( B the base, E the Lie algebra of G). This is a bundle invariant and is natural with respect to homomorphisms of Z-bundles (Theorems I11 and IV, sec. 8.13). T h e analogy between the Weil and the characteristic homomorphism is made precise in article 7: T o each Z-bundle corresponds an associated principal bundle with G as fibre. Moreover, there is a one-to-one correspondence between Z-connections and principal connections; and the characteristic homomorphism of a Z-connection coincides with the Weil homomorphism of the associated principal connection. Finally, a Z-bundle with compact carrier (article 6) is a Z-bundle, ( f , Zt),together with an explicit trivialization, a, of f outside a compact subset of the base, such that under a the cross-sections ad become constant functions. Such a bundle determines a characteristic homomorphism, h; : (V+E*)I --+ H,(B), which is an invariant of the triple (,f, Zt, a). Chapter IX. Pontrjagin, Pfaffian, and Chern Classes. For a real vector bundle, f = ( M , T,B, F ) , the Lie algebra of the structure group is simply L, . Canonical elements, C : E (Vp L,*), , (corresponding to the coefficients of the characteristic polynomial of a linear transformation of F ) are defined in Appendix A. These give rise, via the characteristic

Introduction

xix

homomorphism, to cohomology classes in H ( B ) , called the Pontrjagin classes of [. Their properties are established in article 2. In article 3 we consider pseudo-Riemannian bundles, and use the Riemannian metric to obtain simplified representatives for the Pontrjagin classes (Proposition VIII, sec. 9. I 1). T h e Pfaffian of a skew transformation of an oriented pseudo-Euclidean space, F, (cf. Appendix A) determines an element of (VSk,*), . This yields a characteristic class, the Pfaffian class, for an oriented pseudoRiemannian vector bundle. Its properties are established in secs. 9.12 and 9.13. In articles 4 and 5 we consider complex vector bundles (i.e., vector bundles whose fibre is a complex space, F ) . T h e characteristic coefficients of a complex linear transformation determine elements of (VL,*), and Chern classes of the so we again obtain characteristic classes-the complex vector bundle. If 4 is a real vector bundle, then the Pontrjagin classes of [ coincide with the Chern classes of C @ [. According to Theorem I, sec. 9.21 (which ends the chapter), the Chern classes satisfy four basic axioms, and are uniquely determined by them. Chapter X. Gauss-Bonnet-Chern Theorem. This chapter consists of an exposition of Chern’s proof of the Gauss-Bonnet theorem, which asserts that the Pfaffian class of an oriented Riemannian vector bundle (of even rank) coincides with the Euler class of the associated sphere bundle. This theorem, combined with the theorems in Chapters VIII and X, volume I, implies that the integral of the Pfaffian class of the tangent bundle of a compact oriented manifold of even dimension is equal to the Euler-PoincarC characteristic of the manifold.

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Contents of Volumes I and I11 Volume I: De Rham Cohomology of Manifolds and Vector Bundles

0 Algebraic and Analytic Preliminaries I Basic Concepts I1 Vector Bundles I11 Tangent Bundle and Differential Forms IV Calculus of Differential Forms V De Rham Cohomology VI Mapping Degree VII Integration over the Fibre VIII Cohomology of Sphere Bundles IX Cohomology of Vector Bundles X The Lefschetz Class of a Manifold APPelndix A. The Exponential Map Volume 111: Lie Algebras, Algebraic Connections, and Characteristic

C1asses Part I I

Lie Algebras and Graded Differential Algebras

I1 Cohomology of Lie Algebras I11 IV V VI VII VIII IX

The Weil Algebra Operation of a Lie Algebra in a Graded Differential Algebra Algebraic Connections Cohomology of an Operation Subalgebras Operation of a Lie Algebra Pair The Classical Groups

Part I1 X Spectral Sequences XI Koszul Complex of a P-Algebra XI1 Koszul Complex of a P-Differential Algebra xxi

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Chapter 0

Algebraic and Analytic Preliminaries

SI. Linear algebra 0.0. Notation. Throughout this book iX denotes the ide:itity map of a set X . When it is clear which set we mean, we write simply L. If U., (i = 1,..., r ) are subsets of X , then Ua,.2....,denotes their intersection. T h e empty set is denoted by O . T h e symbols N, Z, Q, R, and C denote the natural numbers, integers, rationals, reals, and complexes.

0.1, We shall assume the fundamentals of linear and multilinear algebra. We will consider only real vector spaces (except for the occasional complex space). A pair of dual vector spaces is denoted by E*, E and the scalar product between E* and E is denoted by ( , ). If F C E, then

FL =

{JJ*

T h e dual of a linear map cp: E spaces EP is denoted

E

E* I ( y * , F )

+F

=

O}.

is denoted by cp*. A direct sum of

T h e determinant and the trace of a linear transformation rp: E-+ E are denoted respectively by det cp, tr cp. A determinant function in an n-dimensional vector space is a nonzero skew-symmetric n-linear function. Every nonzero determinant function A , in a real vector space defines an orientation. Given two vector spaces E and F, we shall denote by L ( E ; F ) the space of linear maps E -+ F. L ( E ; E ) will also be denoted by L, . Finally if E, , ..., Ep , and F are vector spaces, L(E, , ..., Ep ;F ) denotes the x Ep -+ F. space of p-linear maps E, x T h e group of linear automorphisms of a vector space E will be denoted by GL(E). 1

2

0. Algebraic and Analytic Preliminaries

A Euclidean space is a finite-dimensional real space, together with a positive definite inner product (also denoted by ( , )). A Hermitian space is a finite-dimensional complex space together with a positive definite Hermitian inner product (also denoted by ( , )). If F is a real vector space, make F" = C @ F into a complex space by setting

P(oI@x)=/~oI@x,

/3,ci~C, X E F .

F" is called the complexijication of F. If ( , ) is a positive definite inner product in F, then (a

0x, /3 @ y > c

=

&X,Y>,

01,

B E c,

X,Y EF

defines a Hermitian metric in F'. An indeJnite inner product in a finite-dimensional real vector space E is a non degenerate symmetric bilinear function ( , ). If E, is a maximal subspace in which ( , ) is positive definite, then E = E+ @ E i . T h e integer dim E+ - dim E $

is independent of the choice of E , , and is called the signature of ( , ). T h e symbol @ denotes tensor over R (unless otherwise stated); for other rings R we write OR. 0.2. Quaternionsand quaternionicvector spaces. Let H be an oriented four-dimensional Euclidean space. Choose a unit vector e E H , and let K = e l ; it is a three-dimensional Euclidean space. Orient K so that, if el , e2 , e3 is a positive basis of K , then e, e, , e2 , e3 is a positive basis of H . Now define a bilinear map H x H + H by P4= -(P,4>e+P pe

=p =

xq,

ep,

P,4EK PEH,

where x denotes the cross product in the oriented Euclidean space K. In this way H becomes an associative division algebra with unit element e. I t is called the algebra of quaternions and is denoted by W. T h e vectors of W are called quaternions and the vectors of K are called pure quaternions. Every quaternion can be uniquely written in the form p=Ae+q=X+q,

AER,

~ E K .

3

1. Linear algebra

X and q are called the real part and the pure quaternionic part of p . The conjugate p of a quaternion p = he q is defined by p = he - q. T h e map p + j5 defines an anti-automorphism of the algebra W called conjugation. T h e product of p and p is given by p p = I p e = 1 p 12. Multiplication and the inner product in W are connected by the relation

+

/,

(PY, Q Y )

=

P,4 , y E w.

(P,4) ,

I n particular,

lPl= IPIlrl,

P,YEW.

A unit quaternion is a quaternion of norm one. A pure unit quaternion q satisfies the relation q2 = -e. If ( e l , e, , e3) is a positive orthonormal basis in K , then e,e,

=

e3 ,

e2e3 = el

e3el = e2 .

,

0.3. Algebras. An algebra A over R is a real vector space together with a real bilinear map A x A --t A (called product). A system pf generators of an algebra A is a subset S C A such that every element of A can be written as a finite sum of products of the elements of S. A homomorphism between two algebras A and B is a linear map q ~ :A -+B such that

d X Y ) = 4 4 dY),

x, Y

E

A.

A derivation in an algebra A is a linear map 8: A =

@)Y

-+

A satisfying

+ XO(Y).

A derivation which is zero on a system of generators is identically zero. If O1 and 8, are derivations in A , then so is O1 o 8, - O2 o 8, . More generally, let v: A + B be a homomorphism of algebras. Then a ?-derivation is a linear map 8: A -+ B which satisfies O ( X Y ) = %4dY)

+ Y(X)O(Y)*

A graded algebra A over R is a graded vector space A = CP>,, Ap, together with an algebra structure, such that AP * AQC AP+Q.

If ~y =

(--l)P*y~, x E AD, y

E

Aq,

4

0. Algebraic and Analytic Preliminaries

then A is called anticommutative. If A has an identity, and dim Ao = 1, then A is called connected. If A and B are graded algebras, then A @ B can be made into a graded algebra in two ways:

Y d X 2 O Y Z ) = X l X 2 0YlY2 (x1 OYl)(”Z O Y 2 ) = ( - 1 ) Q I P Z X 1 X 2 O Y l Y 2 where x1 , x2 E A , y 1 ,y 2 E B, deg y1 = q1 , deg x2 =p , . The first algebra is called the canonical tensor product of A and B , while the second one is called the anticommutative or skew tensor product of A and B. If A and B are anticommutative, then so is the skew tensor product. An antiderivation in a graded algebra A is a linear map a: A + A , homogeneous of odd degree, such that (1) (2)

(x1 O

+ (-1)’

‘ Y ( x ~= ) O~(X)Y

x E A’,

X’Y(Y),

y

E

A.

If a1 and a2 are antiderivations, then a2 a1 + a1 0 a2 is a derivation. If a is an antiderivation and 8 is a derivation, then a 0 6 - 6 a is an antiderivation. T h e direct product A, of algebras A, is the set of infinite sequences {(xu) I x, E A,}; multiplication and addition is defined component by component. The direct sum C, A, is the subalgebra of sequences with finitely many nonzero terms. 0

0

n,

0.4. Lie algebras. A Lie algebra E is a vector space (not necessarily of finite dimension) together with a bilinear map E x E --+ E, denoted by [ , 1, subject to the conditions

[x,X I

=

0

and

0.5. Multilinear algebra. T h e tensor, exterior, and symmetric algebras over a vector space E are denoted by

0E = 1 O P E ,

AE

=

(If dim E

= n,

AE

1 APE, P>O

P>O

= C:=,

A pE.)

VE =

C Pa0

V”E.

5

1 . Linear algebra

If F is a second space, a nondegenerate pairing between E* @ F * and E Q F is given by X* E

(x* @ y * , x @ y ) = (x*, x)(y*,y),

E*, y* E F * , x E E, y E F .

If E or F has finite dimension, this yields an isomorphism E* @ F* ( E @ F)*. I n particular, in this case (OPE)* O P E * . Similarly, if dim E < co, we may write ( A P E ) * = APE*, ( VqE)* = VqE* by setting (x*1

A

..-A x*p, x1 A

A

xs) = det((x*i, xi))

and (y*l v

v y*P, y1 v

v y,) = perm((y*i, yJ),

where “perm” denotes the permanent of a matrix. T h e algebras of multilinear (resp. skew multilinear, symmetric multilinear) functions in a space E are denoted by

T h e multiplications are given respectively by (@

0Vl)(Xl

1

--a,

X,+J

= @(Xl

,

**a,

x,) U‘(X,+l,

...,x,+,)

and

Here S* denotes the symmetric group on p objects, while E, = & I according as the permutation a is even or odd. If dim E < co, we identify the graded algebras T ( E ) and @E* (resp. A ( E ) and AE*, S ( E ) and V E * ) by setting @(xl Y(xl

, ..., xP) = (0, x1 @ , ..., x,) = (Y, x1 A

@ x,), -*-

A

x,),

@E OPE*

Y E APE*

0. Algebraic and Analytic Preliminaries

6

and X ( X , ,...,x P ) = ( X , x1 v

***

v xB),

X E VPE*.

A linear map y : E +F extends uniquely to homomorphisms @p:

@E

+O

F,

AT: AE --+ AF,

V y : VE -+ VF.

These are sometimes denoted by y o , y,, , and y v . T o each x E E we associate the substitution operator i ( x ) : A ( E )--+ A ( E ) , given by (i(x)Q,)(xl

...I

xp-1)

= @(x,

3

xs-l),

Q, E

Ap(E), P >, 1,

Q, E Ao(E),

i(x)Q, = 0,

and the multiplication operator p(x): A E 3 A E given by p(x)(a) = x

A

a,

a E AE,

i ( x ) is an antiderivation in A ( E ) and is dual to p(x).

$2.

Homological algebra

0.6. Rings and modules. Let R be a commutative ring. If M , N are N is again an R-module (cf. R-modules, then the tensor product M 8, [4, p. AII-561 or [5, $8, Chap. 31). If Q is a third R-module and if y : M x N -+ Q is a map satisfying the conditions

d x + Y , u ) = d x , + 94% u ) + v ) = d x ,u> + T(X, v )

(1)

(2) d x , u and

(3) d h x , u> = d x ,Au) for x,Y E M , u, V E N , A E R, then there is a unique additive map 1+4: M OR N Q such that -+

P)(x, U ) = +(x @ u),

x E M,

uEN

(cf. [4, Prop. I(b), p. AII-511 or [5, $8, Chap. 31). If (iii) is replaced by the stronger cp(xX,

U) =

X ~ ( X , U) = P)(x,XU),

x E M,

u E N , X E R,

then t,b is R-linear. T h e R-module of R-linear maps M -+ N is denoted by Hom,(M; N ) . Hom,(M; R) is denoted by M*. A canonical R-linear map a: M*

ORN - +

HOmR(M; N )

is given by a ( f 0U)(.)

=f ( x ) u ,

x E M,

U E

N , f E M*.

A module M is called free if it has a basis; M is called projective if there exists another R-module N such that M @ N is free. If M is projective and finitely generated, then N can be chosen so that M @ N has a finite basis. If M is finitely generated and projective, then so is M*, and for all R-modules N , the homomorphism a given just above is an isomorphism. In particular, the isomorphism M*

ORM

HomR(M;M ) 7

0. Algebraic and Analytic Preliminaries

8

specifies a unique tensor tME M*

Q R

M such that

(Y(tM) = LM

.

It is called the unit tensor for M . A graded module is a module M in which submodules Mp have been distinguished such that M =

C

Mp,

P>O

The elements of

Mp

are called homogeneous of degree p . If x

p is called the degree of x and we shall write deg x

E MP,

then

=p .

If M and N are graded modules, then a gradation in the module M QRN is given by (MORN)"=

C

MP@RN*.

P+n=T

An R-linear map between graded modules, q: M + N , is called homogeneous of degree 12, if v ( M p )C Np+k,

p 30

An R-linear map which is homogeneous of degree zero is called a homomorphism of graded modules. A bigraded module is a module which is the direct sum of submodules M p q p 2 0, q 2 0). An exact sequence of modules is a sequence * * a

11-1 1t -+ Mi-l --+ Mi +

-

.*.,

where the qi are R-linear maps satisfying ker vi = Im vi-l.

Suppose

iw1--,M , --+ 11

1 2

M , 5 M~ -%M~

is a commutative row-exact diagram of R-linear maps. Assume that the maps a1 , az , a4 , a5 are isomorphisms. Then the Jive-lemma states that a3 is also an isomorphism.

2. Homological algebra

9

On the other hand, if 0

0

0

1

1

1

1

1

1

1 1

1 1

0

0

1 10

is a commutative diagram of R-linear maps with exact columns, and if the middle and bottom rows are exact, then the nine-lemma states that the top row is exact. An algebra over R is an R-module A together with an R-linear map A ORA + A. I n particular if M is any R-module, the tensor, exterior, and symmetric algebras over M are written ORM , A, M and, V RM . If M is finitely generated and projective, there are isomorphisms, (8: M ) * 0 sM * , (A: M ) * A: M * , (V: M ) * V: M*, defined in exactly the same way as in sec. 0.5. 0.7. Differential spaces. A dzflerential space is a vector space X together with a linear map 6: X -+ X satisfying a2 = 0. 6 is called the dzflerential operator in X . T h e elements of the subspaces

Z ( X ) = ker 6

and

B ( X ) = Im 8.

are called, respectively, cocycles and coboundaries. T h e space H ( X ) = Z ( X ) / B ( X )is called the cohomology space of X . A homomorphism of differential spaces rp: (X, 6,) -+ ( Y ,6,) is a linear map for which v 6, = 6 , o v. I t restricts to maps between the cocycle and coboundary spaces, and so induces a linear map 0

rp* : H ( X ) -+ H ( Y ) .

A homotopy operator for two such homomorphisms, rp, $, is a linear map h: X -+Y such that

If h exists then y e

= +#

9,-4 = h 0 8 + 6 0 h . .

10

0. Algebraic and Analytic Preliminaries

Suppose 0 -+

f x -+

g Y -+ 2

40

is an exact sequence of homomorphisms of differential spaces. Every cocycle z E 2 has a preimage y E Y. I n particular, g(6y) = 62 = 0

and so there is a cocycle x E X for whichf(x) = Sy. T h e class E H ( X ) represented by x depends only on the class 5 E H ( 2 ) represented by z. T h e correspondence 5 ++ f defines a linear map

a: H ( Z ) + H ( X ) called the connecting homomorphism for the exact sequence. T h e triangle

f*

H(X)

is exact. If O-+X-Y--+Z---tO

is a row-exact diagram of differential spaces, then

at

(a,

xti

= Fa

a

3' the connecting homomorphisms).

0.8. Graded differential spaces and algebras. A graded space = &,>o X p together with a differential operator 6 homogeneous of degree 1 is called a graded differential space. In such a case the cocycle, coboundary, and cohomology spaces are graded:

X

+

P ( X ) = Z ( X ) n Xp,

Bp(X) = B ( X ) n Xp

and H q X ) = ZqX)/BqX).

11

2. Homological algebra

A homomorphism of graded dtffaential spaces is a homomorphism of differential spaces, homogeneous of degree zero. Now assume that X has finite dimension and let v: X - + X be a homomorphism of graded differential spaces. Let @':

XP

Xp

-+

and

(y#)P: HIJ(X)+ Hr'(X)

be the restrictions of p and p# to X p and H p ( X ) . T h e algebraic Lefschetx formula states that

= L , we obtain the

I n particular, if

C P>O

(-l)P

dim XP =

Euler-Poincare'formula

1

(-l)P

dim HP(X).

P20

A graded differential algebra A is a graded algebra together with an antiderivation, 6 , homogeneous of degree one such that li2 = O.7n this case Z ( A ) is a graded subalgebra and B ( A ) is a graded ideal in Z(A). T h u s H ( A ) becomes a graded algebra. It is called the cohomology algebra of A. If A is anticommutative, then so is H ( A ) . A homomorphism of graded differential algebras y : A -+ B is a map which is a homomorphism of graded differential spaces and a homomorphism of algebras. It induces a homomorphism between the cohomology algebras, q # : H ( A ) + H(B). Next let A and B be graded differential algebras and consider the skew tensor product A Q B. Then the antiderivation in A Q B , given by 6(Xoy)=6X~y+(-l)Pxosy,

X E A P ,

YEB,

satisfies a2 = 0. Thus A Q B becomes a graded differential algebra. T h e tensor multiplication between A and B induces an isomorphism

-

-

N ( A )@ H ( B )4 H ( A @ B )

of graded algebras. It is called the Kiinneth isomorphism.

s3. Analysis and topology 0.9. Smooth maps. Let E, F be real, finite dimensional vector spaces with the standard topology. Let U C E be an open subset. A map v: U +F is called dzfferentiable at a point a E U if for some #a E L(E;F )

h E E. I n this case 4,is called the derivative of We shall write

v

at a and is denoted by ~ ' ( a ) .

y'(a; h) = rp'(~)h= $a(h),

h E E.

If is differentiable at every point a E U , it is called a dzyeerentiable map and the map y': U + L(E;F) given by a I+ ?'(a) is called the derivative of v. Since L(E;F ) is again a finite dimensional vector space, it makes sense for v' to be differentiable. In this case the derivative of v' is denoted by v";it is a map y": U

-

L(E;L(E;F))= L(E,E;F).

More generally, the kth derivative of y'k)':

v (if it exists) is denoted by T ( ~ ) ,

U -+ L(E,..., E;F). k

terms

For each a E U , v ( k ) ( a )is a symmetric k-linear map of E x x E into F. If all derivatives of cp exist, v is called infinitely dzyerentiable, or smooth. A smooth map v: U -+ V between open subsets U C E and V C F is called a di@omorphism if it has a smooth inverse. Assume now that y : U + F is a map with a continuous derivative such that for some point a E U y'(a): E 5

-F

is a linear isomorphism. Then the inverse function theorem states that there are neighbourhoods U of a and V of v(a) such that v restricts to a diffeomorphism U -% V . 12

3. Analysis and topology

13

We shall also need the basic properties of the Riemannian integral of a compactly supported function in Rn (linearity, transformation of coordinates, differentiation with respect to a parameter). T h e theory extends to vector-valued functions (integrate component by component). Finally, we shall use the Picard existence and uniqueness theorem for ordinary differential equations as given in [6, p. 221. 0.10. The exponential map. Let E be an n-dimensional real or complex vector space and let u: E + E be a linear transformation. It follows from the standard existence theorems of differential equations that there is a unique smooth map T : [w +LE satisfying the linear differential equation +=a07

and the initial condition ~ ( 0= ) L . T h e linear transformation ~ ( 1 )is called the exponential of u and is denoted by exp u. In this way we obtain a (nonlinear) map exp: LE +LE. I t has the following properties:

(0) (1) (2) (3)

e x p o = L. If u1 o u2 = u2 o u1 , then exp(ul + u2) = exp u1 o exp u2 . exp(ku) = (exp u ) ~k, E Z. det exp u = exp tr u. (4) If a Euclidean (Hermitian) inner product is defined in the real (complex) vector space E and if u* denotes the adjoint linear transformation, then exp a* = (exp u)*.

(All these properties are easy consequences of the uniqueness theorem for solutions of differential equations.) Relations (0) and (1) imply that exp u is an automorphism with (exp u)-' = exp(-u). In particular, if u is self-adjoint, then so is exp u and if u is skew (resp. Hermitian skew), then exp u is a proper rotation (resp. unitary transformation) of E. I n terms of an infinite series we can write

0.1 1. General topology. We shall assume the basics of point set topology: manipulation with open sets and closed sets, compactness, Hausdorff spaces, locally compact spaces, second countable spaces, connectedness, paracompact spaces, normality, open coverings, shrinking of an open covering, etc.

14

0. Algebraic and Analytic Preliminaries

T h e closure of a subset A of a topological space X will be denoted by

A. If A and B are any two subsets of X , we shall write A -B

= {x E

A 1 x 4 B}.

A neighbourhood of A in X will always mean an open subset U of Xsuch that U 3 A. An open cover of X is a family 0 of open sets whose union is X . It is called locally finite if every point has a neighbourhood which meets only finitely many elements of 0.0 is called a rejinement of an open cover @ if each 0 E 0 is a subset of some U E a. X is called paracompact if every open cover of X has a locally finite refinement. A basis for the topology of X is a family 0 of open sets such that each open subset of X is the union of elements of 0. If 0 is closed under finite intersections, it is called an i-basis. If X has a countable basis, it is called second countable.

s4. Summary of volume I 0.12. Manifolds and vector bundles. All manifolds are smooth (i.e., infinitely differentiable), second countable, Hausdorff, and finite dimensional. T h e set of smooth maps between manifolds M and N is written Y ( M ;N ) . If y : M .+ N has a smooth inverse, it is called a dzgeomorphism. Y ( M ) denotes the algebra of smooth real-valued functions on M . If A and A’” are Y(M)-modules, then A BMA’”, Horn,(&; A’”), and A denote the obvious linear and multilinear constructions, taken over Y(M). A vector bundle is a quadruple f = ( E , n,B, F ) where: (1) n: E+ B is smooth; (2) F and each set F, (= n-l(x)) is a finite-dimensional vector space; and (3) there is an open cover { U,> of B and a system of diffeomorphisms 4,:U, x F 5 n-lU, such that $, restricts to linear isomorphisms $ a , x : F 5 F, (x E U,). E, B, and F are called the total space, base space, and typical fibre of 5; n is called the projection. F, is called the fibre at x. T h e dimension of F is called the rank of 4. T h e collection {( U , , $=)}is called a coordinate representation for 5. If E = B x F and n is the obvious projection, f is called trivial. Let 8’ = (E‘, r’,B‘, F’) be a second vector bundle. A bundle map or homomorphism 4 + 5‘ is a smooth map y : E + E‘ that restricts to linear maps y,: F, F&(,) , x E B. T h e correspondence x tt $(x) defines a smooth map $: B + B‘. If $ = L , then y is called a strong bundle map. The Cartesian product of 5 and 4‘ is the vector bundle f x 5‘ = ( E x E , n x n’, B x B‘, F @ F‘). If F ‘ = 0 (so that E’ = B’, n’ = L ) , we write simply 5 x B’. A vector bundle 4 determines vector bundles f * , o p t , A 5, V q f , whose F, , A F, and Vq F,. If is a second vector fibres at x are the spacesF2, bundle with the same base and with typical fibre H , then 5 @ q , 4 8 q , and L(5;q ) denote the vector bundles with fibres F, 0H , ,F, O H , , and L(F, ; H,). 6 @ q is called the Whitney sum of 4 and q. T h e bundle L( f ; 5) is written L , . T h e cross-sections in f are the smooth maps u: B + E which satisfy r 0 u = L. T h e carrier of u, carr a, is the closure of the set of x E B such that u(x) # 0. T h e operations ---f

of’

(0

+4 4

=).(.

+ 44,

(f.4 4 = f(44 .)

make the cross-sections into an Y(B)-module; it is denoted by Sec 4. 15

0. Algebraic and Analytic Preliminaries

16

Apseudo-Riemannian metric in ( is a smooth assignment to the fibres F, of inner products g ( x ) (also written ( , ), , or simply ( , )). Thus g E Sec V2(*. If each g ( x ) is positive definite, g is called a Riemannian metric. Suppose rank ( = r . An orientation in ( is an equivalence class of nowhere vanishing cross-sections in A'(* under the equivalence relation: d, A , if d, = f * A , for some f E Y ( B ) , with f ( x ) > 0, x E B. A cross-section in one of these classes is called a determinant function in 5 which represents that orientation.

-

0.13. Tangent bundle and differential forms. Let M be an n-manifold. T h e tangent space, T,(M), at x E M is the space of linear maps E : Y ( M )-+ R, which satisfy ((f g ) = ((f) * g(x) f (x) * C(g). T h e tangent bundle of M , written 7, = ( T , , T,M , R.), is the vector bundle whose fibre at x is T J M ) . The derivative of a smooth map v: M -+ N is the bundle map d y : T , ---f TNwhose restriction to T J M ) is given by

+

((ddZO(f1 =

E(f0F-41

fE

ww

If each ( d q ) , is surjective, is called a submersion; if also 9 is surjective, then N is called a quotient manifold of M . If d v is injective, ( M , v) is called an embedded manifold; if in addition y is a homeomorphism onto ?( then & M isI called ),a submanifold of N . Let a ( t ) ( t o < t < t l ) be a smooth path in M. Then &(t)E T a ( t ) ( Mis) defined by

A vector field on M is a cross-section X in r M ; the module of vector fields is denoted by X ( M ) . An orbit of X is a smooth path a ( t ) such that 4 ( t ) = X ( a ( t ) ) . T h e Picard theorem asserts that for each EM there is a unique orbit of X through x. If X E X ( M ) and f E Y ( M ) , then X ( f ) E Y ( M ) is defined by ( X (f ) ) ( x ) = X ( x ) (f ) . T h e Lie product, [ X , Y ] ,of X , Y E % ( M ) is the unique vector field satisfying

[X,Y l ( f )

=

X(Y(f)) - Y(X(f))*

Vector fields X E X ( M ) , Y E X ( N ) are called ?-related with respect to a smooth map y : M -+ N if (d?),X(x) = Y(v(x)),x E M . In this case we write X 7 Y. If g, is a diffeomorphism, v * ( X ) denotes the unique vector field on N which is prelated to X . A dzflerential form on M is a cross-section, 0,in A T & . If each @(x) E APT,(M)*, then @hasdegreep. T h e differential forms are a graded

4. Summary of volume I

17

algebra, A ( M ) = C, A p ( M ) ,with multiplication given by (@ A Y)(x) = A Y ( x ) . Ap(M) can be regarded as the space ofp-linear (over Y ( M ) ) skew-symmetric maps X ( M ) x x % ( M ) Y ( M ) ,via the equations

@(x)

--f

(@(&

9 ...I

XPM4

= @(x; - w x ) ,

..., X,(X)).

Then

.’.,X,,,)

(@ A V(X1 9

A q~*:

smooth map q ~ :M -+ N A ( M ) +- A ( N ) defined by (cp*@)(x; E l

, *.*, 6,)

determines the

= @(cp(x); (dcp) E l ,

**., (dP)[ P I ,

homomorphism

P21

(cp*f)(x) = f(cp(xN.

T h e maps M , N + M x N given by x t+ (x, b) and y t+ (a, y ) are called the inclusions opposite b and a. Their derivatives define an isomorphism T,(M) @ T,(N) % T(a+b)(M x N ) ; these isomorphisms in turn identify T~ x T~ with T M ~ .N In particular, X E X ( M ) determines the vector field, i,X, in X ( M x N ) given by (x, y ) t-t ( X ( x ) ,0); it is also denoted by iLX . Moreover, the induced isomorphisms, AT(a,b)(Mx N ) * g

1 APT,(M)* @ AQT,(N)*, P,Q

define a bigradation in A ( M x N ) : A ( M x N ) = &,gAP$Q(Mx N ) . E A Q ( N ) then , 0 x Y E Ar’,Q(M x N ) denotes q)-form given by T$@ A T$Y ( r MM : x N M, the ( p rN: M x N -+ N are the obvious projections). T h u s (@ x Y ) ( a , b) = @(a) 0Y ( b ) . T h e substitution operator i ( X ) , the Lie derivative O(X),and the exterior derivative 6 are the operators in A ( M ) , homogeneous of degrees -1, 0, and 1, defined, respectively, by

-

If@ E & ( M ) and Y

+

(i(X)@)(&,.’*,X,) = @ ( X ,X , , ..., X,), D

(&q@)(X, , ..., X,)

=

X ( @ ( X ,, ..., X,)) -

1 @(Xl, ..., [ X ,XJ, ..*,X,), j=1

(8@)(X,, ..., X,) =

9

1 (-l)jX,(@(Xo , ..., xi,..., X,)) i=O

0. Algebraic and Analytic Preliminaries

18

and

They are respectively an antiderivation, a derivation, and an antiderivation. These operators satisfy the relations

and 62

= 0.

Moreover, if v: M -+ N is smooth and X T Y , then i ( X )o p)*

=

v* o i ( Y )

and

O(X)o v*

=

v* o B(Y).

I n any case v* 0 6 = 6 0 v*. Let F be any finite-dimensional vector space. T h e cross-sections in the bundle L(AT$ ; M x F ) (respectively, L(APT$ ; M x F)) are called differential forms with values in F (respectively, p-forms with values in F); these modules are denoted by A ( M ;F ) and AP(M;F). If Q E AP(M;F), then Q(x) is a skew-symmetric, p-linear, F-valued function in T,(M). An isomorphism A ( M ) @ F + A ( M ; F ) is given by (@

0a)(x; t1 , *..,

ED)

= @(xi

t1

,

..a,

69)

. 0.

T h e operators i(X) @ L, O(X) @ L, and 6 @ 6 in A ( M ;F ) are denoted simply by i(X), O(X),and 6; they satisfy the relations given above in the case F = R. A smooth map v: M --+ N induces a map lp* = (v* 0L): A ( M ;F ) + A ( N ;F ) .

If a : F + H is linear, we define a*: A ( M ; H ) --t A ( M ; H ) by 6,) = a(@(x;f 1 ,..., f,)). a* commutes with the operators i(X), O(X),6, and v*. An orientation of M is an orientation of T~ ; thus it is an equivalence class of nowhere vanishing n-forms. A smooth map v: M --t N (dim M = dim N ) is called orientation preserving (respectively, orientation reversing) if v*d (respectively, -v*d) represents the orientation of M when d represents that of N . T h e space A,(M) of differential forms with compact carrier is an ideal ( a * @ ) ( x ; f 1 ,...,

4. Summary of volume I

19

in A ( M ) . Assume M oriented and of dimension n. Then the integral is defined; it is a linear map J M : A t ( M ) -+ R, natural with respect to orientation preserving diffeomorphisms, and satisfying

where A , is the positive normed determinant function of an oriented Euclidean space E. T h e integral extends to the linear map J M = JM @ L : A,(M; F ) +F. If a: F + H is linear, then a o J M = J M o a* . I n particular, suppose dim E = n 1, and Snis the unit sphere in E. Then T,(Sn) = XI, x E Sn, and the n-form, Q, on Sn given by Q(x; h, ,..., h,) = A E ( x ,h, ,..., h,) orients Sn.Its integral is called the volume of Snand is given by

+

2m+l 1 . 3 ... (2m 2 ,m+l

-

1)

nm,

n

=

2m, m 2 1

n=2m+1,

m!

m>O.

0.14. De Rham cohomology. Let M be an n-manifold. Then ( A ( M ) ,6) is a graded differential algebra; its cohomology is denoted by H ( M ) = C= :, HP(M)and is called the de Rham cohomology algebra of M . T h e homomorphism v*: A ( M ) +- A ( N ) determined by a smooth map induces a homomorphism v#: H ( M ) t H ( N ) . If dim H ( M ) < 00, then the pth Bettinumber, b, ,of Mis dim Hp(M).

T h e polynomialf(t) number

=

Ep bptp is called the Poincare'polynomial and the

c (-l)Pbp n

x,

=

V=O

is called the Euler-PoincarC characteristic of M . If M is compact, then d i m H ( M ) < 00. Smooth maps y , #: M -+ N are homotopic if there is a smooth map H : R x M ---f N such that H(0, x) = ~ ( x and ) H(1, x) = #(x). H is called a connecting homotopy. T h e operator h: Ap(N) -+ Ap-l(M) given by

0. Algebraic and Analytic Preliminaries

20

is called the homotopy operator induced from H ; it satisfies

#*

-p,*

=h

o S +Soh.

In particular, if v and $I are homotopic, then rp# = $I*. T h e ideal A,(M) is stable under 6 and the corresponding cohomology algebra is denoted by H,(M). Multiplication of differential forms makes H,(M) into a left and right graded H(M)-module, and these structures are denoted by ( a , 8) ++ a

*B

(B, a )

and

B * a,

+F

aE

H(M),

BEW M ) .

Assume M is oriented. Then J”,, 0 6 = 0 and so J, induces a linear map J”;: Hg(M) R. T h e Poincare‘ scalar product is the bilinear map PM:H(M) x H,(M) -+ R, given by --f

B)

=

P)M(a,

J# cx * B,

H”(M),

a

B E

ff:-D(M),

M

and deg a

PM(a, p) = 0,

+ deg B # n.

I t induces an isomorphism D M :H(M) 5 H,(M)*, called the Poincart isomorphism. In particular, if M is connected, ;1 is an isomorphism; i.e., ker J

=

ImS.

M

T h e unique cohomology class u,,,E HE(M) such that ;J w,,, = 1 is called the orientation class. If M is compact, then H,(M) = H(M) and SO bp = b,, . T h e map @ @ Y I--, @ x Y (cf. sec. 0.13) defines homomorphisms K:

A ( M )@ A ( N )

-+

A(M x N )

and

K

~

A: c ( M )@ Ac(N)-+ Ac(M x N ) .

These induce the Kiinneth homomorphisms K#:

H ( M ) @ H ( N ) - + H ( M x N ) and

H c ( M )@ H c ( N ) + Hc(M x N ) .

(K~)#:

( K J # is always an isomorphism, while K # is an isomorphism if either H(M) or H(N) has finite dimension. Suppose rp, $: M N are smooth maps between compact connected oriented n-manifolds. T h e degree of 9 is the integer, deg y, defined by ---f

1

M

p,*@ =

degp,

1,

@,

@ E A”(N).

4. Summary of volume I

21

Let @ restrict to ~ ( pin) H p ( N ) and let $ p ) : HP(M) --+ H p ( N )be the dual of I,U~-~) with respect to the PoincarC scalar products. Then the coincidence number of and $ is the alternating sum n

~ ( r p ,+) =

1 ( - 1 ) ~ tr(rp(9)

0 @,)I.

9=0

If M

=

N , then the Lefschetz number of

cp

is the alternating sum

n

L(rp) =

1 ( - 1 ) ~ tr

rp‘p).

p=o

0.15. Smooth fibre bundles. A smooth fibre bundle is a quadruple = ( E , T , B, F ) where (1) E, B, F are manifolds (total space, base space, typical fibre) and T : E + B is smooth, and (2) there is an open cover {Urn}of B and a family of commutative diagrams

93’

U, x F

$a

\=/

r-’U, ($, , a diffeomorphism).

urn For x E B, n - l ( x ) is a closed submanifold of E ; it is denoted by F, and is called theJibre at x. Thus $, restricts to diffeomorphisms +,: F 5 F, . T h e family {( U , , $,)} is called a coordinate representation for @. If 9‘= (E’, T ’ , B’, F ) is a second bundle, a smooth map 9:E + E’ is calledfibrepreserving if it restricts to smooth maps q~,: F, 4F;(%)(x E B). T h e induced map $: B -+ B’ is smooth. Fix @ = ( E , T , B, F). T h e spaces T,(F,) ( z E F, , x E B ) are the fibres of a subbundle of T , ; it is called the vertical subbundle and is denoted by ( V , , p, E, Rr) ( r = dim F ) . T h e fibre at z E E is written V s ( E ) and called the wertical subspace at z ; thus V,(E) = ker(dn),. A horizontal subbundle is a subbundle HE of r Esuch that r E = HE 0 V , . Its fibre at z is called the horizontal subspace (with respect to the choice of HE)and is written H,(E). If 9 is a vector bundle, then T,(F,) = F, . These identifications define a bundle map a : V , --t E inducing T : E -+ B, and restricting to isomorphisms in the fibres. An orientation in V , is called an orientation in 3;thus an orientation of a is a smoothly varying orientation of the fibres F, . If Y E A‘(E) and its restriction to each F, represents the orientation of F, , then Y is said

0. Algebraic and Analytic Preliminaries

22

to represent the orientation of g.If @ orients the manifold B, then r*@ A Y orients E ; this orientation depends only on the orientations of g and B , and is called the local product orientation. If 33' is a vector bundle, the definition above is a second definition of an orientation in 9#;in this case, we use the map a, above, to identify orientations in g as defined in sec. 0.12 with orientations in the vertical bundle. T h e space A,(E) of diflerential forms with fibre compact carrier consists of those @ such that carr @ n +(K) is compact whenever K is a compact subset of B. If F is compact, then AF(E)= A ( E ) ;while if B is compact, then AF(E)= A,(E). Let Q E Ap+'(E) ( r = dimF). Fix vl ,..., q,, E T,(F,); then, for tiE T,(E), Q(z; t1 ,..., t p ,vl ,..., qr) depends only on the vectors ti= (d7r),t4(E T,(B)). Thus a APT,(B)*-valued r-form, Q, on F, is given by

(a,(z;71 , ..., Tr), 5,

A

'*'

A

6,)

= Q(z; 51

9

..., 5,

3

Ti

3

...,77)~

(dr)ti = t i *

. If s2 E A,(E), then each Q, has compact carrier. is oriented; then an orientation is determined in each Suppose manifold F, . T h e fibre integral is the linear map .jF:A,(E) -+A(B), homogeneous of degree -r, given by

Q, is called the retrenchment of Q toF,

It is surjective and satisfies jFli*@ A

a=@ A

i,Q

and

j;S

.

= S o j F

If B is oriented and E is given the local product orientation, then the Fubini theorem asserts that D E Ar(E), m

=

dim E.

0.16. Sphere bundles. An r-sphere bundle is a smooth bundle with fibre the r-sphere. If 4 = (E, T , B, F) is a vector bundle with a Riemannian metric, then the unit spheres S, C F, are the fibres of a sphere bundle 4, = ( E , , T ~B ,, S ) called the associated sphere bundle. An orientation in 8 defines an orientation in the fibres F, ; the induced orientations in the spheres S, (cf. sec. 0.13) define an orientation in

es.

4. Summary of volume I

23

Suppose 28 = ( M , 7r, B, S ) is an oriented r-sphere bundle. Then there are differential forms SZ E Ar(M),@ E Ar+'(B)such that .fsQ = - 1 and 7r*@ = 6Q (thus 6@ = 0). T h e cohomology class represented by @ (in Hr+'(B)) depends only on the oriented bundle 28. I t is called the Euler class of 28 and is written X, . = ( M , T , B , S ) be an oriented sphere bundle with dim B = Let n = dim S 1. Assume B is oriented. A cross-section in 5@ withfinitely many singularities a, ,..., ak is a smooth map a: B - {al ,..., ak} -+ M such that 7r 0 u = i. (Such cross-sections always exist, if k 1.) Using the local product structure we obtain, from a, smooth maps

+

ui:U i- { a i }4 S ,

where U, is a neighbourhood of ai . T h e orientation of U, determines an orientation in a 'sphere' Siabout a, . Let T , be the restriction of u, to Si; then the degree of ri is independent of the various choices. It is called the index of u at a, and is written j J u ) or simplyj,(o). T h e sum j(u) = x,ji(u) is called the index sum of u. It satisfies the relation -4t

J - x,

=j(u).

B

Moreover, if 28 is the associated sphere bundle of the tangent bundle of a compact oriented n-manifold B, then X, E H"(B) and

Chapter I

Lie Groups

SI. Lie algebra of a Lie group 1.1. Definition: ALiegroup is a set G which is both a group and a smooth manifold, and for which the following maps are smooth:

-

(i) T h e multiplication map p: G x G 4 G given by (x, Y )

(ii) T h e inversion map

V:

XY *

G -+ G given by x

H x-1.

T h e unit element of a Lie group is denoted by e. A homomorphism of Lie groups 9:G -+ H is a smooth homomorphism of groups. An isomorphism o j Lie groups is a map that is both a homomorphism and a diffeomorphism. Let G be a Lie group. Each a E G determines smooth maps A, , pa: G -+ G, given by ha(%)= ax

and

p a ( x ) = xu.

They are called left and right translation by a. T h e group axioms yield the relations /\a ' / \ b = /\ab > Pa 'PO = Pba and ha o Pb = p b 0 h a . he = p e = L , 3

I n particular, A, and p b are diffeomorphisms, with inverses X,-I and Pa-1 We shall denote the derivatives of A,, Pb by La = dh,: T G + TG

and

Rb = dp,: T G + T G .

T h e relations above yield the relations Ra 0 Rb = Rb, , La OL, = Lab, La O R ,= R b "La R = L = L T ~ , and 24

I

.

1 . Lie algebra of a Lie group

25

If cp: G -+ H is a homomorphism of Lie groups, then

v

s,

and

v

dp, oLa = La(,)0 ds,

and

dp, 0 Rb = Rm(b) 0 dp,.

0

ha

h a )

1

0

0

Pb

= Pm(b)

0

V*

Hence I n particular, each ( d ~ ) )Tz(G) ~ : -+ T q ( z ) ( H(x ) E G ) is injective (respectively, surjective) if and only if (dcp), is. Now consider the multiplication and inversion maps. Their derivatives are bundle maps dp:

TG

x

TG

--f

T,

and

dv:

T,

--f

TG

.

and

Proof: (1) Let j u : G ---t {a} x G and j b : G -+ G x {b} denote the inclusions opposite a and b respectively. Then

dd'!,7)= ( d p

dib)('!)

+ (dp

dj,)(T)

= Rb(t)

+ La(T).

(2) Since x w p(x, v ( x ) ) is the constant map G -+ e, we have

4 4 5 , d V ( 0 ) = 0. Now (2) follows from (1).

Q.E.D. 1.2. Invariant vector fields. T h e left and right translations of a Lie group G induce automorphisms (A), and (p,)* of the real Lie algebra, %(G), of vector fields on G (cf. sec. 0.13). A vector field X on G is called left invariant if L,(X(x)) = X(ax), a, x E G ; i.e., if (A,),X = X , a E G . In view of Lemma I, sec. I . 1, this is equivalent to iRX

-

X

Y

(iRX(X, Y ) = (0, X(Y)>* Since each (A,)* preserves Lie products, the left invariant vector fields

form a subalgebra, Z L ( G ) ,of %(G).

I. Lie Groups

26

A strong bundle isomorphism

Proposition I: given by

01

x Te(G)

TGis

-w).

(a, A )

Proof: by

a: G

restricts to isomorphisms in the fibres. Moreover it is given

&(a,A )

,A )

= 440,

(cf. Lemma I, sec. 1.1) and hence it is smooth.

Q.E.D. Corollary I:

An isomorphism XL(G) 5 Te(G) is given by X

M

X(e).

In particular dim XL(G) = dim G. Corollary 11: An isomorphism of Y(G)-modules

-

XL(G) 0Y ( G )5 T ( G )

is given by

X

@ f tt f

X.

Definition: Let h E Te(G). T h e unique left invariant vector field X such that X ( e ) = h is denoted by X,,and is called the left invariant vector field generated by h.

Similarly, a vector field Y is called right invariant if (pb)*Y= Y , b E G. T h e Lie algebra of right invariant vector fields is denoted by T R ( G ) The . same proof as given in Proposition I shows that Y w Y(e)

defines an isomorphism XR(G) 5 Te(G). T h e right invariant vector field corresponding to h E Te(G)under this isomorphism is called the right invariant vectorfieldgenerated by h, and is denoted by Yh. Proposition 11:

If X

E

XL(G) and Y E %R(G), then [ X , Y ] = 0.

Proof:

Define i L Y E %(G x G) by i L Y ( x ,y) iRX

N

U

X

and

iLY -,

U

=

Y,

( Y ( x ) ,0). Then

27

1. Lie algebra of a Lie group

and it follows from Proposition IX, sec. 3.14, volume I, and Proposition VIII, sec. 3.13, volume I, that 0

=

[Z'RX,iLY] 7[ X , Y ] .

Since p is surjective, [ X , Y ] = 0.

Q.E.D. Finally, consider the inversion map v: x t-t x-l of G. Since v2 a diffeomorphism. Clearly, v

0

ha = pa-'

0

I n particular,

dv 0 La = Ra-l 0 dv,

V,

v*

and

v*

0

=

(ha), = (pa-&

L,

v is

0

v.+.

restricts to an isomorphism

--

TdG)

TAG)

of Lie algebras. In view of Lemma I (2) sec. 1.1, we have v*Xh = -Yh ,

and hence, for h, k

E

h E Te(G),

T,(G), [Xh

, XkIM

= -[yh

>

YkI(4

1.3. Lie algebra of a Lie group. T h e Lie algebra of a Lie group G is the vector space, T,(G), together with the Lie algebra structure induced from S L ( G )by the isomorphism of Corollary I to Proposition I, sec. 1.2. Thus, for h, k E T,(G), [h, kl = [Xh Xkl(4 9

(Note that the isomorphism S,(G) 5 T,(G) determines a second Lie product [ , 3" in T,(G). In view of sec. 1.2, we have [h, k]

=

h, k

-[h, k]",

E

Te(G).

Thus the map h ++ -h defines an. isomorphism between these Lie algebra structures.) Now consider a homomorphism of Lie groups, s,: G H . Since y(e) = e ( e denotes the unit of both groups), the derivative ds, restricts to a linear map --f

(dq)e: Te(G)

+

This map will be denoted by s,'.

Te(H).

28

I. Lie Groups

Proposition 111:

v' is a homomorphism

of Lie algebras.

Proof: I t follows from sec. 1.1 that

Hence [ X h, X,] 7 [X,,, , Xvtk]. Evaluate this relation at e to obtain v"h, k] = [v'h, v'kl.

Q.E.D.

If #: H

+K

is a second homomorphism of Lie groups, then

(4 lp)' = 4'

0 I$.

0

1.4. Examples: 1. The vector group: If V is a finite-dimensional real or complex vector space, vector addition makes V into a Lie group. 2. The group GL(V): Consider the group GL(V) of linear automorphisms of an n-dimensional vector space V (real or complex). It is an open subset of the vector space L, = L ( V ; V), and hence a manifold; moreover, multiplication and inversion are smooth and so GL(V) is a Lie group. Since GL( V) is an open subset of L, , its tangent bundle is the restriction of the tangent bundle of L, ,

In particular, the underlying vector space of the corresponding Lie algebra is L, . Next, observe that the left translations A, , T E GL(V), are given by &(a) = T

0

a,

T, a

E

GL(v).

I t follows that &(u, a ) = (T 0 U, T

0

a),

uE

G L ( v ) , a EL".

Hence the left invariant vector field generated by Xa(7)= (T,T 0 a),

T

E

01

EL, is given by

GL(v).

To determine the Lie product, let f be a linear function in L, and denote its restriction to GL( V) also by f. Then

29

1. Lie algebra of a Lie group

and so ([Xu P X,lf)(T)

Since

T E

GL( V ) and f

=

E L*y were

[Xu > X,l

f(T

8 - B 4).

(a O

O

O

arbitrary, we obtain =

Xu,s-o,a

*

I n particular, the Lie algebra structure of L , induced from the Lie group structure of GL( V ) is given by

[a,8] = a o / 3 - p o a . 3. The group of invertibles of an associative algebra: Let A be an associative finite-dimensional algebra over [w, with unit element. For a E A , define p(u): A + A to be left multiplication by a. Then a has an inverse in A if and only if p ( a ) is a linear isomorphism; i.e., if and only if det p(a) # 0.

T h e invertible elements of A form a group G ( A )under composition; the condition above shows that G ( A )is open in A. Hence G ( A )is a Lie group. T h e same argument as given for GL( V ) in L , shows that the Lie algebra of G ( A )is A , with Lie bracket given by a, 8 E A.

81 = a8 - 8%

[a,

4. Direct products: Let G, H be Lie groups. T h e product manifold G x H can be made into a Lie group by setting (x, y )

*

(x’, y’) = (X

. x’, y . J J ’ ) ,

X, X’

E

G y , y’ E H .

This Lie group is called the direct product of G and H . T h e projections nG: G x H -+ G and nH: G x H -+ H , and the inclusions G, H -+ G x H , opposite e, are all homomorphisms of Lie groups. T h e Lie algebra homomorphisms nb , are given by ~ k ( hk,)

=h

&(A, k )

and

=

k.

It follows that the Lie product in T,(G x H ) is given by [(A, k), (h’, k’)] 5.

=

([A, h’], [k, k‘l),

h, h’

E

Te.(G), k , k’ E Te(H).

Tangent bundle: If G is a Lie group, then the map dp:

TG

x TG+

TG

I. Lie Groups

30

makes T , into a Lie group, with inversion map dv. (The associative law is obtained by differentiating the relation p 0 ( p x L ) = p 0 ( L x p).) T h e zero cross-section 0 : G -+ T , is a homomorphism of Lie groups. 6. The I-component: Let G be a Lie group, and let Go denote that connected component of the manifold G which contains e ; it is an open submanifold. Since p, u are continuous and Go x GO, Go are connected it follows that p(Go x GO) C

Go

and

v(G0)C Go.

Similarly, aGOa-l C GO, a E G. Thus Go is a normal subgroup of G. I t is clearly a Lie group and is called the I-component of G. T h e quotient group GIGois called the component group of G. 7. T h e nonzero reals R. = [w - (0) and the nonzero complex numbers @. = @ - {0} are each a Lie group under multiplication. If V (res-

pectively, W) is a real (respectively, complex) vector space, then the maps det: GL(V)+ R.

and

det: GL(W)+ @.

are Lie group homomorphisms. Their derivatives are given, respectively, by tr:LY+ R and tr:Lw+ @; i.e., det'

=

tr (cf. sec. 1.3).

s2.

The exponential map

1.5. One-parameter subgroups. A 1-parameter subgroup of a Lie group G is a homomorphism, a, of the additive group of real numbers into G, a: R + G .

I n other words, a 1-parameter subgroup is a smooth map a: R -+ G such that a(s

+ t ) = a(s) a(t),

s, t E R.

I n particular, a(0) = e and a(-t) = a(t)-l. Suppose a: R -+ G is a 1-parameter subgroup. Then (cf. sec. 0.13) [Y determines a path &: R -+ T,:

I n particular, &(O)

E

TJG).

Proposition IV: Let or: R -+ G be a smooth map such that a(0) = e and let &(O) = h. Then the following are equivalent: (1) a is a 1-parameter subgroup. (2) a is an orbit of xh . (3) a is an orbit of Yh. Proof: (1) * (2): Denote the vector field t M d/dt on R by T; it is the left and right invariant vector field generated by T(0). Hence if a is a 1-parameter subgroup,

i.e., a is an orbit of xh . (2) => (1): Assume a is an orbit of t ++a(s

+ t)

and 31

xh

and fix s t

H a(s)

E

R. Then

a(t)

I. Lie Groups

32

are both orbits of xh (use the left invariance of Hence (cf. Proposition X, sec. 3.15, volume I) a(s

xh), and

agree at t = 0.

+ t ) = a(s) .(t).

(3) o (1): Same proof as (2) o (1).

Q.E.D.

Proposition V: T o every vector h E T,(G) corresponds a unique I-parameter subgroup, a , such that i ( 0 ) = h. Proof: T h e uniqueness is immediate from Proposition IV. Now we prove existence. According to Proposition X, sec. 3.15, volume I, for some E > 0 there is an orbit 010

: (-6,

6)

-+

G,

for X , , satisfying ao(0) = e. Now fix to E (0, c). Define smooth maps a :,

(pto - 6,Pto

+4

+

G,

P E z,

by

4)= ao(to)pao(t- P o ) . Since X , is left invariant, these maps are all orbits f o r . __ %-1(Po)

.n

oreover,

= ao(to)* = %J(Pto)*

Hence app-land ap agree in the intersection of their domains. It follows that a smooth map a : R + G is given by

is an orbit for X , satisfying a(0) = e; thus by Proposition IV it is a 1-parameter subgroup. Q.E.D.

(y.

T h e 1-parameter subgroup, 01, that satisfies &(O) = h is called the 1-parameter subgroup generated by h, and is denoted by a h . I n particular, the 1-parameter subgroup generated by 0 is the constant map t t-+ e. Example: Let C. be the multiplicative group of nonzero complex

2. The exponential map

33

numbers: C. = {x E C I z # O}. Then the corresponding Lie algebra is @, considered as a real vector space. T h e 1-parameter subgroup generated by a vector h E C is given by q , ( t ) = exp th.

1.6. The exponential map. Let G be a Lie group with Lie algebra E (= T,(G)).Define a set map #:R x E + G

by # ( t , h) = ah(t),

t

R, h E E.

E

Lemma 11: $ is a smooth map. I t satisfies #(st, h) = # ( t , sh),

S,

t

E

R, h E E.

Proof: T h e equation holds because both sides define the l-parameter subgroup generated by sh (cf. sec. 1.5). T o show that $ is smooth, define a vector field 2 on the manifold E x G b y Z(h, 4

=

(0,&(a)).

I n view of Theorem 11, sec. 3.15, volume I, there are neighbourhoods I of 0 in R, V of 0 in E and U of e in G, and there is a smooth map

?:I x ( V x U ) + E x G such that +(t, h, a )

=

Z(&, h, a)),

d o , h, 4

=

(A,

t

E I,

hE

v, a E u,

and

4.

Now write

Then ;PE(t,h) = 0, yE(O,h)

=

h, and so

rpE(t, h) = h,

It follows that

t €1, h E

v.

I. Lie Groups

34

Hence y G ( t ,h) = ah(t) = $(t, h) and so $ is smooth in I x V . Now the functional equation

implies that $ is smooth in R x V . Finally, applying the equation h ) = $(t, sh), we see that $ is smooth in R x E. Q.E.D.

$s(t,

Definition: T h e exponential map for G is the smooth map exp: B -+ G given by

exp h

= +(l,

h)

= q,(l).

It follows from Lemma I1 that the 1-parameter group generated by h E E can be written as ah(t)= exp th,

In particular exp ph Proposition VI:

=

(exp h ) p , p

E

R.

Z, h E E.

T h e exponential map satisfies exp 0

Proof:

E

t

=

and

e

(d exp),

= 6.

Fix h E E. Then h

= A*(O) =

(exp th)'(O) = (d exp),(h). Q.E.D.

Corollary I: There are neighbourhoods V of 0 in E and U of e in G such that the exponential map restricts to a diffeomorphism N

exp: V A U. Corollary 11: Let E = E, @ @ E, be a decomposition of E as a direct sum of subspaces. Define y : E -+ G by

v(hl @

@ h,)

= exp

Then ( d q ~= ) ~L , and so maps onto a neighbourhood of e.

a

h, ... exp h,

,

hi E Ei

neighbourhood of 0 diffeomorphically

2. The exponential map

35

) ~ to the identity in each Ei; hence it is Proof: Clearly ( d ~restricts the identity in E . Q.E.D. Corollary 111:

If G is connected, then exp(E) generates G.

Proof: By Corollary I, exp(E) contains a neighbourhood of e. Thus the corollary follows from Lemma I11 below. Q.E.D. Lemma 111: If G is connected, and U C G is a neighbourhood of e, then U generates G. Proof: U generates an open subgroup H of G. Thus each coset Ha (a E G) is open and G=HuUHa &H

partitions G into two disjoint open sets. Since G is connected, G = H . Q.E.D. Examples: 1. Consider the case G = GL( V ) ,E 2, sec. 1.4). Then exp is the map given in sec. 0.10.

=L,

(cf. Example

2. Let H be a second Lie group with Lie algebra F. Then the exponential map for G x H is given by

h E E, K E F.

exp(h, K) = (exp,(h), exp,(k)),

1.7. Homomorphisms. Proposition VII: Let T: G + H be a homomorphism of Lie groups. Then the induced homomorphism, T', of Lie algebras satisfies cp o exp, = exp,

o

cp'.

Proof: Fix h E T,(G). Then a: t

H T(exp,(th))

and

/3: t

are 1-parameter subgroups of H. Moreover,

i(0) = v'(h)= j(O),

++

expH(tT'(h))

I. Lie Groups

36

and hence (Proposition V, sec. 1.5) 01

=

8. I n particular Q.E.D.

Corollary I: Assume +: G --f H is a second homomorphism of Lie groups and that cp' = #'. If G is connected, then cp =

+.

Proof: Proposition VII implies that cp and

+ agree in exp,(T,(G)).

By Corollary I11 to Proposition VI, sec. 1.6, this set generates G. Since cp and

# are group homomorphisms,

it follows that cp

=

#.

Q.E.D. Corollary 11: T h e homomorphism cp is injective if and only if

is injective. I n this case cp embeds G into H. Proof: If dy is injective, then certainly p is injective. Conversely, assume cp is injective. Let V be a neighbourhood of 0 in Te(G) such that the restriction of exp, to V is injective. Then since exp, o cp' = 0 exp, , the restriction of expH 0 cp' to V is injective. I n particular, the restriction of p' to V is injective. Since cp' is linear and V is an open subset of Te(G),it follows that cp' is injective. Since

each (dcp), is injective. Hence so is dcp.

Q.E.D. Corollary 111: If cp is bijective, then it is a diffeomorphism and hence an isomorphism between Lie groups. Proof: Since cp is injective, Corollary I1 implies that dcp: TG+ THis injective. Now Proposition IV, sec. 3.8, volume I, implies that cp is a diffeomorphism.

Q.E.D. Proposition VIII: A continuous group homomorphism p: G --f H between Lie groups is smooth.

2. The exponential map

31

Proof: Consider first the case that G = R. It has to be shown that a continuous map a : R -+H which satisfies a(s

+ t ) = a(s) a@),

s,

t

€

R,

is smooth. In view of Corollary I to Proposition VI, sec. 1.6, there is a neighbourhood V of 0 in T,(H) which exp,, maps diffeomorphically onto a neighbourhood U of e in H . Without loss of generality we may assume that a@)€

Define a continuous map

u,

It I

< 1.

/3: I -+ V ( I = {t E R I I t I

< I}) by

B ( t ) = expi' a(t).

Since a is a homomorphism,

Hence q P(t) E V if and only if

Fix q # 0. Consider the set

0 E (I/dlI 4

*

E

v>*

The above relation shows that this set is both closed and open in (1/q)I, and hence equal to (l/q)I. Thus 4 * B(t)

= P(qt),

I 4t I

< 1%

I. Lie Groups

38

Since 01 is a homomorphism (as is t t+ exp,(tP(l))) and the interval (- I, I ) generates the additive group R, it follows that

4)= exp,(W)),

2E

R,

and so 01 is smooth. Finally, consider the general case, g,: G + H . Choose a basis e, of T,(G) and consider the smooth map +: Rn + G given by (b(tl,

..., tn) = expG(tlel) ...

,..., e,

expc(tnen).

+

By Corollary I1 to Proposition VI, sec. 1.6, maps a neighbourhood V of 0 diffeomorphically onto a neighbourhood U of e. On the other hand, the maps t t-t y(expG(tei)) (i = I, ..., n) are continuous homomorphisms [w + H ; thus they are smooth by the argument above. Since g, is a homomorphism, we have (P’

and so g, o

#)(tl P

...)tn) = dexpG(tlel))

’”

dexPdtnen))

+ is smooth. In particular, y is smooth in U. But for any a P’(4 =

E

G,

d4P ’ ( 4

Thus g, is smooth in a neighbourhood of a and hence in G.

Q.E.D.

s3. Representations In this article G denotes a fixed Lie group with Lie algebra E. 1.8. The derivative of a representation. A representation of G in a finite-dimensional vector space W (real or complex) is a homomorphism of Lie groups P : G -+ GL( W ) .

Since the Lie algebra of GL( W )is the space L , of linear transformations of W (cf. Example 2, sec. 1.4), the derivative of the homomorphism P is a homomorphism of Lie algebras,

(cf. Proposition 111, sec. 1.3). P' will be called the derivative of the representation P. A Lie algebra homomorphism 8: E -+LfiJis called a representation of E in W. Thus P' is a representation of E in W. A representation, P , of G (respectively, 8 of E ) is called faithful if ker P = e (respectively, if ker 8 = 0). If P is a representation of G in W, then the invariant subspace of P is the subspace W,=, (or simply W I )given by W, = {W E W 1 P(x)w = W , x E G}.

Similarly, if 8 is a representation of E in W, then the invariant subspace for 8 is the subspace W,=, (or W,) given by

w,=,= {w E w 1 O(h)w = 0, h E E } . A subspace I'C W is called stable for P (respectively, stable fw 8) if each of the operators P (x), x E G (respectively 8(h), h E E ) maps V to itself. Now fix h E E. Then P(exp th), and P'(h) are linear transformations of W. In particular, we regard the 1-parameter group P,: t

H

P(exp th) 39

40

I. Lie Groups

as a path in the vector space i),(t) inL,

.

Lw . Thus differentiation yields a path

On the other hand recall from Example 2, sec. 1.4, that T,,(,,

GL(W) x Lw . Moreover,

x p * ( h ) ( ~= ) (7, T

Applying this formula with

0

T

P‘(h)), =

T

=

GL(w), h E A!?.

E

Ph(t)gives

Proposition IX: (1) The invariant subspaces W, and W, for P and P‘ are related by

w,c w,.

If G is connected, then W, = W,. (2) If V C W is stable for P , then it is stable for P . If V is stable for P’ and G is connected, then V is stable for P. Proof: ( I ) Suppose h E E and w

E

W, . Then Ph(t)w = w , and

SO

= (P*(t)w)’ = 0.

&)W

Now Lemma IV yields P’(h)w = 0. Thus W, C W,. Conversely, let h E E and assume w E W, , Then Lemma IV implies that Ph(t)w = w , t E R. It follows that P(exp h) w

=

w,

h E E.

Now if G is connected we can apply Corollary I11 to Proposition VI, sec. 1.6, to obtain P(x)w = w , x E G. (2) is proved in the same way. Q.E.D. 1.9, Examples: In this section P (respectively, 6 ) denotes a fixed representation of G (respectively, E) in W.

1.

Contragredient representation: T h e representation, P Q ,of G in

W* contragredient to P is defined by P ~ ( x= ) (P(x)-’)*,

x E G.

T h e representation 6Qof E in W* contragredient to 6 is defined by eyh)

=

+A)*,

h E E.

41

3. Representations

Evidently (Pb)’ = (P’)? 2. Multilinear representations: Representations @P, A P and V P of G i n @W, AW, VWaregiven by (@P)(X)

=

(AP)(x) = A

@P(x),

W

and ( V P ) ( X )= VP(X), x E G, (cf. sec. 0.5). Representations Bo , 8, , 0, of E in @ W, A W, and V W are given by

c w1 0*..e(h)wi ... P

Bo(h)(wl 0 ... 0w p ) =

@wp,

i=l

ev(h)(wl

P

... v w,)

1

=

W1

v

... e(hlwi ...

p 3 I,

w p ,

i=l

and e,(h)h

= 0,

o,(~)A= 0,

e,,(h)A = 0,

A

E

R.

Evidently, (@P)’

= (P’)@,

(AP)’ = (P’),,

(VP)’ = ( P ’ ) ” .

and

3. Recall that Tp(W) denotes the space of p-linear functions in W. Define a representation, Pp, of G in TP(W) by setting (P”(X)@)(Wl, ..., w p ) = @(P(x-l)w,, ..., P(x-’)w,),

w iE W ,

X E

G,

@E

TP(W).

Then the derivative of Pp is given by P

[(Pp)‘(h)](@)(wl , ..., w,)

=

-1 @(wl , ..., P’(h)w, , ..., w P ) ,

h E E.

i=l

4. DiSferential spaces: Let (W, d ) be a differential space (cf. sec. 0.7) and denote its homology by H ( W). Assume that P is a representation of G in W such that

P(x) d 0

=

d 0 P(x),

x

E

G.

I. Lie Groups

42

Then P(x) determines a linear map P(x),: H( W )+ H( W )

and P,: x F+ P(x), is a representation of G in H ( W ) . On the other hand, the representation, P’, of E satisfies P’(h) o d

=

hEE

d o P’(h),

(differentiate the relation above). Hence P’(h) determines an operator P’(h), in H ( W ) and (P‘),: h i--t P’(h)#

is a representation of E in H( W). I t follows immediately from the definitions that (P’)# is the derivative of p , , (Pel’ = (P’)# * 1.10. The adjoint representation. Each a

E

G determines the inner

automorphism, r, , of G given by T,(x)

x E G.

= ax&,

Hence the derivative, r; , of r, is an automorphism of the Lie algebra E. It is denoted by Ad a. Since r, = A, 0 p i ’ , Ad a

= La 0 RL’,

a E G.

Proposition X: T h e correspondence Ad: a representation of G in E.

Proof: Evidently r,

0

A d a defines a

rb = r a b ,and so

AdaoAdb =Adab.

Thus Ad is a group homomorphism. I t remains to show that Ad is smooth. Define a smooth map T : G x G -+ G by setting

w, 4

= TdX),

y , x E G.

Its derivative, dT,is smooth. But (dT)(l/.e)(O, h) = AdY (h).

Hence, for each h E E, the maps y Ad is smooth.

I-+

Ad y ( h ) are smooth. It follows that

Q.E.D.

43

3. Representations

T h e representation Ad is called the adjoint representation of G. On the other hand, a representation, ad, of the Lie algebra E in the vector space E is given by h, k

(ad h)(k) = [h, k ] ,

E

E.

I t is called the adjoint representation of E. Proposition XI: Lemma V:

ad is the derivative of Ad.

Fix a

E

G, h

E

E. Then Y*da(h)(a).

&(a) =

Proof: Recall that Ad a = R;’

0

L, . Hence

Proof of the proposition: Fix h E E and let el Then functions f i on G are defined by

,..., e,

be a basis for E.

n

Ads@) =

c f4-4 ez

I

x

E

G.

k

E

E.

i=l

They satisfy n

Ad’k(h) =

c ( X k (f l ) ) ( e )

ei

,

i=l

On the other hand, we can apply Lemma V to obtain

i=l

Since [X,, Ye,]= 0 (cf. Proposition 11, sec. 1.2), it follows that

c X,(fi)Ye*. 72

[Xk Xhl I

=

1=1

Evaluate this at e to obtain [k, h]

=

Ad’k(h).

Q.E.D. Corollary: Ad(exp h) = exp(ad h), h

E

E.

Proof: Apply Proposition VII, sec. 1.7.

Q.E.D.

s4. Abelian Lie groups

An abeliun Lie group is a Lie group G satisfying xy = yx E such that [h, k] = 0 (k,h E E ) . Let G be a Lie group with Lie algebra E and consider the following conditions: 1.11.

(x, y E G). An abelian Lie algebra is a Lie algebra

(1) G is abelian. (2) T h e adjoint representation of G is trivial: Ad a = L ( a E G ) . (3) T h e left and right invariant vector fields coincide, xh = Y h

,

h E E.

(4) T h e adjoint representation of E is trivial: ad h = 0 ( h E E). ( 5 ) E is abelian. Proposition XII:

T h e conditions above satisfy (1)

* (2)

(3)

a (4) + (5).

If G is connected, they are all equivalent. Proof: T h e sequence of implications is an immediate consequence of the relations 7; = Ad a and Ad' = ad, together with Lemma V, sec. 1.10. If G is connected and E is abelian, then Corollary I to Proposition VII, sec. 1.7, shows that Ad

=y

and

ra = L

(aEG),

where y : G + L is the constant homomorphism.

Q.E.D. Examples: 1. 1.4) are abelian. 2.

Vector spaces under addition (cf. Example 1, sec.

Consider the unit circle of the complex plane S={zE@I[ZI

=I}.

It is an abelian Lie group under multiplication, T h e tangent space, Te(S1), 44

45

4. Abelian Lie groups

is given by Te(S1)= (1)l;i.e., it is the pure imaginary axis. We identify R with Te(S1)by the correspondence t F+ 2rrit. With this identification the exponential map exp: R -+ S' is given by exp h

h E R.

e2nih,

I n particular, exp-'(1) = Z. 3. Tori: Recall that Rn is an abelian Lie group under addition. Consider the closed subgroup Zn C [wn consisting of n-tuples of integers. I n Example 3, sec. 1.4, volume I, the factor group T n = Rn/Zn was made into a smooth manifold in such a way that the projection T":

R" -+ T"

was a local diffeomorphism. With this smooth structure T n becomes a connected abelian Lie group. It is called the n-torus. If n = 1, then T1 is the circle S1 and rr is the exponential map (Example 2 above). Since [w" = [w

x ... x R,

Z" = Z x

... x z

(as Lie groups), it follows that T " s S1 x ." x S'

(as Lie groups). I n particular, Tn is compact. Moreover, rn = rr

x ..* X r

=

exp

X

...

X

exp

= expTn.

Thus we may identify rrn with the exponential map for Tn. 1.12. Proposition XIII: Every connected abelian Lie group G is isomorphic to the direct product T p x RQ (for somep, Q E N). I n particular, a compact connected abelian Lie group is a torus.

T h e proposition follows at once from Lemmas VI and VII below. Lemma VI:

Let E be the Lie algebra of G. Then exp(h

+ k) = exp h . exp k,

i.e., exp is a Lie group homomorphism. Proof

Since G is abelian, 01:

t

t-+

exp th . exp tk

h, k E E ;

46

I. Lie Groups

+ k ; hence exp th . exp tk = a ( t ) = exp t(h + k ) .

is a I-parameter subgroup. But &(O)

Now set t

=

=

h

1. Q.E.D.

Corollary: exp is a surjective local diffeomorphism. exp-'(e) is a closed discrete subgroup of E. Proof: Apply Proposition VI, sec. 1.6, and its third corollary.

Q.E.D. Lemma VII: Let K be a closed discrete subgroup of Rn. Then there are linearly independent vectors e, ,..., e?, E Rn such that K consists of the integral combinations of the e,:

i=l

Proof: Clearly, we may assume that K contains a basis of Rn, and we argue by induction on n. Fix a positive inner product in Rn. Choose el E K so that el # 0 and 1 el I 1 x 1 for x E K. Then (R el) n K consists of the integer multiples of el . Now consider the projection

-


0,

it

If G is unimodular (in particular, if G is compact), this formula reduces to

I. Lie Groups

50

1.15. Integration over compact groups. Let G be a compact ndimensional Lie group with Lie algebra E. Give G the left orientation induced by an orientation of E. Let d be the unique left invariant n-form such that jGA

=

1.

Let f E Y ( G ; W ) ( W , a vector space). Then the vector, Jcf * d, is independent of the orientation. It is called the integral o f f , and we write

I n particular, JG da = 1. Since G is unimodular, the relations in sec. 1.14 give

j f ( u b )du 1 f ( a ) da / =

c

=

c

G

b E G.

f ( b u ) du,

More generally, assume y~ is a diffeomorphism of G such that A

=E .

IE 1

=

cp*

where E : G -+ Iw is smooth and

A,

1. Then for f E Y ( G ; W )

I n particular (apply Lemma I, sec. 1.1) this condition holds for y the inversion map. Thus

1

c

f(a-') du =

1

f ( a ) du.

= v,

(1.3)

G

Finally, if a : W -+ V is a linear map, then (cf. sec. 0.13)

: -+ G be a smooth map. I t induces the smooth map Next let y ~ G $: G -+ LE given by

Y w = (-Gtd)m(r)(dv), O

Proposition XIV: deg

.

(4Je

1

x

6

G.

If G is compact and connected, then

j f ( x ) dx G

O

= G

f ( q ~ ( x ) ). det $(x) dx,

f E Y ( G ;W ) .

5. Integration on compact Lie groups

Proof:

Evidently, cp*d deg p) .

1

=

det $ d. Hence

f ( x ) dx =

G

51

s

p)*(fA) = G

s

p)*f.

det #

.d

G

Q.E.D. Corollary:

deg cp

Examples:

1.

#(x) =

=

If cp

lE,

jG det $(x) dx. =

, pa or v, then $ is given by

A,

#(x) = Ad a-l,

or

$(x) = -Ad x,

respectively. I n this case the Proposition yields formulae (1.2) and (1.3) above, in turn.

+

2. rp(x) = x2. Then #(x) = L Ad x-l. Now the Corollary to Proposition XIV, together with formula 1.3, yields deg p)

3.

=

s

det(1 G

+ Ad x) dx.

rp is a homomorphism. Then #(x) = cp', x

deg p)

.

1

f ( x ) dx = det p)'

G

.

1

f(p)(x)) dx,

E

G, whence

f~ Y ( G ; W ) .

G

Setting f = 1, we obtain deg cp = det cp'. I n particular, det cp' is an integer. Now assume that deg cp # 0. Then the relations above yield

Moreover, in this case cp' is a linear isomorphism. Hence each map (dcp), (x E G) is an isomorphism. Thus cp is a local diffeomorphism, and so the set cp-I(e) is finite. Now Theorem I, sec. 6.3, volume I, implies that the integer I det cp' I is equal to the number of points in cp-'(e). 1.16. Invariant subspace of a representation. Let P be a representation of a compact Lie group in a finite-dimensional vector space W. Since P is a smooth, map G +L, , we can form the integral, Po =

s

P ( x ) dx, G

I. Lie Groups

52

to obtain a linear transformation of W. (Note that Po is not in general a linear automorphism of W.) With the notation and hypotheses above

Proposition XV:

Po P ( x ) = Po = P ( x ) P o , x E G. (2) Pi = Po (3) If P Qdenotes the contragradient representation, then (Ph)o= P$. (4) A vector w is invariant (i.e. w E W,) if and only if Pow = w . (1)

0

0

Proof: We rely throughout on formulae (1.2), (1.3), and (1.4) of sec. 1.15. T o prove (1) observe that for x E G,

Similarly, P(x) 0 Po = Po and so (1) follows. This relation yields P:

=

Po 0

P ( x )dx G

=

1 Po P(x)dx 0

Po dx

=

G

=

Po

G

Next note that

j

(PQ)o=

T o prove (4) let w

E

Pow

P(x-l)* dx G

=

(s,

P(x) dx)

*

= P,*.

W , . Then

=

[

G

(P(x)w)dx=

wdx

= w.

JG

On the other hand, if Pow = w , then (1) yields P(x)w = ( P ( x )0 P0)w = Pow = W ,

and so w

E

x E G,

W ,. Q.E.D.

Corollary I: T h e dimension of W , is given by dim W, = J tr P(x) dx.

5. Integration on compact Lie groups

53

I n particular, W, = 0 if and only if

s Proof: Since Pi

=

tr P ( x ) dx

=

0.

G

Po and Im Po = W, ,

dim W, = tr Po =

1 tr P(x)

dx.

G

Q.E.D. Corollary 11: If WF is the invariant subspace for P Q ,then dim WF = dim W,.

Corollary 111: Consider the induced representations, AkP, in A k W fork = 0,...,r ( r = dim W ) ,and let Ck

=

k

dim(AkW)I,

=

0 , 1, ..., r.

Suppose G is connected. Then det(P(x)

+ A,)

c k P k=

dx =

G

k=O

ckP. k=O

Proof: Corollary I gives ck

=

I

tr A k P ( x ) dx.

G

in the polynomial det(P(x)

But tr AkP(x) is the coefficient of (cf. sec. A.2). Thus

s

det(P(x) C

+ hi) dx = k=O

[I

+ Xc)

tr A k P ( x ) dx] hr-k

G

T o establish the other equality, note that because G is compact and connected, the homomorphism det 0 P : G + R. has a compact connected image; i.e., det P(x) = 1, x E G.

I. Lie Groups

54

It follows that for X # 0, det(P(x)

+ hi) = AT det(X-h + P(x-l)).

Integrating over G , we obtain

1.17. Invariant inner products. Let P be a representation of a Lie group in a real (respectively, complex) vector space W. A Euclidean (respectively, Hermitian) inner product ( , ) in W is called invariant with respect to P, if it satisfies (P(x)u,P(x)v)

= (u,v),

x E G,

U, v E

W.

If ( , ) is such an inner product, it follows that for each h map P'(h): W -+ W is skew.

E

T,(G), the

Proposition XVI: Every representation of a compact Lie group admits an invariant inner product. Proof: Let ( , ) be any Euclidean (respectively, Hermitian) inner product. Define ( , ) by setting (u, v) =

1

c

(P(u)u,P(u)v) du.

Then ( , ) has the desired properties.

Q.E.D. Corollary: Let G be a compact connected Lie group. Then the map 9:x w x2 is surjective. Proof:

Recall from Example 2 sec. 1.15, that deg

=

1,

det(a + Ad x) dx.

Now choose an inner product in T,(G) which is invariant under the adjoint representation. Thus each Ad x is a proper rotation, and it follows from elementary linear algebra that det(i

+ Ad x) 2 0,

x E G.

5. Integration on compact Lie groups

55

Since det(r

we obtain deg y

+ Ad e) = det(2~)= 2",

> 0. Hence y

(n

=

dim G),

is surjective.

Q.E.D. Remark: In sec. 2.18 it will be shown that, for a compact connected Lie group, the map x I-+ x p is surjective for every integer p # 0.

T h e following example of Hopf shows that the map x ++ x2 is not necessarily surjective if G is not compact. Let G be the group SL(2; R) consisting of linear transformations a: R2-+ R2 with det a = 1. (It follows from Theorem I, sec. 2.1, of the next chapter that G is a Lie group.) T h e Cayley-Hamilton theorem yields a2 - (tr a) a

+

L

=

0,

aE

G,

whence tr 2 - (tr a)z

Hence tr 012 3 -2 E G given by

if

a E

B(e1) =

has trace

< -2,

G. In

-2%

9

+ 2 = 0. particular,

the transformation

P ( 4 = -4%

and so is not the square of any cy. in G.

A representation of a Lie group in a vector space W is called semisimple, if every stable subspace W, C W has a stable complement; i.e., if W, C W is stable, then there is a stable subspace W , such that W = W, @ W,. Proposition XVII: Every representation of a compact Lie group in a finite-dimensional vector space is semisimple. Proof: In view of Proposition XVI there exists an invariant inner product in W. Now let W, C W be stable. Then

w = w,@ w; and W: is also a stable subspace.

Q.E.D.

Problems

G is a Lie group with Lie algebra E, 1.

Show that a I-parameter subgroup is either R or S1.

2. Construct a nonabelian Lie group with trivial adjoint representation and abelian component group. What is the minimum number of components of such a group ?

3.

Let h, k

E

E and f E Y ( G ) .

(i) Show that

(ii) Use the fact that a2

---f(exp at aT

a2

~k . x . exp th) = -f(exp aT at

to conclude that [ X , , Y,J

=

~k . x exp th)

0.

(iii) Show that

and

4.

Let q: M

+

N be a smooth map such that (dq),

(i) Show that a bilinear map defined by

P(55 7Nf)

56

0.

p: T,(M) x T,(M) + Tq(,)(N)is

= X(Y(q*f))(a),

where X ( u ) = .$ and Y ( u ) = 7.

=

f E Y(W

57

Problems

(ii) Show that (iii) and a

is symmetric.

Determine /3 in the case M e x e.

=G

x G, N = G, q(x, y ) = ~ y x - ~ y - ~

=

5. Show that the upper-triangular real (n x n)-matrices with 1’s on the main diagonal form a Lie group G. Show that G is nonabelian if n > 2. Find the Lie algebra of G, and prove that the exponential map is a global diffeomorphism. 6. Use the Cayley map (cf. Example 9, sec. 1.5, volume I) to make the group of proper rotations of Euclidean space into a Lie group.

Let T be an n-torus with Lie algebra L T . T h e subset of L, given by r T = exp-l(e) is called the integer lattice of L T .

7. Tori. r T

(i) Show that r T Zn (Zn = Z @ @ h,n terms, cf. sec. 1.12). (ii) If y : T + S is a homomorphism into another torus, show that rp’: L T +L , restricts to a group homomorphism y r : -+ r, . Show that this defines a bijection between the set of homomorphisms T + S and the set Hom(Zn; hm) ( a = dim T , m = dim S). (iii) Show that a subspace L C L T is the Lie algebra of a subtorus if and only if L is generated (over R) by vectors in r T . (iv) Given a subtorus S, of T , find a second subtorus S , such that the map S, x S, + T given by (x, , x,) -+xlxz is an isomorphism of Lie groups. 8. Power maps.

Define Pk: G + G by Pk(x) = xk, (k E Z).

(i) Show that

where, if k 3 1, k-1

Qk(x) =

C (Ad x-l)j.

3=0

if K < 1. (ii) Fix x E G. Show that det Q k ( x ) an h E E such that Find

Qk

(Ad x),h

=h

and

=

0 if and only if there exists

(Ad x)h # h.

I. Lie Groups

58

(iii) If G is compact and connected, show that det Ok(x) 3 0, x E G. Conclude that the maps Pk are all surjective. Use this to show that the exponential map is surjective. {iv) If G is compact and connected, show that deg P ,

=

dim(AE*)I,

where (AE*), denotes the subalgebra of AE* invariant under the representation A Ado. 9. The group RP3. Fix a Euclidean inner product, and an orientation in R3.

(i) Show that the cross product makes R3 into a Lie algebra. Let $(h)(x) = h x x and show that $ is an isomorphism from R3 to the Lie algebra of skew transformations of R3. (ii) Show that

h, x E R3.

(iii) Let B be the closed ball of radius r in R3. Regard RP3 as the quotient space of B under the equivalence relation x - y if and only if either x = y or I x I = r and x = -y. Use (ii) to obtain an embedding RP3 -+ GL(R3)whose image is the set of proper isometries of R3. (iv) Conclude that RP3 is a Lie group with Lie algebra R3. Write down the exponential map explicitly. Obtain expressions for the left and right invariant vector fields. 10. Let (xl, x2) = x and ( y l , y z ) = y belong to R2. Set xy = (xl yle?2, xz y 2 ) and show that this makes R2 into a Lie group. Find the I-parameter subgroups, the left and right invariant vector fields, and the Lie algebra.

+

+

11. 1-parameter subgroups. (i) Show that quaternionic multiplication makes S3 into a Lie group. Show that the I-parameter subgroups are the great circles through e. (ii) Let x ( t ) be the 1-parameter subgroup of GL(R3)generated by a skew transformation u. Show that x is periodic with period 2n/(- 4 tr u2)1/2.

59

Problems

12. Representations. Let V , W be complex vector spaces. Two representations P: G -+ GL( V )and Q: G -+ GL( W )are called equivalent if there exists a linear isomorphism y : V 3 W such that q~ 0 P ( x ) = Q(x)

0

q ~ ,

xE

G.

A representation P in V is called irreducible, if V is not the direct sum of nontrivial stable subspaces. T h e character of P is the complex-valued function X, on G given by X,(x) = tr P(x). (i) Let P, Q be representations of G in V and W, respectively. Show that a representation R of G in the space L( V ; W )is given by R(x)# = Q(x) 0 4 0 P(x)-l,

# E L ( I/; W ) .

Show that R is equivalent to the representation P Q @ Q in V* @ W. Show that P and Q are equivalent if and only if the space L ( V ; W ) , contains a linear isomorphism. If P and Q are irreducible, show that they are equivalent if and only if L( V ; W ) , # 0. (ii) Show that equivalent representations have the same character. Prove the relations (the last only if G is compact)

(where

xp is the complex conjugate of X,).

13. Representations of compact Lie groups.

Let G be compact.

(i) Show that each representation of G is the direct sum of irreducible representations. (ii) Let P and Q be irreducible representations of G in complex vector spaces. Show that

f, ""

=

I

1 0

if P and Q are equivalent

otherwise.

Conclude that P and Q are equivalent if and only if X, = X, . (iii) Assume that {(PA, V,)}is a collection of inequivalent irreducible complex representations such that every irreducible complex representation is equivalent to some PA. Define a canonical G-linear isomorphism

where the representations are @,(PA @

L)

and P, respectively.

I. Lie Groups

60

14, Finite groups. Let r be a finite group and let I F 1 denote the be the complex vector space with the elements of r order of F. Let @(r) as basis.

into an algebra. (i) Show that the multiplication of r makes @(r) (ii) Iff is a complex-valued function on show that

r

(iii) Show that left and right multiplications determine equivalent representations L and R of r in the space C ( r ) . They are called the left (respectively, right) regular representations of I'. Show that

(iv) If P is a representation of that

r in a complex vector space V , show

tr(R(x) @ P ( x ) ) = 0,

x # e.

Conclude that t r R(x) @ P ( x ) dx = dim V .

(v) Show that L determines a representation L, of F in the invariant subspace [ C ( r ) @ V ] ,(with respect to R @ P).Show that a linear map 'p: C ( r ) @ V -+ V is given by

cp(xoV)= P(+, Show that

'p

XE

r,

v.

restricts to an isomorphism

and that 9 is an equivalence between the representations L @ I and P. Conclude that the right regular representation is a direct sum of irreducible representations, and that each irreducible representation occurs p times, where p is the dimension of its representation space. 15.

Let A be a real finite-dimensional associative algebra.

(i) Show that the group of units, GA, of A is dense in A.

Problems

61

(ii) Show that left multiplication defines a representation of G, in A. What is its derivative ? (iii) Define the adjoint representation of G, in terms of the multiplication in A. 16, Local homeomorphisms. Let Q be a second countable Hausdorff space and let 7 r : Q -+ M be a local homeomorphism into a smooth manifold. Show that there is a unique smooth structure on Q which makes x into a local diffeomorphism.

17. Covering spaces. Let (Q, x , M , F ) be a smooth bundle and assume that x is a local diffeomorphism. Then Q is called a covering manifold of M and x is called a covering projection.

(i) If 7 r : Q -+ M is a covering projection, show that the fibre consists of finitely or countably many points. (ii) Show that the composite of two covering projections is a covering projection. 18. Universal covering manifold. Let M be a connected manifold and fix a point xo E M . Let X denote the set of continuous maps q: [0, 13 + M satisfying q(0) = xo . For each open subset U of M and each compact subset C of [0, 11, set Xc,v = (9'E X I q ( C )C U}. Give X the weakest topology such that each Xc,v is open. Define an equivalence relation, -, in X as follows: 4p $ if ~ ( 1 )= #(1) and if there is a continuous homotopy q lconnecting q and $ such that ~ ~ ( = 0 xo ) , ql(l) = q(l) (0 t 1). Let A? be the set of equivalence classes with the quotient topology.

-

<


v>,

u, v EF.

T h e Lie algebra of Sp(n; 88) is the subalgebra, Sy(n; R), of L(2n; R) consisting of those transformations, y , satisfying

.> + +

(du),

I t has dimension n(2n

(u,

d.)> = 0,

u, EF.

1). Hence

dim Sp(n; R)

= n(2n

+ 1).

Sp(n; R) is called the real sympzectic group. Note: I n the literature the terminology “symplectic group’’ and notation Sp(n) is frequently used for the compact group defined in Example 4 below. 2 . Thegroup Sp(n; C): Let F be a complex vector space of dimension 2n and let { , ) be a nondegenerate, skew-symmetric, complex bilinear function in F. Then the complex linear automorphisms, T, of F which satisfy >

= (u,

v>,

u, CJ EF

2. Linear groups

73

form a closed subgroup of GL(F).It is denoted by Sp(n; @)and is called the complex symplectic group. The corresponding Lie algebra consists of the complex linear transformations, a, of F which satisfy

(44, u)

+ ( u , +)>

= 0.

It is a complex Lie algebra of complex dimension n(2n dim Sp(n; C) = 2428

+ 1). Hence

+ 1).

3. The group of unit quaternions: Let W be the algebra of quaternions with quaternionic norm. Then

141= I a I l B I ,

“,FEW,

and so the unit quaternions form a compact Lie group (nonabelian) whose underlying manifold is the three-sphere S3 (cf. sec. 0.2). S3 is a closed subgroup of the Lie group of nonzero quaternions (under multiplication). Since this latter group is the group G(W) of units of W , it follows from Example 3, sec. 1.4, that its Lie algebra is W , with Lie bracket given by [a,

PI

= aB

a, B E

- pa,

w.

Now S3 is a subgroup of G(W). Thus its Lie algebra E is the subalgebra given by E = T,(S3)= (e)l; ie., E is the Lie algebra of pure quaternions. 4. The quaternionic group Q ( n ) : Let V be an n-dimensional Euclidean space, and consider the quaternionic space

F

=

Define a real bilinear m a p F x F

W @w V .

W by

-+

( q @ u, q’ @ u ’ ) = qq’(u, u’),

q, q’ E

w,

u, u‘ E

v

(I’ is the conjugate of q‘, cf. sec. 0.2). F is a left vector space over W,with scalar multiplication given by q.(p@~)=qp@u,

qrPEW,

U E V

The underlying real space F , has dimension 4n, and Re( regarded as a Euclidean inner product in F R .

, ) may be

11. Subgroups and Homogeneous Spaces

14

A real linear map a : F

---f

F is called quaternionic linear if 4 E W, x E F .

n ( p ) = p(x),

If, further, ( 4 4 , n ( y ) )= ( X Y Y ) ,

X,Y

EF,

then a is called a quaternionic linear isometry. These isometries form a closed subgroup of O(F,); it is denoted by Q ( F )or Q(n) and is called the quaternionic group. Q(n) is a compact Lie group; in sec. 3.6 it will be shown that Q(n) is connected. T h e (real) Lie algebra of Q(n)consists of the W-linear transformations, a , of F satisfying

+ ( u , .(v))

(n(u), v)

u, v E F .

= 0,

It is denoted by Sk(n; W ) and has dimension n(2n dimQ(n) = n(2n

If n

=

1, t h e n F

=

+ 1). Thus

+ 1).

W and ( , ) is given by

(P,4 ) A quaternionic linear map by some p E W . Since

P,4 E W.

= PP,

v of W

is simply multiplication on the right

(P(41), v(42))

= 41PF!% 9

it follows that y is an isometry if and only if p has norm I ; i.e., Q(1) is the group of unit quaternions. 2.8. The groups. SO(2), S0(3), SO(4). 1. SO(2): Regard C as a Euclidean plane. A proper rotation of the plane is then multiplication by some eis. Thus (cf. Example 3, sec. 2.6)

SO(2) = U(1)

=

s1.

2. SO(3): Consider a three-dimensional Euclidean space F. Recall that SO(F) has Lie algebra Sk(F) (Example 3, sec. 2.5). Orient F, and make F into a Lie algebra by setting [x, Y1 = X

x y,

x, y E F

(cross-product). Then an isomorphism a : F % Sk(F) of Lie algebras is defined by m(a)(x) = a x x.

2. Linear groups

75

Next we shall establish a diffeomorphism SO(3) RP3, where RP9 denotes the three-dimensional projective space. Let W be the space of quaternions, and identify F with the orthogonal complement of the unit element e E W. Let S3 be the unit sphere in W. Then every unit vector p E S3 determines the proper rotation rP of F given by T,(x) =

pxp-l,

x EF.

I n this way we obtain a homomorphism of Lie groups T:

S3+ SO(3).

(S3is given the Lie group structure defined in Example 3, sec. 2.7.) T is surjective, and its kernel consists of the vectors e and - e (cf. [7, p. 327)]. Since p E s3, T p = 7-,, the map T factors over the canonical projection commutative diagram

n:

S3 + RP3 to yield a

RP3

The induced map u is a diffeomorphism. 3. SO(4): Let S3 be the unit sphere in the space of quaternions. Define T: S3 x S3 + SO(4) by

.(A q)(x) = P F '

PI

q E s3,x E En.

Then T is a surjective homomorphism of Lie groups, and the kernel of T consists of the pairs (e, e) and ( - e , -e) (cf. [7, p. 3291). I t follows that T induces an isomorphism of groups p: (S3x

S3),'Z2-+ S0(4),

where B, denotes the normal subgroup of S3 x S3 consisting of (e, e) and ( - e , -e). Restricting T to the normal subgroup S3 x e of S3 x S3, we obtain a Lie group isomorphism of S3 x e onto a normal subgroup HI of SO(4). Similarly, the restriction of T to e x S3 determines a second normal subgroup H , of SO(4) that is isomorphic to S3.

76

11. Subgroups and Homogeneous Spaces

We finally note that S0(4), as a manifold, is diffeomorphic to the product RP3 x S3. In fact, let $: Ss x S3 -+ SO(4) be given by

P,4 E s3,x E w.

$ ( A4 ) ( 4 = P W 4

Then $ factors over the projection yield a diffeomorphism

7~

-

-

x

c:

S3 x S3 .+ RP3 x Ss to

RP3 x S3 _C SO(4).

s3. Homogeneous spaces In this article G denotes a fixed Lie group with Lie algebra E. 2.9. Definitions: Let K C G be a closed subgroup. Consider the set G/K of left cosets; i.e., an element of GIK is a subset of G of the form aK, (a E G). The projection a t,aK defines a surjective map x:

G -P G/K.

I t will be convenient to write n(a) = 5. We make GIK into a topological space by calling 0 C G / K open if and only if n-l(O) is open. If U C G is open, then x-+T(U)) =

u

uu.

aEK

Hence it is open in G. Thus n( U ) is open and T is an open map. It follows that the topology of G/K is second countable. Moreover, since K is closed, G/K is a H a u s d o d space. In this article we make G/K into a smooth manifold. In fact we prove Theorem 11: There is a unique smooth structure on G/K such that n is smooth and G/K is a quotient manifold of G. The dimension of G / K is given by dim GIK = dim G - dim K.

The manifold G/K so obtained is called a (smooth) homogeneous space. The uniqueness of the smooth structure on G/K follows immediately from the Corollary to Proposition V, sec. 3.9, volume I. T o construct the smooth structure on GIK, denote the Lie algebra of K by F , choose a subspace L C E so that E = L @ F and define r : L x K -+ G by ~ ( ky ,)

= exp

K .y

K E L , y E K.

Lemma V: There is a neighbourhood W , of 0 in L such that restricts to a diffeomorphism T:

W, x K Z O

onto an open set 0 C G. 77

T

11. Subgroups and Homogeneous Spaces

78

Proof: According to Lemma IV, sec. 2.2, there are neighbourhoods V , of 0 in L, V , of 0 in F and U of e in G with the following properties: (i) (k,h) w exp k * exp h defines a diffeomorphism

v$+

VL x

lp:

u

and (ii) U n K = exp(V,). For any subsets A, B C G define A-'B C G by A-'B

= {a-lb

1 a E A , b E B}.

Because the map (x, y) ct x-ly is continuous we can find a neighbourhood W , of 0 in L such that W,C V ,

and

(exp W,)-l * (exp W,) C U .

I t will be shown W, satisfies the conditions of the lemma. We prove first that the restriction of T to W , x K is injective. In fact, assume that ~ ( k ,y,) , = ~ ( k , y2) for some k, , k, E W, and y l , ya E K. Then (exp k1)YI

= (exp k,)Yz

and so yay;1E [(exp WJ'

Thus, for some h It follows that

E

*

exp W,] n K C U n K = exp V F.

V, ,y2yy1= exp h.

.

p(K1 , 0) = exp k, = exp K, exp h = l p ( K , , h).

Since y is injective,

k,

= K,

and

h = 0.

Hencey, = y, , and so T is injective in W, x K. I t remains to be shown that T is a local diffeomorphism at all points of

W, x K. T h e commutative diagram

3. Homogeneous spaces

79

implies that T is a diffeomorphism in WL x ( U n K ) . But ~ ( kb) , = 7(k, e) b for each b E K . It follows that T is a local diffeomorphism at all points of W, x K. Q.E.D.

-

Choose WLas in Lemma V. Set

2.10. Proof of Theorem 11. ci

= rr

o

exp: W L-+ G/K.

Then the diagram

commutes. We shall show that 01 is a homeomorphism onto the open set x ( 0 ) . Denote x ( 0 ) by 6. a is obviously continuous, and it follows from the diagram that 01 is open (because T is). T h e diagram also shows that a( W,) = 6. T o prove that 01 is injective, assume

4,)= +2),

kl

9

k2 E

WL *

Then, for some b E K ,

Since 7 is injective, k, = k, . Note also that 0 = OK and so n-l(6) = 0. We shall now construct a smooth atlas on GIK indexed by the points of G. First define a smooth structure on 6 via the homeomorphism a. Then a becomes a diffeomorphism, and the diagram implies that the restriction of x to 0 is smooth. Next, define maps T,: GIK -+GIK ( a E G ) by T,(bK)

=u

It will be convenient to write this as 6 I+

~K. a

it follows that each T , is a homeomorphism.

6. Since

11. Subgroups and Homogeneous Spaces

80

Now set

0, = a 0 *

and

a, = T,oa,

aEG.

We shall show that { 0, , a;', WL}is a smooth atlas for G / K . I n fact, let a E G and b E G be two points such that 0, n 0, # Then the identification map,

0.

is given by

=

a-1 o

Tb-', o 7~ 0 exp

= a-l o

T

0

Xb-,,

o

exp.

Since n is smooth in 0 = n-lo, it follows that the map z+, Hence this atlas makes G / K into a smooth manifold. Finally, in the commutative diagrams

is smooth.

all the horizontal maps are diffeomorphisms. It follows that n: G -+ G / K is smooth and that each ( d ~ )T,(G) ~ : -+T,(G/K)is surjective. Therefore n makes G / K into a quotient manifold of G . As regards dimension, observe that dim G / K = dim W L = dimL =

= dim

E - dimF

dim G - dim K .

Q.E.D. 2.1 1. Consequences of Theorem 11. Corollary I: ( d ~ )E~-+: T,(G/K) induces a linear isomorphism

EIF

N

T,(G/K).

Proof: Observe that n ( K ) = t?; since F

F C ker(dn), .

T h e derivative

=

Te(K), it follows that

3. Homogeneous spaces

81

On the other hand, dim Im(dr)e = dim T,(G/K) = dim E - dim F

and so ker(dn), = F.

Q.E.D. Corollary 11:

such that rr

o

For each a

ua =

E

G, there is a smooth map

0,:

0, -+ G

C.

Next, define a map T : G x GIK -+ GIK by setting

T(a,bK) = T,(bK) = UbK.

-

We also write T(a, 6)= a 6. T h e diagram

commutes. Since GIK is a quotient manifold of G, the smoothness of p implies that T is smooth (cf. sec. 3.9, volume I). T satisfies the relations (ab) . x = a . ( b . X)

and

e

.f

=

x

a, b E G,

P E GIK.

T will be called the action of G on G/K (cf. sec. 3.1 for general actions of Lie groups). Finally, note that if we consider the quotient space of right cosets Ka ( a E G), then Theorem I1 remains true. T h e proof differs from that given above only insofar as elements of K must be written on the left instead of on the right in all formulae. T h e space of right cosets will be denoted by K\G. 2.12. Factor groups. Suppose K is a closed normal subgroup of a Lie group G. Then the coset space GIK becomes a group, with multigiven by plication, ,ii, ,E(T(u), r ( b ) ) = r ( ~ b ) ,

U,

b E G.

Since G / K is a quotient manifold of G, it follows (as above for T ) that ,ii is smooth. Similarly, inversion in G/K is smooth. Thus G/K is a Lie group. It is called the factor group of G with respect to K.

82

11. Subgroups and Homogeneous Spaces

Proposition I: Let v: G + H be a surjective homomorphism of Lie groups and let K be the kernel of y . Then K is a closed normal subgroup of G and the map $: G / K ---t H defined by the commutative diagram,

GAH

is an isomorphism of Lie groups. Proof: Since GIK is a quotient manifold of G the map $: G / K+H is smooth. On the other hand, a,h is a bijective homomorphism. Hence, Corollary I11 of Proposition VII, sec. 1.7, implies that $ is an isomorphism of Lie groups.

Q.E.D. Example: T h e group SU(n) is a normal subgroup of U(n) and the factor group is isomorphic to S': U(n)/SU(n)

s 1 .

In fact, regard the determinant as a surjective homomorphism det: U(n)+ S1.

Then ker det

=

SU(n), and the result follows from Proposition I.

s4. The bundle structure of a homogeneous space

In this article G denotes a fixed Lie group of dimension n with Lie algebra E. K is a closed r-dimensional subgroup with Lie algebra F and i: K + G denotes the inclusion map. 2.13. The bundle (C, x , C / K , K ) . By Theorem I1 (sec. 2.9) the projection x : G + G/K makes G/K into a quotient manifold of G. I n this section we shall show that (G,x , G/K,K ) is a smooth fibre bundle. I n view of Corollary I1 to Theorem 11, sec. 2.1 1, there exists a covering of G/K by open sets V , and a family of smooth maps a,: V, + G such that x o u, = 1 . Define smooth maps V, x K -+ G by (CIo(x,Y) = a,(x)y,

v,

x

7

Y E K.

Then x(CI.(x,y)

XEV,, ~ E K .

= x,

Moreover, each y5, is a diffeomorphism onto x-l(V,) with smooth inverse v,: n-l( V,) + V, x K given by

v,(z) = (x, u,(x)-lz),

z E n-l( V,),

x

= xz.

It follows that (G, x , G/K,K ) is a smooth fibre bundle with coordinate representation {( V , , +,)}. Remark: T h e coordinate representation (( V , , + J } satisfies +a@*

Yl) . Yz

YlYZ) = (CI&

9

x E v a 9 Y1 * Yz E

K.

2.14, Orientations and fibre integration. Choose elements A,€AnE* and A , E hrE* such that A , # 0 and A , restricts to a nonzero determinant function in F. Recall that a strong bundle isomorphism G x E Z T ~ is given by (a, h) ++ U

h )

(cf. Proposition I, sec. 1.2). Hence differential forms A , A , E A'( G) are defined by &(a; L,hl

9

a**,

M,)= A E ( h 1 , .*, 83

A,)

E

A"(G) and

11. Subgroups and Homogeneous Spaces

84

and

Evidently, A,*AG = A ,

and

X,*AK= A K ,

aEG.

Lemma VI: With the notation above

(1) A , orients the manifold G. (2) d e orients the bundle (G, T , G/K,K ) (cf. sec. 0.15). (3) If K is connected, then G / K is orientable. Moreover, an orientation in TLGIK) extends to a unique orientation in G/K such that the diffeomorphisms T, (u E G ) are all orientation preserving. Proof: ( I )

This is clear (cf. sec. 1.13).

(2) First note that i*d, is a left invariant r-form on K whose restriction to the Lie algebra F orients F. Hence i*d, orients K . Now fix a E G and let j,: K , --f G be the inclusion map ( K , denotes the fibre over a). There is a unique diffeomorphism q,: K % K , , such that j , 0 9, = A, 0 i. Thus

and so j z d , orients K , . (3) Fix an orientation in T,(G/K). For each y E K , the bundle map dT, restricts to a linear automorphism of T,(G/K);since K is connected, these automorphisms all preserve the orientation. Thus, for each ii E G / K , the orientation of T,(G/K) induced by the isomorphism, dT,: T t ( G / K )-% T,(G/K),

is independent of the choice of a in K , , These orientations determine the desired orientation of G/K. I t is obviously unique.

Q.E.D. I n view of Lemma VI, the fibre integral

4. The bundle structure of a homogeneous space

85

is defined (cf. sec. 0.15). Since, by construction, the left translations A, all preserve the bundle orientation, we have (cf. Proposition VIII, sec. 7.12, volume I)

Now assume that both G and K are compact and connected. Then AK(G) = A(G). Choose A , and A , so that the corresponding forms A , , d K satisfy /,AG

Set d,/K

=

=

1

and

I K i *A , = 1 .

jKKdG. 8

Proposition 11: With the notation and hypotheses above: = A C / K a E G. ( I ) TL$AG/K (2) A c l K orients GIK, and JGIKAGIK = 1. (3) A , = T*A,/K A A,. 9

Proof: (1)

This follows from the relation (cf. formula (2.1))

~ ~ =A : ,A ,j= , ~

A,,

T Z , ~ K

K

~ E G .

K

(2) In view of Lemma VI, (3), GIK is orientable. Orient it so that A , represents the local product orientation in G (cf. sec. 0.15). T h e n

(cf. sec. 0.15, or Theorem I, sec. 7.14,volume I). Thus, for some @ E GIK, ~ l ~ / ~ is ( istrictly f) positive. Since the T, are all orientation preserving, it represents the orientation of G / K . now follows from (1) that Observe that jKAK is a function on G / K , and that for E GIK (3)

i.e., jKAK = 1 . Now set m* A G / K A A K =

@.

11. Subgroups and Homogeneous Spaces

86

Then in view of sec. 0.15 (cf. also Proposition IX, sec. 7.13, volume I)

lG@=j

GIK A G / K j K A K = l *

On the other hand, A,*@ = 7r*Tz d G / K A

A K = T* d G / K A A K =

These relations imply that CP =

@,

a E G.

. Q.E.D.

§5. Maximal tori In this article G denotes a compact connected n-dimensional Lie group. Its Lie algebra, E , is equipped with a fixed Euclidean inner product, ( , ), with respect to which the transformations Ad x ( x E G) are isometries (cf. Proposition XVI, sec. 1.17). 2.15. Maximal tori. Let T be a closed connected abelian subgroup of G. Then T is compact and (cf. Theorem I, sec. 2.1) a Lie subgroup. It follows from Proposition XIII, sec. 1.12, that T is a torus. I n particular, the closure of a 1-parameter subgroup is a torus. A maximal torus in G is a torus that is not properly contained in another torus. Clearly, an automorphism of G carries a maximal torus onto a maximal torus. Let T be a torus in G and denote by F the Lie algebra of T. Then F C E and hence we can write

E

=FL O

F,

where F L denotes the orthogonal complement of F in E. Proposition XII, sec. 1.11, shows that the adjoint representation of T in F is trivial. Moreover, FI is stable under Ad y ( y E T ) .Thus we can write Ady=AdLy@L,

YET,

(2.2)

where Ad1 y denotes the restriction of Ad y to FI . Lemma VII: Let T be a torus with Lie algebra F and let a E T be a generator (cf. sec. I. 12). Then F C ker(c - Ad a)

and equality holds if and only if T is maximal. Proof: Let S be any torus in G such that T C S and denote the Lie algebra of S by H. Then FCHCker(6 -Ada).

Thus F C ker(c

-

Ad a ) and if equality holds, then T is maximal. 87

11. Subgroups and Homogeneous Spaces

88

Conversely, assume that T is maximal. Let L = ker(c. - Ad a). Formula (2.2) shows that L=FLnL@F.

Now, if h E FI n L , then the 1-parameter subgroup H generated by h centralizes u and so it centralizes T . I t follows that the closure S of the group H T is compact, connected, and abelian; i.e., it is a torus in G with h in its Lie algebra. Since T is maximal, and S 3 T, we have S = T. Hence

-

heFnFI

i.e., ker(r - Ad u )

= (0);

= F.

Q.E.D. 2.16. The Weyl group. Let T be a maximal torus in G and consider its normalizer, N( T ) . N( T ) is a compact Lie group (cf. Example 4, sec. 2.4) and T is a closed normal subgroup of N ( T ) . T h e factor group W , = N ( T ) / T is called the Weyl group of G (with respect to the maximal torus T ) .

Remark: It follows from Theorem 111, sec. 2.18, that the Weyl groups of G with respect to any two maximal tori are isomorphic. Example: Let U(n) be the unitary group. T o find a maximal torus, let ei ( j = I, ..., n) be an orthonormal basis of @" and consider the unitary transformations given by r e j = cjej,

j

=

I,

..., n,

E ~ E @ ,

1 ~j I

= 1.

Clearly these transformations form a torus subgroup, T , of U(n). T o show that T is maximal assume that S is an abelian subgroup of U(n) such that T C S. Then every u E S and T E T commute. Choose T E T such that the ej are distinct, Then the transformations that commute with 7 preserve the one-dimensional spaces (e,): aej = h i e j ,

aE

S.

Thus S C T. T o determine N( T ) , let a E N( T ) . Then a-lTa C T and so we have, for T E T , ((Y-%(Y) ei = ciei , i = 1,2,...,n;

5. Maximal tori

89

i.e., the vectors a(eJ are eigenvectors for T. Choosing T E T such that the ei are distinct we see that each a(ei) must be a scalar multiple of one of the vectors e, ,..., e, . Since a is injective, it follows that a(e,)

=

i = 1,..., n,

Aieu(i) ,

where w is a permutation of ( 1 , ..., n). Conversely, every unitary map of this form normalizes T . In particular, the correspondence a H w defines a surjective group homomorphism p: N ( T )+ S" (S" the group of all permutations of the set {1,..., n}). Evidently ker p = T , and so p induces an isomorphism N

sn.

wu(n) z

Proposition 111: Let T be a maximal torus in G. Then T is the 1-component of N ( T ) .I n particular, the Weyl group W , is finite. Proof: Let H denote the Lie algebra of N( T ) . We show that H = F. Since F C H , we have H = ( F L n H)@F.

Since T is a subgroup of N( T ) , H is stable under the transformations Ad y ( y E T ) and so F I n H is stable under these transformations. Since ad = Ad' (cf. Proposition XI, sec. l.lO), this implies that FJ-n H is stable under ad h, [h,k]~F'nH,

hcF, kEFLnH.

On the other hand, since T is normal in N( T ) ,F is stable under the maps Ad x, x E N ( T ) . Hence it is stable under the transformations ad k (k E H ) . I n particular, for h E F and k E FL n H , we have [h, R] E F n (FL n H ) = 0,

whence (Ad y)k = K, y E T , k E FI n H. Since T is maximal, it follows from this relation and Lemma VII, sec. 2.15 that FL n H = 0. Thus F = H. In particular, T and N( T ) have the same dimension, and so T is a connected, open subgroup of N ( T ) ;i.e., T is the 1-component of N ( T ) . It follows that the compact group W , is discrete, and hence finite.

Q.E.D.

11. Subgroups and Homogeneous Spaces

90

Since T is normal in N( T ) , a left action, @, of N ( T )in T is given by @(x, y ) = XJJX-~,

y

E

T, x E N(T).

Because T is abelian, @(x, y ) depends only ony and the coset % of x in W, . T h u s a left action, Y,of the finite group W , on T is given by V(3, y )

= @(X,

Y),

X E

NV), y

E

T.

2.17. The map 9. Let T be a maximal torus in G with Lie algebra F. Define a map, p:G x T+G,

by p(x, Y ) = x ~ x - ' ,

x E G,

JJ E

T.

Since T is abelian, it follows that dxz,Y ) = d

X ,

y),

z E T.

Hence cp factors over the projection, rr

x L : Gx T - + G / T x T ,

to yield a smooth commutative diagram

GXTAG

G/T x T

It follows from the definitions that the diagram,

commutes, where i a n d j are the inclusions ( j is induced by the inclusion map N ( T ) + G).I n other words, Y is the restriction of t,h to N( T ) / T x T. 2.18. Degree of 9. Orient F l and F and give E the orientation induced by the decomposition E = FI @ F. Orient T,(G/T)so that

-

(drr),:F L -% T,(G/T)

5. Maximal tori

91

preserves the orientations. T h e orientations of F and E determine left orientations in T and G ; because T and G are connected, these orientations are also invariant under right translations. Finally (cf. Lemma VI, sec. 2.14) the orientation of T,(G/T)determines an orientation of G/ T. Since t,h is a map between oriented, compact manifolds of the same dimension, the degree of t,h is defined (cf. sec. 0.14). Proposition IV: With orientations as described above, the degree of t,h is equal to the order of the Weyl group,

I n particular, t,h is surjective. Lemma VIII: T h e derivative of commutative diagram,

where a,(h, R,k)

=

4 at

R,((L - AdLy)h

(F, y) (y E T ) is given by the

+ k),

h EFI,k

Proof: Observe (via Lemma I, sec. 1.1) that

civ

E

F.

is the restriction of

Q.E.D. Lemma IX:

Let a be a generator of T and let 3 G G/ T. Then

is an orientation preserving isomorphism. Proof: I n fact, the diagram

*

G/T x T - G

11. Subgroups and Homogeneous Spaces

92

commutes, where T denotes the left action of G on G / T and T&)

= XYX-'.

T h e vertical arrows are orientation preserving diffeomorphisms. Thus we can reduce to the case Z = t?. Since a generates T , Lemma VII of sec. 2.15 implies that the map L

- Ad'a :FL+F'

is a linear isomorphism. Hence a, is an isomorphism, and so Lemma VIII shows that (d$)cc,a, is an isomorphism. Moreover, since Ad1 a is a proper rotation, det(r. - AdL a) > 0, and so a, preserves orientations. Thus so does ~I,/J(~,,,

,

Q.E.D. Proof of the proposition: Regard W, as a subset of G/T. First we show that, for a generator a E T , $-1(a) = ((2,x-lax) I 2 €

In fact, if x

E

G, y

E

WG}.

T , then a = qJ(x,y) = xyx-'

holds if and only if x-'ax = y . Since a generates T and y E T, this implies that x E N( T ) . Thus p-l(a) = {(x, x-lax) I x E N ( T ) } ,

whence $-1(a)

= ((2, x-lax)

I % € WG}.

According to Proposition 111, sec. 2.16, W , is finite. Moreover, for W, , x-lax is again a generator of T.Thus Lemma I X shows that dt,h is an orientation preserving isomorphism at each point (5,x-lax). Now Theorem I, sec. 6.3, volume I, implies that ff E

deg $ = cardinality of $-'(a)

=

I W , I. Q.E.D.

Theorem 111: Every element of a compact, connected Lie group G is contained in a maximal torus, and any two maximal tori are conjugate.

5. Maximal tori

93

Proof: Let T be a maximal torus and let a E G. Since $ is surjective, there are elements b E G and y E T such that a = byb-l. Hence a is in the maximal torus bTb-'. If S is any maximal torus, let a be a generator of S. Then, for some b E G, a E bTb-'. This implies that S C bTb-'.

Since S is maximal, it follows that S

=

bTb-l; i.e., S is conjugate to T. Q.E.D.

Corollary I: For every compact connected Lie group G the map exp: E + G is surjective. Proof: Given a E G choose a torus T such that a E T , and observe that the map exp: F + T ( F is the Lie algebra of T ) is surjective (cf. the corollary to Lemma VI, sec. 1.12). Q.E.D.

X

Corollary 11: For every compact connected Lie group G, the maps = f l , *2 ,... are surjective.

+xp, p

Proof: Note that exp ph = (exp h ) P and apply Corollary I.

Q.E.D. 2.19. The Weyl integration formula. We retain the notation of the preceding sections. In particular, G, T , and G / T are oriented as described in sec. 2.18 ; E and F are the Lie algebras of G and T ; and if y E T , A d l y denotes the restriction of Ad y to F I . A central function f on G is a smooth function such that

f(xyx-'1

= f(Y),

x, y

E

G,

or, equivalently, f(XY) = f(Y4

x, Y E G.

I n this and the next section we establish Theorem IV: Let f be a central function on a compact connected Lie group G. Then f ( x ) dx

=

1 W , 1-l

f(y) det(6 - AdL y) dy.

11. Subgroups and Homogeneous Spaces

94

Before proving the theorem we establish some notation. Let A , , be the differential forms as constructed in sec. 2.14; thus in A,, particular A , orients G, i*A, orients T , A G p orients G / T ,and

Proposition V:

The map $ of sec. 2.17 satisfies $* A ,

= AGl,

x g . i* A T ,

where g E Y (T ) is given by g ( y ) = det(c

- AdL y ) .

Proof: Since A,,, x i*AT orients G/T

for some g,

E

x T , we can write

Y ( G / T x T ) . Combining the relations

we find that ( T ,

x

t)*g, = g, , a

$*A,

E

G, It follows that

= AGIT X

g * i* A T ,

where g ( y ) = gdg, Y ) . On the other hand, using Lemma VIII, sec. 2.18, we find that

where k, E F ~ h,, EF. Since A , = .rr*A,/, A A , (cf. Proposition 11, sec. 2.14) and since pci*A, = i*AT ( y E T),we obtain

95

5. Maximal tori

Combining these two relations shows that g ( y ) = det(i - Adly).

Q.E.D.

2.20. Proof of Theorem IV: Since f is a central function, we have #*f = 1 x i*f and so Proposition V yields

#*(f A,) '

=

#*f.$* AG = A G I T

X

g ' i*(fd~).

Applying the Fubini theorem (cf. Proposition XIII, sec. 4.13, volume I), we obtain

=

j T f ( y ). det(i - Adly) dy.

On the other hand

(cf. Proposition IV, sec. 2.18) and thus it follows that

s

C

f ( x ) dx = 1 WGj--l

1 f ( y )det(i T

- Ad'-y) dy.

Q.E.D. Corollary:

1 WG1

= JT det(c. - Adly)

dy.

Problems 1. Centre. (i) Show that the centre of a Lie group is a closed normal subgroup and that the centre of a Lie algebra is an ideal. (ii) Find the centres of the groups O(n), SO(n), U(n), SU(n) and of their Lie algebras. 2. The derived group. Let G be a Lie group. A commutator in G is an element of the form xyx-’y-’; it is denoted by [x, y]. T h e derived group G‘ is the subgroup generated by the commutators. Let G(p)= G x --.x G ( p factors) and define cp: G@P)-+ G by cp(x1 ,y1,*’.) x?,,Yp) = P l

up

, n l ’ -.*

[x,

,YDI*

I m cp . (i) Show that G’ = Construct a distribution 4 on G (i.e., a subbundle of T ~ with ) (ii) fibre F, at x E G and satisfying the following conditions: (a) L,(F,) = F, ; (b) Im(dc,), C F+) ; and (c) for each x E G’, there is some p and some x E G(2p)such that Im(dcp), = F, . Show that these conditions uniquely determine 4. (iii) Show that the distribution 4 is involutive, and that the integral manifold through e is G‘.Conclude that G‘ is a Lie subgroup of G (cf. problem 8, Chap. 111, volume I). (iv) Show that if G is connected then so is G‘. Construct an example where Go is abelian and equal to G’, but G # Go. (v) Construct an example where G is compact and G’ is a proper, dense subgroup of G. 3. The Lie algebra of G’. Let E be the Lie algebra of a Lie group G. Let F be the subspace spanned by vectors of the form (Ad x - ~ ) ( h ) (x E G, h E E). (i) Show that F is the Lie algebra of G’.(Hint: First show that L,(F) = R,(F), x E G.Then compute the derivative of cp and conclude that Im(dc,), C LCp(,)(F), z B G(2P).Finally, by considering the paths x * exp th x-l exp ( - t h ) , show that T,(G’) C F ) . (ii) Let L be any Lie algebra. T h e derived algebra L‘ is the space spanned by vectors of the form [h, k ] , h, k E L .Show that E‘ C T,(G’). (iii) If G is connected, show that E’ is the Lie algebra of G’.

-

-

96

Problems

97

4. Homomorphisms. If rp: G -+ H is a homomorphism of Lie groups, show that ker rp is a closed normal subgroup of G. Show that G/ker rp is a Lie subgroup of H and conclude that Im y is a Lie subgroup of H .

5. The Lie subgroup associated with a subalgebra. Let E be the Lie algebra of a Lie group G and let F be a subalgebra of E. (i) Show that there is a unique connected Lie subgroup K of G with Lie algebra F. (Hint: Show that the spacesL,(F) (x E G) define an involutive distribution E on G and take K to be the maximal connected integral manifold of 6 through e (cf. problem 8, Chap. 111, volume I).) (ii) Show that the closure R of K is a Lie subgroup containing K as a normal subgroup. (iii) Show that the derived group (R)'is contained in K . 6. Connected subgroups. Let G be a Lie group and let K be a subgroup (i.e., a subset closed under multiplication and inversion). Assume that, for each y E K , there is a smooth path y t in G, joining e t o y and such that each y t E K . Prove that K is a Lie subgroup of G.

Hint: Let E be the Lie algebra of G and let F be the subset of E whose elements are tangent vectors at e to smooth curves which are contained in K . Show that F is a subalgebra. Then apply problem 5. 7. Killing form. T h e Killingform of a Lie algebra E is the bilinear function given by

K(h, k)

=

tr(ad h 0 ad R),

h, k E E.

(i) Show that K is symmetric and that the transformations ad h ( h E E) are skew with respect to K . (ii) If E is the Lie algebra of G, show that the transformations Ad x (x E G) are isometries with respect to K. (iii) Suppose E is the Lie algebra of a compact group G. Show that = 2, @ E'. Show that 2 , is the null space of the Killing form, and that the restriction of K to E' is negative definite. Show that (E')' = E'.

E

8. The groups Aut E, Aut C. (i) Let E be a Lie algebra. Show that the group Aut E of automorphisms of E is a closed Lie subgroup of GL(E). Show that the Lie algebra of Aut E is the Lie algebra of derivations of E.

11. Subgroups and Homogeneous Spaces

98

(ii) Let E be the Lie algebra of a connected Lie group G. Denote the group of automorphisms of the Lie group G by Aut G. Show that cr w cr' defines a group homomorphism rpc: Aut G -+ Aut E. 2: G be the universal covering group (cf. problem 19, (iii) Let Chap. I). Show that rpe is an isomorphism of groups, and hence make o y c maps Aut G injectively onto Aut into a Lie group. Show that the subgroup of Aut consisting of those elements which normalize kerp. (Hint: cf. problem 20, Chap. I). (iv) Conclude that Aut G is a Lie group and that rpc is an isomorphism of the Lie group Aut G onto a closed subgroup of Aut E. What is the Lie algebra of Aut G ? (v) Show that a homomorphism of Lie groups T: G + Aut G is defined by .(a)(.) = axa-l (a, x E G). Show that (rpc 0 ..)(a) = Ad a. Conclude that I m T and Im Ad are normal Lie subgroups, and that G/Z, .

Im Ad

Im 7

What is the Lie algebra of Im Ad ? (vi) Consider a vector space V as a Lie group (under addition). Find Aut V . 9. Semidirect products. (i) Let H and K be Lie groups. Obtain a bijection between Lie group homomorphisms rp: H + Aut K and smooth maps #: H x K -+K that satisfy + ( v z

5

4

= $(a1

3

+(a2

9

and

b))

+(a, b l h ) = +(a,

4 )+(a9 4-

(ii) Let rp: H -+ Aut K be a Lie group homomorphism. Define a multiplication in H x K by (a,

4

*

(a1 9

bl)

= (a.1

9

[da;')(@l . 4).

Show that this makes H x K into a Lie group; it is called the semidirect product of H and K (via rp), and written H x , K . (iii) Let F and E be Lie algebras. An action of F on E by derivations is a homomorphism 8 o f F into the Lie algebra of derivations of E. Given such a homomorphism show that the multiplication in F 0E defined by [(hl

9

h)7

(hz *

MI

=

(Pl

5

hzl, [kl

1

h1 + Wl)(kZ) - W Z ) ( M )

is a Lie product. This Lie algebra is called the semidirect product of F and E (via e) and is written F E.

o8

Problems

99

(iv) Show that the Lie algebra of a semidirect product of Lie groups is a semidirect product of the Lie algebras. (v) Show that a Lie group G is a semidirect product of Lie groups H and K if and only if: (a) H and K are closed Lie subgroups of G, (b) K is normal in G, (c) H n K = {e}, and (d) every element in G is of the form ab, a E H , b E K . (vi) Tangent group. Show that the Lie group TG is the semidirect product G x Ad E. (vii) Afine group. A map y : Rn --t Rn is called afine if it is of the form q ( x ) = +(x) a, where is linear and a E R". Show that the affine bijections of Rn form a group under composition. Identify this with the group GL(Rn) x T Rn, where T is the standard representation of GL(Rn)in Rn.

+

+

10. The group SU(2). Make C2 into a Hermitian space.

(i) Show that the complex linear transformations y of C2 that satisfy y = AT ( A E R, T E SU(2)) form a real four-dimensional subalgebra A of L(2; C). Show that A is isomorphic to W. (ii) Obtain an isomorphism SU(2) Q(1) of Lie groups. Conclude that SU(2) is diffeomorphic to S3. 11. The group SO(4). (i) Show that SO(4) contains two normal subgroups H , and H , each isomorphic to SO(3). (ii) Show that SO(4) = ( H , x H2)/&. (iii) Let T E SO(4) and suppose T # i t . Write R4 = F M F I , where F and F I are planes, oriented so that R4 has the induced orientation, and stable under T. Let T~ and T ~ be I the restrictions of T to F and F I , and let 9 and 91 (-n < 9, 91 n) be the corresponding rotation angles. Show that T E H , if and only if 9 = 8 1 and 7 E H , if and only if 9 = -91.


0, then O( p , q) is not compact. (iii) Find the Lie algebra of O( p , q). 13. The Lorentz group, T h e Lie group O(3, 1) is called simply the Lorentx group.

(i) Show that O(3, 1) has four components. Characterize its I-component, OO(3, 1). (ii) Show that an inner product of type (3, 1) is defined in the space, S, of selfadjoint mappings of C2 by (u, T) = +(tr u

0

T

- tr u

*

tr T),

u, T E

S.

Conclude that O(3, 1) is the group of isornetries of S. (iii) Show that SL(2;@) is the universal covering group of OO(3, 1 ) and that the covering projection is given by (..a)(.)

Find the kernel of

=

01

0

u 0 d,

01

E SL(2; @), u E

s.

7.

14. The Mobius group, (i) Show that the fractional linear transformations

+

az b T(z)= -

cz+d’

a , b, c, d E @,

ad - bc =

1,

form a group M of smooth transformations of the Riemann sphere S2. (ii) Show that M is a Lie group with SL(2; @) as universal covering group. (iii) Show that M is isomorphic to the Lorentz group OO(3, 1). (iv) Show that the map M x S2-+ S2 given by ( T , x) )--t T ( x ) is smooth. 15, Elliptic isometries: Let M E denote the subset of M consisting of the transformations of the form

+

az b T ( z ) = -62 + a ’

I a l2

+ I b l2 = 1.

101

Problems

(i) Show that Me is a closed subgroup of M , diffeomorphic to RP3. (ii) Define a Riemannian metric, g, in S2 such that g(z; C1, 5,) = (1 I z /2)-2 it is called the elliptic metric in S2.Show that ME acts via isometries on S2 with respect to g. (iii) Let u be the stereographic projection of the 2-sphere of diameter 1 from the north pole to the tangent plane T, at the south pole. Identify T, with the complex plane with elliptic metric. Show that (T is an isometry. Conclude that ME SO(3).

+

16. Hyperbolic isometries: Let MH be the subset of M consisting of the transformations of the form

(i) Show that M H is a closed subgroup of M , diffeomorphic to the manifold obtained from the hyperboloid x: x i - x i - x i = 1 in R4 by identifying antipodal points. (ii) Show that M His a group of isometries of the unit discQ( I z I < 1) with respect to the Riemannian metric given by

+

T h e unit disc with this Riemannian metric is called the hyperbolic plane. Show that a fractional linear transformation is in M H if and only if it maps Q onto Q. (iii) Give R3 an inner product of type (2,l). Consider the hyperboloid H of vectors x satisfying (x, x) = -4. Show that the inner product induces a positive definite Riemannian metric in H . Show that hyperbolic stereographic projection is an isometry of the lower shell of H with Q. Conclude that M H = OO(2, I). 17. (i) Show that every homomorphism y : U ( n ) -+ S1has the form V(T) = (det .r)*(det T)‘J ( p , q E Z).

(ii) Show that the Lie groups S1 x SU(n) and U(n) are diffeomorphic, but not isomorphic. Construct a covering projection S1x S U ( n ) + U ( n ) .

102

11. Subgroups and Homogeneous Spaces

(iii) Show that the Lie groups Z, x SO(n) and O(n) are always diffeomorphic, but isomorphic if and only if n is odd. (iv) Let y : U(n)-+ S1be the homomorphism given by V ( T ) = det T . Show that there is no homomorphism 4: S1-+ U(n)such that y 0 t,h = L. (v) Let y : O(n)-+ Sobe the homomorphism given by P ( T ) = det T . Show that there is a homomorphism t,h: So-+ O(n) such that y 0 t,h = L if and only if n is odd. (vi) Show that U(n) = SO(2n) n Sp(n; W) = SO(2n) n GL(n;C).

(vii) Show that Q(n) = SO(4n) n GL(n;W). 18. Let P be a representation by isometries of a compact connected Lie group G in a Euclidean n-space W. Fix a vector vo E W and let K = {a E G I P(a) v,, = vo}. Assume that K is connected and that G * vo contains an orthonormal basis.

(i) Construct a smooth map a : G/K+ W, such that &(a)= P(a)ao. (ii) Show that there is an r-form d on G / K (r = dim G / K )such that T,* A = A ,

acG,

and

s,/K

A

=

”

where T is the action of G on GIK. (iii) If y : W + W is a linear map show that

wheref(x) = (ya(x), ax), x E G/K. (iv) Apply this to the natural representation of SO(n) in AW to obtain an integral formula for the characteristic coefficients of a linear transformation of W. 19. Clifford algebras. Let ( E , ( , )) be an n-dimensional space with a symmetric bilinear function ( , ). Let 9 C BE be the ideal generated by elements of the form x @ x - (x, x), x E E. T h e factor algebra C, = BE/$ is called the CZiSford algebra of ( E , ( , )); the canonical projection is written T : BE -+ C, .

(i) Obtain a Z,-gradation CE = C i @ Ci of CE from the decomposition B E = C, even OPE 0C podd OPE.

103

Problems

(ii) Show that 7~ , 1 is injective and identify E with n(E). (iii) Show that CE satisfies a universal property that determines it uniquely. (iv) If the bilinear function ( , ) is zero, show that C, = AE. (v) Assume a direct decomposition E = F @ H such that ( y , z ) = 0, y E F , z E H . Prove that C, C, @ CH (as &-graded algebras), where the right-hand side is the anticommutative tensor product. Conclude that dim C, = 2". (vi) Let C, be the subspace of C, spanned by the vectors 1, k). Show that C, C, C C,,, . Obtain an algebra ... .xi ( x y E E,j structure [n @, C,JC,-, ,and show that this algebra is isomorphic to AE. (vii) Let C+, (respectively, C,) denote the Clifford algebra of an n-dimensional space with a positive (respectively, negative) definite inner product. Establish isomorphisms


.

and

Then

T h e case k is obvious.

< - 1 can be treated in the same way and the case k = 0 Q.E.D.

4.15. The Lefschetz class. I n this section we assume that G is oriented. Denote its orientation class by wG E H"(G) (cf. sec. 0.14). 3efine the quotient map

by q(a, b) = a% Proposition VII:

T h e Lefschetz class, A , , for G is given by (1,

= q*wG

(cf. sec. 10.3, volume I). Proof: Let 7rL , nR:G x G -+ G be the left and right projections. It has to be shown (cf. Corollary I to Proposition I, sec. 10.3, volume I) that

Let y be the diffeomorphism of G x G given by v(a, b)

= (a,

4.

y is a fibre preserving and orientation preserving map of the trivial

4. Cohomology of compact connected Lie groups

171

bundle (G x G, r L ,G, G). Moreover, it induces the identity map in the base. Hence Proposition VIII, sec. 7.12, volume I, yields

whence

01

=

J#

vqT#RO1' T#RwG,

01

E

H(G).

G

Recall that we identify H ( G ) 0H(G)with H(G x G) via the Kunneth isomorphism K # (cf. sec. 0.14). It follows from Example 2, sec. 5.17, volume I, that if y E H ( G x G), then

wherej, : G -+ G x G is given byj,(a) this yields y

Now set y

. T#RwG

= @ria.

=j T y

= (a, e).

'# 0W G = rLjly

Since

wG

*

H+(G) = 0,

#

Observing that

T ~ w G .

rR 0 p o j l = L

we find that

(cf. Example 2, sec. 7.12, volume I).

Q.E.D. Corollary I: Let M be a compact connected oriented manifold and let p, #: M -+G (dim M = dim G = n) be smooth maps. Then the coincidence number (cf. sec. 0.14) for p and # is given by

L(v, 4

=

deg(v-'

*

ICI),

IV. Invariant Cohornology

172

where

v-l * #: M -+G is given by (q-1

*

*)(x) = p'(x)-'

x E M.

. *(x),

Proof: Apply Proposition VII, sec. 10.7, volume I, noting that p-l

where A , : M

.

*

=

qO

(p'

x *) A,, O

x M is the diagonal map.

+M

Q.E.D. Corollary 11: Let y : G -+G be a smooth map and denote by q+p) the restriction of @ to H p ( G ) . Then the Lefschetz number of q~ is given by n

L(p')

=

(-1)p tr p'p' = deg v1 ,

p=O

where

vl = q r l

L.

Corollary 111: Let k EZ. Then the Lefschetz number of the power map Pk is given by

L(P,)

=

deg

I n particular, the Euler-PoincarC characteristic of G is 0 (set K = 1).

T

4.16. The spaces HJG). Let T be a maximal torus in G and let T. Recall that a smooth map $: G / T x T + G is given by

= dim

-

+(nu,y ) = uyu-l,

u E G, JJ E T ,

where n: G G / T denotes the projection (cf. sec. 2.17). Clearly, the diagrams G/T x T-G

dkI G/T x T commute, where

rj,

*

*

kk G,

(4.5) KEZ,

denotes the power map for T.

4. Cohomology of compact connected Lie groups

173

These yield the commutative diagrams H ( G / T )@ H ( T ) z H(G)

&P;

t

H ( G / T )@ H ( T )

*"

tP:

H(G),

k E Z.

Proposition VIII: Let H,(G) denote the eigenspace of the linear map P: : H(G)+ W(G) corresponding to the eigenvalue 2 p . Then

(1) H ( G ) = Z;=n HJG). (2) For every k # 0, H J G ) is an eigenspace of the linear map, P$ , corresponding to the eigenvalue k p . Proof: Recall from the example of sec. 4.14 that, for PE(01)

= kP

cy

E

HP( T),

' 01.

Thus H ( G / T )@ H ( T ) is the direct sum of the eigenspaces H ( G / T )@ HP(T) of L @ & corresponding to the eigenvalues k p ( p = 0, ..., r ) . I n view of the diagram above, I m I,P is stable under the map L @ P i . This implies that T

Im * ) I n [ H ( G / T )@ HP(T)].

Im I,$*= p=n

+

Next observe that, according to Proposition IV, sec. 2.18, deg # 0 and so $J* is injective (cf. Corollary I to Proposition 111, sec. 6.5, volume I). Hence the relation above shows that

and that Pk#restricts to it follows that

kp

*

L

in (+*)-'(H(G/T)@ H p ( T ) ) . I n particular,

HAG) = (*#)-'(H(G/T)0H P ( T ) )

(consider the case k = 2 ) , and so both parts of the proposition are obvious.

Q.E.D.

174

IV. Invariant Cohomology

Corollary: p* restricts to linear maps p#: H,(G) -+

1H,(G) 0Hj(G).

i+i=p

Proof: Apply Proposition VI (3), sec. 4.14.

Q.E.D. Lemma VII:

Each space H J G ) is graded, n

HAG)

=

c H3G),

9=0

where HS(G) = H,(G) n Hq(G). Moreover, if H,P(G)= 0.

01

p

f q (mod

2), then

Proof: T h e first part of the lemma is obvious. Now assume that (l), sec. 4.14, yields

E H i ( G ) . Then Proposition VI,

Thus, if p $ q (mod 2),

01

=

0. Q.E.D.

Proposition IX:

T h e dimension of H J G ) is given by dimH,(G)

where r

=

Proof:

=

('), P

O0 wK

if and only if G and K

1.

Proof: T h e first formula follows from Proposition XI1 and Theorem V, and shows that X G j K >, 0. Let S be a maximal torus in K , and let L, F, and E denote the Lie algebras of S, K , and G. Write E

=F

OFL = L @ ( L l n F ) OFL.

Let Ad:, Adl, and Adl, denote the representations of S in L l A F, and ( L l n F ) @ FI, induced by the adjoint representation of G; thus

FL

Adk(y) = A M Y )

0AdL(y),

y

E

8.

T h e Weyl integration formula (cf. Theorem IV, sec. 2.19) yields

J

det(r - AdL(x)) dx

=

Is

1 W K1 -' det(r - Ad'(y))

K

On the other hand, since Adh(y)

=

*

det(4 - Adi(y)) dy.

Adi(y) @ AdL(y), it follows that

det(r - AdL(y)) * det(r - Adl,(y))

=

det(r - Ad&(y)),

y

E

S.

Thus the first formula in the proposition applied to both G / S and GIK gives XG,,

=

1 W z I-'

S

det(r - Adk(y)) dy = I WK 1-l

XG,s.

Now assume that rank K < rank G. Then S is not a maximal torus in G. Hence the corollary to Proposition XI1 implies that X, = 0; it follows that X G / K = 0.

5. Homogeneous spaces

183

On the other hand, if rank K = rank G, then S is a maximal torus in G. Now the first formula in the proposition (applied when K = S ) together with the corollary to Theorem IV, sec. 2.20, yields X G / s = I W, I. This shows that X G , K = I WGlil W K1.

Q.E.D.

Problems 1. Left invariant p-vector fields. A p-vector field on an n-manifold M is a cross-section in the vector bundle APT,,,, . Denote the space of p-vector fields on M by A,(M) ( p = 0, ..., n).

(i) Given a Lie group G, use left multiplication to define a left action of G on rc , APT, and on A,(G). Ap-vector field Qj is called left invariant, if it is invariant under this action. T h e space of left invariant p-vector fields on G is denoted by Ak(G). (ii) Show that the map, r L :A,L(G)--+ APE, given by evaluation at e, is an isomorphism. n) (iii) Consider the space D,(G) of p-densities on G (0 p and let a: DJG) -+ D,-,(G) denote the divergence operator (cf. problem 8, Chap. IV, volume I). Show that an isomorphism,

<
be a coordinate representation for 6. T h e iso-' morphisms #, :F 7F, determine set bijections

xE

%.x

: GL(F)4 G,

,

u,

xE

9

by V,,z(T) = k x O

v E GL(F).

v,

Thus set bijections rp, : U , x G L ( F )+ n-l( U,) are given by %AX,

V) = $,,x

O

w

X E

u,

I

rp E GL(F).

Evidently (Pi' O %>(Xl

= (X?

Ki x

E

O

h

x

O

PI>

U, n U, , y

E

GL(F).

It follows that F '; o yo is a diffeomorphism of ( U , n U,) x GL(F). Hence (cf. Proposition X, sec. 1.13, volume I), there is a unique smooth structure on the set P such that ( P ,7 ~ B, , GL(F))becomes a smooth bundle.

1 . Principal bundles

195

Finally, define a right action of GL(F) on each set G, by setting

These actions define a right action of GL(F) on the set P. Moreover,

It follows that the action of GL(F) on P is smooth and that B = (P,T , B, GL(F))is a principal bundle. Fix a basis el ,..., e, of F. Then a bijection from G, to the set of bases (or frames) of F, is given by

For this reason B is often called the frame bundle associated with t. Frame bundles are discussed again in article 5 of this chapter, and then extensively in article 7 of Chapter VIII. 5.2 Elementary properties. Let B = ( P , T , B, G ) be a principal bundle admitting a cross-section u over an open set U C B. a determines the homomorphism 9 ~ :U x G -+P of principal bundles, given by ~ ( x u, ) = u ( x ) . a,

x

U , a E G.

E

9~ may be regarded as a strong isomorphism from the trivial bundle to the restriction of B to U. In particular, if 9’ admits a cross-section, it is the trivial bundle. If T is a second cross-section over a second open set V , then there is a unique smooth map

guv: U n V + G

such that p)(x, guv(x)) = T ( x ) . We have T(X)

= U(X) . g,(x),

and this equation determines g,,

x E U n V,

.

Lemma I: Let B = ( P , 7,B, G ) be a smooth bundle. Let T be a smooth free right action of G on P, whose orbits coincide with the fibres of the bundle. Then B is a principal bundle with principal action T.

196

V. Bundles with Structure Group

Proof: Let {U,} be an open cover of B such that each U, admits a cross-section u, : U, -+ P. Define $, : U, x G 3 n-l( U,) by setting

Then {( U , , $,)} is a coordinate representation satisfying condition (iii). Q.E.D. Next, let 4 = (p, ii, 8,G) be a principal bundle, and let $: B --f I? be a smooth map. We shall construct a principal bundle (P, T , B, G) together with a homomorphism, v: P-+ P , of principal bundles which induces $. I n fact, let P be the disjoint union, p

=

u ((4x

GLd,

XEB

and define T by setting T({x> x G*Lo)= x. Define a right action, T,of G on the set P and an equivariant set map 9): P P by ---f

T ( ( x ,z), a ) = (x, z * a )

and

y(x, z ) = z,

Z E G ~ ( , ) ,X E B , U E G .

Give P a smooth structure, as follows. Choose an open cover { V,} of I? such that each V , admits a cross-section D , : V , --t P. Set U , = $-’( V,) and define bijections X, : U , x G + n-l( U,) by Xv(x, 4 = (x, % M x ) )

Then for x E U , n U,

*

a).

,

(Xi1

O

Xv>(x,

4 = (x, g u ” ( w ) a > ,

where g,, : V, n V , -+ G is the smooth map satisfying

We can thus apply Proposition X, sec. 1.13, volume I, to obtain a unique smooth structure on P such that 9 = (P, T , B, G) is a smooth bundle with coordinate representation {( U , , X”)}. Since the maps X, are equivariant, T is a smooth action and (9, T ) is a principal bundle. Moreover, is a homomorphism of principal bundles. 9’ is called the pull-back of 6@ to B via r/, and it is often written $*@.

1. Principal bundles

197

Let 9, = (P,, n1 , B, G) be a second principal bundle over B which admits a homomorphism v,: P, + P of principal bundles inducing $: B -P I?. Then a strong isomorphism qz:P 5 PIis defined by v2(4

Note that

v1 v2 = v. 0

= ((F1)2

O

Yd4,

z E n-1(4.

s2.

Associated bundles

Notation convention: I n this article 9 = (P,n, B, G) denotes a fixed principal bundle with principal action T. Moreover, S:GxF+F

will denote a fixed left action of G on a manifold F. 5.3. Associated bundles. Consider the right action, Q, of G on the product manifold P x F given by QJz, y ) = (z,y ) * u = (Z *

*y),

U,

z E P, y E F , u E G.

Q will be called the joint action of G. T h e set of orbits for the joint action will be denoted by P x ,F and q: P x F + P x c F

will denote the corresponding projection; i.e., q(x, y ) is the orbit through (x,Y). T h e map q determines a map p : P X , F 4 B via the commutative diagram

P x

F A P X,F

P-B. 7r

where

np is

the obvious projection. Denote p-l(x) by F, , x E B.

Proposition I: that

There is a unique smooth structure on P

,

X,

F such

( 1 ) $. = ( P x F , p, B, F) is a smooth fibre bundle. (2) q: P x F -+ P x F is a smooth fibre preserving map, restricting to diffeomorphisms N

qz: z

x F aF,,(z) ,

on each fibre. 198

z E P,

199

2. Associated bundles

(3) ( P x F,q, P X, F,G) is a smooth principal bundle with principal action Q. (4) r pis a homomorphism of principal bundles. Definition: $. is called the jibre bundle with jibre F and structure group G associated with 8 ;q is called the principal map. Proof of Proposition I: If a smooth structure satisfies 3, it makes X, F into a quotient manifold of P x F under q. Hence, by the corollary to Proposition V, sec. 3.9, volume I, it is uniquely determined.

P

We construct a smooth structure on P x GF for which

Proof of (1):

5 is a smooth bundle. Let {Urn}be an open cover of B and consider cross-sections

u,:

U , --+ P. These are related by uBB(x)

=

Ua

*

-

uB

7

where #& U, n U, -+G are smooth maps. Define set maps, va: u a

xF

by setting P)&,

Y ) = q(%(x), Y ) ,

p-YUJ,

xE

u,

9

Y

EF.

Then p(ya(x,y ) ) = x and so y a restricts to set maps

va,s:F + p-'(x),

x E U,

.

Moreover, to each orbit in p-l(x) there corresponds a unique y E F such that the orbit passes through (a,(x),y). Hence is bijective, and so y , is bijective. Further, the relations q(x a, y ) = q(zl a * y ) imply that -1

%

P)dX,Y ) = (x,gadx)

'

Y)?

ua

UB

7

Y F.

Thus Proposition X, sec. 1.13, volume I, yields a smooth structure on P xc F for which 5 is a smooth bundle with coordinate representation 9

va)).

,

Proof of (3): T o show that ( P x F,q, P x F,G), is a smooth principal bundle with principal action Q consider the commutative diagrams, U, x G x F I

% - ..-I( -

U,) x F I

V. Bundles with Structure Group

200

where &(x, a ) = a,(x) 4-1(

*

V,)

a. Set V, = p-l( V,);then =

(U,)

(77 0 q ) - l

= 7r-1 ( U,)

x F.

Thus diffeomorphisms X,: V , x G 5 q-l( V,) are given by Xa(Va(X,

Y ) ,4

=

($a(%

4, a-l . Y ) .

They satisfy the relations X,(w,ab) = Q(X,(w,a ) , b),

and

(q o X,)(w,a ) = w,

w E V, , a, b E G

(cf. diagram 5.2). (3) follows. Proof of (2): T h e commutative diagram (5.1) shows that q is fibre preserving, while the commutative diagrams (5.2) imply that the maps

-

N_

42:

F

Fnk)

are diffeomorphisms. Proof of (4):

This is obvious.

Q.E.D. 5.4. Equivariant maps. Assume L@ = ( p ,6,8, G) is a second principal bundle and that is a left action of G on a manifold P. Suppose further that y:P+fj and wF+P

s

are smooth equivariant maps. x a: P x F + P x P is equivariant with respect Then the map to the joint actions of G; i.e., it is a homomorphism of principal bundles. Thus it induces a smooth map, ~ , x G ~ : P x G F + P x G P ,

which makes the diagram, P x F Z :

commute.

PxF

201

2. Associated bundles

Let

#: B -+

be the smooth map induced by a

P x G F '-f xPG

p'.

Then the diagram,

x,P

commutes; i.e., p' x 01 is a fibre preserving map between the associated bundles. T h e commutative diagrams

show that, if 01 is a diffeomorphism, then so is each (p' X, T h e case that B = @ and p' = L , is of particular importance; in this case we obtain a fibre preserving map, (I

x c ~ l )P: X G F + P

x~P,

which induces the identity map in B. 5.5. Examples: 1. F = {point}. Then P x F = B and the principal bundle (P x F,q, P x F, G) coincides with 9'. 2. Assume the action of G on F is trivial. Then E = ( B x F,p, B, F) is trivial. Also, if the principal bundle B is trivial, then so is 4.

3. Suppose y E F is fixed under the action of G: a * y = y , a E G. Then the inclusion j : { y } + F is equivariant. It induces (sec. 5.4) a smooth commutative diagram p

XG{Y)

thus cr is a cross-section in

--L-+ p XGF

[.

4. A-extension: Let A: G + K be a homomorphism of Lie groups. Then G acts from the left on K by u

*

Y

uE

= X(u)y,

Thus we obtain a bundle PA= ( P

X,

G, y

E

K.

K , p, B , K ) .

202

V. Bundles with Structure Group

O n the other hand, the multiplication map of K determines a right action ( P x K ) x K - P x K. This map factors over q to yield a free right action T A : ( P x G K ) xK - P x G K .

T h e orbits of T,, are precisely the fibres of P x GK. Thus it follows from Lemma I, sec. 5.2, that (8,,T,,)is a principal K-bundle. I t is called the A-extension of 8. Next, define a smooth map rpA:P --t P x K by setting rpA(z)= q(x, e). T h e diagram, L X A

P x G

* P x K

commutes (cf. diagram 5.1, sec. 5.3). This shows that rpA is a fibre preserving map from P to P x K , inducing the identity in B. I n particular, consider the case that G = K and h = L ; thus G acts on itself by left multiplication. I n this case rp,, is a strong isomorphism of principal bundles, and the diagram shows that (P x G, q, P X , G, G) is the trivial principal bundle.

,

5. Reduction of structure group: Again, let A : G -+ K be a homomorphism of Lie groups. Assume that @ = ( p ,73, B, K ) is a principal bundle. A reduction of the structure group of @from K to G via h is a principal bundle 8 = (P,T , B, G) and a smooth fibre preserving map rp: P-+ p, inducing the identity in the base, and satisfying rp(x * a) = q ( x ) * A(u),

a E G.

Such a reduction induces an obvious isomorphism of principal bundles from the A-extension of 9' to @ (cf. Example 4). Conversely, if 8 = (P , r , B, G) is any principal bundle with A-extension 8,, = (P x GK, p, B, K), then the homomorphism rpA of Example 4 is a reduction of the structure group of PAfrom K to G.

2. Associated bundles

203

5.6. Associated vector bundles. Assume now that F is a finitedimensional (real or complex) vector space and S is a representation of G in F. I n this case P x F is a vector bundle. In fact, €or each x E B, z E .TT-~(x), the diffeomorphisms q2:F 5 F, are connected by qz.a = qz 0 S(U),

uE

G.

Since each map S(a) is a linear isomorphism, there is a unique linear structure in F, for which the maps q2 are linear isomorphisms. T h e zero vector of F, is given by 0, = q(z, 0),z E n-l(x). Each pa,, of the coordinate representation {( U , , pa)} for 5 defined in sec. 5.3 is a linear isomorphism. Hence 5 is a vector bundle with vector bundle coordinate representation {( U, , tpa)}. Since q restricts to isomorphisms in the fibres, the trivial bundle (P x F, m p , P,F ) is the pull-back of f to P via 7-r (cf. sec. 2.5, volume I). T o the trivial representation S corresponds the trivial vector bundle. Next, consider a representation of G in a second vector space H a n d let a:F -+H be an equivariant linear map. Then the induced map (cf. sec. 5.41, L X ~ O L P:xGF+P x G H ,

is linear in each fibre, and so it is a (strong) bundle map. Denote the vector bundles corresponding to F and H by consider the induced representations of G in the spaces F@H,

F@H,

L(F;H),

F',

6 and 7 and

AF.

T h e associated vector bundles corresponding to these representations are given, respectively, by

T h e various canonical maps between these vector spaces, such as evaluation : composition: projection: trace:

--

L(F; H ) @ F LF @LF -+ F @ H LF ---f

H, LF, F,

R

commute with the representations of G. Thus they induce maps between

V. Bundles with Structure Group

204

the corresponding vector bundles. For the four examples above we have (cf. sec. 2.10, volume I). evaluation: composition: projection: trace:

7)

05

L, Q L, so7

L,

-

---t

7,

L, ,

s,

Y(B).

s3. Bundles and homogeneous spaces In this article K denotes a closed subgroup of a Lie group G. Their Lie algebras are denoted by F and E (FC E ) . T h e corresponding principal bundle (cf. Example 2, sec. 5.1) is denoted by Y K= ( G ,n K ,G / K ,K ) and we write e = 7rK(e) (e, the unit element of G). T h e left action of G on GIK is denoted by T .

'= (P,T,B, G ) 5.7. Bundles with fibre a homogeneous space. Let 9 be a principal bundle with principal action R. T h e left action of G on G / K determines an associated bundle

(cf. sec. 5.3). T o simplify notation we shall write P x (GIK) = P/K.

Consider the commutative diagram,

P

n

B,

and define p : P --f PIK by p(z) = q(x, e). Proposition 11: With the notation above, (P,p , PIK, K ) is a principal bundle whose principal action is the restriction of R to K . Proof: It is sufficient to show that each w E PIK has a neighbourhood W ) ,p , W, K ) is a principal bundle. W such that Let {( U, , yU))be the coordinate representation for 4 defined in sec. 5.3. Set W = p-l( U,), where 01 is chosen so that w E W. Then

V. Bundles with Structure Group

206

Finally, let j : G -+ G x GIK be the inclusion opposite 8. From diagram (5.2) of sec. 5.3 we obtain the commutative diagram exj

U, x GLx

U, x GjK

U, x G x GIKS(IXL.TT-~U, ,., x GIK -

- 1'.

T U, x GjK

L

N

lq

p-l(U,).

VU

It restricts to the commutative diagram,

where I) = (#, x L ) ( L x j ) . Now PKis a principal K-bundle, and I) is equivariant with respect to the given actions of K. It follows that (p-l( W ) ,p , W , K ) is a principal K-bundle. 0

Q.E.D. Next, fix z E P and write n(z) = x. Then the fibre inclusion, j,,, : G / K PIK, for the bundle E is given by ---f

N

jG,K

= qz : G / K

P-'(x).

Let j,: G -+ P and j,: K -+ P denote the fibre inclusions given by j&) = z * b

b E G, a E K ,

jK(a) = z * a,

and

and let i: K -+G be the inclusion map. Then the diagram, K

=

K

- -

G~K

jClK

P~K

P

B,

commutes. Moreover j, is a homomorphism of principal K-bundles.

3. Bundles and homogeneous spaces

207

5.8. Subgroup of a subgroup. Assume now that G is a closed subgroup of a Lie group H , and apply the results of sec. 5.7 to the principal bundle B = (H, T,H/G, G). We obtain an associated bundle,

5

=

( H x c GIK, p , HIG, GIK),

and a principal bundle

4 = ( H ,p , H x c GIK, K ) . T h e left action of H on HIK restricts to a smooth map, H x GIK + H / K ,

which factors to yield a diffeomorphism H

X~G H II K KZ

(equivariant with respect to the left actions of H, cf. sec. 5.9). We identify these manifolds via this diffeomorphism and write

5

= (HIK,p ,

4 = ( H ,p , HIK, K ) .

HIG, GIK),

Then 4 is the standard principal bundle, while p is given by p(uK) = uG,

u E H.

Moreover, diagram (5.4)reads K

I

=

G-

K

I

H

5HIG

GIK +H / K

P

(5.5)

II

H/G.

Now suppose that K is normal in G. Then a smooth free right action of the factor group GIK on HIK is given by Z * & = F T ,

XEH, UEG.

T h e orbits of GIK under this action coincide with the fibres in the bundle 5 = (H/K, p , HIG, GIK). It follows from Lemma I, sec. 5.2, that 5 is a principal GIK-bundle.

208

V. Bundles with Structure Group

5.9. Bundles with base a homogeneous space. Let K act from the left on a manifold N . There is a unique left action, A : G X ( G x K N ) - + GX K N ,

of G that makes the diagram, G x G x N z G x N

commute. Clearly A , together with T , is an action of G on the bundle [ = (G x K N , p, G / K ,N ) associated with Y K ; i.e., G acts on the total and base spaces and the projection is equivariant: poA

=

T o ( &x p).

-

Let N , = p-'(i?). Since a B = I ( a E K ) , it follows that A restricts to a left action K x NC+Nc. T h e projection q restricts to a K-equivariant diffeomorphism, qe : N

N

z Nz

(cf. Proposition I, (2), sec. 5.3). Conversely, assume that 7 = ( M ,pM , G/K,Q ) is a smooth bundle over G/K and that A (with T ) is a left action of G on 7. Then we can construct the bundle,

6 = ( G xKQerP,GIK,Qt), via the induced action of K on Qz. A restricts to a smooth map G x Qz-+ M . This factors over q to yield an equivariant fibre preserving diffeomorphism,

which induces the identity map in G/K. 5.10. Vector bundles. In this section we apply the results of sec. 5.9 to vector bundles. Each representation of K in a vector space N yields

3. Bundles and homogeneous spaces

209

a vector bundle over GIK associated with 9,(cf. sec. 5.6) in which G acts by bundle maps. Conversely, if G acts by bundle maps in a vector bundle 7 over GIK so as to induce the standard action in GIK, then the action restricts to a representation of K in the fibre over 8. If these two constructions are applied consecutively, starting off with a representation of K (respectively, a vector bundle over GIK acted on by G), we obtain a representation (respectively, a vector bundle acted on by G) which is equivariantly (respectively, equivariantly and strongly) isomorphic to the original. Examples:

If the representation of K in N is trivial, then

1.

G xKN

and

=

G/K x N

6 is trivial.

2. Assume that the representation of K in N extends to a representation of G in N . Define a diffeomorphism rp of G x N by setting

~ ( by ,)

=

(b, b-l * y ) ,

b E G, y

E

N.

Then (letting Q denote the joint action of K in G x N ) rpo(pa

x

agK

L) = Q a o r p ,

(pa denotes the right translation of G by a ) . It follows that rp induces a

diffeomorphism

4: G / K x Evidently

# is a strong vector

N

-L G x K N .

-

bundle isomorphism. Moreover,

# ( b * z , b * y=) b * + ( ~ , y ) , ~ E G z, E G / K , Y E N ,

(where G acts on G

X,

N as defined in sec. 5.9).

5.11. Tangent bundle of a homogeneous space. Recall that the Lie algebras of K and G are denoted by F and E. T h e adjoint representation of G restricts to a representation, Ad,,, , of K in E. Since the Lie algebra F is stable under the maps Ad,,,(a), a E K , we obtain a representation, Adl, of K in EIF. T h e sequence 0 -+ F

+E

-+

E/F

-+

0

is short exact and K-equivariant with respect to the representations Ad, Ad,., , and AdL of K.

V. Bundles with Structure Group

210

Now form the vector bundles

5

= (G

and

x K (EIF),P E , G/K,E / F )

7 = (G X K F ,p0 3 GIKF).

G acts on both 4 and r). On the other hand, the left action, T , of G on G / K induces a left action, d T , of G on the tangent bundle rClK(cf. Example 7, sec. 3.2). Proposition 111: With the hypotheses and notation above is strongly and equivariantly isomorphic to T ~ / (1) The vector bundle 4 @ r ) is trivial. (2)

.

~

~ ) a~ linear isomorphism Proof: ( I ) According to sec. 2.1 1, ( d ~induces

,.,

-

E/F 2 T,(G/K).

Since nK 0 ha = T,

0 TK

and

TK

0

pa = TK ( a E

( d ~o Ad,,,(a) ~ ) ~ = dT,

0

K ) , we have ,

( ~ T K ) ~

u E K.

Thus this isomorphism is equivariant with respect to A d l and dT. Now apply sec. 5.10. (2) Since the sequence F -+ E -+ E/F is K-equivariant, it determines a sequence of strong bundle maps , ~ G X , E : ~ .

For each z E G / K , the restriction, 0 + F,

-+

E, + (E/F), + 0,

is short exact. Hence, there is a strong bundle map CT: f -+ G X K E such that p 0 D = 6 (cf. Lemma 111, sec. 2.23, volume I). Thus a strong bundle isomorphism,

-

-

v: t B 7 - G

xKE,

is defined by ?(u, V )

=

U(U)

+ i(w),

u E (E/F),, v E F,

, z E GIK.

On the other hand, the representation Ad,,, of K i n E is the restriction of a representation of G. Hence, by Example 2 of sec. 5.10, G X K E is a trivial bundle over G/K. Thus 5 @ r) is trivial. Q.E.D.

3. Bundles and homogeneous spaces

21 1

5.12. Tori. Suppose now that G is compact and connected, and that K is a torus in G. Then the adjoint representation of K is trivial and

hence, so is the bundle 7 = (G x K F , pn , G / K , F ) *

Thus, byPropositionII1, sec. 5.1 1, the Whitney sum of rGIKwith a trivial bundle is trivial. This implies (as will be shown in sec. 7.19) that the Whitney sum of r G / K with the trivial bundle of rank one is trivial, T G I K@

d

e'fl,

Y

=

dim G / K .

(5.6)

Now we distinguish two cases:

Case I: K is a maximal torus (cf. sec. 2.15). Then the EulerPoincarC characteristic of G / K is positive (cf. sec. 4.21). Hence Theorem 11, sec. 10.1, volume I, implies that every vector field on G / K has at least one zero. In particular, the tangent bundle of G / K is nontrivial. Case II: K is not maximal. Then K is properly contained in a maximal torus, T. Since T is compact and connected, the factor group T / K is again a torus (cf. Proposition XIII, sec. 1.12). Thus according to sec. 5.8 we can form the principal T/K-bundle

9 = ( G / K ,v , G / T , TIK). Write TG/K= HG/K@ V G I Kwhere , VGIKis the vertical subbundle and HG/K is a horizontal bundle (cf. sec. 0.15). Since B is a principal bundle, the vertical subbundle is trivial (as will be shown in sec. 6.1), VG/K = ern,

m

=

dim T/K.

By hypothesis, K is properly contained in T and so we have m 2 1. On the other hand, HGIKis the pull-back of rClTunder rr. It follows that r G / K is the pull-back of rGlr@ ern. In view of relation (5.6), with K replaced by T , the bundle rGlT@ d is trivial. Hence so is r G / K . Thus if K is a nonmaximal torus, then the homogeneous space GjK has trivial tangent bundle.

s4. The Grassmannians 5.13. The Grassmann manifolds. Let r be one of the fields R, @, @ r. Introduce a or W and consider the vector space rn= @ positive definite inner product ( , ) in rnwhich is Euclidean, Hermitian, or quaternionic according as r = R, @, or W. I n the case I' = Iw also choose an orientation in P. A k-plane in r n is a r-subspace of r-dimension k. T h e set of all k-planes in rnis denoted by G,(n; k). An oriented k-plane in Rn is a k-plane F together with an orientation of F. T h e set of oriented k-planes in Rn will be denoted by G(n;k) if k < n. Finally, we define G(n;n) to be the set consisting of a single element, namely the oriented vector space Rn. This article deals with each of the four cases listed below. I n each case, I', I(n), G(n;k) is to be interpreted as described below.

r

Observe that in each case the Lie algebra of I(n) consists of the r-linear transformations of rnthat are skew with respect to the inner product ( , ). T h e Lie algebra of I(n) is denoted by E(n). T h e set G(n;k) is made into a manifold in the following way: First define a transitive left action of the Lie group I ( n ) on G(n;k) by setting

This yields a surjection, a:I(n) -+ G(n;It), given by

(where rkis regarded as the subspace of whose last n - k components are zero). 21 2

rn consisting of those vectors

4. The Grassmannians

213

rk

Denote (rk)l by rn-k: rn = @ P - k .This decomposition determines an inclusion, I(k) x I ( n - k) -+ I(n), and clearly a - l ( P ) = I(k) x

I(n - k ) .

Hence a induces a commutative diagram, I(n)

I(n)/(Z(k)x-I(n - k ) )

8-Gin; k), c= -

and /3 is an equivariant bijection. Give G ( n ; k) the unique manifold structure such that /3 is a diffeomorphism. T h e manifold so obtained is called the Grassmannian of k-planes in P. Since /3 is equivariant the action of I(n) on G ( n ; k ) defined above is smooth. Observe that the canonical isomorphism

I(k) x I(n - k ) -=+I(n - k) x I(k) 31

induces, via /3, a diffeomorphism

If F E G ( n ; k), then Q(F) is the orthogonal complement of F in

rn.

5.14. Examples: 1. The Grassmannian of k-planes in Rn: Assume that 0 < k < n. Then an involution, w , of c ( n ; k) is defined as follows: If F is an oriented k-plane, then w(F) is the same k-plane with the opposite orientation. On the other hand, a projection,

p : G(n;k)

-

G,(n; k),

is defined by forgetting the orientations of the elements of G ( n ; k). Evidently, p is a double covering and w is the involution that interchanges the two points in each p-l(F). T o see that p and w are smooth note that S O ( n ) acts transitively on Gw(n;k), and that the isotropy subgroup at Rk is the group K

= SO(n)n (O(k) x

O(n - k)).

V. Bundles with Structure Group

214

This group consists of two components, KO= { ( y ,(CI) I det y

=

1, det 3

3) I det y

=

-1,

=

l}

= SO(k)

x SO(n - k),

and

Kl

= {(y,

det (CI

=

-1).

T h e commutative diagram, SO(n)/(SO(k)x SO(n - k ) ) +G(n;k ) N_

shows that p is smooth, a local diffeomorphism and a double covering. Hence w is also smooth. T h e dimension of Gw(n;k) is given by dim Gw(n;k ) =

)(;

k

n--k

- (2) - (

) = k(n - k).

2. Real projective space: Assume that n 2 2 and consider the manifold c ( n ; 1). Its points are the oriented lines in R" through the origin. Identifying each such line with its positive unit vector, we obtain an SO(n)-equivariant bijection between G(n; 1) and S"-l. Since SO(n) acts smoothly on Sn--l,the commutative diagram (cf. Example 2, sec. 3.6)

SO(n)/SO(n- 1)

shows that this identification is a diffeomorphism. Moreover, the involution, w , in G(n; 1) defined in Example 1 corresponds under this diffeomorphism to the antipodal involution of Sn-l. Thus we obtain a diffeomorphism Gw(n;1)

--

[WPn-l

(cf. Example 2, sec. 1.4, volume I). Hence Gw(n;1) is diffeomorphic to the real projective space of dimension n - 1.

3. Complex and quaternionic projective space: Let n 2. T h e manifolds Gc(n; 1) (respectively, G,(n; 1)) of complex (respectively,

4. The Grassmannians

21 5

quaternionic) lines in Cn (respectively, W n ) through the origin are called complex (quaternionic) projective space and are denoted by CPn-l and WPn-' respectively. 4. Complex and quaternionic projective lines: We shall construct diffeomorphisms

@P1

-

S2

and

Y

WP1 --+ S4.

Define a map C -+ CPl by sending z E C to the one-dimensional complex subspace of C2 generated by the pair (1, z). This is a smooth embedding. Since dim C = 2 = dim CP', it is a diffeomorphism onto an open subset of @P1.T h e only point which is not in the image is the one-dimensional subspace of C2 generated by (0, 1). Since CP' is compact, it is the onepoint compactification of @; i.e., CP' is diffeomorphic to S2. Similarly, WP' is the one-point compactification of W and hence it is diffeomorphic to S4. 5.15. Canonical vector bundles over G(n; k). Recall that in secs. 2.1 and 2.22, volume I, we defined real and complex vector bundles. Quaternionic vector bundles are defined in a similar way, and the definition of all three may be given simultaneously as follows: A r-vector bundle is a smooth bundle 8 = ( M , n, B, F ) , in which F and F, ( x E B ) are r-vector spaces, and which admits a coordinate representation { ( U ,,4,)) such that each map, F

5F , ,

is a r-linear isomorphism. We shall construct canonical r-vector bundles over G(n;k). I t will be important to distinguish between a k-plane, F,as a subspace of P , and as a point in G(n;k). Consider the disjoint union M =

(J F. F€G(n;lr)

Thus a point of M is a pair (F, v ) with v E F. Let p : M + G(n;k) be the projection given by p(F, V ) = F. Observe that a left action of I(n) on the set M is given by

V. Bundles with Structure Group

216

rk)

We shall make tk = ( M ,p, G(n; k ) , into a r-vector bundle so that this action becomes a smooth action. Consider the representation of I ( k ) x I ( n - k ) in rkgiven by

(v, M u )

= p(u),

vE

W ,

4 GI(.

-

4, * E rk.

It determines a r-vector bundle (cf. sec. 5.10), &k

= (I(n)x I ( k ) X I ( n - k ) rk,

I(n - k ) ) , '?,

8,

which admits a canonical left action of I(n). Now define a surjective set map, 0:I(n) x r k + M , by setting

v E w,

q v , 4 = (v(rk), do)),

vE

rk*

Factoring through the joint action, we obtain the commutative diagram, Y

I(n) X I ( k ) X l ( n - k ) rk M -

4

1.

N

I(n)/(I(K)x I(n - k ) )

7G(n;k),

where fl is the equivariant diffeomorphism of sec. 5.13 and Y is an I (n)equivariant bijection restricting to linear isomorphisms on the fibres. Give M the manifold structure for which Y is a diffeomorphism. Then f k becomes a vector bundle acted on by I(n) and E[k = -+

is an equivariant isomorphism. Similarly, we obtain a vector bundle by setting MI =

tk ti

=

( M I , p, G ( n ; n

- k ) , I'n-k)

FL.

FEG(n;k)

It admits an action of I ( n ) and is equivariantly isomorphic to the bundle

(I(n)x I(k)XI(n-k) rn-k, 8, I(n)/(z(K) I(n - k)), (Replace rkby rn-k = (rk)l in the discussion above.) tk and t i are called the canonical k-plane and ( n - k)-plane bundles over G ( n ; k). T h e direct decomposition, r k

0r n - k -%r n ,

4. The Grassrnannians

217

determines a strong bundle isomorphism

Finally, the actions of I(n) on (k and & defined above, together with the standard actions of I(n) on G(n;k) and rndefine actions on the bundles (k @ (f and G(n;k) x rn.Moreover, the isomorphism defined above is equivariant. 5.16. The tangent bundle of C(n;k ) . Given two r-vector bundles ( and 7) over the same base B, we can form the (real) vector bundle Lr(5; 7) whose fibre at x E B consists of the r-linear maps between the fibres of 4 and 9 at x.

T h e tangent bundle of G(n;k) satisfies

Proposition IV:

7G(n;k) % L d 6 k ;

6).:

Proof: Identify G(n;k) with I ( n ) / ( I ( k )x I ( n - k)). According to sec. 5.11 its tangent bundle is obtained from the representation A d l of I(k) x I(n - k) in E(n)/(E(k)@ E(n - k)). On the other hand, Lr([k'k; ( i )is obtained from the representation of I(k) x I(n - k) i n L r ( r k ; P - k ) given by (0, T)('p)

=

7 0

'p

0

I J

U-l,

Ez(k),

7

E z ( n - k),

'p

ELy(rk;rn-k).

Thus we must construct an (I(k) x I(n - k))-linear isomorphism L y ( r k ;Ppk) E(n)/(E(k)@ E(n - k)).

Recall that E(n) is the realvector space of r-linear skew transformations of rn, and that rn-k = T h e Lie algebras E ( k ) ofI(k) and E(n - k) of I(n - k) (considered as subalgebras of E(n))are given by

(rk)l.

E(k) = {a E E(n) I u ( Z ' ~ - ~ )= 0)

and

E(n - k )

1

= {a E E(n) a ( P ) = 0).

Define a subspace L C E(n) by setting L

E(n) 1 a ( P ) C rn-k and

= {a E

a(Z'n--k)C P}.

Then E(n)

=

E ( k ) @ E(" - k ) @ L.

Moreover, since the adjoint representation of I(n) is given by (Ad u)a

=u

o

a

o

u-1,

u E I(n), a E E(n),

V. Bundles with Structure Group

218

it follows that L is stable under I(k) x I(n - k). In particular, there is an isomorphism of (I(k) x I(n - k))-spaces L

0E(n - k)).

E E(n)/(E(k)

Finally, define a linear isomorphism, D: L r ( r k ;

m - k )

2L,

by setting D(+

~

EL,(r*; r n - k ) ,

-qy),

y=)

x Er

k ,

y Em-k,

where d denotes the adjoint of a. Since I(k) and I(n - k) consist of isometries, it follows easily that @ is (I(k) x I(n - A))-equivariant. Hence Ly(rk; rn-k

) E L s E(n)/(E(k)0E(n - k ) ) ,

which completes the proof.

Q.E.D. Corollary: There are isomorphisms rCR(n;k)

6k

OR 6 Ik

9

TG(n:k)

6k

OR 6;

9

and

rGC(n;k)

* @C

6k

I 6k

(where tkis interpreted as a vector bundle over the appropriate manifold, and @ is the complex dual of f k ) .

s5. The Stiefel manifolds

We continue the notational conventions of article 4. 5.17. Stiefel manifolds. An orthonormal k-frame in of k vectors, ( u l ,..., uk),such that ( U i ,Uj) =

rnis a sequence

621. . '

An n-frame in the oriented space, Rn, is called positive, if it represents the orientation of Rn. We extend the conventions of this article by letting V(n;k) denote any one of the sets Plw(n;k), Va(n;k), Vc(n;k), and VH(n; k) defined by:

Case I Case I1 Case 111 Case IV

P,(n; k) Vmg(n;k) Vc(n;k) L''w(n; k)

Orthonormal k-frames in

R".

Orthonormal k-frames in R" if k < n; positive orthonormal n-frames in R" if k = n. Orthonormal k-frames in

@".

Orthonormal k-frames in

W".

A transitive left action of I(n) on V(n;k) is given by

I n particular, write rn= rk@ Tn-k and let (el ,..., ek) be a fixed orthonormal basis of rk.Then the subgroup of I(n) which fixes the k-frame (el ,..., ek) is exactly I(n - k) (cf. sec. 5.13). Thus the action of I(n) on V(n;k) determines an equivariant bijection Z(n)/l(n- k )

--

V(n;k ) .

Assign V(n;k) the unique manifold structure such that this bijection is a diffeomorphism. (Then the action above is smooth.) T h e manifold V(n;k) is called the Stiefel manifold of orthonormal k-frames in n-space. 21 9

220

V. Bundles with Structure Group

5.18. The universal frame bundle over C(n; k). A canonical principal bundle, 9 ( n ;k )

= (V(n;A), rk

9

G(n;k ) , I(k)),

is defined as follows: If (ul ,..., uk) E V(n;k), let r k ( u l ,..., uk)be the (oriented) k-plane with u1 ,..., u k as (positive) basis. Then rk: V ( n ;k ) + G ( n ; k) is a well defined map. Moreover, we have the smooth commutative diagram, I(n)/I(n- k )

N

V(n;k ) N

I-

I(n)/(I(k)x I(n - k ) ) A G(n;k),

where the horizontal diffeomorphisms are defined in sec. 5.17 and sec. 5.13 respectively, and ~ ( * uI(n - k ) ) = u (I(k) x I(n - k)), u E I(n). We can apply sec. 5.8 to obtain a smooth principal bundle

Thus the diagram above shows that rk is the projection of a smooth principal bundle, B (n; k) = ( V ( n ; k ) , rk , G ( n ; k), I(k)). Note that, if F E G(n; k) then rk1(F) consists of the (positive) orthonormal k-frames in F. For this reason B ( n ; k ) is called the universal frame bundle over G ( n ;k ) . T h e inclusion maps,

determine smooth commutative diagrams,

which are, in fact, homomorphisms of principal I(k)-bundles. T h e vector bundle, q k , associated with 9 ( n ; k) via the action of I(k) in rk,is canonically isomorphic to the bundle Sk = ( M ,p, G ( n ; k), rk) of sec. 5.15. Indeed, fix a (positive) orthonormal basis (e, ,..., ek) of rk.

221

5. The Stiefel manifolds

Define a map, q: v(n;k )

x

r k

-+ M ,

as follows:

I t is easy to check that q induces an I(n)-equivariant, strong isomorphism

5.19. The manifolds Z(n; k ) . Let I(n;k) denote the set of isometric inclusions r k 3 rn(except in case I1 when k = n; then I(n;n) will denote the set of orientation preserving isometries of Rn). Note that I(n) and I(k) act, respectively, from the left and right on I(n;k) via

v.*=vo* and * ‘ u = * o a ,

vEI(n), *€I(n;k), ,El(k).

Now fix a (positive) orthonormal basis, el I(n)-equivariant bijection, I(n; K) .-+ ‘(n; k),

,..., ek , of

rk.Then

an

is given by v I+ (ve, ,..., Ve,). We use this bijection to make I(n;k) into a smooth manifold, and to identify it with V(n;k). I n particular, we may write g ( n ;k )

= (I(n;k), mk

9

G(n;k),I(k)).

Then T k ( v ) = q(rk), rp €I(n; k). Moreover the principal action of I(k) is the right action given above. Finally, observe that the isomorphism I(n;k) X , ( k ) rk% ( k of sec. 5.18 is induced by the map, q : I(n; k) x rk+ M , given by q(p, w) = v(4,

4,

p

ZJ E

rk.

Proposition V: Let B = ( P , T , B, I(k)) be a principal bundle. Then, k, there is a homomorphism of principal bundles for some n

P -2 I(n;k) B

4

G(n;K).

V. Bundles with Structure Group

222

Definition: t,b is called a classifving map for the principal bundle 9.

According to the proposition, 9 is the pull-back of 9 ( n ; k ) to B via $. Before proving the proposition we establish Lemma 11: Let f = ( P x , ( k ) r k , p c , B, rk) be the vector bundle Then, for some n >, k, associated with 9 via the action of I(k) on there is a strong bundle map cr: .$ + B x rn restricting to r-linear injections on the fibres.

rk.

Proof: This lemma is proved in sec. 2.23, volume I, in case I and case 11. T h e same argument holds in cases I11 and IV, using Hermitian and quaternionic inner products. Q.E.D. Proof of the proposition: Let u be the bundle map constructed in Lemma I1 and let q: P x

r k +

P x ~ (r ~ k )

be the principal map(cf. sec. 5.3). Then a smooth map defined by the relation

v: P +I(n; k ) is

(p(z),9J(z)u)= u(q(4 U)),

z E P,

u E Tk.

= 9J(Z)(T(u)) = (9J(Z) 0 T ) U ,

z E P,

7

Clearly, 9J(Z ' T)U

E

I(k),

Hence rp(z * T ) = rp(z) 0 T and so rp is equivariant; i.e., morphism of principal bundles.

uE 9)

rk.

is a homoQ.E.D.

5.20. Examples: 1 . Hopf jiberings: A point of V ( n ; 1) is just a unit vector in P. Thus, if n > 1,

Vrw(n; 1)

= sn-1,

V,(n; 1)

V&; 1)

= S2n-1,

= S4*-1.

Moreover, the left action (cf. sec. 5.17) of O(n), U ( n ) ,and Q ( n ) on these spheres is the standard one. Next observe that in cases I, 111, and IV, I(1)can be identified with the unit sphere of r (r= R, C, or W) as follows: For each unit vector a E r, define pa E I( 1) by pa(z) = zu-l,

E

r.

5. The Stiefel manifolds

223

Then a e pa is an isomorphism of Lie groups (cf. Example 2, sec. 2.6 and Example 3, sec. 2.7). Thus the universal 1-frame bundles become (Sn--l, T,RPn-', SO),

(S2,-l,T,CPn-l, a), and

(S4n-1, T ,WPn-l, S3).

Notice that the first bundle is simply the double covering of Example 2, sec. 5.14. Moreover CPl = S2and WP1 = S4 (cf. Example 4, sec 5.14). Thus the bundles 8,(2; 1 ) and PH(2;1 ) can be written (S3,T,S2,Sl)

and

(S7,T , S4,S3).

Consider the right action of So (respectively, S1,S3) on Rn (respectively, C",W") given by (zl, ..., z,)

.z

=

(z-lzl , ..., z-lz,)zi , E r.

This action restricts to an action of So (respectively, S1, S3) on Sn-' (respectively, S2"-',S4"-I).We shall show that these actions are the principal actions of So,S', and S3 on the 1-frame bundles. In fact, let u E I ( n ; 1) and write u( 1) = (zl, ..., zn). Then u( 1) E Vr(n; 1) and the principal action of I(1) is given by (cf. sec. 5.19)

.

u(1) z =

(u 0 &)(I) = u(z-1) = (z-'zl

, ..., z-12,).

2 . The Stiefel manifold V R ( n ;2): Let r = R and consider the Stiefel manifold V R ( n ;2). Its points are the isometries a: R2-+ Rn. An embedding cp: V R ( n ;2) -+ Rn @ Rn is defined as follows: Choose an orthonormal basis e, , e2 in R2 and set ?(a) = (a(el),a(e2)). T h e image of cp consists precisely of the pairs (x, y ) satisfying Ixl=l,

IyI=l,

<x,y)=O.

On the other hand, consider the bundle ( M , 7,Sn--l, S1L-2),of unit tangent vectors of Sn-l. Then the map,

4: z I-+

( T ( Z ) , z),

z E M,

defines an embedding of M into Rn @ Rn and the images of q5 and cp coincide. Composing cp with the inverse of i,h yields a diffeomorphism of V a ( n ;2) onto M .

$6 The cohomology of the Stiefel manifolds and the classical groups T h e notation conventions of articles 4 and 5 are continued in this article. We shall frequently make the identifications V,(n; k) = SO(n)/SO(n- k).

and

Vc(n;k) = U(n)/U(n- k)

T h e tensor product of graded algebras is always the anticommutative tensor product. 5.21. Complex and quaternionic Stiefel manifolds. Theorem I: T h e cohomology algebras of the manifolds Vc(n;k) and I;/w(n;k) are exterior algebras over oddly graded subspaces (i.e., subspaces whose homogeneous elements all have odd degree). T h e PoincarC polynomials are given by k

fvc(n; k)

=

n

(1 $. tz'n-k+i)-l1

n

(1

i=l

and k

fvw(n;

k) =

i=l

+ t4(n-k+i)-11

Corollary: T h e PoincarC polynomials for U(n) and &(n) are respectively given by (since Vc(n;n) = U(n) and J"w(n;n) = Q(n)) n

fo(*) =

fl (1 +

n

and

f =1

focn,

=

n

i=l

(1

+ t4i-1).

Proof: We consider the complex case; the argument in the quaternionic case is identical. T h e proof is by induction on k (for fixed n). Since Vc(n,1) = S2n-1,the theorem is clear for k = 1. Suppose it holds for some k. From sec. 5.8, we obtain a bundle

6

= (U(n)/U(n- k

- I), p, U(n)/U(n- k), U(n - k ) / V ( n- - 1)).

Since U(n - k ) / U ( n- k - 1)

=

Sz(n-k)-l, 6 is a sphere bundle.

224

6. Cohomology of Stiefel manifolds and classical groups

225

Moreover, since U(n - k) acts on the sphere by orientation preserving diffeomorphisms, the bundle is orientable. Thus its Euler class,

x, E H2'"-k'(

U(n)/U(n- k ) ) ,

is defined. On the other hand, by our induction hypothesis, the theorem holds for k, and so the formula in the theorem shows that H2("-k)(U(n)/U(n- k)) = 0.

Thus X, = 0. Now it follows from Corollary I1 to Proposition IV of sec. 8.4,volume I, that H(U(n)/U(n- k - 1))

H(U(n)/U(n- k ) ) @ H(S2(n-k)-1 1

(as graded algebras). This closes the induction.

Q.E.D. 5.22. The Stiefel manifolds Vw(n;2). Proposition VI: homology algebra of Va(n;2) (for n 2 3) is given by

H( VR(2m;2)) g H ( S 2 m - 1 )

0H(S2rn-2) and

H ( Va(2m

T h e co-

+ 1 ; 2 ) ) g H(S4rn-1),

Proof: Recall from Example 2, sec. 5.20, that the sphere bundle associated with the tangent bundle of Sn-l is given by

5

=

(Vw(n;2 ) , T , Sn-1,

S"-2).

Moreover (cf. Example 1, sec. 9.10, volume I)

x

109 '- 2w,-,

- 1 odd n - 1 even, 12

,

where wn-l denotes the orientation class of Sn-'. Case A : n = 2m, m > 1. Then since X, = 0 there is a class w E H2m-2(VR(2m; 2)) such that w = 1 (cf. sec. 8.4, volume I). Moreover, the map, 01

@1

+p @

WZrn-2

+ T#01

+ T"/3

*

w,

01,

p E H(SZrn-'),

defines a linear isomorphism H(S2rn-1)

@ H(S2rn-2) 5 H ( V,(2m; 2)).

I n particular, H4m-4(Vw(2m; 2)) = 0, and so u2= 0. It follows that this isomorphism is an isomorphism of graded algebras.

V. Bundles with Structure Group

226

Case B:

n

= 2m

J.

+ 1, m

1 . T h e Gysin sequence for 5 reads

zHi( Vw(2m +

HE'(S2rn)

1" HZ-zmfl(S2m) 1 ; 2) 9 Hi+l(SZ") --f ,

(cf. sec. 8 . 2 , volume I). This shows that, for i # 0, 2 m - I, 2m, 4m - 1, Hi(V,(2m

Since D(1)

= X, = 2

+ 1; 2 ) ) = 0.

~ D restricts ~ ~ to , an isomorphism

Now the exactness of the Gysin sequence yields H2"( V,(2rn

+ 1 ; 2))

=0 =

H2-y Vm(2m

+ 1 ; 2)). Q.E.D.

+ +

5.23, Bundles with fibre VR(2rn 1; 2). Let 7 = ( E , T,B, F ) be an oriented bundle with F = V R ( 2 m 1; 2). In view of sec. 5.22, H(F)

H( s4m--1).

Now the proofs of the results for sphere bundles established in article 1, Chap. VIII, volume I, depend only on the cohomology and compactness of the fibre; in particular, the identical results hold for 7. This implies that there is a class X, E H4"(B), depending only on 77, and determined by the following condition: Let @J E represent X, . Then, for some 52 E A4m-1(E), d@

=

SQ,

and

j,"

= -1.

Moreover there is a long exact sequence,

-

where Da!= a! X, . If X, = 0, then there is an isomorphism of graded algebras, H ( E ) H ( B ) @ H(S4m-1).

6. Cohomology of Stiefel manifolds and classical groups

221

5.24. The real Stiefel manifolds VR(n;k). Theorem 11: T h e cohomology algebra of V,(n; k) (k < n) is an exterior algebra over a graded vector space. T h e PoincarC polynomials are the polynomials given below.

n

=

2m,

k

=

21

+ 1,

(1

120

+

n 1

t2m-1)

(1

+

t4m-4t-y

,=I

n = 2m

+ 1,

k

=

21,

12 1

fi

(1 t t4m-41+3 )

(I

+ t2m-21)(1+ tZm-') fi (1 +

*=I

k

n = 2m,

=

21, m > 1 2 1

t4m-4,-1)

,=1

n

=

2m

+ 1, k

=

21

1-1

+ 1, m > I > o

(I

+ tzm-zl)n

(1

+

t4m-**+3)

,=l

Theorem 111: T h e PoincarC polynomials for the groups SO(n) are given by

+ tZm-') n (1 + t4i-1) m-1

fSo(zm) =

(1

i=l

and m

fSo(zm+l) =

n

(1

i=l

+ t4i-')*

Proof of Theorem 11: Since VR(n,1) = Sn--l, the theorem is correct for k = 1. If 2 = k < n the theorem is contained in Proposition VI, sec. 5.22. Now we use induction on k. Assume the theorem holds for V,(n; i), i < k, and consider two cases separately. Case A : n - k isodd. Write n

-

k

=

2q - 1.

(SO(n)/SO(n- k), p, SO(n)/SO(n- k

Consider the bundle

+ 21, I.',(2q + 1; 2)). +

By induction the theorem holds for SO(n)/SO(n- k 2). I t follows that H ~ ~ ( S O ( ~ /-S KO+ ( ~2 ) ) = H ~ - + ~ ( s o ( ~ )-/ ks o (2)) ~ = 0.

+

Hence it follows from sec. 5.23 that H(SO(n)/SO(n- A ) )

and the induction is closed.

H(SO(n)/SO(n- k

+ 2))@

H(SZn-zki-1)

228

29

V. Bundles with Structure Group

Case B : n - k = 29 and q > 0. Since (always) k 3 I, we have < n. Now consider the sphere bundle (SO(n)/SO(n- k),p , SO(n)/SO(n- k

+ l), 2Pk).

Since n - k is even, we have a linear isomorphism,

-

-

H(SO(n)/SO(n- k)) -=+ H(SO(n)/SO(n- k

+ 1)) 0H(Sn-”),

of graded vector spaces (cf. Corollary I1 to Proposition IV, sec. 8.4, volume I). It follows from our induction hypothesis that H2“+”(SO(n)/SO(n - k

+ 1)) = 0.

This, as in the proof of Proposition VI, implies that the isomorphism is an isomorphism of graded algebras.

Q.E.D. Proof of Theorem 111: Let (vl ,..., wn-J be an orthonormal (n - 1)frame in Kin. Then there is a unique vector, v, E Fin, such that (vl ,...,wn) is a positive orthonormal n-frame. This provides a diffeomorphism, Vw(n;n - 1) 5 SO(n). Now apply Theorem 11.

Q.E.D.

Problems 1. Free actions. (i) Let G be a Lie group that acts freely and properly on a manifold M (cf. problem 5, Chap. 111). Show that ( M , n, M/G, G) is a principal bundle (cf. problem 6, Chap. 111). (ii) Apply this when G is discrete and the action is discontinuous (problem 21, Chap. 111). Show that the universal covering projection for any connected manifold is the projection of a principal bundle (problem 18, Chap. I). 2. (i) Show that the closed proper subgroups of S1are finite, and are in 1 - 1 correspondence with the groups Zp = Z/pZ, p = 1, 2, ... . Z,) where Z, acts by (ii) Construct principal bundles (Sl,n, S1, multiplication. Let (Sl xz,S1, p , S1, Sl) be the associated bundle (same action of Z),. Identify it as a principal S1-bundle, and show that it is the trivial bundle. (iii) Construct a principal bundle (R2, n, S1xz,S1, Z x Z), where H x Z acts on R2 by

(X,Y)'(m,n)=(x+pm+n,Y+n),

%YER, m,neZ.

is not diffeo(iv) Let Z, act on S1via eiS I+ e-(@.Show that S1x $*S1 morphic to S1 x S'. 3. Let M(n, m ;k) denote the set of linear maps from R" to Rm of rank k.

(i) Make M(n, m ;k) into a smooth manifold. (ii) Show that composition defines a smooth map p: M(n, K ; K )

x M(K, m ;K ) + M(n, m ;K ) .

(iii) Show that p is the projection of a principal bundle. n, B, G) be a principal bundle. Let G act on itself by 4. Let 9 = (P, conjugation. Show that the resulting associated bundle is a bundle over B with fibre G. Construct an example in which this bundle cannot be made into a principal bundle.

229

230

V. Bundles with Structure Group

5. Let ( P , T,B , G) be a principal bundle. Assume that G acts on an r-manifold, Y , and let ( M , p , B, U ) be the associated bundle. Let 6 = ( V , ,p , M , W) be the vertical subbundle of the tangent bundle r M .

(i) Show that ( V M,p 0 p , B, Ty)is a smooth bundle. Identify it with the bundle P x Ty-+ B. (ii) Assume that Y = G / K , where K is a closed subgroup of G. Identify 6 with the bundle ( P x KEIF, p , , P / K , EIF), where E and F denote the Lie algebras of G and K. 6 , (i) Let f = ( M , T,B, F ) be a Riemannian vector bundle. Construct a principal O(n)-bundle whose fibre at x is the set of isometries F -+ F, . Show that 6 is the associated vector bundle. (ii) Make similar constructions for real vector bundles, oriented real bundles, oriented Riemannian bundles, complex bundles, Hermitian bundles, and quaternionic vector bundles. (iii) Apply (i) and (ii) to the tangent bundle of a manifold. Show that the resulting principal bundle has trivial tangent bundle. 7. Flag manifolds. of subspaces,

Ajlag in 02" (respectively, C", W") is a sequence 0 CF, CF, c

***

CF, = w,


0).Then, since r is invariant, the corollary

i.e.,

Substitution of the relation Q

= 6w

+ * [ w , u]yields the formula

(The calculation is long but elementary except for the observation that 7

(- 1)L

C=o (I) p+l =

1

so

1 - x)' dx

=

7!

p *..(P

+

7)

*)

6.21. Formal power series and the Taylor homomorphism. Consider the infinite sequences

r = ( T o ,rl , ...)

with

r k

E

VkE*.

Define addition and multiplication by

(r+ p)k= rk+

f k

and

( r .p)k= C

Ti v

pj

( k = 0, I,...).

itj=k

T h e associative algebra so obtained is called the algebra of formal power series in E* and is denoted by V** E*. Next, recall from sec. 1.9, volume I, that Y o ( E ) denotes the algebra of smooth function germs at 0. That is, an element of Y o ( E ) is an equivalence class of functions f E Y ( E ) under the following equivalence relation: f -g iff - g is zero in a neighbourhood of 0. If U is a neighbourhood of 0 in E and g E Y ( U ) , then there is a unique germ, [gl0E Y o ( E ) ,such that any f E [gl0 agrees with g sufficiently close to 0. We say g is a representative of [gl0.

270

VI. Principal Connections and the Weil Homomorphism

Now let f E 9 ( U ) ( U , a neighbourhood of 0 in E). Then the kth derivative off is the smooth m a p f ( k )E Y ( U ; VkE*) defined inductively by f ' 0 ) z= f

and

(Note that we identify VkE* with S k ( E )via is as described in sec. 6.18.) T h e Leibniz formula states that (jg)(k)=

c

f'i'

f,g E 9 ( U ) ;

v g'j),

if+k

i.e., the map,

f k-+

(f(O), f'(O), f"(O),..*),

is a homomorphism of Y ( U ) into V**E*. Since the derivatives o f f at 0 depend only on the germ off at 0, this homomorphism determines a homomorphism Tay: q ( E ) + V**E* called the Taylor homomorphism. Next recall that G acts on E by the automorphisms Ad a. Thus an action of G on Y o ( E )is defined by u

- [flo = [(Ad u - ' ) * f l o ,

f~ Y ( E ) ,

uE

G.

T h e corresponding invariant subalgebra is denoted by Y o ( E ) ,. On the other hand, we have an induced action of G on V**E*. Clearly, the Taylor homomorphism is equivariant with respect to these actions and hence it restricts to a homomorphism, Tay,: Y0(E), -+(V**E*), ,

called the invariant Taylor homomorphism. 6.22. The homomorphisms h$* and s,. Let B = (P,n, B, G) be a principal bundle over an n-manifold B and consider the Weil homomorphism hg: (VE*), + H(B).

Since Hp(B) = 0, p

> n, hg extends to a homomorphism h$*: (V**E*),

--f

H(B).

Clearly the image of h$* coincides with the image of h 9 .

6. The Weil homomorphism

27 1

On the other hand, we have the invariant Taylor homomorphism (Tay),:

-+

( V * *E*)/

Composing these homomorphisms we obtain a homomorphism sg: Y0(E),+ N ( B ) .

Explicitly, s g [ f ] , = C,"=, h9(fcP)(0)). If y : P -+ P is a homomorphism of principal bundles inducing +: B 3 l?, then yP 0 hs* = h$* and yP o sg = sg as follows from Theorem 11, sec. 6.19, and the definitions. Remark: T h e advantage of using h$* or s9 rather than h9 is the following: Let [f],E Y,(E), , r E (V**E*)/ , a E H(B). These elements are invertible in their respective algebras if and only iff(0) # 0 (respectively # 0, a, # 0, wherea, is the component of a in Ho(B)).Moreover, iff(0) # 0, then

r,

%4[f10') = (s9([f10))-1.

r,,

On the other hand, an element r E (VE*), is only invertible if #0 and Pi = 0, i > 0, while h B ( r )is invertible whenever r, # 0. Hence, if r, # 0, and Ti # 0 for some i > 0, then (hg(r))-l exists but it is expressible in the hg(ri) only via a complicated polynomial. T o obtain simple expressions it is necessary to introduce ( V**E*), .

s7. Special cases 6.23. Principal bundles with abelian structure group.

Let

9'= ( P , x , B, G) be a principal bundle whose structure group G is abelian. Let w be a connection form in 9'with curvature form 52. Then i(h)Q = 0

TZQ

and

= Q,

aEG.

(6.1)

Moreover, the Maurer-Cartan equation (Proposition XI, sec. 6.14) reduces to 6w = SZ. I n particular, it follows that 652 = 0. I n view of Proposition 111, sec. 6.3, relations (6.1) imply that there is a (unique) E-valued 2-form 52, on B such that 52 = x*52,. Since x * 6QB =

&*QB

= 8Q = 0,

it follows that 6QB = 0. Next observe that, since G is abelian, (VE"), y B become homomorphisms y:: VE* + AB(P)

=

VE* and so y, and

ye: VE* -+ A ( B ) .

and

Evidently (cf. sec. 6.18) yB(r)

1

=

7r(QB

P.

,**',

r E WE*.

QB) ,

In particular, yB(h*) = (h*, QB),

Proposition XV: For every h*

E

(6.2)

h* E E*.

E*, let X,* denote the 1-form on

P given by Xh*(Z)

Then Proof:

= 1.

Then y#(@ A G) = I/*@

A

y#G,

@ E A(B), 52 E A(B; [).

T h e map #I* makes A ( B ) into a module over Y(B), the module multiplication being given by @

*

f

=@

**f,

f. 9 ( B ) , @ E A@),

where the scalar has been written on the right for notational convenience. Thus we can form the tensor product A ( B ) @gl Sec 5. Proposition I: With the hypotheses and notation above, an isomorphism of graded A(B)modules,

A(B)

oB Sec [ 5 A(B; 0,

is given by @ @ o H @ A ~ # ~ ,

@ E A ( B ) ,oESec[.

Proof: In view of the isomorphism (7.2) this is immediate from the isomorphism Y ( B ) Sec [ Sec f of sec. 2.26, volume I. (Use the associative law for tensor products.)

Q.E.D. Example: Let 8" be the restriction of ( to U (open in B). T h e inclusion, j : 8" --+ 5, induces a restriction map j # : A( u; 6")

+-

A(B; 6).

1. Bundle-valued differential forms

Identifying the fibres of

309

4 and 4” , we can write

(j#.n)(x)

xE

= Q(x),

u.

Type 111: The map y* . Finally, assume that the vector bundles f and [ have the same base B , and assume that y is a strong bundle map (t,h = 1 ) . Then a homomorphism of A(B)-modules,

v*: A@; 0 is given by (y*Q)(x; hl

I

-a*,

h,)

= v,(Q(x;

hl

x E B,

and

,

A(B; f ) ,

*‘a,

hp)),

h, E T,(B), Q E Ap(B; E), p 2 1,

(v*u)(x) = vz(o(x)),

xE

B,

(J

E

Sec E.

T h e restriction y * : Sec 8 --+ Sec f of y* coincides with the map defined in sec. 2.15, volume I. On the other hand, if 4 and [ are trivial bundles, and y is given by v(x, 4

44, x E B, a EF, F -+p, then y* coincides with

= (x,

for some fixed linear map

01:

the map

a*: A ( B ;F ) -+A ( B ;P )

of sec. 4.7, volume I. Note that, if y : morphism, y # = (rp*)-’.

5 -5 [ is

a strong bundle iso-

7.4. Multilinear bundle maps. Recall that a multilinear bundle map 01 E Hom(fl, ..., f m ; 7)assigns to each x E B an m-linear map,

in a smooth way (cf. sec. 2.4, volume I). Such an 01 determines a map, a*:

A(B; 11)x ..* x A ( B ;Em)

e).

+

A ( B ;v),

as follows: Let Qi E APi(B; Then a,(sZ, p , and is given by p = p, +

+

a*(-%

,

*.a,

Qm)(x;

hl ,

--a,

h,)

,..., Q,)

has degree

VII. Linear Connections

310

With the identification, (7.2), a* is given by &*(@I

@ 0 1 , ..., am@ 0,)

Note as well that, if sZi

where qi = p , “*(QI

E

&(B;

== (@I A

“*

e)and X

A

Gm)@ “*(Ox

E

%(B),then

a*.,

0,).

(7.4)

+ + pi-l . Moreover, if, in addition, @ E Ag(B),then **.

..., @ A Gi , ...,Qm)= (-

@

A

a*(G1 , ...,Q,).

(7.6)

$2.

Examples

In this article we consider examples of multilinear maps

..., 5”;

a E Hom(tl,

7.5. Dual bundles.

7).

Suppose (* is dual to ( and let aE

Horn([*, 5; B x F )

be the scalar product ( , ). We write,

,*(a*, Q) = ,

Q* E A(B; [*),

Q E A(B; 5);

< , > extends the scalar product

thus E A ( B ;r).Note that { , ) between Sec (* and Sec 5.

Lemma I: Let 9:4 + [ be a bundle map inducing #: B -+ B and restricting to isomorphisms in the fibres. Assume [*, [* are dual to (, [. Then <rp*Q*,

rp#Q>

=

Q * E A(B; p), Q E A(B;[).

#* = ,

Q * E A(B;7*), Q E A ( B ; 5). 311

Q.E.D.

312

VII. Linear Connections

Consider a bilinear map

7.6. Bilinear maps and algebras. E

If

01

Horn([, 5; 5).

is symmetric, then

.*(a,,

Q2)

=

Q, E A V ; 8, Q, E A

, Q,),

(--1)P*a*(Q2

V;

8,

while if a is skew-symmetric, then

,Q,) = ( - - l ) D q + l O 1 * ( Q ,

,521).

Next, observe that the map a E Hom(.f, E; 6) makes each fibre F, into an algebra, while a* makes A ( B ; f ) into an algebra. We write .*(GI ,Q2)

= Q,

Q2

,

Q, E

A(& 5),

for the algebra multiplication. If the algebras F, are all associative, so

In this case we define the pth power of bundle-valued form Qp E A ( B ; f ) given by = Q.

QP

...

.

Q,

( P factors)

D (DE A(B; 6)) to be the

P 2 1, t

and D

= 1.

Bundle-valued forms SZ, E Ap(B; 5) and D, E Aq(B; 6) will be said to commute (with respect to the map a*) if, for each x E B and hi E T,(B), the vectors Qn,(x;h, ,...,h,) and Q2(x; h,,, ,..., h,,) commute in the algebra F, . If SZ, and D, commute, then Q,

.R,

Q2 =

.

Thus if in addition, p or q is even, then D, !2, = Q, Q, and so the binomial formula, k (Q1

+

Q2Y

=

c

i +j =k

( ) Q:

Q5,

holds.

.

Recall (sec. 2.10, volume I) that L, is the vector 7.7. The bundle L, bundle whose fibre at x is the space of linear transformations of F , . Evaluation and composition are the bilinear maps, E E Horn&,

f ; 6)

and

o

E

Horn& , L f ; L f ) ,

2. Examples

313

given by and

E(OI,Z))=~(Z))

ci,/3€LF,, n s F X , x c B

o(01,/3)=010/3,

(cf. sec. 2.10, volume I). T h e induced maps of bundle-valued forms are denoted by E*(Q, @)

SZ, SZ, , SZ,

E

and

= Q(@)

,Q,)

o.+(Ql

= Q, 00,,

A ( B ;LE),@ E A ( B ; 4). They satisfy = QI(Qd@)).

P I O Q,)(@)

In particular, each SZ

E

A ( B ;L,) determines the Y(B)-linear map,

0:Sec 4 -+ A(B; 0, given by

0(a)

= Q(u).

This correspondence defines an isomorphism N -

A(B;LE)

Hom,(Sec 5; A ( B ;5)).

Let y : 4 -+ [ induce 4: B B and restrict to isomorphisms in the fibres. Then a bundle map $: L, -+ L , is given by --f

$x(a) = rpx

0

a € L F , , x E B,

a 0 rp;’,

and $ restricts to isomorphisms in the fibres. Lemma 111: With the hypotheses and notation above rp#(Q(@))

$#(Q1 0 Q,) = $#Q, 0

= ($#Q)(p”@),

$#sz,,

and @#(L,)

= L(

,

sz, szl , Q,

E

A(& Lt), @ E A(B; [)

( t c and l g are the cross-sections in L, and Lg assigning to each x and f the identity transformation in FZ and p2).

Finally, define tr

E

Sec LF by

(try 9),

= t r ~x

,

px ELF,

9

x E B.

This cross-section determines the linear map,

r),

tr: A(B;LE)-+ A ( B ;

314

VII. Linear Connections

given by (tr@)(x;h, , ..., h,)

=

tr(@(x;h, , ..., A,)),

Ap(B;&),x E B, hi E T@).

@E

7.8. Tensor products. Let a E Hom(fl, ..., tensor product, a2(d,

If QiE A(B;

..., ZI")= ZI'

@

e),then cu,(Q,

tm;(l

..., 0,)E A ( B ; f1 =

tm)be the

w i E F ,~ x E B.

@ TI",

a*(52, , ...,52,)

@

@

@ tm);we write

@

sz, @ ... @ f2, .

Note that this extends the tensor product between cross-sections (cf. sec. 2.24, volume I). Note also that @ is not a tensor product unless the Qi's are of degree zero. (Use the isomorphism 7.2.) Now recall that @" (* and O m( are dual with respect to the scalar product given by (w: @

... @:JZ , 21,

@

***

@ ZJ")

= (TI,*, Vl)

Thus, if 0 E A ( B ; @P (*) and QiE A ( B ; (), i @ Q,>. the ordinary differential form (@, Q, @ denoted by @(Ql, ,.., Q p ) .

(i

Lemma IV: Let Q1 E Sec @P = 1, ...,p q). Then

+

( 90@2

3

= (@I,

=

4*, Q2E Sec @ q [*

0 *..0 Ql 0 ..* 0 52),

Ql

.*.( v 2 , wm). 1, ..., p , we obtain This will often be

and let Q,

E

A(B;[)

% I + , >

A (@2

9

Q,+l0

.*.0 Q,,,).

Proof: The isomorphism (7.2) and formula (7.4) allow us to reduce to the case that Q, E Sec [. But in this case the lemma is obvious.

Q.E.D. Let

[plq

denote the tensor product

( p factors [*, q factors f ) . We write A(B; 0E*)

= @fa

A(B; P

(*

@

O ) ;

-

0

.

@

(*

@ [Q

05 ) = @=a;

..* Q [

A(&

and A(B;(0 E*)

0(0 0) = @,,

6".".

These are associative algebras with multiplication 0 induced by the bilinear maps @ ; (?",a x 5'4 + ( P + r , q + a .

2. Examples

315

T h e identity element is the scalar 1, and A ( B ) = A ( B ) @ 1 is a subalgebra of each. Moreover, these algebras contain (respectively) the subalgebras Sec

0[*

=

0, Sec

Sec

["q0,

0[ = 0, Sec [O~Q

and

set((@ [*) @ (0 5)) = @,, Sec P,*. Next, define a strong bundle map 8: L , x: E F Z , xi E F , , x E B, O(a)(zl* @

'.. @ 2;

c

--f

LEP,l by setting, for 01 E L F Z ,

... @ z,)

@ z1 @

P

=

-

(2:

... @ a*(.')

@

@ z,* @ z1 @ *.. @ z,)

@

i=l

and 8(a)(1) = 0. T h e map, 8, induces a map 8,: A ( B ;L,) + A ( B ; L f P , p ) . 7.9. The bundle AE. Recall that A ( is the bundle with fibre AF, at x. T h e exterior multiplication in the fibres determines an associative multiplication in A ( B ; A t ) (cf. sec. 7.6), which we denote by (Q, , Q,)

* Ql A Q, ,

Qz

E

A(B; A t ) .

zp.*

We write A ( B ; A t ) = AP(B; A.0; this makes it into a bigraded algebra, the 1-element of which is the constant function 1. T h e isomorphism (7.2) reads A(B;A[)

A(B) @ B Sec A (

g

A(B) @B A, (Sec ()

and under this isomorphism the multiplication is given by (0@ U) A (Y@ 7) = (@

A

Y)@ (U A

T),

@, Y E A(B), u, 7 E Sec A[.

In particular A ( B ; A t ) is the canonical (not the anticommutative) tensor product of the algebras A ( B ) and Sec A t . Observe as well that A ( B ) = A(B) @ 1 are subalgebras of A ( B ; A t ) .

and

Sec A[

=

1 @ Sec A[

316

VII. Linear Connections

Next note that the skew-symmetry of the multiplication in A 6 implies that

and p

Thus, if 52 E Ap(B; A")

+ q is odd, then QAa=o.

Moreover, since A t is associative, we can form the kth power of SZ E A ( B ; A t ) (cf. sec. 7.6). If SZ E A1(B;f ) , then

Now define (as in sec. 7.8) I3 :L,

e(cu)(zlA ... A z,)

--t

LA,by setting B(cu)( 1)

V

=

1 z1

A

i=l

..- A

= 0 and

... A zv ,

a(zi) A

a € L F Z , z ~ E F , , X E B , p > 1.

Let 8,: A ( B ; L,) -+ A ( B ;LA,) be the induced map and note that we use the same notation as in sec. 7.8. Finally, let cr* E Sec .$* ([*, a bundle dual to 8). Define an operator i,(cr*)in A ( B ; A t ) by setting (i,(o*)Q)(x;hl

,

.*a,

A,)

= i(o*(x))(Q(x;hl

xE

1

.*a,

4J)),

B, hi E T,(B), Q E AP(B; At),

where i(o*(x)) is the substitution operator in AF, . i,(o*) is homogeneous of bidegree (0, -1) and satisfies

This formula yields, for cr i,(u*)(U A

E

Sec 5, 52

E

A ( B ; At),

Q) = ( u * , ~7)52 - 0 A &(U*)Q.

I n particular, if 52 E A ( B ; Art) ( r = rank the formula above reduces to

t),then cr A SZ = 0, and

( u * , u)Q = u A i,(u*)Q.

2. Examples

317

7.10. The bundles VpS. Recall that V p t is the vector bundle whose fibre at x is the p t h symmetric power V P F , , of F, (cf. sec. 2.12, volume I). T h e multiplication maps,

v.5 x v q

--f

vp+q,

determine maps A(B; VP5) x A(B; V.6)

which we denote by (Q,

I

Q,)

-

+ A(B; VP+Pf),

Q, v Q,

.

T h e direct sum of the spaces A ( B ; ‘ 4.0 is denoted by A(& V5)

A(B; VPt)

=

and the maps above make it into an associative algebra. T h e strong + V p t given by bundle maps vs: rrs(zl @

.*.@ z,)

= X, v

1..

v zP ,

determine a homomorphism

(4*: A(&

05 )

-

ziE F, , x E B,

A(& Vt),

of bigraded algebras. Finally, note that Q, v 9, = (-1)””’Q2

(cf. sec. 7.8).

v GI,

In,E A”‘(B; Vf),

Z’

= 1, 2,

s3. Linear connections 7.11, Definition: A linear connection in the vector bundle, f , is a r-linear map, V: Sec E -+ A1(B;t),

which satisfies the condition

V(f

U)

= 6f A u

f~ Y ( B ) ,

VU,

+f

u E Sec f ,

If f is complex, V is sometimes called a complex linear connection. A cross-section CT is called parallel with respect to V if Va = 0. Let V be a linear connection in f . Then, for each vector field X on B , an operator, V, , in Sec f is given by V, = i ( X ) 0 V. These operators satisfy the relations V x ( f . u)

= X(f)* u

+ f . Vxu,

ueSec [, f~ Y ( B ) ,

and Vf.,(O)

Similarly, if h given by

E

. Vxu.

T,(B), we can form the operator V,: Sec 4 -+F, VhU

Clearly, ( V , 4 ( x )

=f

=

= i(h)((Vu)(x)).

VX(Zd4

Suppose f is trivial: M = B x F. Then (cf. sec. 7.1) A ( B ;F ) . I n this case the exterior derivative,

Examples: 1,

A ( B ;f )

=

6: Y ( B ;F ) -+ A1(B;F),

is a linear connection in f . I t is called the standard connection. 2. Let V, be a linear connection in f and suppose Y Then the map, Sec f -+ A1(B;f ) , given by 0

-

V,(a)

E

A1(B;Lc).

+ Y(4

is again a linear connection in f . Conversely, if V, is a second linear connection in f , then the map, V, - V, : Sec f 318

-+

A1(B;t),

3. Linear connections

319

is Y(B)-linear. Hence there is a unique Y E A1(B;L,) such that V,a - V2a = !P(a),

uE

Sec 5

(cf. sec. 7.7). 3,

Let {U,} be a locally finite open cover of B, and assume that

(p,} are smooth functions on B such that carr p , C U , , and C, p , Let 5, be the restriction of 5 to U , . Assume that V, is a linear connection in V: Sec 5 --t A1(B;5)) by setting Va

=E

5,.

=

1.

Define a r-linear map,

P , . Va%,

dI

where

U,

is the restriction of u to U, . Then V is a linear connection in .$.

Assume

4.

5

@ 7 = B x P. Then we have the strong bundle

maps

i: 5 + B x P (inclusion)

and

p: B x

TQ+ 8 (projection).

A linear connection V in .$ is given by = p*

6i,(a),

aE

Sec 6.

5 . Tangent bundle: A linear connection in a manifold B is a linear connection, V, in the tangent bundle, r B. Given such a connection we define a map S: X ( B ) x X ( B ) + %(B)by setting

S ( X , Y ) = V,Y - V y X - [ X , Y ] S is a 7,-valued 2-form: S E A 2 ( B ;7,). I t is called the torsion of V. S determines the 1-form Y E A1(B;LrB)given by Y ( X ) ( Y )= S ( X , Y ) ,

X , Y E%(B).

By Example 2 above, Q = V - Y is again a linear connection in B. 9 is called the conjugate connection of V and satisfies Q,Y

=

V,Y - S ( X , Y ) .

It follows that the torsions for V and 9 are related by S

=

-3.

Proposition 11: Every vector bundle .$ admits a linear connection.

3 20

VII. Linear Connections

Proof: Example 1, above, shows that trivial bundles admit linear connections. I n view of Example 3, a partition of unity argument now shows that every ( admits a connection. Example 4, above, provides a second construction of a linear connection. We give a third, purely algebraic proof. Let m

t, =

C a: 0at i=l

(ai* E

Sec [*, ai E Sec [)

be a representation of the unit tensor for ( (recall, from p. 81, volume I, that t, E Sec((* @ 6) corresponds to 1 , E Sec L , under the isomorphism (* @ ( L,). Define a r-linear map V: Sec ( .--t A1(B;6) by setting

va =

m

?a (,:

a> A

(Ti

a

i=l

(If r = C recall, from sec. 7.2, that A(B; () is a module over A ( B ; C)). T h e relation (p. 81, volume I), m

C ui = a,

aE

Sec 5,

implies that V is a linear connection. Note that V depends on the particular representation of C.

Q.E.D. 7.12. Induced connections. In this section V and V, are linear connections in (, V, is a linear connection in 7 and Vi is a linear connection in These connections determine linear connections in the associated linear and multilinear bundles, as described in the examples below.

c.

Examples: 1. Dual bundles: Let (* be dual to unique linear connection, V*, in (* such that (V*a*,

a)

+ (a*, VO)

= 8(0*,

a),

u* E Sec

(. Then there is a [*,

a E Sec 6.

V* will be called the dual of V. I n fact, fix CT* E Sec (*. A simple computation shows that the map Sec f + A1(B)given by a tt &(a*,

a)

- (a*, Va)

321

3. Linear connections

is Y ( B ;F)-linear. Hence there is a unique element, V*u* such that

=

dpt

=

f~ Y ( B ) ,

DMu,

Of,

uE

Sec 6.

f~ Y ( B ) .

tER.

Proof: A straightforward computation shows that

Since, if

R

E

V,(M),

it follows that (1) is equivalent to (2). Condition (3) is simply condition (2) in the case f = t . Thus (2) implies (3). T o prove that (3) implies (2) observe that if k E T J M ) and x = n(z), then

340

VII. Linear Connections

Proposition X: Let D M be the general connection in 4 corresponding to a horizontal bundle H M , Then, with the hypotheses and notation above, D M is a linear connection if and only if

+

T, cr, 7 E Sec I . Thus, in Proof: a* is given by a*(u @ T) = u view of Examples 1 and 2 above, condition (1) is equivalent to

DM(u

+

T)

= DMu

+

DMT,

u, T E

Sec 5.

Now the proposition follows from Lemma VI.

Q.E.D.

§7. Riemannian connections I n this article r = R. T h e vector bundle .$ is equipped with a fixed Riemannian metric ( , ). T h e Riemannian metric can be considered as a cross-section in V 2 t * , and will then be denoted by g. 7+23. Riemannian connections. Let V be a linear connection in and let V denote as well all the induced connections. Then

6,

Definition: V is called a Riemannian connection if Vg = 0, or, equivalently, if a

=



+

,

vector

5.

bundle

admits

a

Proof: Let V, be any linear connection in the Riemannian bundle Define Y E A1(B;L,) by

(6, ( , )).



Then V,

= (V,g)(u,

71,

0

,

E~Sec 5.

+ 4Y is a Riemannian connection. Q.E.D.

Remarks: 1. Recall from sec. 2.21, volume I, that Sk, denotes the subbundle of L, whose fibre at x is the space of skew transformations of F, . If V, and V, are Riemannian connections in then V, - V, is a 1-form on B taking values in Sk, . Conversely, if Y E A1(B;Sk,) and V, is a Riemannian connection in 6, then so is V, Y.

e,

+

2. Let q ~ :[ +8 be an isometry of Riemannian bundles. Then the pull-back, via q ~ , of a Riemannian connection in 6 is a Riemannian connection in [. 341

342

VII. Linear Connections

7.24. Basic properties. Fix a Riemannian connection, V, in that a strong bundle isomorphism,

5. Recall

is defined by the equation,

(cf. sec. 2.17, volume I). I t is immediate from the relation above, and the defining relation for the dual connection V* (cf. Example 1, sec. 7.12) that f is connection preserving, f*oV=V*of*.

Henceforth we identify I* and 5 under this isomorphism. Next recall that ( , ) induces Riemannian metrics in the bundles o p e , Ape, V p t ; they are given by (Z10

.**

(zlA

0z, , w1 0 0w,) *.*

a . 1

A

= (z1 , Wl)

*..(z, , w,),

... A wD) = det((zi , wj)),

zl,, w1 A

and (ZI V

... v z, , w 1 v

v w,) = perm((zi, wj>),

x i , wj E F,

,

x E B.

On the other hand, V determines linear connections in o p e , A*(, V P t (cf. sec. 7.12). Evidently these connections are Riemannian with respect to the induced metrics. Lemma VII:

Suppose u

E

Sec 4 and

f E .

Further,

$u,z

VIII. Characteristic Homomorphism for Z-bundles

384

Next, let {( U, , $,)} also denote the induced coordinate representations and V P f g . Then, for x E U , , Faex restricts to isomorphisms for c=

*,,,:

(ODE*), -=+(@”E&),

,

(V”E*), 5 (V’E,*,,),

x,,x:

*

It follows that the spaces (OPE&,),and (VPEG,), are the fibres of subbundles, (@”t$), C @”t,* and ( V ” t z ) ,C V’t;, with coordinate representations { ( U , , $,)} and {( U, , X,)}, respectively. Definition: (OPE;),and ( VPtZ), are called the p t h invariant tensor bundle, and the p t h invariant symmetric bundle associated with (6, Zc). A cross-section of (Opt:), (respectively, (Vp[g),) is called invariant. Proposition 111: There are unique strong bundle isomorphisms,

4: B x ( W E * ) , 5 (@”[:),

X: B x (V”E*), 5 (V’f;), ,

,

with the following property: If {( U , ,y,)} is any Z-coordinate representation for 5, and and X,,x are the maps defined above, then $,

=

A,,

and

X,

= Xu,,,

XE

U,.

and X,,x are independent of Proof: I t is sufficient to show that the choice of U, and of the choice of coordinate representation. Since the union of two Z-coordinate representations is again a Z-coordinate representation, it is sufficient to show that

#,, Write y;:

0

= *B,X

and

= T.

Then r

$;,\

0

xu.2

E

= XB.,

Ad r : E

II

_-t

I t follows that the induced isomorphisms satisfy 0

xE

u, n u,

*

G and

$B,x =

$is\

,

$B,x =

E.

$a,z, $B,x:

OPE* 5 OPE&,

0” Ad(.r)h.

Restricting this equation to (OPE*),yields

*i,;a. *BOX Similarly, X,,x = XB,x.

= 1,

XE

u, n up. Q.E.D.

3. Invariant subbundles

385

Identify (ODE*), and ( V P E * ) , with the constant cross-sections of the trivial bundles. As in secs. 7.8 and 7.10, write m

m

(0 E*), = 1(@”I?*),

,

(VE*), =

p=o

2 (V”E*),

p=0

and m

S e c ( 0 tf), =

0sec(@”t;),

m

,

Sec(Vt;),

=

p=o

0Sec(Vpt;), . p=0

These spaces are all graded associative algebras. Moreover, induce canonical homomorphisms $*:

(0 E*), + Sec 0ti

#

and

x

X,: (VE*), -+ Sec V f z .

and

Remark: T h e constructions of this article depend only on the isomorphism class of the pair ( F , ZF).I n fact, suppose ( F l , ZF1)is a second pair, in the same isomorphism class, with corresponding Lie algebra El C LF1 . Then there is an isomorphism a: F 3 F , carrying ZF to ZF,. I t induces isomorphisms

0E* - 0E: = -+

and

VE* -% VE:.

and

( VE*),

These restrict to isomorphisms,

(0 E*), 5 (0 E:),

c?=

(VE:), ,

which are independent of the choice of u. Moreover, the diagrams,

(0 E*)l

(0 E31

(VE*),

,

YE:),

commute; this shows that, up to canonical isomorphism, independent of the choice of F and ZF.

,

#* and

X , are

8.9. 8-connections. Recall that the connection, 9, in L , induced by V restricts to a linear connection in f E (cf. Proposition 11, (I), sec. 8.7). Hence it determines a linear connection, V, in the bundles and v p g .

VIII. Characteristic Homomorphism for Z-bundles

386

Proposition IV:

#: B x

T h e inclusions,

(ODE*), -+ @”(:

X: B x (V’E*),

and

--f

V”(;,

are connection preserving with respect to the standard connection, 6 , and V. Proof: It is sufficient to consider the case that ( = ( B x F, T ,B , F )

and

u i ( x ) = ( x , vi),

xE

B, i

=

1,

..., m.

+

I n this case V = 6 Y, where Y E A1(B;E) (cf. Example 5, sec. 8.6). Moreover the total space of f E is B x E and the induced connection in f E is given by

% = 87

+ [Y, T I ,

T

E

- Y ( B ;E ) .

is given by V = 6 Thus the induced connection in is the LOPE*-valuedI-form defined by

+

Yp,

where

Y p

-c z1 9

Y f ( x h)(x, ; @ .. @ x”)

=

@

... @ ad*(Y(x; h))xi @ ... @ x p ,

i=l

xE

B , h E T,(B), xi E E*.

I n view of Proposition IX, sec. 1.8, it follows that v

Y’(x; h)(v) = 0,

E

(@”I?*),

.

This proves the proposition for I); the proof for X is identical. Q.E.D. Corollary: T h e inclusions,

#*: (0 E*), -+

Sec

0

and

X,: ( VE*)I-+ Sec V(,*,

of sec. 8.8 are isomorphisms into the graded subalgebras of invariant, V-parallel cross-sections. If B is connected they are surjective. + (7,Z,,) be a homomorph8.10. Homomorphisms. Let q ~ :(5, Zt) ism of Z-bundles (cf. sec. 8.1). It induces a bundle map, Y E : f E -+ 7 E , whose fibre maps (vE)$ are isomorphisms. Thus the linear isomorphisms,

and

V”((q~~)z)-’,

3. Invariant subbundles

define bundle maps bundle maps,

QPeg

-+ Q

-

P q g and

(O”t*,),(OD&

and

VP&$

387

+VP7;.

(V”t,*),

-

These restrict to

(VP&

9

and all these bundle maps will be denoted by y E . Moreover, all the vEinduce the same map vB:B -+ P (B, the base of T ) . The same argument as that given in Proposition 111, sec. 8.8, shows that the diagrams,

commute. Thus the diagrams,

also commute.

s4. Characteristic homomorphism

I n this article we continue the notation conventions of article 3. 8.11. The homomorphisms Pe and y C . Define a linear map,

(recall that R

E A2(B;f E )

is the curvature of the Z-connection V).

Lemma I: /3( is an algebra homomorphism. I t factors over the -+ Sec Veg to yield a commutative canonical projection (rS)*:Sec 0s; diagram,

of algebra homomorphisms. Proof: Let A iE Sec

(i = 1,2). Then (cf. Lemma IV, sec. 7.8)

T h u s Bt is a homomorphism. 388

4. Characteristic homomorphism

389

Since R is a 2-form, Im BE is a graded subalgebra of the commutative algebra C, A2p(B;F ) ; in particular,

PP(4AP s ( 4

=P

P(4

A

4 E See 06:

PE(fll),

*

Hence Bs factors as desired. Q.E.D. Extend

/3(

and y f to homomorphisms,

by setting and

Ps(@ A A ) = Q, A Pc(A)

(1E

ys(@ A

9)= Q,

A

ys(E),

Sec @,*, 9E Sec V e f ,

@ E A(B).

T h e analogue of Lemma I holds and

Lemma 11: T h e maps

pE and y s satisfy

Proof: T h e second relation follows trivially from the first. T o prove the first, fix A ~ S e @cp , $ g . According to the Bianchi identity (cf. Proposition VI, sec. 7.15), V R = 0. Thus it follows from Example 4, sec. 7.12, that PP(V(l)= rank 6.

t7; = 0,

2. Group actions: Let T: G x M -+M be an action of a compact connected Lie group on a manifold M . Recall that every point x E M determines the smooth map A,: G + M given by &(a) = a

*

aE

X,

G, x

E

M.

Assume that the subspaces Im(dA,), C T,(M) all have the same dimension (equivalently, all the orbits of G have the same dimension). Then these spaces are the fibres of a distribution, v, on M . Moreover, the module Sec 9 is generated by the fundamental vector fields. It follows that 71 is involutive. Give T~ a G-invariant Riemannian metric (cf. Example I, sec. 3.18), and let ( be the orthogonal complement of 7. Then the fundamental vector fields 2, ( h E E ) satisfy Z,((X, Y>)= ( [ Z , > XI, y>

+ ( X , [ Z , , YI),

x,y E qw.

I t follows that [Z, , XI E Sec f

if X

E

Sec f .

Next, construct a G-invariant connection, VI , in be any linear connection and set (V,X)(x;Z(x))

=

a-'

9

as follows: Let V,

(V,(a . X ) ( a x; a . Z(x))) du.

G

Then

-

2 E T(M), X E Sec 5, u

(V,)a.z(a X ) = a ((V,),X),

E

It follows that (set a = exp th and differentiate with respect to t )

+

( V r ) ~ z ~ . z d X )(Vr)z([Zh XI) =

[z,, (VJz(X)l,

(23, the Lie algebra of G).

9

h E E,

E

T(M), X

E

Sec t

G.

2. Real bundles: Pontrjagin and trace classes

6 by

Finally, define a linear connection, V, in VYX

=

425

setting

X , Y E Sec 5

(VdYX,

and X E Sec [,

V y X = (pc)*[Y,XI,

Y

E

Sec 7.

Then the corresponding curvature satisfies i ( Y ) R = 0,

Y

E

Sec 7.

(9.11

I n fact, according to Example 1, R( Y , , Y,) Thus we have only to show that R( Y , X )

Y E Sec 7, X

= 0,

E

= 0,

Yl , Y , E Sec 7.

Sec 5.

Since the fundamental fields span the fibres of 77, it is sufficient to show that X E Sec f , h E E. R(Za, X ) = 0,

as follows from the relation above. Next observe that V,.,(a

X ) = a . V,X,

Z

E

%(M), X

E

Sec 5, a

E

G.

It follows that R(a . 2, , a * Z,)(a * X ) =a

*

R(Z, , Z,)(X),

a E G,

ZiE T ( M ) , X

Let @ E ( V P L Z ) , (F, the typical fibre of imply that the differential form, 1 P ' =-

p!

E

Sec 5.

(9.2)

6). Relations (9.1) and (9.2)

-1 -1 @(r R ,..., 27ri R ) ' 7rE

is both horizontal and invariant with respect to the action of G. Thus i(Za)Y = 0

and

T,*Y

= Y,

~ E E U , EG.

IX. Pontrjagin, Pfaffian, and Chern Classes

426

Hence h EE.

O(2,)Y = 0,

It follows that i(Y)Y=O=O(Y)Y,

YESecq,

and so P ! = 0 if 2p > rank 4. Let A,(M)i=o denote the subalgebra of horizontal invariant forms. T h e remarks above show that the homomorphism 7:: (VL;), + A ( M ) determined by R is in fact a homomorphism into A,(M)i=o.Thus it determines a modified homomorphism (VL;), H(C @ A,(M)+,), and the diagram, --f

commutes. In particular, if @ E (VpL;), and 29 &(@)

> rank [, then

0.

9.4. Pontrjagin classes. In sec. A.2 of Appendix A the characteristic coefficients CE E ( VkLg),are defined. They satisfy

det(p,

+

r At)

v EL,,

r

C,'(g,,..., v),

k

C:(v) hr-k,

=

= dimF,

k=O

where

C,'(v) = 1,

Cc(v) =

I

=

1,..., r .

T h e cohomology classes p,(() given by P k ( 8 =

Rp(G),

0 d 2k

,4)= 0 = Pf(5, ( 1, 4). 7

9

If the index of ( , ) is 2q, then Pf('!,

< , >,4) = (-I)*

Pf(59 ( ), 4). 9

Proof: Recall, from sec. 9.9, that (4, ( , ), ( , )) is a Z-bundle and that 5 = f + @ 6-. T h e corresponding Lie algebra is the subalgebra of LFconsisting of the transformations which are skew with respect to both ( , ), and ( , )F; this is exactly Sk,+ @ Sk,- . Thus the associated Lie algebra bundle is Sk,+ @ Sk,- . Since (f, ( , ), ( , )) is a 2-bundle, it admits a 2-connection V. V is both a Riemannian and a pseudo-Riemannian connection; thus its curvature, R, which takes values in Sk,+ @ Sk,- , can be used to obtain both pf(6, ( , ), A,) and pf(5, ( , ), A,), Now the lemma follows from Proposition IX, sec. A.7. Q.E.D. Proposition IX: T h e Pfaffian class has the following properties:

(1) Suppose rp: 6 -+ 7 is a bundle map between oriented pseudoRiemannian bundles which restricts to orientation preserving (respectively, orientation reversing) isometries in the fibres. Then Pf(5,


, A t ) = 4" Pf(7,


>

4J,

IX. Pontrjagin, Pfaffian, and Chern Classes

438

where (CI denotes the induced map between the base manifolds and E = + 1 (respectively, E = -1).

(2) pf(5, ( , ), A , ) depends only on index of ( , ) F .


,4))'= (- l)"pm(S) (2m = rank S). (4) Suppose is the orthogonal direct sum of oriented pseudoRiemannian subbundles 7 and [, and assume .$ has the orientation induced from the orientations of 7 and [, and the decomposition 5 = 7 @) 5. Then


,4)PfK, < '

t >I

4.

This follows from Theorem IV, sec. 8.13.

(2) I n view of Lemma I, it is sufficient to prove that any two Riemannian metrics ( , and ( , )' in 5 determine the same Pfaffian class. According to Proposition VI, sec. 2.17, volume I, there is a strong to (5,( , )'). It is evident from the bundle isometry from (5,( , construction that this isometry preserves the orientation of the fibres. Thus (2) follows from (1). ( 3 ) Proposition VII, sec. 8.17, implies that the diagram,

commutes. Moreover, according to Proposition VI, sec. A.6, j;(C&)

= PfF v

PfF.

Now (3) follows.

(4) Choose pseudo-Riemannian connections in 7 and 5 and give 5 the induced pseudo-Riemannian connection. Its curvature takes values in Sk, @) Sk, . Thus (4) follows from the corollary to Proposition VIII, sec. A.6. Q.E.D.

3. Pseudo-Riemannian bundles: Pontrjagin classes and Pfaffian class

439

9.13. The Pfaffian class of a Riemannian vector bundle. I n view of Proposition IX, sec. 9.12, the Pfaffian class of an oriented Riemannian vector bundle, 5, is independent of the choice of the Riemannian metric; this class will be called the Pfafian of 5 and will be denoted simply by pf(f). In Chapter X it will be shown that the Pfaffian class coincides with the Euler class of the associated sphere bundle (Gauss-Bonnet-Chern theorem). T h e Pfaffian class of the tangent bundle of an oriented Riemannian manifold M will be written pf(M) and called the Pfufian class of M. Let d be the positive normed determinant function in 7,; it is called the volume form for M and J, A , is called the volume of M . Moreover if R' is the Riemannian curvature of a Riemannian connection in M , then

,

(-')" < A w , (R')") m !(27~)"

represents pf(M) (2m = dim M ) . Now let K E Y ( M )be the function determined by the equation (- 1)" vol S2" < A , , (R')") (2a)"m!

=

2K * A , .

This function is called the Gaussian curvature of the connection. Evidently pf(M) is represented by (2/vol S2")K A , . Proposition I X specializes to Proposition X: T h e Pfaffian class of an oriented Riemannian vector bundle, f , has the following properties: (1) It changes sign if the orientation is reversed. (2) If v: 4 ---f r) is a bundle map restricting to orientation preserving isomorphisms in the fibres and inducing I,!Ibetween the base manifolds, then *#

5

(3) Suppose f

Pf(d

= r) @ 5, where

=

Pf(0.

and is given the induced orientation. Then r)

5 are oriented

subbundles and

I n particular, if f admits an oriented subbundle of odd rank, then pf(f) = 0.

440

IX. Pontrjagin, Pfaffian, and Chern Classes

9.14. Decomposable curvature. Suppose 8 is an oriented Riemannian vector bundle that admits a Riemannian connection whose Riemannian curvature is of the form

(cf. Example I , sec. 9.8). Assume that < A , , (R')")

= f"

5

has even rank r

= 2m. Then

. < A , , Yzm).

I t follows that p f ( 0 is represented by the differential form P

(3 1!

= -

-f"., 2 (cf. problem 29). (iii) Reverse WP'.

32. Kodaira class. Let 8 = (P, r, B, T ) be a principal torus bundle, and let 5 be the complex vector bundle associated with a representation, @, of T in a complex space F. Let f have a linear connection V induced from a principal connection in 8.

(i) Interpret the curvature, R, of V as a 2-form with values in E: R E A2(B;E ) , where E is the Lie algebra of T.Show that 6R = 0.

476

IX. Pontrjagin, Pfaffian, and Chern Classes

(ii) Let % = {U,} be a simple cover of B, and let {( U, , +,)} be a 0 +,,z, principal coordinate representation for 8.Let g,,(x) = x E U,, . Show that there are smooth functions feD: U,, 3 E such that exp Of,, = g,, * is constant in UaOy . If hmaYis the (iii) Show that f,, + j o y+ f Y , constant value of this function, prove that the eigenvalues of O’(haOy) are integral multiples of 2ni. Interpret the correspondence UdOy ++hapyas a simplicial 2-cocycle, Q, on the nerve Jv; of %, with values in E. (iv) Assume f is a function which assigns to each non-void intersection UE0n *..n UWp an element of AP( U,, n ..*n U , ; E) ( p and q fixed). Then f is a q-cochain of E-valued p-forms. Make these functions into a space C**Qand introduce an operator, V = 6 D,in the direct sum, C = @ CP,Q,exactly as in article 7, Chap. V, volume I. Regard A ( B ; E) and C ( N ;E ) as subspaces of C. (v) Show that R - Q = V(Y) for some Y E C. (vi) Consider the case that d i m F = 1. Show that an integral simplicial 2-cocycle, 8, on JV is defined by @’(ha,,,) = -2ni8(&)~. Prove that the class represented by 8 in H(M)(the Koduira class) corresponds to cl(f) under the isomorphism H ( N ) g H(B). @ F, , where the Fi are (vii) For general F write F = F , @ 1-dimensional 7‘-stable subspaces. Let 8, E C 2 ( N ) be the cocycle corresponding to Fi . Set 8 = ni=,(1 Oi). Show that 8 is an integral cocycle in C ( N ) whose class corresponds to c( 6) under the isomorphism H ( N )g H ( B ) .

:+;

+

+

33. Vector bundles over S4. Let R4 be the space of quaternions and let S4 be the one-point compactification of R4. Give S4 the orientation induced by R4. Let M be the 8-manifold obtained from two copies of S4 x R4 via the identification, +: k4 x R4 a4 x R4, given by

where p E E and q E Z. Construct a vector bundle, and show that P I ( [ ) = 2 ( P - 4)w,

8

= ( M , T , S4,R4),

where w denotes the orientation class of S4. Hint: Use problem 17, Chap. VI.

Chapter X

The Gauss-Bonnet-Chern Theorem 10.1. I n this section [ = ( M , T , B, F ) denotes a fixed Riemannian vector bundle of rank r = 2m. Thus, if 5 is oriented, the Pfaffian class pf(5) is defined. On the other hand, the Euler class x, of the associated sphere bundle (via a Riemannian metric) is also defined. This chapter is centered around the following theorem:

Theorem I (Gauss-Bonnet-Chern): Let [ = ( M , T , B , F ) be an oriented Riemannian vector bundle of rank r = 2m. Then the Pfaffian class of 5 coincides with the Euler class of the associated sphere bundle:

As an immediate consequence of Theorem 111, sec. 9.9, volume I, and Theorem I, sec. 10.I , volume I, and formula 9.6, sec. 9.12, we obtain Theorem 11: Let [ be an oriented Riemannian vector bundle of rank 2m over a compact connected 2m-manifold B. Let u be a cross-section with finitely many zeros and let V be a Riemannian connection in [ with Riemannian curvature R*.Then the index sum of u is given by

Moreover, if f is the tangent bundle of B , then these numbers coincide with the Euler-PoincarC characteristic of B: 2m

j(u) =

~

m!(2n)"

B

(-1)p

< d E , F m )= P=O

dim HP(B) =

1' X, B

Remarks: 1. I n the theorem above, (R')" denotes the mth power of R' in the algebra A ( B ; A f ) and A , is the positive normed determinant function in f . 477

X. The Gauss-Bonnet-Chern Theorem

418

2. In view of Lemma I, sec. 9.12, there are almost identical analogues of Theorems I and I1 for pseudo-Riemannian bundles.

Corollary I: T h e Euler class of the Whitney sum of two oriented Riemannian vector bundles is given by

Proof: Apply Proposition X, (3), sec. 9.13. (Recall from sec. 8.2, volume I, that xe = 0 if .$ has odd rank). Q.E.D. Corollary 11: An oriented Riemannian vector bundle with nonzero Euler class contains no oriented subbundle of odd rank. Proof: Again apply Proposition X, (3), sec. 9.13.

Q.E.D. Corollary 111: T h e tangent bundles of the even dimensional spheres contain no proper nonzero orientable subbundles. Corollary IV: Assume that B is a compact oriented n-manifold (n = 2m) whose tangent bundle admits a Riemannian connection with decomposable curvature R' =f * (!PA Y) (cf. sec. 9.14). Then the Euler-PoincarC characteristic of B is given by

where $,(h)

=

Y(x; h), h E T,(B).

Proof: Apply the Gauss-Bonnet theorem and sec. 9.14.

Q.E.D. 10.2. Normed cross-sections. Let 5 = ( M , T , B, F ) be an oriented Riemannian vector bundle of rank Y, equipped with a Riemannian connection V. Let R' denote the corresponding Riemannian curvature and let d, be the unique positive normed determinant function in 5. Assume that $, admits a normed cross-section u; i.e., ("(X),

"(X))

=

1,

x E B.

(Observe that this is equivalent to assuming that without zeros.)

6 admits a cross-section

X. The Gauss-Bonnet-Chern Theorem

Define differential forms @k @k

= .

z

Finally, observe that the cross-section, u, , is given by ua(z>= (z, z), S, . Hence, the corollary to Proposition XIII, sec. 7.27, yields

E

(A,,

U, A

(SU,)'-')

==

(2m

1s

- I)!

V O ~S2m-'.

It follows that

( j s@ ) ( a )= 1, izQ = (-1)m (cf. sec. 0. I3 for vol S2m-1).

nt!(27r)m,

a E B,

Q.E.D.

Probfems 1. Angle function. Let M be an oriented Riemannian 2-manifold with metric tensor g and normed determinant function d. Let a : [0, 11 -+ M be a path on M . A vector field along a is a crosssection, X , in the pull-back, ( Y * T ~ of , r Munder a (equivalentIy, X is a smooth map from the unit interval to T , such that X ( t ) E 7'a(t)(M), t E [0, 11). Let V be the Levi-Civita connection in M and let V r be the induced connection in a*rMin the direction of the vector field T = d / d t . T h e covariant derivative of a vector field along a, denoted by V , , is defined by V , X = V r X .

(i)

Let X

E

% ( M )and set X ( t ) = X ( a ( t ) ) .Establish the chain rule

(ii) Let X and Y be vector fields along a such that I X ( t ) l = 1 and I Y(t)l = 1. An angle function is a smooth function 8 of t (0 < t < 1) satisfying cos q t ) = g(.(t);

q4,Y(t)),

sin 8(t) = d(a(t);X ( t ) , Y(t)).

Construct an angle function for X and Y . Show that if O1 and O2 are angle functions then, for some k EZ, tI2 - = 2k7r. Conclude that the difference O(1) - e(0) does not depend on the choice of the angle function. (iii) Show that ~ ( l) W )=

J

1

[ - ~ ( a ( t ) ~; ( t )vaX(t)) ,

0

+ d ( a ( t ) ;~ ( t )vay(t))l , dt*

Hint: Use the identity A x ; h, hl) . 4 x ; h2 ,A,)

+ g(.;

+ A x ; h, A),

.

h, 9 h,)

h, h3) * d(x; h, , 12,) = 0.

(iv) Show that 8 changes sign if the orientation of M is reversed. 485

X. The Gauss-Bonnet-Chern Theorem

486

(v) Let X i (i = 1,2, 3) be vector fields along a and denote the corresponding angle functions by Oij . Show that, for some k E Z,

+

=

Conclude that

el'&)

+ 2nkG

e23(t)

+ e,,(t)

=

2nk.

(vi) Assume that a is a closed path homotopic to the constant path a(0).Let y : Q --f M be a homotopy from a to a. , where Q is the unit square 0 t 1, 0 T 1. Construct a cross-section Y in the bundle c ~ ' * Tsuch ~ that I Y ( t ,T)I = 1 (0 t 1, 0 T 1). If X is a vector field along a show that 8(1) - 6(0) is independent of the choice of Y. Write a,,:t w

<
(X,

A .’*

A XD+Q)

O € L A g F , ? P € L A Q F ,X ~ E F .

These bilinear maps make the spacexi=oL A P F into a graded algebra, C(F).

It is called the characteristic algebra for F. On the other hand, make the direct sum d ( F ) = C;=O(APF* Q APF) into a commutative and associative algebra by setting (U*

0U)

*

(V*

@ V ) = (U*

A V*)

@ (U

A fl),

U*, V * €

m*,

I(, V

E

m.

Then the canonical linear isomorphisms APF* Q ApF 5 LA,, define an algebra isomorphism c=

d ( F ) JC(F).

I n particular, it follows that C ( F )is commutative and associative. Henceforth we shall identify the algebras d ( F ) and C(F)under the isomorphism above. T h e p t h power of an element @ E C(F) will be denoted by Qm, 0 m

=

*.. 0 0.

0

( D factors)

I n particular, qm

=p!

A’p,

ELF.

More particularly, if L denotes the identity map of F and the identity map of APF, this formula becomes Lm

= P!LD.

493

L~

denotes

Appendix A. Characteristic Coefficients and the Pfaffian

494

It follows that

Next, recall the substitution operators i ( x ) : AF* + AF* and i(x*): AF --*. AF determined by vectors x E F and x* E F*. They are the unique antiderivations that satisfy and

i(x)y* = ( y * , x)

y* E F*, y

i(x*)y = (x*, y),

E F.

An algebra homomorphism i : d ( F )-+ LA(F) is defined by i(x*l

A

-.. A

x*p @ x1 A

A

x p ) = i(x,)

o

-.+i(xl) @ i ( x * p ) o

o

o

i(x*l).

With the aid of the identification above we may regard i as a homomorphism i : C(F)---* L C ( F ) . Finally, note that the spaces L A , are self-dual with respect to the inner product given by (@, Y ) = tr(@ 0 Y) = ( L ~ @ , 0 Y ) = i(@)Y.

It satisfies (u* @ U, V * @ V )

=

u*,V *

(u*,v ) ( v * , u),

E

APF*,

Moreover i ( @ )is dual to multiplication by @, @ E L,,tF

U, v E

APF.

.

A.2. Characteristic coefficients. T h e pth characteristic coefficient for an n-dimensional vector space F is the element C ~ ”VPL$ E given by Ct = 1 and

ci(v1

)*.*i

PD) = tr(pl 0 “ *

pp)

= (Lg

,

0 “‘ 0p p > ,

p b 1, ppi E L F *

Note that Cp”= 0 if p > n. C,F will be denoted by DetF. T h e homogeneous functions, Cz , corresponding to CL are given by Cr(p) = tr App,

(cf. sec. A.0). We shall show that n

p eLF

1. Characteristic and trace coefficients

495

I n particular, det p

1 n!

...,p).

= - DetF(p,

T o prove formula (A.l) we argue as follows. Let el of F. Then det(p

+ X L ) el ...

A

A

T h e elements eil writing

A

AVp(ei,

en = (p

-*.

A

+ X L ) el

Aeiv(il

A


n. Q.E.D.

Corollary 11: The subalgebras of V L $ generated respectively, by Cc, ..., CE and by Trg, ..., TrE, coincide and contain all the trace coefficients and characteristic coefficients. Q.E.D.

s2.

Inner product spaces

I n this article F denotes an n-dimensional vector space and ( , ) denotes an inner product in F. It induces a linear isomorphism F 3 F* which we use to identify F with F*. Further, ( , ) extends to an inner product in each space ApF. Sk, denotes the Lie subalgebra of LF consisting of the linear transformations which are skew with respect to ( , ). A.4. Multiplications in A F @ AF. I n the vector space AF 0AF we introduce two algebra structures: the first is the canonical tensor product of the algebras AF and AF; the second is the anticommutative tensor product of AF and AF. The first algebra contains d ( F )as a subalgebra (cf. sec. A.l) and so its multiplication is denoted by 0:

T h e second algebra is canonically isomorphic to A(F OF), and so multiplication is denoted by A : (u

@ v)

A

(ul 0w l )

=

(-l)q'(u

A

ul) @ w

A

v, ,

w E AqF,

u1 E ArF.

The two products are connected by the relation @

0Y

= (-I>"'@

A

U,

@E

A F @ AqF, Y EArF @ AF.

This implies that p1

..* 0pp = ( - l ) P ( P - l ) / z p l

A

A

pp ,

p, E F @ F.

I n particular,

where L~ is regarded as an element of APF @ APF. Now define an inner product in F @ F by (x

0Y , x1 0Y l ) = (x, Y l ) + ( Y , 31). 500

(A.2)

2. Inner product spaces

501

(This is not the usual inner product!) Extend it to an inner product in A(F OF). T h e induced inner product in AF 0AF (via the standard algebra isomorphism (AF 0AF, A ) A(F @ F ) is given by (APF @ AqF, ArF @ AsF) = 0,

p

unless

and q

=s

= r,

and (U

@ b, u

0U)

= (-1)Pq(u,

~ ) ( b u, ) ,

U, u E

APF, b, u

E

A'F.

Remark: U p to sign, this inner product agrees with the inner product in C(F) defined in sec. A.1.

Next, identify F @ F with (F @ F)* under the above inner product. Let T : F @ F -+ F @ F be the linear isomorphism given by T(X,

Its dual,

y ) = (X

+y ,

X

-Y),

X,

Y E F.

r*, is given by T*(X,

y)

=(

y - X, y

+

X,

X),

y EF.

r and r* extend to algebra automorphisms r, and r* of (AF @ AF, which are dual with respect to the inner product defined above. Observe that

~ , ( 1 0%~ y =)(X

A

y ) 01 - x @ y + y @ x

+ 1 @ (x

A

A)

y).

Lemma 11: r has the following properties:

(I) (2)

p

rA(x

T"(LP)

-Y = 2PLP

.

0)'

y) 0 -

= 2((x

0(x A Y))*

Proof: (1) is immediate from the formula above as is (2) in the case 1. T o obtain (2) in general observe that

=

1

~~(6,)

= (-l)P(D-1)/2

= (-1),(p-1)/2

- T*(L

A

... A

1 (& -

A

***

P! P!

A

L)

&)

==

2P6,

. Q.E.D.

502

Appendix A. Characteristic Coefficients and the Pfaffian

A.5. Characteristic coefficients for F. Let canonical isomorphism given by B(X A

p: A2F 5 SkF be

the

Y ) ( 4 = (X, 2)Y - ( Y , Z>X*

Proposition IV: Let cp E Sk,

.

Then the characteristic coefficients

CpF(cp) are given by C,F(P) = 0,

P odd,

and

Proof:

Let cp E Sk, . Then det(v

+ AL) = det(y* +

At)

= det(-p

+ AL).

It follows that C;k+l(q) = -C[k+l(cp), whence C&+l(cp)= 0. T o establish the second formula, regard the inclusion j : Sk, a linear map from Sk, into F @ F. Then

--t

LF as

Thus Lemma 11, (1) shows that

Now let ( , ) be the inner product in AF @ A F defined in sec. A.4. Then, for cp E Sk, (cf. Lemma I1 and formula (A.2), sec. A.4),

Q.E.D.

2. Inner product spaces

Next, define elements B,

E

503

VZkSk$by

Then, as an immediate consequence of Proposition IV, we have Let j : SkF +LF be the inclusion. Then

Proposition V:

A.6. Pfaffian. Suppose F has even dimension n = 2m and let a E AnF. Then the Pfajian of the pair (F,a) is the element, Pf; E VmSk$, given by pf:(~i

j...,

pm)

= ( a , P-l(~i)A

***

A

P-YVm)),

~p

E

-

S ~ F

I t determines the homogeneous function Pfz given by 1 pff(p) = a p f f ( p , * * *V),,

SkF

a

The scalar PfE(F) is called the P'uflun of F with respect to a. We extend the definition to odd-dimensional spaces by setting the Pfaffian equal to zero in this case. Proposition VI:

Let u E AnF and b E AnF. Then Pf: v Pf:

=

( a , b)j'(Det),

where j : Sk, -+ LF denotes the inclusion. In particular, (Pf:(p))2 Proof: In fact,

= (a, a )

det p,

p E SkF .

Appendix A. Characteristic Coefficients and the Pfaffian

504

(since a E AmF and b E A W ) . This shows that Pf,f v P f l

= (a,

b ) Bm .

Now apply Proposition V, with k = m. Q.E.D. Next, let

T:

F -+F be an isometry; i.e., (7x9

Then det T

=

f l . If det T = 1, T is called proper.

If

Proposition VII: (1) Pf:(r

o

q1 o

x, Y EF-

TY> = (x, y>,

7-l

,..., 7 vm 0

is an isometry of F, then

T

0

T - ~ )=

det 7 Pf:(yl ,...,pm),

pi

E

SkF

(2) If $h E SkF, then

Proof: In fact, since

P(TX A

TY) = T

0

p(X A

y ) 0 T-',

X, JJ E

F,

it follows that Pf:(.r

o

v1 o T - ~ ..., , T vm 0

0

7-l)

= det

7

Pf,f(vl ,...,vm),

which establishes (1). Similarly, for $h E Sk, B((Cx

A

Y

+

X A

(CY) = r h P b

A

r)l,

whence m

C pf:(~i

19, ~ i l , . . .Pm) ,

= tr

4

*

Pf,"(R

**-**

v m > ==

0.

a 4

Q.E.D. Let H be a second inner product space and give F @ H the induced inner product; i.e., <x

0Y , X1 0JJJ= (X,

Xl>

+ .

2. Inner product spaces

T h e inclusion map j : Sk, @ Sk,

505

+ SkFoHinduces

p : V Sk; 0 V Sk;

+V

a homomorphism

Sk*,,,

Moreover, multiplication defines a canonical algebra isomorphism, AF

0 A H -% A(F 0H ) ,

which preserves the inner products. We shall identify the algebras AF @ A H and A(F @ H ) under this isomorphism. Proposition VIII: Let a E AnF and b E A'H, where n r = dim H. Then, with the identification above,

jv(Pf,FgOf) = Pf:

n

Proof: If n + r is odd both + r = 2k. Then we have, for q~

E

= dim F

and

0PfF.

sides are zero. Now assume that Sk, and E Sk, ,

If n and r are odd, it follows that

Corollary: Pf:$,H(p,

0$) = Pft(v) Pff(+),

9~ E SkF,

4E SkH .

A.7. Examples: 1. Oriented inner product spaces: Let F be a real inner product space of dimension n = 2m (note that we do not require the inner product to be positive definite). Let e E AnF be the unique element which represents the orientation and satisfies I(e, e)l = 1. Then Pf,F is called the Pfafian of the oriented inner product space F,

506

Appendix A. Characteristic Coefficients and the Pfaffian

and is denoted by PfF . Reversing the orientation changes the sign of the Pfaffian. Proposition VI implies that det p

=

.

p E SkF

(e, e)(PfFp)z,

Next let F = F f @ F- be an orthogonal decomposition of F such that the restriction of the inner product to Ff (respectively, F-) is positive (respectively, negative) definite. Define a positive definite inner product ( , ) in F by setting

+ x-, Y+ + y - )

(x'

x+, y+ E F+, x-, y- E F-.

= <x+, Y+> - <x-* y-),

Let p be a skew linear transformation of F that stabilizes F+ and F-, p-; F-

p+:F+ + F+,

p = p+ @ p-,

+ F-.

Then 'p is skew with respect to both of the inner products ( , ) and ( , ) and so that Pfaffians Pf: , )(p)) and Pf? , )(v)are defined. Proposition IX: Suppose 'p satisfies the conditions above. Then:

(1) If dim F- is odd, PfP, >(V)

(2) If d i m P -

pf:. ,(p) = 0.

= 0,

= 2q. Then

Pf8. >(d= (-~)"ff*

)(d*

Proof: T h e corollary to Proposition VIII, sec. A.6, shows that, for suitable orientations of Ff and F-,

and

Pf? , )(P) = Pf?f )(97+)

*

pf7: Ap-1

Pff, )(d= PffT dP+) * PfK )(v-)*

Since ( , ) and ( , ) coincide in F+,it follows that Pf?: AT+)

= PfPf

dv+) ( , ) is

negative definite; i.e., We are thus reduced to the case that F=F-andp,=p)-. In this case, ( , ) = -( , ) and so the linear isomorphisms ,!?