Geometry of Manifolds

Geometry

of Manif0Ids

PURE A N D APPLIED MATHEMATICS A Series of Monographs and Textbooks

Edited by

PAULA. SMITHand SAMUEL EILENBERG Columbia University, N e w York

I: ARNOLDSOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) 11: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 111: HERBERT BUSEMANN AND PAULJ. KELLY.Projective Geometry and Projective Metrics. 1953 AND M. SCHIFFER. Kernel Functions and IV: STEFANBERGMAN Elliptic Differential Equations in Mathematical Physics. 1953 V: RALPHPHILIPBOAS,JR. Entire Functions. 1954 VI: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 VII: CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 Hu. Homotopy Theory. 1959 VIII: SZE-TSEN IX: A. OSTROWSKI. Solution of Equations and Systems of Equations. 1960 X: J. DIEUDONNB. Foundations of Modern Analysis. 1960 XI: S. I. GOLDBERG. Curvature and Homology. 1962 HELGASON. Differential Geometry and Symmetric XII: SIGURDUR Spaces. 1962 XIII: T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963 XIV: SHREERAM ABHYANKAR. Local Analytic Geometry. In preparation. XV: RICHARDL. BISHOPAND RICHARDJ. CRITTENDEN. Geometry of Manifolds. 1964 XVI: STEVEN GAAL.Point Set Topology. In preparation.

Geometry

of Manifolds

Richard L. Bishop

Richard I . Crittenden

Department of Mathematics University of Illinois Urbana, Illinois

Department of Mathematics Northwestern University Evanston, Illinois

1964

ACADEMIC

PRESS

NEW YORK AND LONDON

COPYRIGHT 0 1964,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED.

NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l LIBRARY OF CONGRESS CATALOG CARDNUMBER:64-20317

PRINTED I N THE UNITED STATES OF AMERICA

PREFACE Our purpose in writing this book is to put material which we found stimulating and interesting as graduate students into book form. I t is intended for individual study and for use as a text for graduate level courses such as the one from which this material stems, given by Professor W. Ambrose at M I T in 1958-1959. Previously the material had been organized in roughly the same form by him and Professor I. M . Singer, and they in turn drew upon the work of Ehresmann, Chern, and E. Cartan. Our contributions have been primarily to fill out the material with details, asides and problems, and to alter notation slightly. We believe that this subject matter, besides being an interesting area for specialization, lends itself especially to a synthesis of several branches of mathematics, and thus should be studied by a wide spectrum of graduate students so as to break away from narrow specialization and see how their own fields are related and applied in other fields. We feel that at least a part of this subject should be of interest not only to those working in geometry but also to those in analysis, topology, algebra, and even probability and astronomy. In order that this book be meaningful, the reader’s background should include real variable theory, linear algebra, and point set topology. T o get an idea of the scope of this book we refer to the table of contents and the introductory paragraphs to the chapters. We have not included the study of integration theory, for example, the de Rham’s theorems and the Gauss-Bonnet theorem, because we did not wish to get involved in the theory of topological invariants. However, the background for these topics is thoroughly treated, and Morse theory is carried to the point where topology takes over from analysis. T he theorems, lemmas, propositions, and problems are numbered consecutively within each chapter. Our use of these numbers in cross references should be transparent. Th u s in the text of Chapter 6, “theorem 7” refers to the seventh theorem in Chapter 6, while “problem 5.4” refers to the fourth problem in Chapter 5. Definitions are generally distinguished only by italics. T h e word “section” is usually omitted in this usage ; that is, an unmodified number reference is to the corresponding section. In this case, the chapter number is always given. V

vi

PREFACE

T h e problems range from trivial to very difficult, from essential to the text to clearly tangential. T h e subjects of holonomy groups and complex manifolds are developed exclusively in problems. Some problems almost certainly will require recourse to the reference given, namely, problems 1.11, 2.7, 2.13, 2.14, and 8.15. A brief appendix is provided with a statement of the theorem on existence and uniqueness of solutions of ordinary differential equations most appropriate for our needs. T h e reader is referred to [33]and [50] for their extensive bibliographies as well as to their very fine treatment of much of the subject matter of the present text. Italic numbers in brackets are, of course, references to entries in the bibliography. R.L.B. R. J.C.

April 1964

C O NT E NT S Preface

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CHAPTER 1

Manifolds 1.1

1.2 1.3 1.4 1.5 1.6

Introductory Material and Notation Definition of a Manifold . . Tangent Space . . . . Vector Fields . . . . Submanifolds . . . . Distributions and Integrability .

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25 26 28 29 30 34

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CHAPTER 2

Lie Groups 2.1 2.2 2.3 2.4 2.5 2.6

Lie Groups . . . . . . Lie Algebras . . . . . . Lie Group-Lie Algebra Correspondence Homomorphisms . . . . . Exponential Map . . . . . Representations . . . . .

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CHAPTER 3

Fibre Bundles 3.1 3.2 3.3 3.4

Transformation Groups . . . Principal Bundles . . . . Associated Bundles . . . . Reduction of the Structural Group

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CHAPTER 4

Differential Forms 4.1 Introduction . . . . . 4.2 Classical Notion of Differential Form 4.3 Grassmann Algebras . . . . vii

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4.4 4.5 4.6 4.7 4.8 4.9 4.10

CONTENTS

Existence of Grassmann Algebras . Differential Forms . . . . Exterior Derivative . . . . Action of Maps . . . . . Frobenius’ Theorem . . . . Vector-Valued Forms and Operations Forms on Complex Manifolds . .

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57

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68 70 71 72

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CHAPTER 5

Connexions 5.1

5.2 5.3 5.4 5.5 5.6

Definitions and First Properties . . . . . . . Parallel Translation . . . . . . . . . . Curvature Form and the Structural Equation . . . . . Existence of Connexions and Connexions in Associated Bundles Structural Equations for Horizontal Forms . . . . . Holonomy . . . . . . . . . . .

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74 77 80 83 84 87

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CHAPTER 6

Affine Connexions 6.1 6.2 6.3 6.4

Definitions . . . . . . . . T h e Structural Equations of an Affine Connexion T h e Exponential Maps . . . . . Covariant Differentiation and Classical Forms

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122 127 129 132

CHAPTER 7

Riemannian Manifolds 7.1 7.2 7.3 7.4

Definitions and First Properties T h e Bundle of Frames . . Riemannian Connexions . . Examples and Problems . .

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CHAPTER 8

Geodesics and Complete Riemannian Manifolds 8.1 Geodesics . . . . . 8.2 Complete Riemannian Manifolds 8.3 Continuous Curves . . .

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ix

CONTENTS

CHAPTER 9

Riemannian Curvature 9.1 9.2 9.3 9.4 9.5

Riemannian Curvature . . . . Computation of the Riemannian Curvature Continuity of the Riemannian Curvature Rectangles and Jacobi Fields . . . Theorems Involving Curvature . .

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161 165 166 172 178

CHAPTER 10

Immersions and the Second Fundamental Form 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

. . . . . . . Definitions T h e Connexions . . . . . . Curvature . . . . . . . T h e Second Fundamental Form . . . Curvature and the Second Fundamental Form . . . . . T h e Local Gauss Map Hessians of Normal Coordinates of N . . A Formulation of the Immersion Problem . Hypersurfaces . . . . . . .

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CHAPTER 11

Second Variation of Arc Length 1 1 . 1 First and Second Variation of Arc Length 11.2 T h e Index Form . . . . . 11.3 Focal Points and Conjugate Points . I I .4 T h e Infinitesimal Deformations . . 1 I .5 T h e Morse Index Theorem . . . 11.6 T h e Minimum Locus . . . . 1 I .7 Closed Geodesics . . . . . 11.8 Convex Neighborhoods . . . . 1 I .9 Rauch’s Comparison Theorem . . 11.10 Curvature and Volume . . . .

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APPENDIX: Theorems on Differential Equations

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BIBLIOGRAPHY .

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SUBJECT INDEX

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Geometry

of

Manifolds

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CHAPTER 1

M an i fo Ids In this chapter the basic tools of manifold theory are introduced and the main theorems are stated without proof. Lie derivatives are discussed via local one-parameter groups of transformations, and various interpretations of the bracket of vector fields are given. Frobenius' theorem on the integrability of p-plane distributions is given in outline form [4, 24, 25, 33, 50, 78, 831. 1.l Introductory Material and Notation

+

If is a map of M into N andJ! , I a map of P into T , then I,!J o will denote their composition, that is, J!,I o is followed by 4. Here M , N , P, T are any sets and we understand that the domain of I,!J o is #-l(P) n M (in particular, $J c may have an empty domain). T h e same sort of convention, namely, that the domain is the largest meaningful set, will be used in the formation of sums, products, and other combinations of maps. If U c M we use +ILI for the restriction of to u.

+ +

+

+

+

FIG. 1.

1

2

1. MANIFOLDS

T h e d-dimensional Euclidean space will be denoted by Rd,provided with the usual coordinate functions, {ui}, that is, if t = ( t l , ...,td) E Rd, then ui(t) = t i . In the case d = 1 we write R1 = R and u1 = u. If U is open in Re, then a map 4 : U -+ Rd is said to be of class C" (written 4 E C") if the real-valued functions ui o 4, i = 1, ..., d, have all kth order continuous partial derivatives for every non-negative k. C" maps are not necessarily analytic, as is shown by the example: f(x) = exp(-l/x2) if x # 0, f(0) = 0. In fact, there exist nontrivial C" real-valued functions on Rd which vanish outside a given compact set. (See [85], pp. 25 and 26 for the construction of C" Urysohn functions.) Problem 1.

Define exp(-I/.)

if x > O if x 0.


bt(g) (b) t(f-1 = t ( f )g ( m ) f ( m ) t(g), for a, b E R, f , g E q M , m).

+

T h e tangents at m form a linear space, denoted by M , . If c is a g are constant function, then t ( c) = 0. We recall that fg and f the usual product and sum, but defined only on the intersections of the domains o f f and g. Letting l u be the function defined only on U and there equal to 1, we see from (a) and (b) that t(flu) = t ( f ) and hence t ( f ) = t ( f lu); that is, t( f ) depends only on the local behavior off.

+

Prove that for constant function c and neighborhood U 1”) = 0. T h e problem includes proving t ( c ) = 0. If = (xl, ..., x d ) is a coordinate system the partial derivative a t m with respect to x i , Dzi(m), is the tangent defined by (DZi(m))f = ( a ( f o+-‘)/aui) (+(m)), which is also denoted by Dzif ( m ) . When the coordinates are {ui} on Rd we shall write Di instead of D U i . Problem 8.

of m that t(c

FIG.6.

I t is easily seen that Dzixi(m) = aij (Kronecker delta), and, hence, {D5i(m)} is linearly independent, as can be seen by evaluating a linear combination on each of the functions xi in turn. Problem 9 . Give an example to show that DZi depends on xl, and not just on x i .

..., xd

9

1.3. Tangent Space Tangents are completely characterized by the following:

If (x, , ..., x,) is a coordinate system at m E M , t a tangent at m, then t = C (tx,) D,i(m).

Theorem 1.

For the proof, we assume the: Lemma.? If f € F [ R d a), , a = ( a l , ..., a,), then there are functions g, , ..., g, € F ( R da) , such that f = f ( a ) C (u, - a,) g, in a neighborhood of a [72; 94, p. 2211. Note. From this it follows that g,(a) = Dif(a).

+

+

Proof of Theorem 1. Let f € F ( I M , m), a coordinate system at m. Then by the lemma, there are g, such that if a = +(m), then giE F(Rd,a), f o 4-l = f 3 +-'(a) C (u, - a,) g, in a neighborhood of a, and g,(a) = D i ( f o +-')(a). Hence, f = f ( m ) C (xi-xi(m)) hi in a neighborhood of m, where h,(m) = D r i f ( m ) , hi E F ( M ,m). Therefore,

+

tf = 2 t ( x d U r n ) + 2 =

Corollary. (Proof:

2 t(xi) oz,(m)(f).

+

-

.,(m)) (4

QED

T h e dimension of Mrn is d, the dimension of M.

{Dzi(m))is a basis.)

We have already defined the tangent vector y * ( t ) for parameter value t of a C" curve y in M . We point out that every tangent at m is such a y*(t). For if xl, ..., x, is a coordinate system at m and s = C a,Dz,(m), then s is clearly the tangent to the curve given by: y ( t ) = that point whose coordinates are x,(m) ta, .

+

+

If : M + N is C", we define the dtfferential of 4, d+ : Mm -P N+(nL,, by: if t E M , , f € F ( N ,+(m)), then d + ( t ) ( f )= t ( f o 4). d+ is clearly a linear map. Problem 10. Prove that the following is an alternate definition of d+: for every C" curve y in M and parameter value t, d+(y*(t))= (+oy)*(t). t In the case of Ck manifolds (which we have not defined) the corresponding lemma is not true, since the gi's will not always be Ck.In fact, for Ckmanifolds the space of derivations at m is infinite in dimension, so the tangent space is defined to be the space spanned by {D,i(m)} [64A].

1. MANIFOLDS

10

+

If : M -t N , m E M , (xl , ..., x d ) is a coordinate system at m, (yl , . . . , y e )is a coordinate system at +(m), then the matrix of d+ with respect to the bases { D J m ) } and {Duj+(m))} is the Jacobian (Dzj(yio +)(m)).

lacobian matrices.

FIG. 1.

Chain rule.

If

+ : M - t N , i,/J : N-+ P are C", then d($ o +) = di,/Jo d+.

We can rephrase our previous definition of the tangent y * ( t ) to a C" curve y as: y,(t) = dy(D(t)). (D = D , = d/du.) Later (3.3) we shall give the collection of all tangents to M , T ( M ) ,a C" structure, so that y* will become a C" curve in T ( M ) . Tangent space of a product. If M , N are manifolds, then there is a natural isomorphism T between M x N,,,,) and M , N,, (direct sum). If p : M x N - t M and q : M x N --f N are the ) dp(t) dq(t). projections and t E M x N(,,,), then ~ ( t= Now let : M x N - + P be C". For (m,n) E M x N , define C" maps +,,,IN + P @:M-tP

+

+

+

by +,,(n) = # ~ ~ ( m =)+(m,n). Let s E M,, , t E N , as an element of ( M x N)(,,,,,, omitting T . Theorem 2.

d+(s

+ t)

= d+%(S)

+ d+m(t)*

T h e proof is left as an exercise. (Fig. 8.)

, and view

s

+t

1.3. Tangent Space

11

A dz..eomorphism of M onto N is a one-to-one map 4 ; M - P N such that 4 and 4-l are C". Existence of a diffeomorphism is the natural equivalence relation for manifolds. Difficult results of Milnor and others [44, 52, 53, 58, 841 show that this equivalence relation is not the same as topological equivalence, at least for manifolds of dimension greater than six. For manifolds of dimensions one, two, and three it is known that both equivalence relations are the same, as is true allegedly also in dimensions four through six according to unpublished work of Cerf.

FIG. 8.

It is easy to give examples to show that a C" homeomorphism need not be a diffeomorphism, that is, that the condition that 4-' be C" is independent. For any integer n > 1 the map : R -+ R is such an example, since the inverse does not have a derivative at 0. If we change our viewpoint and consider the C" homeomorphism as the identity map on an underlying topological manifold, we get examples of different C" structures on the same space, although the resulting manifolds may be diffeomorphic under another map. T h e example given then shows that if we take { P - l } as a basis for a C" structure on R we get different structures for different n's. (However, these manifolds are equivalent in the above sense.) Characterization of diffeomorphism. 4 : M -P N is a diffeomorphism if and only if 4 is a C" homeomorphism with range N and for every m E M , y5 a coordinate system at +(m),y5 o is a coordinate system at m [25, p. 751.

+

Inverse function theorem. Let (xl, at m E M , fl , ..., fd E F ( M ,m). Then

..., x d ) be a coordinate system 4 = ( fl , ..., fd) restricts to a

12

1 . MANIFOLDS

coordinate system at m if and only if det(DXjf i ( m ) ) # 0, that is, d# is nonsingular on M , [25, p. 701. Corollary. I f # : M -+ N , # E C", m E M such that d# : M , -+ N4(,, is one-to-one into, then there is a neighborhood of m in M which is mapped homeomorphically (into) under #. Moreover, if (yl , ..., ye) is a coordinate system at #(m), then a coordinate system at m may be chosen from restrictions of y1 o +, . . . , y eo # [25, p. 791.

If # : M -+ N , # E C", m E M such that d+ : M , -+ N4(,, is onto, then the image under # of every neighborhood of m in M is a neighborhood of #(m) in N . Moreover, if (yl , ...,ye) is a coordinate functions x , + ~ ,..., xd defined in a system at #(m), then there are neighborhood of m such that (yl o #, . . . , y eo #, x ~ , . ..., ~ , xd) is a coordinate system at m [25, p. SO]. Corollary.

c"

# : M -+ N , is a diffeomorphism of an open neighborhood of m onto an open neighborhood of #(m) if and only if d# is an isomorphism onto at m [25, p. SO]. Corollary. (Inverse function theorem for manifolds.) If

# E C", m E M , then #

Corollary. If # : M-+ N , 4 E C", d# = 0 everywhere, and M is connected, then # is constant [25, p. SO]. Problem 11.

Measure zero is a sensible notion on manifolds:

S C M has meczsure zero if for every coordinate map #, # ( S )C Rd has

measure zero. Prove Surd's theorem:

If # : N + M is C", then S = { m E M I m = #(n) for some n such that d# : N , -+ M , is not onto} has measure zero [79]. Problem 12. Prove the following generalization of the first corollary above. Let C be a compact subset of M , # : M -+N , # E C", such that # is one-to-one on C and for every m E C, d&, is one-to-one into. Then there is a neighborhood of C which is mapped homeomorphically under # [83,p . VI-451. Differentials of functions. Every element f of F ( M , m ) gives rise, via its differential, to an element of the dual space M,* of M , as follows: we may identify R,(nl)with R [ a D ( f ( m ) )is identified with a ] , and hence df : M m -+ R,(,,, e R. Note that under this identification, if

13

1.4. Vector Fields

t E M , , then l

Y y m ) - 2 D 1 ( Y f o h,)(O) - 2 D 2 ( Y f 0 h,)(O) D,"(fo h,)(O) 2 D , D d f o h,)(O) D z 2 ( f ch,)(O) YZf(m) 2 X Y f ( m ) - 2D1(Y f 0 h,)(O) - 2D,(Yf 0 h,)(O)

+ + + + + X2f(m) 2 D d X f o M O ) - 2DZ(XfO hl)(O) + Dl"f hl)(O) + 2D,Dl(f hl)(O) + 4Yf hl)(O) -

0

0

+

0

Y2f(m) 2 X Y f ( m ) - 2Y2f(m) - 2 X Y f ( m ) X2f(m) - 2 Y A p ) - 2X2f(m) Y 2 f ( m ) 2 X Y f ( m ) X2f(m) = 2XYf(m) - 2YXf(m). QED

=

+

+

+

+

Problem 20. Let h be a C" curve such that h,(O) = 0, and define g on small positive numbers by g(P) = h(t), and let m = h(0). Show that: (1) T h e map defined on F ( M , m) by f + ( f c h)"(O) is a tangent at m. (2) T h e same tangent is 2 liml+o+g*(t),where the meaning of lim is the same as above. If we call this the 2nd order tangent to h at a point where the 1st order tangent vanishes, generalize (1) so as to define an nth order tangent to h at a point where the 1st order, 2nd order, ..., (n - 1)st order tangents vanish, and prove a result analogous to (2).

20

1. MANIFOLDS

Problem 21. If X is a C" vector field and X ( m ) = 0, show that the integral curve of X starting at m is the constant curve: y ( t ) = m for every t. Hence if y is a C" curve with y*(O) = 0, then y* does not have an extension to a C" vector field unless y is a constant curve. Problem 22. For X as in problem 19 and f = u p show explicitly that the Lie derivative of df with respect to X is d ( X f ) = df.

Use theorem 3 and also theorem 4 to show that [ X , Y ] = 0, where X and Y are as in problem 19. I t is a consequence of the following theorem that if [ X , Y ] = 0 in a neighborhood of m then the broken curves of theorem 4 are actually closed for sufficiently small t, that is, g ( t ) = m for all t near 0. Problem 23.

Let X i , i = 1, ..., e, be C" vector fields defined and linearly independent in a neighborhood of m E M , where M is a , = 0 in that neigh&dimensional manifold, and such that [ X iXj] borhood for every i , j . Then there is a coordinate system (x, , ..., xd) such that Xi coincides with Dz, in the coordinate neighborhood (i e ) and xi(m) = 0, i = 1, ..., d.

Theorem 5.


,

T E Gl(d,K).

Then the map J : g+gI(d, R), given by < J ( X )u, w) = X(e)fi.,w for every v, w , is the isomorphism. Equivalently, we may do as follows: every element v of Rd may be considered as a C" map Gl(d, R ) + Rd if we set v( 7') = T(v).Then for each X E g we get a linear transformation J ( X ) on Rd by defining J ( X )ZI = X(e)v. Then J is a Lie algebra isomorphism J : g + gI(d, R). From now on we make the identification J ( X ) = X , that is, we shall write X v for X ( e ) u . On the other hand, if we consider Gl(d, R ) as matrices, and g1(d, R ) as the space of all d x d matrices, then g may be identified with gr(d, R ) as follows: if { x i j } are the coordinate functions on Gl(d, R ) ,

28

2. LIE GROUPS

that is, xij(g)

=

ijth entry of g as a matrix, then for X

E

g,

Xij =

x(e>(xii>* Problem 1.

(b)

Show that: (a) xii o Lg = C,x,,(g)

dLg(DzhAl))

xpj.

xjk = xjh(g) sik.

(c) If Eii is the matrix with 1 in the i j position, 0 elsewhere, then the left invariant vector field Xij corresponding to Eij is

(d) Use this formula for Xij to verify directly that the brackets are preserved under the identification of g with gI(d, R ) [25]. 2.3 Lie Group-Lie

Algebra Correspondence

Theorem 1. Let G be a Lie group. Then there is a one-one correspondence between the connected Lie subgroups of G and the subalgebras of the Lie algebra of G.

Outline of proof. T h e correspondence is given as follows. If H is a Lie subgroup, then the left invariant vector fields on H can be extended uniquely to left invariant vector fields on G. T h e set of extensions form a subalgebra of the Lie algebra of G which is isomorphic to the Lie algebra of H . T h e existence of the subgroup corresponding to a subalgebra is established by considering maximal integral manifolds of the involutive distribution obtained from the subalgebra. T h e one containing the identity of G is easily seen to be an abstract subgroup of G. I t then follows trivially from problems 28 and 31 of Chapter 1 that it is a Lie subgroup (cf. [25], pp. 107-109). For every Lie algebra the one-dimensional subspace generated by a nonzero element is a subalgebra. Thus we have:

For every X E g, the Lie algebra of G, there is a onedimensional subgroup with Lie algebra generated by X ; that is, there is a curve y : R + G such that Corollary.

+

(1) r ( S t ) = Y ( 4 At) (2) X ( Y ( 4 ) = Y*(+

2.4. Homomorphisms

29

y ( R ) is called the one-parameter subgroup corresponding to X . It is the integral curve of X which passes through e. T h e group of all orthogonal transformations of Rd, O(d), is a Lie subgroup of GZ(d, R). Under the isomorphism J of 2.2 the Lie algebra of O(d) is identified with the subalgebra o(d) of gl(d, R ) consisting of all skew-symmetric transformations. T h e fact that O ( d ) is a Lie group follows from theorem 1 as applied to the component of the identity in O(d),that is, the rotation group R, or SO(d), as a subgroup of the component of the identity of GZ(d, R).

Orthogonal group.

Lemma 1.

Let J : G -+ G by J ( g ) = g-l. Then if t E G,

,

d ] ( t ) = -dR,-, o dL,-,(t).

(R,is right multiplication by g.) Proof. Let D : G -+ G x G by D(g) = (g,g),

/3 : G x

G + G by /3(g, h )

Then /3 o (1 x J ) o D Let t E G, , then

=

= gh.

constant, so d/? o d(1 x J ) o dD

=

0.

0 = dP o d(1 x ]) o d D ( t ) =

dP

0 (dl(t)

= dP(t =

Corollary.

+ dl(t))

dR,-,(t)

If X

+ dJ(t))

E g,

+ dL,(dJ(t))

(theorem 1.2)

QED

then d J X is right invariant.

There is a parallel formulation of the Lie algebra of a Lie group in terms of right invariant vector fields and d J gives the connection with our formulation. 2.4 Homomorphisms

A homomorphism of one Lie group into another is a mapping which is both a homomorphism of the underlying groups and a C" mapping of the underlying manifolds. We assume for the remainder of this section that our groups are connected.

30

2. LIE GROUPS

If j : G + H is a homomorphism and t is a tangent at the identity of G , then it is easily verified that the left invariant vector fields corresponding to t and dj(t), respectively, are j related. Thus j gives rise to a Lie algebra homomorphism dj : g -+ t,. T h e converse of this last result is not true, unless we replace the concept of homomorphism by that of “local homomorphism” in which case the correspondence is one-to-one. [25, p. 1121. However, if G is simply connected, a local homomorphism can be extended to a homomorphism, so we obtain: Let G and H be Lie groups with G simply connected. Then the correspondence j t)d j is one-to-one between homomorphisms of G into H and homomorphisms of g into b.

Theorem 2.

T h e kernel of a homomorphism is a closed normal subgroup, and the kernel of the corresponding Lie algebra homomorphism is an ideal, and it is easily seen that this ideal belongs to the subgroup. Conversely, if H is a closed normal subgroup of G, then the set of left cosets G / H can be given a natural manifold structure in such a way that the projection G -+ G / H is a homomorphism of Lie groups. More generally we have: If H is a Lie subgroup of G , a necessary and sufficient condition for H to be normal is that its Lie algebra f~ be an ideal in the Lie algebra g of G . I f , moreover, H i s closed, the Lie algebra of G / H is naturally isomorphic to the factor algebra g/t, [25, pp. 115 and 1241.

Theorem 3.

If G is simply connected and t, is an ideal of g, the Lie algebra of G , then by Ado’s theorem there is a Lie group having Lie algebra g / t , [42]. Then by theorem 2 there is a homomorphism of G corresponding to the projection g -+g/t,, from which it follows that the normal subgroup H belonging to t, is closed. Thus, if a Lie subgroup of a simply connected group is normal, then it is closed. It is known that a closed subgroup of a Lie group is a Lie group (Cartan’s criterion, [25, p. 1351).

2.5 Exponential Map

Let G be a Lie group, X E ~Let . y x be the integral curve of X starting at the identity. Then the exponential map Q + G is the map

31

2.5. Exponential Map which assigns yx(l) to X ; we write exp t + exp t X is just y x .

X = yx(l).

Clearly the map

FIG. 13.

Commutativity with homomorphisms.

then the diagram

I7

If j : G +H is a homomorphism,

dj

exp

exp

is commutative.

G-H

Proof. This follows immediately from the fact that y d j ( x ) = j o y x for any X E ~ . From this and the following theorem we see that the exponential map gives the correspondence between subalgebras of g and subgroups of G.

Theorem 4. T h e map exp is everywhere C" and in a neighborhood of 0 in g , it is a diffeomorphism.

32

2. LIE GROUPS

Proof. Assuming for the moment that exp is C" we first show that it is a diffeomorphism in a neighborhood of 0. I t is sufficient, according to the inverse function theorem, to prove that d exp is onto at 0. If s E G, , then there is a left invariant vector field X such that X(e) = s. By definition, exp takes the ray generated by X (that is, the curve in g given by r -+ r X ) into the integral curve of X through e . Hence d exp takes the tangent to this ray at 0 into the tangent to the integral curve of X at e, that is, into s. We blame the differentiability of exp on theorems in differential equations which say that solutions depend in a C" manner on parameters entering in a C" manner into the functions defining a system of differential equations (see appendix). I n this case we have the system of differential equations determined by a left invariant vector field X and X depends linearly on a system of linear coordinates of g. QED Let X , , ..., X, be a basis of g; then we identify g with Rd as a manifold by the correspondence CciXi tf (c, , ..., c,). I n a sufficiently small neighborhood U of 0 in g, exp is a diffeomorphism and hence can be taken as a coordinate map. T h e coordinates and coordinate neighborhood obtained in this way are designated as canonical.

For matrix groups there is another exponential map, which turns out to be the same as the one already defined if we identify the Lie algebra of the matrix group with the right subalgebra of the Lie algebra of all matrices of the given order; namely, if A is a d x d matrix we define

M a t r i x exponential.

" 1

eA = z g A k

(Ao= I ) .

0

This series can be shown to converge in the norm induced on the space of d x d matrices through identification with Rd2,and in fact, convergence is uniform on bounded sets. Furthermore, it is evident that if A B = BA, then eA+B= eAeD; in particular, e(s+t)A= esAeLA, s, t real numbers, and eAe-' = eo = I . Thus e A E Gl(d, R ) for every A, and the curve s -+ esAis a one-parameter subgroup; it is easy to see that the tangent at I of this one-parameter I n this subgroup is naturally identified with A = (d/du)(O)(eUA). way it is shown that the two exponential maps are essentially the same.

33

2.5. Exponential Map

The following commutative diagram for Gl(d, R ) with the identification indicated by " w'' will clarify relations.

G4d, R ) Problem 2. By showing that B = (-'/,'-:) has no square root, prove that exp does not map the Lie algebra of SZ(2, R)(= 2 x 2 real matrices with determinant 1) onto ,542, R). Orthogonal group. A canonical coordinate neighborhood of the identity in O(d) is obtained by taking the exponentials of skew-symmetric matrices lying in a sufficiently small neighborhood of 0. T h e coordinates for a skew-symmetric matrix may be taken as the elements of the matrix lying above the main diagonal. Thus the dimension of O(d) is +d(d - 1). For example, O(2) is the one-dimensional group having as neighborhood of the identity the matrices

and this correspondence is a diffeomorphism for 1 B 1

< 7.

Let A E gr(d, R), x E Rd, and define a C" curve u in (exp sA) x. Prove that u*(s) = A (exp sA) x, where tangents to Rd are identified with elements of Rd by the map C a,D,(y) -+ ( a , , ..., ad). Problem 3 .

Rd by

a(s) =

Let ( , ) be the usual inner product on Rd and let o l , be two C" curves in Rd. Then a C" function 5 is defined by U(S) = (ul(s), a,(s)). Using theorem 1.2 or otherwise, prove that Problem 4.

oz

o.'(s) = (01*(4

4)) +

(

4

)

1

u,*(s)>.

Problem 5. If A E gl(d, R ) is skew-symmetric, x E Rd, show that ((expsA) x, (exp $A)x) is a constant function of s, thus equal to (x, x), the value at s = 0; thus exp sA is orthogonal for every s.

Prove this directly using the map e above.

2. LIE GROUPS

34

Define a complex Lie group. Examples are Gl(d, C), and the complex torus P ( C ) = C d / D ,where D is a group generated by 2d real-linearly independent translations of Cd. Problem 6. Cd,

Problem 7. Show that a connected, compact, complex Lie group is Abelian (use a generalization of the maximum modulus theorem)

~31. Let U ( d ) = { g E Gl(d, C ) 1 ggl = I } . Show that U ( d ) is a compact Lie group (the unitary group) but not a complex Lie group. Problem 8.

2.6 Representations

A representation of a Lie group G is a homomorphism of G into a matrix group. A representation of a Lie algebra g is a Lie algebra homomorphism of g into a Lie algebra of matrices. If instead of matrices we use the group, GI(V ) ,[or Lie algebra gI( V ) ]of linear transformations of a vector space V , then we say we have a representation on V . I n neither case do we exclude matrices with complex entries or vector spaces over the complex numbers. A faithful representation is a representation which is an isomorphism into; if a group has a faithful representation it is thus isomorphic to a subgroup of a matrix group. Problem 9. (a) Show t h a t j : C-+ GZ(2,R ) , C = complex numbers, is a faithful representation of the Lie group by j(x i y ) = (;-:) C* of nonzero complex numbers with multiplication as operation. What is the subalgebra of g l ( 2 , R ) corresponding to j(C*) ? What are the one-parameter groups of C * ? (b) Show that the restriction of exp to the Lie algebra of j ( C * ) corresponds to the usual complex exponential function ex+ig = ex ( c o s y i sin y ) and that C with addition as its operation is the simply connected covering group of C* with e* as the covering map. (c) Construct a map 4 : C* -+ S1 x S1 such that &eis) = (cis, 1) and is a Lie group homomorphism which covers S1 x S’. (d) Taking 1, i as the basis of C as a vector space over R, show that j corresponds to “multiplication by,” that is, if $(a ib) = ( a , b ) E R2 then the matrixj(z) acts on x E R2 asj(z) x = $(z$-l(x)).

+

+

+

Problem 10.

Let Q*

=

nonzero quaternions. T h e regular left re-

2.6. Representations

35

presentation of Q* on Q is the representation q3 of Q* on the real vector space Q given by +(q) q’ = qq‘ (+(q) = “left multiplication by 4”).

(a) Show that with respect to basis 1, i, j , k (with the usual multixi y j zk, plication table) q3 has matrix form, for q = w

+ + +

(b) Show that +(q) is orthogonal if and only if IqI =

w2+xZ+y2+22=

1.

(c) Compute det +(q) by showing +(q)+ ( q ) I is a multiple of I and observing what the coefficient of w 4 is in det +(q). Automorphisms. An automorphism of a Lie group G is an isomorphism of G onto itself. The set of all automorphisms of G form a group A . For every j E A we have the automorphism dj of the Lie algebra g of G, and the diagram

is commutative. Since dj is a nonsingular linear transformation of g we have that the map j + dj takes A into the group of linear transformations of g, and it is evidently a homomorphism, since d ( j o k ) = dj o dk. Speaking loosely, we have a representation of A on g. In case G is connected this representation is faithful, a fact which follows from the remarks of 2.4. However, we prove this directly. Suppose j were in the kernel; this means that for every X E ~d j, X = X . However, since j (exp X ) = exp (djX ) , j must then leave the image of exp fixed, and since this contains a neighborhood of the identity, which generates G, j is the identity automorphism. T h e group A is actually a Lie group, but we do not show this here [25, pp. 137 and 1381.

36

2. LIE GROUPS

T h e set of inner automorphisms of G is a subgroup of A , that is, for every x E G we have the inner automorphism j , : y -+ xyx-l. Moreover, the map x - j , is a group homomorphism. The map Ad : G - + GZ(g) defined by Ad(x) = d j , is called the adjoint representation of G. Adjoint representation.

Proposition.

Ad is a representation of G on

g.

Proof. Ad is evidently a group homomorphism so that it suffices to show that it is C", and in fact, to show that it is C" in a canonical coordinate neighborhood. First we note that for a fixed y in G the map x -+j,(y) is C". In fact, it is the composition of maps involving group operations, which are obviously C": x + (x, x) --+ (x, xy-1)

--f

(x, yx-1)

xyx-1.

4

Now for y in a canonical coordinate neighborhood we have y = exp X, and by commutativity, j,(y) = exp(Ad(x) X ) . If we choose a basis X I , ..., X , of g, then Ad(x) is given in terms of a matrix (aii(x)): Ad(x) Xi = C aij(x) Xi. Now for y = exp( tXj) so that the canonical coordinates we get j,(y) = exp (C taij(x) Xi), of j,(y) are taij(x), i = 1, ..., d, this being defined for t sufficiently small. SincejJy) is C" in x this means that aij(x) is C" in x for all i, j , that is, Ad(x) is C". T h e center. T h e center of G is the set of all elements of G each of which commutes with every other element of G. It is clear that the kernel of the group homomorphism x + J , is just the center of G, and since j , -+ dj, is faithful if G is connected, the center is the kernel of the adjoint representation. As such it is a closed subgroup, and being obviously normal, we have that the adjoint induces a faithful representation of the factor group G/(center of G), whenever G is connected. As a corollary to theorem 3.1 we shall see that the differential of the adjoint representation is the adjoint representation of g, which is given by ad(X)Y = [X, yl,that is, d(Ad)(X) is the restriction of the Lie derivative with respect to X to g.

Problem 11. Show that Q* is the direct product of the positive real numbers and S3, giving a polar decomposition of Q* = R* x S 3 . Since R* is in the center of Q* the adjoint representation has R*

37

2.6. Representations

in its kernel and may be considered to have as representation space the tangent space to S3 at the identity 1. Identify that space with the subspace of Q spanned by i, j, k and show that the adjoint representation on S3is then given by $(q) P = qPq-l, P = xi yj zk. (For q E S3, q-l = 6.) Identify the rest of the center of Q* and the image of the adjoint representation on S3as a subgroup of GZ(3, R).

+ +

Problem 12. Let X I , X , , X , , X , be the left invariant vector fields on Q* which are equal to Di , i = 1, 2, 3, 4, at I = (1, 0, 0, 0). (Here we have identified Q with R4.) Show that

Xi

= UiDi

+

~ $ 2

+

~ $ 3

+u~D, 9

and find the corresponding formulas for X , , X , , X,. Problem 13. Show that an Abelian Lie group G has an Abelian Lie algebra (see problem 3.3) and hence by theorem 2, G M Rd/D, where D is a discrete subgroup. Therefore conclude that

G w Re x Tf, e

+ f = d [55, pp. 83-86)].

Problem 14. Show that a continuous homomorphism of a Lie group is C", and hence show that a Lie group structure is a topological group invariant [25, p. 1281. Problem 15. Prove that an integral curve of a left invariant vector field is also an integral curve of a right invariant vector field. Problem 16. Show that there is a neighborhood of the identity in a Lie group which contains no subgroups except the trivial one. Use this, problem 14, and the Peter-Weyl theorem [55, p. 991 to show that every compact Lie group has a faithful (C") representation. Remark on quotient spaces. I n 2.4 we considered quotients of Lie groups by closed normal subgroups. Now let H be a closed subgroup of G. T h e H has the induced topology and the left cosets G/H have a natural manifold structure such that the projection r : G -+ G/H is C", G acts as diffeomorphisms on G/H, and f: G/H -+ R is C" if and only if f o r is C".

CHAPTER 3

Fibre Bundles In the first section transformation groups are discussed and an interpretation of an important special case of the bracket operation is derived. T h e remainder of the chapter is devoted to principal and associated fibre bundles and reduction of the structural group, the treatment being from the point of view of transformation groups, although coordinate bundles are also defined [4, 55, 66, 8.51.

3.1 Transformation Groups

Let G be a Lie group and M a C" manifold. G acts (dzxerentiably) on M to the left if there is a C" map : G x M -+ M , and we write +(g, m ) = g m , satisfying the following conditions:

+

(a) For each g E G , the map g : M is a diffeomorphism. (b) For all g , h E G, m

-+

M , given by g ( m ) = gm,

M , (gh) m = g(hm). G is said to act effectively if g m = m, all m E M , implies that g = e = identity of G. G acts on M to the right if (b) is replaced by: (b)'

+: M

E

For all g , h E G, m E M , (gh) m x G + M in this case.

=

h(gm). We also shall write

Every Lie group acts on itself to the left by left translation and by inner automorphism, while it acts on itself to the right by right translation. If G acts on M , then to every m E M there corresponds a C" map, denoted by m also, of G into M defined by: m(g) = gm. 38

3.1. Transformation Groups

39

G acts transitively to the left if for every m, n E M , there is g E G such that g(m) = n. I n this case, fixing some m E M , let H = {gE G 1 g(m) = m},the isotropy group of m, then H is a closed subgroup of G and the map GIH (left cosets of H) --+M defined b y g H -+ g m is C“, one-to-one, onto. If GIH is compact, for example if G is compact, then this map is a homeomorphism. Example. Gl(d, R ) acts differentiably to the left on Rd and on Rd - (0). T h e action on Rd - (0) is transitive; the isotropy group g), where of (1, 0, ..., 0) consists of matrices of the form (i

B

E

GZ(d - 1, R), A

E Rd-l,

0 is a column of d - 1 zeros. T h e subgroup H may be identified as the semidirectproduct of Gl(d - 1, R ) and Rd-l, that is, the multiplication is given by ( B , A)(B’, A’) = (BB’, AB‘ + A’). [In general, if group G acts as homomorphisms to the right on a group H, then the semidirect product of G and H is given by defining the products as (g, h)(g’, h’) = (gg’, (hg’) h’).] Conversely, if H i s a closed subgroup of G, then G acts transitively on GIH byg(kH) = (gk)H. A space with a transitive group of operators is called homogeneous. If G acts on M , g the Lie algebra of G, then we define a Lie algebra homomorphism h of g into a Lie algebra of vector fields on M , denoted by g, as follows: if X E g, then (hX)(m) = dm(X(e)). We shall also write X = A X . Problem 1.

of

X is elx.

Prove that the one-parameter group of transformations

If G acts effectively, then h is one-to-one. G acts freely if the only element of G having a fixed point on M is the identity, that is, if for some g E G there is m E M such that g m = m, then g = e. If G acts freely, then the elements of are nonvanishing vector fields on M . Moreover, if N is the orbit of m, that is, N = {gm I g E G}, then for every t E N , there is a unique X E g such that 8(m) = t, since m : G 4 M is a diffeomorphism #J of G with N and hence we may take the XEg such that X(e) = d#J-’t. Problem 2 .

Prove that if X(m)

=

0 then etx(m) = m for every t.

3. FIBREBUNDLES

40

T h e differentials of the transformations making up a transformation group G, as they act on g, are given as follows: (a) If G acts to the left, then d g ( 8 ) = Adg X .

--

(h) If G acts to the right, then d g ( x ) = Adg-lX.

Proof of (a). First we compute the composition of g : M-+ M and g-lm : G -+ M : g c g-lm(h) = g(hg-lm) = ghg-lm = m(j,(h)). Thus (dg x ) ( m ) = dg(x(g-lm)) = dg o d(g-lm) X ( e ) = dm c dj,(X(e)) = dm(Adg X ) ( e ) )= A d g X(m).QED Let W be the linear space of C" vector fields on M , and let 9be a finite-dimensional subspace of W invariant under G, that is, for any g E G, d g ( 9 ) C 9. Hence, we have a representation? j of G in the that is, in GZ(9); group of nonsingular linear transformations of 9, and hence, if g I ( 9 ) , the algebra of all linear transformations of 2, is regarded as the Lie algebra of GZ(9), we have the commutative diagram

-1

G

-lexp i

GZ(Y)

T h e following result has many uses. Theorem 1.

particular,

For every X

[X,Y ]is in 9.

E g,

Y E9, dj(X) Y

=

-[x,yl. In

Momentarily assuming this, we state and prove the corollary on the differential of the adjoint representation: Corollary.

ad(X) Y

=

If X , Y ~ [X,Y].

g the ,

Lie algebra of a Lie group G, then

Proof. Let G act to the left on itself as follows: g E G, g : G + G is defined by g(h) = hg-l; that is, the transformation g is just R,-1, right multiplication by g-l. Since Ad g is dL, o dR,-1 , and X is t Here we are assuming the action is a left action. In the case of a right action, obvious modifications are necessary in the definitions, although the statement of the theorem is the same.

3.2. Principal Bundles

41

invariant under left translations dL, for every X E ~we, may apply the theorem with 9 = g and j = Ad. We have to determine, for X E g, the vector field on G. For m E G= M , we have that m : G+M as above is given by m(g) = g(m) = mg-l, so m =L, o J , where J is the inversing map. Thus x ( m ) = dL, o dJ(X(e)) = dL,(-X(e)) = - X ( m ) , by lemma 2.1. Therefore k = - X . Since dj = d(Ad) = ad, the theorem now gives the result. Problem 3 .

Prove as a corollary to this: if G is Abelian, then

g

is

Abelian. Proof of theorem. Assume the action is to the left. Let X Eg, so y : t+ exp t X is a one-parameter group of diffeomorphisms of M ,

so there is a vector field 2 on M which arises from differentiating functions along orbits. For a real-valued function f on M , and m E M we have d Zf(m)= ( O ) ( f @ 0 Y) = dm(r*(O))f,

so Z(m) = dm(X(e)) = X ( m ) . On the other hand, since the diagram above is commutative exp ( t d j ( X ) ) = exp (dj(tX))= jexp ( t X ) .

Thus, dj(X) Y is the derivative at 0 of the curve t -+ ( j exp ( t x ) ) Y = d(exp(tX)) Y in 2 '.But for m E M , (d exp tX(Y))(m) = d(exp(tX))(Y(exp ( - t X ) m)), that is, the curve in M , which we are differentiating is given by the values of Y along the curve t -+ exp ( t ( - X ) ) m pulled back to m by the action of the one-parameter group t + exp(t(-X)). Thus we are taking the Lie derivative with respect to -8 (see 1.4), so by theorem 1.3 the conclusion follows.

3.2 Principal Bundles A (Cm)principal fibre bundle is a set (P, G, M ) , where P, M are C" manifolds, G is a Lie group such that: (1) G acts freely (and differentiably) to the right on P, P x G -+ P. For g E G, we shall also write R, for the map g : P + P. (2) M is the quotient space of P by equivalence under G, and the projection 7~ : P+ M is C", so for m E M , G is simply transitive on r-l(m).

42

3. FIBREBUNDLES

(3) P i s locally trivial, that is, for any m E M , there is a neighborhood U of m and C" map F , : n-l( U )-+ G such that F , commutes with R, for every g E G and the map of n - I ( U ) + U x G given by p -+ ( ~ ( p F,(p)) ), is a diffeomorphism.

G

U

FIG.14.

P is called the bundle space, M the base space, and G the structural group. For m E M , d ( m ) is called the jibre over m. T h e fibres are diffeomorphic to G, in a special way via the map p : G +r - I ( n ( p ) ) C P, defined by p ( g ) = R,p. We note that in terms of the F , , the right action of G on P is given by right translation, that is, if p 4( ~ ( p , ) F,(p)), then pg + (n(p),F&) g ) . This follows from the fact that F d P d = F d P )g . We now give some examples of bundles. If G is a Lie group, M a manifold, then M x G provided with the right action of G on itself in the second factor, that is, ( m , g) h = ( m , gh), is the bundle space of a principal bundle, the trivial bundle. A bundle is isomorphic to a trivial bundle if and only if there is a C" cross section of n, that is, a C" map K : M - t P such that n o K is the identity on M (cf. [85], pp. 25 and 36). Trivial (product) bundle.

P

=

3.2. Principal Bundles

43

Bundle of bases. Let M be a C" manifold and B ( M ) the set of (df 1)tuples (m, el , ..., ed), where m E M and el , ..., ed is a basis of M,, and let r : B ( M ) + M be given by r ( m , e l , ..., ed) = rn. T h e n GZ(d, R ) acts to the right on B ( M ) by: let g E GZ(d, R), viewed as a matrix, g = (gij); let (m, el , ..., ed) E B ( M ) , and define R,(m, el, ..., ed) = ( m , C gilei , ..., C giaei). If m E M , (xl , ..., x d ) a coordinate system defined in a neighborhood U of m, then we define F , by: if m' E U , Fu(m',fl , ...,fa)= (dxj(fi)) = (gij) E GZ(d, R). Thus the functions yi = xi o 7-r and yii = xii o F , give a coordinate system on r - l ( U ) , where xii are the standard coordinates on Gl(d, R ) (see 2.2). Using the C" structure given to B ( M ) by the local product representation (IT, F,), we see that B ( M ) is the bundle space of a principal bundle, called the bundle of bases of M . It is sometimes convenient to view B ( M ) as the set of nonsingular linear transformations of Rd into the tangent spaces of M , that is, we identify p = (m, el, ..., ea) with the map p : ( r l , ..., r d ) -+c riei. When this is done it is natural to consider GZ(d, R ) as the nonsingular linear transformations of Rd, for we have: pg(rl, ..., r d ) =

that is, pg (as a map)

=p

(as a map) o g.

b = (m, e, , ..., ed) E B ( M ) is in the coordinate neighborhood r-l( U ) show that dr(D,$(b)) = Cj yii-l(b)ej , where (yii-l(b)) is the inverse of the matrix (yii(b)). Problem 4. If

Homogeneous spaces. If G is a Lie group, H a closed subgroup, then there is a principal bundle with base space G / H (left cosets), bundle space G, and structure group H such that IT : G + G / H is the canonical map and the right action is given by (g, h) -+gh (see [85], p. 33).

(1) Let R - (0) = R* act on Rd+l- (0) by scalar multiplication. Then this action is differentiable, free, and simply transitive on orbits. T h e orbit space is Pd, d-dimensional projective space, so is a principal bundle. ( R d f l- {0}, R*, P d ) Examples

3. FIBREBUNDLES

44

( 2 ) If we use the positive reals R+ instead of R* we get (Rd+l - { 0 } , R+,S d )

is a principal bundle.

(3) If we use C* and Cd+l - {0), C = complex numbers, then we get complex projective space CPd as base space: (Cd++'- {0}, C*, C P d )

is a principal bundle.

Problem 5. Let (P, G, M ) be a principal fibre bundle with P connected. Then if Go is the component of the identity in G, there is a unique principal fibre bundle (P, G o ,h?) such that the action of Go on P is the same and h? is a connected covering space of M (cf. examples 1 and 2 above). We give an alternate approach to principal fibre bundles. Principal Coordinate bundles. Let (P,G, M ) be a principal bundle, and be an open covering of M such that wl( U i ) can be represented let { Ui} ( i )-+ G. For i, j such that as a product space via the function Fi: ~ - l U Ui n U j # 0, we define a map Gji : U, n Uj-+ G as follows: if m E Ui n U j , let p E n-l(m), and put Gji(m) = Fj($)(Fi(p))-l. Gji measures how much the cross section over Uicorresponding to U, x {e} under the product structure given by Fi differs from the cross section over U j determined similarly by F j . We want to show that the definition of G j i ( m )is independent of the choice of p . If p' E .rr-l(m) also, then since r - l ( m ) is the orbit of p , there is g E G such that p'=pg, so we have

F,(P')FZ(P')F1= P,F;g) F,(Pg)-l = FAP) F i ( P Y ,

= F,(P) g(Fi(P)g)-l

as desired. T h e functions satisfy the further properties: G,,(m)

=

G,,(m) Gji(m)

for

mE

U in U j n U,.

(*I

T h e functions Gji are called the transition functions corresponding to the covering { Ui},and in fact, with this covering they define a principal coordinate bundle in the sense of Steenrod. Hence, our definition of bundle is an equivalence class of coordinate bundles, which is another way of saying that principal coordinate bundles are equivalent if and only if their right actions agree. Therefore, we have, by [85, p. 141, that any set of functions Gii defined for a covering { U i } ,

3.3. Associated Bundles

45

satisfying (*), uniquely determine a principal bundle whose transition functions relative to the covering { Ui}are the Gji. Problem 6 . If 4 : N - t M is a C" map, ( P , G, M ) a principal fibre bundle, then let P = {(n, p ) E N x P 1 +(n) = ~ ( p ) } . (a) Show that 1? is a submanifold of N x P under the inclusion map. (b) Show that ( P , G, N ) becomes a principal fibre bundle if we define right action by (n, p ) g = (n, p g ) . ( P , G, N ) is called the bundle induced by and (P, G, M ) .

P N

P

M

(c) Show that transition functions for (P, G, N ) may be taken as G,, o 4, so that we could define the coordinate bundle directly. The vector fields of 8. If (P, G, M ) is a principal bundle, then since G acts freely and effectively we have an isomorphism A : g + 8 = Lie algebra of vector fields on P, which consists of nonvanishing vector fields. By remark (6) in 3.1 we have dR,(h(X)) = h(Ad(g-l) X ) , where g E G, X E g.

For the real coordinates u l , ..., uZd+, on Cdfl - {O} ({z, = uzjPl iuZj}is the dual basis to the standard complex basis of Cd+') find the expression for h(aXl + bX,), where X , and X , are the left invariant vector fields on C*: X, = ulDl u2D2, X , = -u,D, ulD, and h is the isomorphism associated with (Cd+l - {O}, c*,CPd). Problem 7.

+

+

+

3.3 Associated Bundles

Let (P,G, M ) be a principal fibre bundle, and let F be a manifold on which G acts to the left, We define the jibre bundle associated to

46

3. FIBRE BUNDLES

(P, G, M ) with Jibre F (it also depends on the action) as follows. Let B’ = P x F , and consider the right action of G on B’ defined by ( p , f )g = (pg, g-lj), where p E P, f~ F, g E G. Let B = B‘/G, the quotient space under equivalence by G, then B is the bundle space of the associated fibre bundle. We have the following structure. . T h e projection 7r’: B -+ M is defined by: ~ ’ ( ( pf, ) G) = ~ ( p ) If m E M , take U a neighborhood of m as in 3.2(3), with F , : T-’( U )-+ G. Then we have F,’: r’-l(U )+F given by

so that 7 r t - l ( U ) is homeomorphic to the product U x F , and hence we define B as a manifold by requiring these homeomorphisms to be diffeomorphisms. Note that now 7r’ E C“, and also the natural projection B’ -+ B is C“. Associated coordinate bundle. If we define transition functions Gji‘ for the associated bundle ( B , F , G, M ) analogously to the definition in the principal bundle case, we have, for a covering { Ui} admitting functions Fit, for m E Ui n U j , (p,f)G E 7rt-l(m):

Gji’(m) = F j ( ~ ) ( F i ( p ) ) -= l Gji(m).

We therefore have that ( B , F , G, M ) is the equivalence class of the coordinate bundle associated to the principal coordinate bundle defined by the transition functions Gji, in the sense of Steenrod [85, Part I, $91. Examples

(1) Let (P, G, M ) be as above, and let G act on itself by left translation. Then (P, G, M ) is the bundle associated to itself with fibre G.

(2) Tangent bundle. Consider the bundle of bases B (M). (Note that we shall often denote a bundle by its bundle space alone.) Now by definition GZ(d, R ) is the group of nonsingular linear transformations of Rd, and hence acts on Rd to the left. The bundle space of the associated bundle with fibre Rd is denoted by T ( M ) , and the bundle is called the tangent bundle to M . T ( M ) can be identified with the space of all pairs ( m , t ) , where m E M , t E M , [the “m” in the pair

3.3. Associated Bundles

+

47

is actually redundant, as it is in the ( d 1)-tuples making up B ( M ) , but is inserted for convenience], as follows: ( ( m ,el

7 *'*I

4,( y 1 ... 1

1

Td))

G W ,R)

-

(m,

z

yiez),

or, if we regard B ( M ) as the set of maps p : Rd +M m ,then for x E Rd this identification is ( p , x ) GZ(d, R ) --t (m,p x ) , where m = ~ ( p ) . With this later formulation it is easy to see that the identification is well defined, for if (p', x') GZ(d, R ) = ( p , x) GZ(d, R), then there is g E Gl(d, R ) such that p' = pg, x' = g-lx, and hence, p'x' = (pg) (g-lx) = p x , since as a map, pg is the composition of p and g. Hence we may view the fibre of T ( M ) above m E M as the linear space of tangents at m, that is, as M m , and T ( M ) itself as the union of all the tangent spaces supplied with a manifold structure. Furthermore, under this identification, the coordinates of T ( M ) may be easily exhibited; namely, let U be a coordinate neighborhood in M , with coordinates x l , ..., xd. We define coordinates y l , ..., Y2d on rr'-l( U ) as follows: if (m, t ) E t-r-l( U ) ,

Y d S h

t ) = d%(t)

A C" vector field may then be regarded as a cross section of T ' . I n particular, the trivial vector field (values all 0) gives an imbedding of M as a submanifold of T ( M ) . Problem 8. Prove that if y is a C" curve in M , then y* is a C" curve in T ( M ) . (3) Tensor bundles. When we replace Rd in example ( 2 ) by a vector space constructed from Rd via multilinear algebra, that is, the tensor product of Rd and its dual with various multiplicities or an invariant subspace thereof, we get a tensor bundle. A cross section which is C" on an open set is called a C" tensor field, and is given type numbers according to the number of times Rd and its dual occur. T h e group of a tensor bundle is Gl(d, R ) ; it acts on the factors of the tensor product independently, and on Rd as with the tangent bundle, on the dual via the transpose of the inverse: if v E the dual, X E Rd, g E GZ(d, R), then g v ( x ) = v(g-'x). Frequently the bundle B ( M ) has its structural group reduced to a subgroup (see 3.4) in which case more subspaces of tensor products of Rd and its dual may become invariant, leading to tensor bundles

48

3. FIBREBUNDLES

of different sorts. For example, this is the case when M has an almost complex structure (cf. problem 11). (4) Vector bundles. These are bundles in which the fibre is a vector space and the structural group is a subgroup of the general linear group of that vector space. T h e tensor bundles are a special case. They are frequently defined with no explicit mention made of the structural group by giving the bundle space as the union of vector spaces, all of the same dimension, each associated to an element of the base space, and defining the manifold structure via smooth, linearly independent, spanning cross sections over a covering system of coordinate neighborhoods. For example, we essentially did this for T ( M ) when we exhibited the coordinates: in that case the cross sections were the coordinate vector fields D,*. Another example of this type is the quotient space bundle of an imbedding, which usually is considered in the case of Riemannian manifolds as the normal bundle (the Riemannian metric, defined in Chapter 7, is employed to get a uniform choice of representatives for the quotient spaces). This vector bundle may be defined as follows: Let i : N+ M be the imbedding of the submanifold N in M . T h e fibre over n E N is the quotient space Mi(Jdi(Nn), and the bundle space is the union of these fibres, so we may consider the bundle space di(N%)), where t E Mi(%). T o get as the collection of pairs (n, t coordinates we first take a coordinate system in M at i(n), say xl, ..., xd, and we may assume that xj o i = yi, j = 1, ..., d‘ give a coordinate system at n, and that Xj(n) = D,.i(n)) di(Nn) are 1, ..., d. Then for some neighborlinearly independent for j = d‘ 1, ..., d are still linearly independent, hood U of n, Xj(n’), j = d‘ so there is a dual basis Vi(n’). Then for (n‘, X ) E ~’-l( U ) , where n’ is the projection from the bundle space to N , we define coordinates z1, ..., zd by:

+

+

+

+

zj(n’, X ) = yj(n’)

if

j = 1,

zj(n’, X)= Vj(n’)(X)

if

j = d’

..., d‘

+ 1, ..., d.

T h e group of this bundle may be taken to be Gl(d - d‘, R). ( 5 ) Grassmann bundles. T h e set of e-dimensional subspaces of Rd can be given a manifold structure so that Gl(d, R ) acts in the obvious manner as a differentiable transformation group on the left. T h e bundles associated to B ( M ) by this action are called the (unoriented)

3.4. Reduction of the Structural Group

49

Grassmann bundles of M ; the bundle space may be regarded as the collection of e-planes in the tangent spaces of M . A C" e-dimensional distribution is a C" cross section of this bundle. The action of a principal bundle on an associated fibre. Let (P, G, M ) be a principal bundle and B an associated bundle with fibre F. Then the quotient projection P x F + B defines, by restriction of the first variable, for every p E P a C" map p :F -+ B, namely, p(f) = (p, f ) G. I t satisfies p(gf) = (pg) f for every g E G. We have already seen this in the case of B ( M ) and T ( M ) (cf. [85], Part I, $8.9). Remark. An associated bundle is trivial if its principal bundle is trivial. This is not equivalent to the existence of a cross section of the associated bundle, but implies the existence of a family of cross sections with pairwise disjoint ranges which fill the associated bundle. Problem 9 . Verify the above, and also show that for a vector bundle with fibre Re triviality is equivalent to the existence of e cross sections linearly independent at each point. It is well known [85] that the tangent bundle to a differentiable manifold admits a nonzero cross section if and only if the Euler characteristic is zero; for example, if the manifold is compact and odd-dimensional. Hence all odd-dimensional spheres have such cross sections, although it is a deep result of Milnor and Kervaire [I41 that only S1,S3,S7 have trivial tangent bundles (i. e., are parallelizable).

3.4 Reduction of the Structural Group

Let (P,G, M ) be a principal bundle. We assume G is separable. Let H be a subgroup of G, then in the sense of Steenrod, the structural group G is reducible to the subgroup H if and only if there exists a coordinate bundle in the equivalence class determined by ( P , G, M ) whose transition functions take their values in H , that is, if and only if there exists a covering { Ui}whose Gji satisfy Gji(Uin U j )C H . In terms of the right action P x G - t P, this definition can be formulated in the following way ([66], p. 20). Let (P, G, M ) , (P', G', M ' ) be principal bundles. A bundle map f : ( P , G, M ) -+(P', G', M ' ) is a set of C" maps (fp,fc,fM) between the obvious pairs, such that fc is a homomorphism, and the following

3. FIBREBUNDLES

50 relations are satisfied:

fiMcx=7r'0fp

f p o R,

=

RfG(g, of p

for every

g E G.

Now we may state the second definition as a theorem. If (P, G, M ) is a principal bundle, H a subgroup of G, then G is reducible to H if and only if there exists a principal bundle (P', H , M ) which admits a bundle map f : (P',H , M ) +(P, G , M ) such that f M is the identity on M , f p is one-to-one, and f G is the inclusion map H C G (Proof omitted.) Theorem 2.

Steenrod proves [85, p. 591 that if (P, G, M ) is a principal bundle, H a maximal compact subgroup of G, then G can be reduced to a

bundle with structural group N.In particular, every principal bundle with Gl(d,R ) as structural group [for example, B(M)]can be reduced to a bundle with structural group the orthogonal group O(d). We shall return to this when we consider Riemannian metrics on a manifold. T h e reduction of a principal bundle induces the reduction of associated bundles, in an obvious sense, since we have given the definition in terms of transition functions only. Problem 10.

Complex manifolds. Let M be a complex manifold,

9 = complex valued differentiable functions defined on a neighga= conborhood of m E M , F a = holomorphic functions in F, jugates of functions in flu (conjugate holomorphic functions),

F T= real valued functions in F, .Ym= complex linear derivations d mcomplex linear extensions of Mm , gm= the annihilator of 9, of F a in Tm. For t E F m , f E 9 , define if = if. Show that

+

(direct sum). (a) Y m= d m i d m (b) f E TwL for every t E Ym. (c) d m = all t E Y m such that tFTC R. (d) Sm= the annihilator of Fa. (e) t = f if and only if t E Am.

(f) Z mn iFm = 0. (g)

y m = *m

+ 2,.

3.4. Reduction of the Structural Group

51

(h) If t E d m, then the decomposition of t from (g) is of the form t = h h, where h E Z v l ,i.e., d m = {h I h €Zm}. This defines a one-to-one real linear map P : d m +Z m , t --t h.

+

+

Let j : Y m -+ Y mbe multiplication by i. Define J = P-l jP. (i) J z = -identity.

(j) Compute J in terms of real coordinate vector fields which come from the real and imaginary parts of a complex coordinate system.

(k) J is defined on T ( M ) and is a bundle map. An almost complex structure on a manifold M is a bundle map

J : T ( M ) + T ( M ) such that

(1) J ( M m ) = M , for every m E M . (2) J z

=

-identity

on each Mm.

T h e J obtained in the above problem is called the complex structure of the complex manifold M . An almost complex structure will be called a complex structure only if it is obtained in this way. Problem 11. (a) If M has an almost complex structure, then M has even dimension.

(b) M has an almost complex structure if and only if the group of the bundle of bases can be reduced to Gl(d/2, C) represented in (which corresponds to Gl(d, R) as matrices of the form (g ):( A iB)E Gl(d/2, C)). Every 2-dimensional orientable differentiable manifold admits a complex structure, so, of course, every 2- dimensional manifold which admits an almost complex structure admits a complex structure. T h e latter result is not true for higher dimensional manifolds [21]. If M has an almost complex structure then Mm iMnlhas a direct iJx for sum decomposition Z m Slnsuch that H m= all x x E Mm. We say that C" vector field X belongs to Z if X ( m ) E Z m for every m, and then write X E 2. I t is not in general true that if X E 8, Y E H then [X, Y ]E 8. However, if J is a complex structure the X v l agrees with our previous definition and [Z,Z]c 2 ;in fact, this latter condition has been shown to be also a sufficient condition that J be a complex structure [64].

+

+

=

+

T h e maximal compact subgroups of R* and C* are So and {eie} = S1,respectively. Show that the reduction

Problem 12.

(1, -1}

+

52

3. FIBREBUNDLES

of the principal bundles of examples (1) and (3) in 3.2 to these subgroups gives principal bundles: ( S d ,So,P d ) and (SZd+l, S1,CP). Carry out the same construction to get principal bundles over quaternionic projective spaces QPd: (S4d+3, S3,QPd). In the case d = 1 the bundle projections become the Hopf maps S3 -+ S 2 = CP1, S7+ S4 = QP'.

CHAPTER 4

Differential Forms Differential forms are defined via Grassmann algebras, and the intrinsic formula for the exterior derivative is derived. Frobenius' theorem, vector-valued forms, and forms on complex manifolds are also discussed [24,25,29,33,36,66,831. For other topics, particularly the use of differential forms in the study of topological invariants, the reader is referred to [12, 30, 781. 4.1 Introduction

In the last chapter the concept of a tensor was briefly mentioned, and differential forms, the subject of the present chapter are just special types of tensors. However, we shall initially introduce differential forms here by means of the more explicit formulation in terms of Grassmann algebras and shall then return briefly to the tensorial approach (4.5). I f f is a C" function on M , then we notice that to every m E M there corresponds the differential of f at m, 1.3, which is a linear functional on M , , and this correspondence is smooth in the following sense. Let X be a C" vector field on M , then df(X)(m)= df,X(m) =Xf(m) defines a C" function Xf on M . Such a smooth assignment of linear functions is called a differential 1-form, although not every differential I-form arises as the differential of a C" function. However, before pursuing this subject further, we must develop some machinery, namely, Grassmann algebras. 4.2 Classical Notion of Differential Form

A differential form at m is something which in terms of a coordinate system can be expressed as C ail,,,i, dxi, dxiy, where the summation 53

4. DIFFERENTIAL FORMS

54

is over all ordered subsets (il, ..., i,) of (1, ..., d } , and the ail,,,i, are real numbers. So a strict definition will involve some kind of multiplication of differentials and then linear combination of these products. Thus M,* will be imbedded in an algebraic system which has both multiplication and vector operations. Furthermore, this shall be done so that if (yl, ...,y d )is a second coordinate system at m then the expression C bil.,.ipdyi;.. dyiD,obtained from the other by the usual rules for change of variable in multiple integrals, will be just the expansion of the same algebraic object in terms of the new basis. T h e required algebraic system for this is a Grassmann algebra, which we now discuss.

4.3 Grassmann Algebras

Let F be a field and V a finite-dimensional vector space over F of dimension d . A Grassmann algebra over V is a set G such that (i) G is an associative algebra with identity e over F. (ii) G contains V . (iii) Every element v E V satisfies v2 = 0. (iv) G has dimension 2d. (v) G is generated by e and V , that is, every element of G is a sum of products of scalar multiples of e and elements of V . Notice that e is not in V because e2 = e # 0, while by (iii) v2 = 0 for all v E V . Also, if u, v E V , then uv = -vu. This is shown by the standard polarization trick: 0

=

(u

+ V)Z = u2 + + vu + v2 = uv + vu. 11v

Property (iv) is a short but poor way of stating that there are no more relations among the elements of G than those which follow from (iii). T o each basis e l , ..., ed of V there corresponds a basis of G . T h e elements of this basis of G are in one-to-one correspondence with the subsets of {I, ..., d } as follows: (a) If the subset is+ = empty set, we let e4 = e . (b) If the subset is s = {il , ..., i,} with i, e, = e, 1 ..- eig*


= P.

Proof. We may assume that y is C" since we may chain the lifts of pieces together, in fact, in only one way. Extend y to a C" map of ( -E, 1 + 6 ) = I into M . Then by problem 3.6, N = { ( r , q ) E Ix P I y ( r ) = n(q)} is the bundle space of a principal bundle ( N , G, T ' , I ) . The map 0 : N -+ P by B(r, q ) = q may

78

5. CONNEXIONS

be used to get a connexion on N ; the 1-form of this connexion is = 8*+ (see problem 3 below).

6

FIG. 17.

Let X be the unique horizontal lift of D on I to N , and let u" be the integral curve of X starting at (0, p ) . T h e domain of u" is all of I since it could be extended about a neighborhood of the upper limit otherwise. We define 7 = 8 o u". It is obvious that 9 is a lift of y , and since +(f*) = +(do o u"*) = +(X)= 0, 7 is horizontal. That 7 is unique follows easily from the fact that any lift can be factored through N via 0 and a lift of u, but the latter is obviously unique. QED Corollary 1. If H is a connexion on (P, G, M ) , y a curve in M as in the theorem, then there is defined a diffeomorphism T, of n-l(y(0)) onto r-l(y( l)), called parallel trailslution from y(0) to y( 1) along y. T,, is independent of the parametrization of y and satisfies T, o R, = R, o T, , all g E G. Further, if y and a are two such curves with o(0) = y(l), then T,, = T, o T,.

Let p E n-l(y(O)),f be the lift of y with P(0) = p , and define T,(p) = 7(1). We use the right invariance property, which is trivial, to prove T, E C" and T, is a diffeomorphism: if po is any fixed element of n-l(r(O)), Pl = T,(Po), then T,(Pog) = T,(Po)g, so T , = p , o pop1: n-'(y(O)) + G + r-l(y( 1)). Proof.

5.2. Parallel Translation

79

T h e other properties are immediate. We remark that a concept of parallel translation is equivalent to a connexion; namely, if we have parallel translation in P given and satisfying right invariance and certain smoothness conditions-specifically, tangent curves give coincident infinitesimal transformations-then the connexion may be recovered by differentiating as follows: let y be an appropriate curve in M , p E r-l(y(O)), and let t E P, such that d r t = y*(O). Then define Ht = T,(p),(O),where we view T,(p) as the curve t 4 parallel translate of p from y(0) to y ( t ) along y. T h e following result provides an interesting interpretation of the connexion form. Let y, 7 be as in theorem2, and let T be any other lift (not necessarily horizontal). Define 01 : I+ G by: a(r) is the unique element of G such that 7(r) = T ( ~ ) O I ( T ) . Then d R ( a ( r ) - ~ ) ~= * ( r- $) ( ~ * ( r ) ) . (We are identifying G, with g by left translation.)

Corollary 2.

P

FIG. 18.

Proof. Let I,: G ---f P by I,(g) = ~ ( rg.) It is trivial to verify that I , o L, = ~ ( rg): G + P. By theorem 1.2, where the map P x G + P is right action, we have

?*(r) == dIra*(r)

+ dKa(r)

(1)

5. CONNEXIONS

80 Now

) ) dLa(,t-l(a*(r)),recalling our identification Therefore, ~ $ ( d I ~ a * ( r= of G, and g. ) , the O n the other hand, 4(dRacr,7*(r))= Ad a(r)-l d ( ~ * ( r ) by equivariance of 4. Hence, since is horizontal, applying 4 to (1) gives

~L,c,t-+*(r))

=

-

Ad

‘Y(r)-l

+b*(r))

Problem 3 . Let ( f B ,f , ,f M ) : ( B , G, M)-+(B’, G’, M’) be a bundle map with df, : g + g’ an isomorphism onto. Show that any connexion on B’ induces in a natural way a connexion on B. In particular, if f:M -+ M’ and ( B , G’, M ) is the bundle induced by f over M , then this will be the case.

5.3 Curvature Form and the Structural Equation

Define, for a form w on P, the form Dw by Dw = dw o H .

where H is a connexion. More precisely, if w is a p-form, then for Note that t , , ..., tptl E P, , Dw(tl , ..., t,+,) = dw(Ht, , ..., Ht,,,). Dw is always horizontal. T h e curvature form @ of a connexion H with l-form4 is the horizontal g-valued 2-form D+.It is easy to verify that @ is equivariant. We need the following lemma for the proof of Cartan’s structural equation. Lemma 2.

horizontal.

If

X E g, V a horizontal vector field on P, then

[x,V ]is

5.3. Curvature Form and the Structural Equation

81

Proof. We cannot apply theorem 3.1 directly, since the horizontal vector fields do not form a finite-dimensional vector space. However, we may derive the result indirectly as follows. Let V, be a right invariant horizontal vector field, that is, a horizontal lift of a vector field on M . Then, taking 9 = {Vi> in theorem 3.1, we have V,] = 0. Now locally we may write V = Z i f i V i , where the Vi are horizontal and right invariant and the fi are C“ functions on P . Then V ] = X i ( z f i ) V , , by problem 1.14, which is certainly horizontal.

[x,

[x,

i(x)

Problem 4. Applying the formula L8 = d + di(x) to the connexion form and evaluating on the horizontal vector field V , show that +( VJ)= 0, thus giving an alternate proof of the lemma. [Hint: Notice that the one-parameter group of transformations associated with is right action by e“. Hence LR(+) is vertical.]

+ [x, x

(structural equation). If on P, @ its curvature form, then

Theorem 3

d+ =

-

4 is the

1-form of a connexion

Q [+,41 + 0-

Notice that if G is a matrix group and 9 is identified with a space of linear transformations, then - &[+,+] = +. We show that the above 2-forms applied to vector fields X , Y on P agree. Since forms are linear, we need only consider the cases for which the X,Y either belong to fi or are horizontal, in all comProof.

binations. (i) X , Y Eg, so there are elements X’, Y’ E g such that = X, P’ = Y , and hence +(X)= X’, +(Y)= Y’. Now by theorem 4.2 W X , Y ) = X + ( Y )- Y+V)- +([X,YI) =

X ( Y ’ ) - Y ( X ’ ) - [X’, Y’]

=

-

=

-

H+>+l(X,Y )

Q [+,+ l W , Y ) + @ ( X ,Y ) ,

(4.9)

as desired, since @ is horizontal. Note that X ( Y ’ ) = X(constant g-valued function on P ) = 0, and Y ( X ’ ) = 0 similarly. (ii) X E g, Y horizontal. Let X‘ E g be as in (i): &(X, Y ) = X + ( Y ) - Y+(W - N X ,YI)= 0,

5. CONNEXIONS

82

since + ( Y )= 0, + ( X ) = constant, and [X,Y ] is horizontal by the lemma. On the other hand, @ is horizontal, so @ ( X , Y ) = 0 as X is vertical, and [+,+ ] ( X ,Y ) = 0 since Y is horizontal. (iii) X,Y both horizontal. + ( X ) = 0, +( Y ) = 0, and @ ( X ,Y ) = d4(HX, H Y ) = d&Y, Y),

which is the structural equation in this case. QED Remark. T h e restriction of the structural equation to vertical vectors is essentially the equation of Maurer-Cartan. Another interpretation is that it says that d+ has only a horizontal and a vertical part with no mixed part.

(the Bianchi identity). If @ is the curvature form of a connexion on a principal bundle P, then

Theorem 4

D@ Proof.

=

0.

From the structural equation,

D@

=

D db, - D[+, 43.

Now Dd+(X,, X , , X,) = ddcj(HX,, H X , , HX,) = 0, since d2=0. Also, D[+,+] = 0, since [+,#I is a vertical 2-form, and so vanishes when one of its entries is horizontal. Hence, D@ = 0. Let H be a connexion on P, @ its curvature form. Then @ = 0 if and only if H is an involutive distribution, which in view of theorem 1.6, means that P admits local horizontal cross sections. In particular, if M is simply connected, then by a standard monodromy argument P must be the trivial bundle. A connexion with @ = 0 is called p a t . Theorem 5.

Proof.

If X , Y are horizontal vector fields on P, then

@(Z Y) = 4 ( X , Y ) = X W ) - Y 4 V ) - N X , Yl) =

-MX, Yl);

so H is involutive if and only if [ X , yl is horizontal if and only if @ ( X , Y ) = - + ( [ X , yl) = 0, which implies @ = 0, as asserted. Problem 5. Let H be a closed subgroup of a Lie group G and consider the principal bundle (G, H , GIH). Let C$ be a connexion and show

5.4. Existence of Connexions

83

that 4 o dR, = dR, o 4. Now assume 4 is invariant under dL, for every g E G. Show that (a) 4 defines a projection$: g -+ @ of the corresponding Lie algebras, (b) if m = ker$, then [m, $1 C m, (c) conversely, if g = m ij (direct sum) with [m, $1 C m, then the projection of g onto $ gives rise to an invariant connexion in the above sense. Hence, there is a one-to-one correspondence between invariant connexions on (G, H , GIH) and reductive complements m of $. An H admitting such an invariant complement m is called reductive in G (the name arises from the fact that the adjoint representation of G restricted to H is reducible to the adjoint representation of H plus the representation of H on m via Ad,, at least if H i s connected). (d) For the connexion 4, show that the curvature form may be considered as defined on g x g, and derive a formula for it.

+

5.4 Existence of Connexions and Connexions in Associated Bundles Existence of Connexions.

C" connexions exist in abundance. I n Chapter

7 we shall establish the existence of Riemannian connexions on B ( M ) . Here we show that any principal bundle (P, G, M ) , with M

paracompact, has a connexion. Let {Ui} be a covering of M such that n--l(Ui) is trivial. Let {fi}be a C" partition of unity subordinate to the covering { Ui}. Let +i be a flat connexion on n--'( Ui) and define 4 = X(fio n-) bi.4 is a not necessarily flat connexion form on P. Problem 6 .

Verify that 4 is a connexion form.

Remark. If (P, G, n-, M ) is a complex analytic principal bundle over a complex manifold M , it of course admits C" connexions, but in general it will not admit a complex analytic connexion. A necessary condition in a special case is given in [3]. However, real-analytic manifolds admit analytic connexions, although the proof is much more difficult. See the remark following theorem 7.2.

Let (P, G, M ) be a principal bundle with a connexion H , and let ( B , G, F , M ) be an associated bundle with fibre

Associated Bundles.

84

5. CONNEXIONS

F (see 3.3). Then in some sense H induces a “connexion” on B. To be precise, there is a distribution H‘ on B which at each point complements the vertical tangent space. Further, there is a notion of parallel translation of the fibres of B , which derives as before from the horizontal lifts of curves. Let y be a broken C“ curve in M , b E ~ ’ - l ( y ( O ) ) . We define a lifting 7 of y into B which will turn out to be horizontal in the sense below. Let f E F and p E P be such that ~ ( p= ) y(0) and pf = 6 , wherep is here the map defined in 3.3. By theorem 2 there is a horizontal lifting y of y into P with y(0) = p . Now define 7 by p(t) = y ( t )f . We then define parallel translation T, along y from n’-l(y(O>) to ~’-l(y(1)) as in P. Hence we have ?‘, = T,(p) o p - l , so parallel translation is a diffeomorphism. Parallel Translation.

The Distribution H’. Take b E B , p E P such that ~ ’ ( b= ) ~ ( p ) We . may view Pp as a subspace of ( P x F ) ( p , f ,, where f E F is such that pf = b. Let A : P x F + B be the natural map (3.3), and define H i = dA(H,). This definition is independent of p in view of the right invariance of H , while it is clear that the lift defined above is horizontal with respect to H ’ , if the definition of the map

p :F + r ’ - l ( r ( ~ ) ) is recalled. Problem 7. Let 4, y5 be connexion forms, H , K their connexions, H’, K’ the corresponding distributions on B. Show that if s, t E B, , H’s = s, K’t = t , and dn’(s)= dr’(t), then rs (1 - r ) t is in the (1 - r ) y5. distribution on B of the connexion belonging to r+

+

Problem 8.

to R.

+

Determine all connexions on T ( R ) ,the tangent bundle

Problem 9 . Show that there exist horizontal distributions on associated bundles which are not connexions. [Hint: Take T(R ) w R2 and define a distribution with slope eu.]

5.5 Structural Equations for Horizontal Forms

We first prove a basic lemma. Let G be a Lie group of diffeomorphisms of M , G x M - t M ,

5.5. Structural Equations for Horizontal Forms

85

and let 4 be a representation of G as nonsingular linear transformations on a vector space V . Then there is an associated representation of g as linear transformations on V , which may be defined as follows. If v E V , X E g, then set $(X) v = X ( e ) v . This makes sense since v is a vector-valued function on G, namely, v(g) = +(g) v ; and so is mapped into a vector by the tangent X ( e ) (see 1.4, 2.2, and 2.6).

4

Let X E and ~ w be a V-valued p-form, and let Yl , ..., Y , be invariant vector fields on M . (i) If w satisfies w o dg = +(g) w for all g E G, then

Lemma 3.

( 8 X ) w(Y1

9

.", Y,)

=

(AX) 4 Y l , *.., Y,),

where AX is the vector field on M defined in 3.1. (ii) If w satisfies w o dg = +(g-') w for all g E G, then -@X) w ( Y , , ..., Y,)

=

(AX) w(Y1

)

..., Y,).

Proof. We shall prove (ii). T h e proof of (i) is similar. I f f € M , then w ( Y l , ..., Y,) o f is a V-valued function on G, and in fact

4Y1,

..., Y,)of(g)

=

=

w(Y,(gf), *'., Y,(gf)

=

w(dgY,Cf),

=

W1)W ( Y l ( f ) )

.a*,

dgY,Cf)) Y,(f))

4 Y l ( f ) , -*.* YD(f)) #(P)l

where +(g) = g-l, since the Yiare invariant and w is equivariant with respect to the representation 4. Therefore we have

86

5. CONNEXIONS

Theorem 6. Let 4 be the 1-form of a connexion on P and let w be a Q-valued, horizontal, equivariant p-form on P. Then we have the structural equation for w : dw =

-[+ , wl

+ DfJJ.

+

p 1 vector fields Y , , ..., Yptl chosen from a set which locally spans the tangent space to P . For this set we choose vector fields (AX}, X E ~to, span the Proof. We prove this by applying both sides to

vertical tangent space. T h e remaining vector fields we choose in various ways. We consider several cases. (i) No Yiis vertical. We may then assume the Y , are all horizontal, and so [$, u](Y1, ..., Yp+,)= 0 since $ is vertical. Also, H Y , = Yi , so D w ( Y i , ..., Yp+,)= d w ( Y l , ..., Yp+,>,which proves the result in this case. (ii) One Yi is vertical. Assume Y,+, = AX. We may choose Y , , ..., Y p so that they are right invariant and so that [Yi, AX] = 0. T o do this in the neighborhood of a point f E P, we choose a coordinate system at f which derives from the local product structure of P. Then the partial derivatives with respect to the variables coming from M suffice, for they are clearly right invariant and they bracket correctly with AX, since AX depends only on the other coordinates. Now H(AX) = 0 implies that D w ( Y 1 , ..., Yp+,)= 0. We also have from theorem 4.2 dw(Y1 , ..., Y,+l) =

2 (- 1)i-1 Y,w(Y,, ..., Y&+Y,+l, ... Y,+l) )

i

since w is horizontal (-I)~Y,+,W(Y , ..., ~ Y,) (-l)P(hX) W(Yl , ..., Y,) = (-l)p+l ad X w ( Y l ,..., Y,)

=

=

4 = Ad and 2.6, since w is equivariant, = (-l)P+"X, W ( Y l ,..., Y,)]

by the lemma with

(-l)P++"X), W(Y, ..-,Y,)l = - 14,WI(Y1,.*.,y , , YV+I), =

using 4.9 and fact that prove.

I

w

is horizontal. This is what we wanted to

87

5.6 Holonomy (iii) Two or more Yivertical. From the fact that clear that everything vanishes. QED Corollary.

If

Q, is the

w

is horizontal it is

curvature form associated with 4, then we have d@ = -[+,

@I.

Proof. @ is horizontal and equivariant, and so the result follows from the theorem and Bianchi’s identity. We shall have several more occasions for employing the above lemma in the ensuing chapters.

5.6 Holonomy [2,5, 51, 66, 77, 821

We develop the material in this section as a series of problems. Let ( P , G, rr, M ) be a principal bundle with connexion 4. Let P E P and define K, C G by K , = {g E Glpg is a parallel translate of p}.

K, is a subgroup of G,the holonomy subgroup of rp at p. Problem 11. If p’ E P is a parallel translate of p, then K p = K,’. Problem 12. If p’ = pg, g E G,K p . Then R i w i i k = @irn(dx Xj , dx X k ) , and rijk= + k j ( d x

where

Xi),

and QinL are the components of these gI(d, R)-valued forms.

6. AFFINECONNEXIONS

118

A knowledge of the rijk for each coordinate system of a family whose coordinate neighborhoods cover M determines the connexion, since 4 is determined on the cross sections [compare the remark following (iii) in 6.4.21. Problem 17. Let x be the cross section as above and let p, +, P , Y be the pulled down forms w o dx, 4 o dx, SZ o dx, and @ o dx. Obtain the following formulas for these forms in the coordinate neighborhood: p = (dx, , ..., dxd)

yirn=

2 Rirnjkdx, dXk. i,k

Thus derive the equation given for Rimikin terms of the rijk by using the pulled-down structural equation d , + $ - L2

[ A *I

+ YJ.

Problem 18. Pull down the formula d@ = -[4, @] (corollary, 5.5) to get the coordinate form of the Bianchi identity Rrnnij.k

Problem 19.

riik(m)

-k

Rmnjk,i

Rmnki,j

=

0.

Prove that if xi are normal coordinates at m, then

+ r j i k ( m )= 0.

Problem 20. Prove that torsion is zero if and only if for every m there is a coordinate system at m such that Tijk(m)= 0. Problem 21. Connexions and action of groups. Let G act on M to the left in such a way that if g E G and dg is the identity on some M , , then g is the identity of G. Choose b E B ( M ) , b = ( m , el , ..., ed) and define Ib : G -+ B ( M ) ,Z,(g) = (gm, dg el , ..., dg e d ) . Show that z b is an imbedding. Let G also act on B ( M )by gb = I&). Then the action is by bundle maps. An affine connexion on M is invariant under G if the connexion form 4 is invariant, 4 o dg = 4, for every g E G.

6.4. Covariant Differentiation and Classical Forms

119

If M is a homogeneous space of G, show that an invariant connexion on ( G , M , H ) (see problem 5.5) induces an invariant affine connexion 5, as before, then the imon M . I n this case, if we write g = m bedding takes into fundamental vector fields, m into all basic vector fields, restricted to I b ( G ) .

+

Problem 22. Product connexions. Let M’, M“ be manifolds with solder forms affine connexions having connexion forms +’, w ’ , w ” , etc. Let M = M’ x M“ and define the bundle of adapted bases over M to be the submanifold of B ( M ) : +It,

B(M’, M ” )

= {((m’, m”),e ,

and

, ..., e d ) I (m‘,e, ,. .., ed,) E R(M’)

(m”, eCtl , ..., ed) E B(M”)}.

T h e group of B(M’, M”) is Gl(d’) x G l ( d ” ) and it is clear that B(M’, MI’) may be identified with B(M’) x B(M”). Define a conon B(M‘, M“) and extend by equivariance to a nexion +‘@

+”

FIG.23.

+

connexion on B(M). M with this affine connexion is called the afine product of affinely connected manifolds M’ and M“. Show that the product connexion has the following properties: (a) If y’ and y” are curves in M’ and M”, X‘ and X” are parallel X” is parallel along y’ x y”, vector fields along y‘ and y ” , then X‘ and, conversely, a parallel vector field on M has this form. (b) T h e geodesics are products of geodesics on M‘ and M”. Hence the affine product of complete connexions is complete.

+

+ T“sp,,p,.

(4

Ts,+s..,t.+t..= T’s,,t,

(4

R,,,,,.,t,+p. = R’sf,tff R”,t,,t,..

6. AFFINECONNEXIONS

120

Problem 23. If i : iV+ M is a covering map, then there is an induced natural covering map i : B(N)+ B ( M ) . (The prolongation of i.) If M has an afine connexion and if y4 is the connexion form, then i*$ is a connexion form on B ( N ) .Describe this connexion in terms of parallel translation. Problem 24.

Connexions on the afine bundle. Let

A(d, R ) =

A x

1) E GZ(d

+ 1, R ) I A E Gf(d,R),x

E Rd

I.

( x is viewed as a column matrix.)

Show that this defines A ( M ) as a principal bundle over M . (See proof of theorem 10.) If we view Rd as the hyperplane of Rd+l which has final coordinate I , then A ( d , R ) acts to the left on Rd:

I)y = Ay + x,

X

for

y E Rd.

Hence there is an associated bundle S ( M ) which has fibres homeomorphic to the tangent spaces of M . Make this correspondence explicit. Define maps 7] : B ( M )+ A ( M ) T~ : GZ(d,R ) + A ( d , R )

by q(b) = (b, 0), qc(A)= (t :), and show that this gives a bundle map of B ( M ) into A ( M ) which induces the identity map on the base space M . Let w be the solder form on B ( M ) and consider a connexion c j on B ( M ) , with @ and 1;2 as the curvature and torsion forms. T h e Lie algebra of A ( d , R ) may be considered to be

6.4. Covariant Differentiation and Classical Forms

121

6

Define an a(d, R)-valued form on ? ( B ( M ) )with the property that (8 :), and show that may be extended by right translation to a connexion form on A ( M ) ,also denoted by+. If 3 is the corresponding curvature form, show that

6

y*$ =

A connexion on A ( M ) arising in this way from a connexion on B ( M ) is a special case of a Cartan connexion (see [35,48, 491). By considering other horizontal, @-valued equivariant forms in place of w , more general Cartan connexions may be defined, and in general all the connexions on A ( M ) whose distributions are disjoint from T ( q ( B ( M ) )arise in this way, since gI(d, R ) is reductive in a(d, R). Returning to the connexion 6,we note that the parallel translation induced in the associated bundle S ( M ) gives rise to affine transformations of the tangent spaces to M which depend on both the curvature and torsion of the connexion 4 on B ( M ) . Infinitesimally, a curvature transformation R may be defined in a way analogous to the curvature transformation of an afine connexion as follows. Let x, y , x E Mm , ( b , t ) E A ( M ) such that ~ ( b t,) = m, 2, jj lifts of x, y to (b, t). Then

R,, z

=

-(b, t ) @@, j q b , t)-'z,

where we are identifying the fibre of S ( M ) at rn with Mm, By choosing (b, t ) E q ( B ( M ) ) ,show that

Rr,z = R,, z + Tzy-

CHAPTER 7

Ri ern a n n i an M an i fo Ids T h e definition of a Riemannian structure on a manifold is given and the corresponding topological metric is shown to induce the same topology. T h e bundle of (orthonormal) frames is defined and the existence and uniqueness of the Riemannian connexion is established. T h e chapter concludes with a large selection of examples [33,50, 831. 7.1 Definitions and First Properties 7.1.1 Riemannian Metrics and Associated Topological Metrics

A Riemannian manifold is a manifold M for which is given at each m E M a positive definite symmetric bilinear form ( , ) on M , , and this assignment is C" in the sense that for any coordinate system (xl , ..., xd) the functions g..= ( D Z i, D,,)E C". Such an assignment 2.3 is called a Riemannian metrzc on M . If we let S y ( M ) be the bundle of symmetric positive definite tensors of type (2, O), then a more elegant version of the above definition is that a Riemannian manifold is one with a preferred C" cross section of this bundle Sy(M). Let M , N be Riemannian manifolds with metrics ( , )M and ( , ) N . Then a C" map f : M + N is an isometry if it is a homeomorphism and preserves the metrics, that is, for t , s E M , , (dft, dfs)N = ( t , s ) ~ .An isometry is a diffeomorphism. f is called a ZocaZ isometry if we relax the requirement that it be one-to-one. If M is an oriented Riemannian manifold then there is a unique d-form % which determines the orientation and such that % ( e l ,..., ed) = h l for every orthonormal basis e, , ..., ed of M m . % is called the Riemannian volume element of the oriented Riemannian manifold. 122

7.1. Definitions and First Properties

123

We defer examples in this chapter to the last section. If (xl, ..., x d ) is a coordinate system on any manifold M with domain 0, then there is a natural inner product on the tangent spaces to 0, namely, the Euclidean inner product (D, , D,,)= aij. We denote by 1 1 11' the Euclidean norm. We also let ( 1 I ( be the Riemannian norm, that is, 1 1 t 1 1 = ( t , t)1/2, t E T ( M ) , so that we have 1 1 Dz, 1 1 = (gii)lI2E C", and in fact for ai E C", 1 1 ZiaiDZ,1 1 E C" at each point at which not all the aivanish. If we require ( , ) to be only nondegenerate instead of positive definite, then M is called a semi-Riemannian manifold. T h e main result of this chapter, namely, the existence and uniqueness of a Riemannian connexion, holds in the semi-Riemannian case. Problem 1. T h e index of a symmetric quadratic form on a real vector space is the dimension of a maximal subspace on which the form is negative definite. Prove that for a connected semi-Riemannian manifold the index of the metric is the same on every tangent space. For nonconnected manifolds we also require that the index be constant for a semi-Riemannian metric. A manifold with index 1 or d - 1 is called a Lorentz manifold. T h e four-dimensional time-space universe of Einstein is a Lorentz manifold.

If y is a broken C" curve in M , then its arc length is defined by I Y I = f l l aY * I I l

where [a, b] is the interval of definition of y . Problem 2. Let y be a broken C" curve defined on [a, b]. Define nondecreasing continuous function f on [a, b] by

f(4= j-1/ I Y* II. (a) Show that f is C" at every x such that y,(x) exists and is nonzero. (b) Show that Y o f-l : [0, 1 y I] + M is a continuous well-defined function even though f-1 may not be a function, and that it is C" at every f ( x ) for which y,(x) # 0. (c) Let (x,y) be a coordinate system on a two-dimensional manifold and define C" curve y on an interval with 0 as an interior point by the equations

124 x(y(t)) =

st

7. RIEMANNIAN MANIFOLDS

1'' exp (- l/s2) sin l / s ds.

[exp (-l/sz) sin 1/sI2ds, y ( y ( t ) )=

0

' 0

Show that for this curve y o f-' is not a broken C" curve so that y cannot be reparametrized with respect to arc length so as to remain broken C". We define a function p : M x M

where

r = set of all broken C"

Lemma 1.

-+

R

v

{+ a} by

curves from m to n.

The function p is a metric on M .

Proof. It is trivial that p is symmetric and satisfies the triangle inequality, so the only thing remaining to be proved is that p(m, n) = 0 implies that m = n. Assume that m # n, and let (xl , ..., xd) be a coordinate system at m with domain U . Let 0 be a ball with respect to the xi such that 0 C U and n $ 0 .Define a function f : Rd x U -+ R by f ( a , .'*> a d 4 = aPl.,(m) ?

7

(1 x

I/

Then f I S d - l X L I is continuous and positive, and therefore there exists a k > Osuchthat 1 - ,

s, t

E

Nn

T h e linear fractional transformations of the Riemann sphere of complex analysis are conformal maps of the Riemannian 2-sphere, and the homometries of Rd combined with stereographic projection give conformal maps of the Riemannian d-sphere. Problem 20. Prove that stereographic projection is a conformal map of Sd - {pt} onto Rd. 11. Action by compact groups. If G is a compact Lie group then G has a unique (Haar) measure which is invariant under left and right translations and assigns 1 to all of G. If G acts differentiably on M , then we can obtain a metric on M so that G acts as a group of isometries. For let { , ) be any Riemannian metric on M , and for s, t E M , , define

Problem 21. Show that G acts as isometries on ( M , ( , )'). Iff is any self-isometry of ( M , ( , )) which commutes with the action of G, that is, fg = gf for every g E G, show that f is an isometry of ( M , ( , )'). Problem 22. Show that the orthogonal complements of the vertical tangents with respect to a right invariant metric on a principal bundle form a connexion distribution. Hence, give an alternate proof that a paracompact principal bundle admits a connexion (compare 5.4). Problem 23. (a) Show that a left invariant metric on a Lie group G corresponds to an inner product on the Lie algebra g. (b) A left invariant metric on G is also right invariant (and so adjoint invariant) if and only if the corresponding inner product ( , ) on is invariant, that is, if X , Y , 2 E g, then

that is, a d X is skew-symmetric. (This is simply a restatement of the fact that the function (Ad elxY, Ad e"2) is constant if and only if its derivative is zero. Compare with problem 2.5.) (c) If G is compact, it always admits such a metric.

7.4. Examples and Problems

137

(d) T h e Riemannian connexions of such invariant metrics on G are all equal to the torsion free connexion of problem 6.6, and hence are complete (see problem 18). (e) T h e Killing f o r m of a Lie group G is a bilinear form k( , ) on g defined by: if X , Y Eg, then R(X, Y ) = tr(ad X o ad Y ) .

K ( , ) satisfies the invariance property of (b), but is not in general

definite or even nondegenerate, for example, if g has a nontrivial center. 12. Riemannian homogeneous spaces. If M is acted upon by a transitive Lie group of isometries, then M is a Riemannian homogeneous space. By example 11, a homogeneous space of a compact Lie group may be given a homogeneous metric. Problem 24. Let H be a closed reductive subgroup of a Lie group G (see problem 5.5), so that g = b + m, where [b, m] C m. Assume m admits an inner product which is invariant under A d H . Then show that G / H is a Riemannian homogeneous space. V d , ? the , Stiefel manifold of ordered sets of r orthonormal vectors in Rd,is a Riemannian homogeneous space of both O ( d ) and SO(d), namely, Vd,T= O(d)/O’(d- r ) = SO(d)/SO’(d- r ) ,

where O’(d - r ) and SO’(d - r ) are viewed as acting on the last d - r components in Rd. If we make the definition O(0) = SO(0) = {I}, then O ( d ) = V d , d S, O ( d ) = V d , d - l ,and Sd-’ = Vd , l are special cases. 13. Flag manifolds. If d , , ..., d, is a partition of d , then we define flag manifold Fl(d ; d, , ..., d,) as the set of n-tuples ( V , , ..., V,), where Vi is a subspace of Rd of dimension di and these subspaces are mutually orthogonal. Alternatively, Fl(d ; dl , ..., d,) may be considered to be the set of increasing sequences (0)

=

W , C W , C W , C ... C Wn-l C W,

of subspaces of Rd with di = dim

=

Rd

W i- dim Wi-l .

Problem 25. (a) Establish a natural one-to-one correspondence between these two sets. (b) T h e orthogonal group O ( d ) acts on n-tuples ( V , , ..., V,) and the general linear goup Gl(d, R ) acts on increasing sequences

138

7. RIEMANNIAN MANIFOLDS

W, c W, c c Wn-, c W , = R d . Find the isotropy group in each case, thus giving FZ(d ; d, , ..., d,) the structure of a homogeneous space in two ways.

FIG. 27.

(c) Since O ( d ) is compact, there is a Riemannian metric on Fl(d ; d, , ..., d,) on which O(d) acts transitively as isometries. For p = ( V , , ..., V J E F Z (;~d, , ..., dn) and 1 i < j n, define mij to be the subspace of FZ(d; d,, ..., d& spanned by tangents to curves on which only Viand V j are varied. Show that these mij must be mutually orthogonal by using the invariance of the adjoint action of the isotropy algebra. 14. Riemannian symmetric spaces [13, 18, 331. If a Riemannian manifold M has an isometry fm for every m E M which leaves m fixed and such that df,, l M m = - identity (fm is called the symmetry at m ) then M is a Riemannian symmetric space. Since an isometry must take geodesics into geodesics, it is easy to see by a step-by-step use of symmetries at points along a geodesic that geodesics are infinitely extendable, so M is complete. If M is connected, then any two points of M may be joined by a broken geodesic (see problem 8), and hence the composition of the symmetries about the midpoints of the geodesic segments is an isometry sending one of the points into the other. Thus the group of isornetries is transitive; and since this group is always a Lie group, M is a Riemannian homogeneous space [50]. Sd is a symmetric space.


0 such that for each m E 8, , exp, maps B(O(m),2c) diffeomorphically onto B(m, 2c), since E, is compact. Let n E B,+, . We show n E E,+, . By lemma 4 there p(m, n), where p(m, , m) = r . exists m E 8,such that p(m, , n ) = r Therefore, p(m, n) c, so there exists a geodesic y from m to n with I y j = p(m, n). Let u be a geodesic from m, to m with I u 1 = p(m, , m) = r . Then u y is a broken C" curve from m, to n with I 0 y I = I 0 I I y I = r p(m,n) = p(m0, n), so by corollary 3 to theorem 2, u y can be reparametrized as a geodesic. Hence, n E E,,, . This completes the proof of the theorem. It has been shown by K. Nomizu and H. Ozeki [67] that every connected paracompact manifold admits a complete Riemannian metric; furthermore, they show that if every Riemannian metric is complete then the manifold is compact. T h e converse, that a compact metric space is complete, is well known.

+


2. (a) This hypothesis is equivalent to: for every x, y E Rd, ( @ ( E ( x ) ,E ( y ) )x, y ) is constant on fibres of F ( M ) . (b) Since the functions depending on x, y in (a) determine the functions ( @ ( E ( x ) ,E(y))z , w ) , the hypothesis is equivalent to @(E(x),E(y)) is constant on fibres of F ( M ) .

( c ) If F@(E(x), E ( y ) ) = 0 for every vertical F , x, y E Rd, then E(z) @(E(x),E(y)) = 0 for every x, y , x E Rd, and hence @ ( E ( x ) ,E ( y ) ) is constant on F ( M ) , K constant on G d , 2 ( M ) . [Hint: Use the Bianchi identity D@(E(x),Q),E ( 4 ) = E ( x ) @(E(Y)Y E ( 4 ) and the fact that

[A,E(x)] = A E ( x ) - E ( x ) A so

E(x)A

=

+ E(y)@(E(z),E ( 4 )

+ E ( 4 @(E(x),E(y))

E(Ax)

for

A

E

=

0,

o(d),

+ E(Ax) = AE(x).]

9.2 Computation of the Riemannian Curvature

We indicate briefly how the Riemannian curvature can be computed in terms of the metric coefficients g i j . In particular, we show the connection between the curvature transformation and the metric. By 6.4.3, if X , Yare vector fields, then

9. RIEMANNIAN CURVATURE

166

where V , is the covariant derivative in the direction of X.However, by [66, p. 771 if X , Y , Z are vector fields, 2 < V ,Y ,Z)= X ( Y , Z ) + Y ( X , Z ) - Z ( X , Y )

+ ax, YI, z>+ w,XI, y >+ ( X , [Z, YI>.

These two formulas give the desired connection. The above formula depends on the following facts: (1) Torsion zero if and only if [ X , Y ] = V,Y - V,X, X , Y any C" vector fields. (2) Parallel translation preserves the inner product if and only if X ( Y , Z ) = (V,Y, 2 ) ( Y , V,Z). Prove these statements and the formula. Derive an explicit formula for K ( D x t ,D x j )in terms of the gii * Problem 5. Use this formula to obtain an alternate proof for problem 7.18. Problem 4.

+

9.3 Continuity of the Riemannian Curvature

K is not a function on M , the Riemannian manifold, but it is a function on the Grassmann bundle of 2-planes of M [3.3(5)],and in fact a continuous function. From this it will follow that the curvature on a compact subset of M is bounded. Let Gd,, be the Grassmann manifold of plane sections (twodimensional subspaces) of Rd.(See problem 7.30.) So Gd.2 =

O(d)/O(2)x O'(d

-

2).

We denote by G,,,(M) the bundle with fibre Gd,, associated to the frame bundle F ( M ) , where M is a Riemannian manifold. Thus Gas,(&') = F ( M ) x O ( d ) G d , If 2 . m E M , we write G,,,(m) for the ) m, then fibre of G,,,(M) over m. If b E F ( M ) such that ~ ( b = b : Gd,,3 G,,,(m) by : P -+ { ( b , P)} = ( b , P ) O ( d ) , the equivalence class of (b, P)in G,,,(M). But we know that b : RdZMm, s o b : Gd,2 {plane sections at m}, and the resulting identification of G,,,(m) with {plane sections at m} is independent of b. Hence, the Riemannian curvature K can be viewed as a real-valued function defined on Gd,,(M). for the space (We are here using the notation F ( M ) x O(d)Gd,z (F(&') x Gd*,)lO(4 of 3.3.)

9.3. Continuity of the Riemannian Curvature

167

Proposition 1. T h e function K : G d , 2 ( M-+ ) R is C", and hence, in particular, continuous. Proof.

Consider the diagram F(M) x O(4

p , 4 identification maps

It is only necessary to show that K o q o p is C". T o define the map p we must first choose an element, say P o , of G d , 2 .Then p(b, g ) = (b, gPo). Hence, K o q o p ( b , g ) = K(b(gPo)),remembering that b : Gd,2-+ {plane sections at m}. Let Po be spanned by orthonormal vectors x, y E Rd. Then K(bgpO)

=

(Rb(g%)b(gyl

=-

b(gx), b(gy))

(b@,(E(gx)(b),- m ) ( 4 ) g x , g y ) ,

which is clearly C" in b and g . QED Since Gd,2is compact, we have the: If C C M is compact, then there exist H , L E R such that for any plane section P at any point m E C , H K ( P ) L.

Corollary.


(u,0 ) du C

ja

+ c1 {<wQ(D2),wQ(D1)>(b,0) - (wQ(D2),w Q ( 4 ) > ( a ,0)). Proof.

According to the last remark above,

D2 I I d W 1 ) I I (u, 0) = ~ 2 < w Q ( D 1 ) ,wQ(Dl)>""%0 ) 1 = (u,0 ) =

=

=

c c1 wQ(D1)>(u, 0 ) c1 (u,0 ) c1 D, (wQ(D2), 0) 1 - c (wQ(D2), GwQ(D1)>(u, 0 ); -

wQ(D1)>(.,

we have used (1) and the fact that +O(D,) is skew-symmetric. Therefore, 1 b Z'(0) = - - (wQ(D2),D1wQ(D1))(u, 0 ) du

J

C a

+ c1 (wQ(D2),wQ(D1)>(.,0)li which is the desired formula.

9

11. SECOND VARIATION OF ARC LENGTH

216

If Q is a broken C" rectangle having the tangent to the base curve a constant C in length, then Corollary 1.

If the base curve of broken C" rectangle Q is a geodesic (unbroken) and (wQ(D2),wQ(Dl))(b,0) = (wQ(D,), wQ(Dl))(a,0), then 1'(0) = 0. I n particular, l'(0) = 0 if the transverse curves are all perpendicular to the base geodesic. Corollary 2.

Proof. T h e base curve is a geodesic if and only if wQ(D,)is C" and DlwQ(Dl)(u,0) = 0 for every u. Under the hypothesis given the integrand in corollary 1 is 0 and the sum telescopes to give

(wQ(U,), WQ(Dl))(b, 0)

- (W0(D2),

w Q ( 4 ) > ( a , 0) =

0.

Corollary 3. Let N, P be submanifolds of M , and let us consider only broken C" rectangles whose initial and final transversals are in

FIG. 36.

N and P. Then a curve T has the property that for all such rectangles with base curve T, Z'(0) = 0 if and only if T is a geodesic from N to P which is perpendicular to both N a n d P. Proof. T h e idea is that we are able to get rectangles with sufficiently arbitrary wQ(D,) that the formula for Z'(0) = 0 will yield DlwQ(Dl)(u,0) = 0, and then that T is smooth at the breaks, perpendicular to N a n d P at the ends.

11.1. First and Second Variation of Arc Length Let

T

217

be a horizontal lift to F ( M ) . If r # ui , define a curve y in

Rdby y(u) = f ( u ) Dw(?*)(u),wheref(u) is a non-negative C" function which is positive at r and 0 at every ui . Then V = f y is a lift of T to

T ( M ) ,so we may define Q(u, v ) for this Q, <W"(D,),

=

exp,,,,vV(u). It is easy to see that

4 w Q ( W ( u ,0 ) =

f(4I I D I W Q ( D 1 ) I l P

(% 01,

and that the terms in the sum of corollary 1 are all 0. Thus I'(0) = 0 gives S:f(u)

/ I DIWQ(D1)1 1 2

( u , 0) du =

0,

which implies D,wQ(D,)(r,0) = 0. T h e terms in the sum are treated similarly. We show that ( t , '*(ui+) - T * ( z L - ) ) = 0 for every t E M T ( u i )0, < i < n, by parallel translating t along T , multiplying the field so generated by a non-negative function positive near ui but 0 outside a neighborhood of ui , and defining Q as above. T h e rectangles to show normality at the ends of T may be taken to be those having longitudinal curves consisting of a short geodesic e ) and the segment T I r a + r , b l , where y segment from y(v) to T ( U is a curve in N with y(0) = T(u);the other end is treated likewise.

+

Problem 2. Show (independently of the arguments in Chapter 8) that a curve which minimizes distance between two points is a geodesic. (Classically, a geodesic was defined as a solution of the calculus of variations problem of finding minimal curves, with self-parallel condition Vv,y* = 0 following as a consequence.)

Our applications of variational theory will only involve variation of a geodesic in a direction perpendicular t o the geodesic at every point, that is, we shall only be using rectangles having a geodesic as base curve and the associated vector fields perpendicular to the base. As further justification for this assumption we prove the following remark which shows that for C" rectangles satisfying the end conditions the assumption entails no loss at all. T h e remark is not true for broken C" rectangles, however, unless the base geodesic is allowed broken linear reparametrizations. Remark. If Q is a C" rectangle having a geodesic base curve and with the initial and final transversals perpendicular to the base curve,

11. SECOND VARIATION OF ARC LENGTH

21 8

then there is a partial reparametrization of Q for which the longitudinal curves are reparametrizations of those of Q and the associated vector field is perpendicular to the base. Let I'be the associated vector field of Q, T the base curve of ( V , T * ) , s o f is a real-valued function on [a, b]. By hypothesis f ( a ) =f(b) = 0. Define map F : [a, b] x [c, c E] + [a, b] x [c, d ] by Proof.

Q, f

=

+

F(u, 71)

= u

i

1

-

- wf(u), a),

k

where

E

>0

is such that the range of F is in [a, b] x [c, d ] , and k Then Q o F is the desired rectangle. Problem 3.

= (T*

,T * ) .

Complete the proof that the vector field associated to

Q o F is perpendicular to T .

Henceforth we shall assume that all broken C" rectangles have geodesic base and associated vector field perpendicular to the base. For convenience we also assume that the base curve is normalized so as to have unit tangent vectors, initial parameter value a = 0, and hence length b = b - a. Lemma 2 (Synge's variation is given by

DZ2 I I w Q P 1 ) I I

=

formula

I I D1wQ(Dz) I- ,1

[86]). T h e unintegrated second &%?(Dl),

dQ(D2)) I I W Q ( D , )

I l2

+ W ( w Q ( D 1 ) , DzwQ(Dz)>+ <wQ(D1),+Q(Dz)wQ(Dz)>h

where all these functions are restricted to [0, b]

x (0).

The proof is by direct computation, always recalling that we are dropping the argument (u, 0):

Proof.

Dz2 I I w Q ( W I I

= D,(~,(wQ(D1)7 w Q ( 4 ) > ?'l =

-(D,wQ(m,

wQ(4)

>, + D , (DzwQ(D1),

wQ(4)>,

since the factors appearing in the denominators are powers of <wQ(D1), wQ(D1)>(u,

0) = 1.

Now using the first structural equation ( I ) and the fact that dQ(D2) is skew-symmetric gives (a)

D',

/ I w Q ( W II

= - .

11.1. First and Second Variation of Arc Length

219

T h e base curve T is a geodesic, so DlwQ(D,)= 0, hence, z= (Dl)2

=

0,

because the associated field is perpendicular to T.

Let V be the associated vector field along base geodesic V' the covariant derivative with respect to T* , and let 7 be the transverse vector field of Q. Then Corollary 1.

T,

I"(())

=

s" ( 1 1 0

V'

Il'(U)

- K ( V ) 11

/I2( .)

du

+

(T*

9

v,p>],b.

( K (v) = K( V,T*), Cf. 9.4.)

This follows from the observation V ( u ) = ? ( D 1 ~ Q ( D 2 )0)) (u, and 0,v = Q(D2 +"(D2))wo(D2) (cf. theorem 6.11).

+

Corollary 2. Let N and P be submanifolds, Q a rectangle with base geodesic T perpendicular to N and P and with initial and final transversals in N and P. Then

220

11. SECOND VARIATION OF ARC LENGTH

where S is the second fundamental form of the appropriate manifold and V is the associated vector field as before. Proof. T h e integral is the same as in corollary 1. By proposition 10.1, if T V o ,is the difference transformation of MT,,, given by N as a submanifold of M , then for vector field W on N TVO,

W ( 0 ) )= &Yo, W - E ” ( 0 , w.

But E V o , Wis perpendicular to T,(O), so for W = , du.

If Y is an N-Jacobi field and V(b) = 0, letting V , = Y , V , = V gives W , = 0, V,(b) = 0, so I( V , , V,) = I( Y , V ) = 0. Conversely, suppose I( V , Y ) = 0 for all V E 3 such that V(b)= 0. Let Y , be the N-Jacobi field such that Y(b) = Y,(b). Then I ( Y , Y - Y,) = 0 by hypothesis, I ( - Y l , Y - Y,) = 0 as just proved, so Z(Y - Y1)= Z(Y, Y

Hence by corollary 1, Y

-

-

Yl)

+ Z(-Y1, Y

-

Y l ) = 0.

Y , = 0.

Corollary 3. Suppose that there is no conjugate point of ~ ( 0 on ) ~ ( ( 0 b]). , For V E there ~ is a unique Jacobi field Y such that Y(0) = V(O), Y(b) = V(b), by lemma 3. Then I ( V ) >, I ( Y ) and equality occurs if and only if V = Y . Proof. Let T , and T, be ( d - 1)-dimensional transverse manifolds at ~ ( 0and ) ~ ( b )respectively, , such that their second fundamental forms

23 1

11.4. T h e Infinitesimal Deformations

and hence I,( V ) - I( V) = I,( Y ) - I(Y ) , so it suffices to prove the result for I , instead of I. Let I2 = I(T(O), T,), I3 = I ( T , , ~ ( b ) )so, that I , , I , , and I3 are the same except for their domains. Y , , where Yl is the T(O)-Jacobi field From lemma 3, Y = Y , such that Y,(O) = V(0). Since V - Y is 0 at both ~ ( 0 )and ~ ( b ) , Il(V - Y , Y,) = I,(V - Y , Y,) = 0 and I,(V - Y , Y,) = 13( - Y , Y,) = 0, by corollary 2 applied to I , and I3 , respectively. Adding these two equations gives

+

11(v - Y , Yl

+ Y,) = I,(V

-

Y , Y) = 0,

or,

Il(V, Y)

=

Il(Y, Y).

But by corollary 1,

= Il(K

V ) - Zl(K Y),

and equality occurs if and only if V

-

Y

=

0.

Problem 15. If V , , ..., V , are C" vector fields along T and are independent except at T(O), and V is a broken C" vector field along T , with V = ZfiVi the expression for V in terms of the Vivalid on (0, b ] , show by an example that the fi need not have continuous extensions to [0, b] even if V(0) = ZuiVi(0). However, if the Viare Jacobi fields then there are unique continuous extensions of the fi .

An infinitesimal deformation F is I-nonincreasing; that is, I ( V ) 2 I ( F ( V ) ) . Moreover, equality is obtained only if V = F( V ) .

Theorem 5.

Problem 16.

Prove theorem 5 by applying theorem 4 and corollary 3.

(For N = point this is due to Jacobi.) minimize distance to N beyond the first focal point.

Corollary 1.

T

does not

Proof. Suppose T ( C ) is a focal point of N , c E (0, b), and let Y be a nonzero N - Jacobi field which vanishes at c. Let V E 2z be given by

11. SECOND VARIATION OF ARC LENGTH

232

w,

V l [ O , C l = y I[O.Cl , I‘ I[C,bl = 0. Let I = I ( N , T ( b ) ) , I1 = T(C)), I2 = ~ ( T ( c ) ~, ( b ) )Then . I( V ) = 11( V ) I,( V ) = 11(Y ) I,(O) = 0 by corollary 2 to theorem 4. Now choose intermediate manifolds for an infinitesimal deformation F so that c is not one of the ui . Then F( V ) is smooth at c, while V has a break at c, hence F( V ) # V and I(F( V ) ) < I( V ) = 0. By theorem 3 there are shorter curves from ~ ( bto) N .

+

+

FIG. 40.

Corollary 2.

T h e first focal point of N is that point

T(C)

such that

T([O,b ] ) fails to minimize arc length to N for b > c, but T([O,b’]) minimizes arc length to N among curves in a neighborhood of T([O, b‘])for b’ < c.

T h e augmented index of a quadratic form is the dimension of a maximal subspace on which the form is negative semidefinite. Corollary 3. T h e index and augmented index of I are the same as the index and augmented index of I Ix , hence are finite. Proof.

of I

Since it is obvious that the index of I is not less than that of

Ix it suffices to prove the inequality the other way, and similarly

for the augmented index. Let 8 be a subspace of 2 on which I is negative semidefinite. is in the kernel of F, then Then F IJy is an isomorphism. For if V € 2 F( V ) = 0, I(F( V ) )= 0 I( V ) = 0, so from equality, V = F( V ) = 0. Since I is negative semidefinite on F ( X ) this proves the desired inequality for the augmented index. For the index the argument is the same except for using “definite” in place of “semidefinite.”


c E and y(c E ) is defined. Then for the corresponding subsequences {ui} of {yi}, {ni} of {mi},and {qi} of {pi},ui is a minimal segment from qi to

+

“i(C

+ €1,so

c

+

E

=

lim p(qi , ai(c

=p =

(lim qi , lim

P(P,

+

E))

U,(C

r(c + El),

+

+

E))

which contradicts the fact that y does not minimize arc length beyond y(c) = m.

T o show lim inf ci = c we consider convergent subsequences; thus we may suppose that lim ci = c’ < c and reach a contradiction. Let E = (c - c’)/2 > 0. Then p i = yi(ci E ) is beyond the minimum point of pi on yi , so there is a shorter curve T~ from pi to qi;by adding E ) we get a to T~ short segments f r o m p to pi and from qi to q = y(c’ curve ui from p to q such that lim 1 ui I = p(p, q). By theorem 8(a), uiconverges to y , and hence T~ converges t o y also.

+

+

240

11. SECOND VARIATION OF ARC LENGTH

Let E = T' x exp : T ( M )-+ M x M . Then E is nonsingular on the compact set {ut I 0 u c' E}, so by problem 1.12 there is a neighborhood U on which E is a diffeomorphism. Let V = E( U ) , so V is a neighborhood of E(ut) = ( p , y(u)). For sufficiently large i both (pi,yi(u)) and (pi,T ~ ( u )will ) be in V . However, E-l(pi, qi) must E) ti and on the other hand have length ci E because qi = exp,i(ci cannot have length any greater than the integral of the radial lengths of r i , which is less than ci E. QED

< < +

+

+

+

T h e distance from a fixed point p to its minimum point in the direction t E M p is a continuous function of t where defined. (This is immediate from the fact that the distance function is continuous.) Assume now that M is connected.

Corollary 1.

Corollary 2. A Riemannian manifold M is compact if and only if for some point p there is a minimum point in every direction from p . Proof. If M is compact then M is complete and bounded. Thus every geodesic ray can be extended indefinitely but cannot minimize arc length beyond the bound on M . Conversely, if p is a point such that every geodesic ray from p has a minimum point, then the function g : S -+ R, where S is the unit sphere in M , , g(t) = p ( p , exp c,t), is continuous by corollary 1. Thus the set = { t E M , I I I t I I du.

Since curvature is positive this is negative and there are shorter nearby curves; y cannot be minimal. QED Problem 25. Let M be compact, even-dimensional, nonorientable, and have positive curvature. Show that the fundamental group of MisZ,. Problem 26. Let M be compact, odd-dimensional, and have positive sectional curvatures. Show that M is orientable. Problem 27. Let M be a compact Kahler manifold which has positive holomorphic curvature. Show that M is simply connected. [Hint: if y is a geodesic and J is the complex structure operator, then J ( y * ) is parallel along y and y* , J ( y * ) span a holomorphic section.]

By using properties of the minimum locus it is sometimes possible to show the existence of closed geodesics with a method of Klingenberg

14.51. Theorem 12. Let M be complete, p E M such that the minimum locus of p is nonempty, and let m be a point on the minimum locus of p which is closest t o p . If m is not a conjugate point of p, then there is a unique closed geodesic with ends at p and passing through m such that both segments are minimal. Proof. By theorem 8, if m is not a conjugate point, then there are at least two minimal segments from p to m. We show that there are exactly two and that they match smoothly at m. Let y1 and y, be any two. If they do not match smoothly at m, then there is a geodesic u starting at m which makes an acute angle with each of y1 and y z .

FIG. 44.

1 1.7. Closed Geodesics

24 5

There will be minimal segments near to y1 , f r o m p to points on u near m,and shorter than y l ; similarly, there will be such minimal segments near to y z . Since y1 and y 2 are distinct these minimal segments will also be distinct when the points on u are sufficiently close to m. Then the points on u will be minimum points of p, by theorem 8, but this contradicts the fact that m is the closest minimum point t o p . Corollary 1. Let M be compact and let (p, m) be a pair which realizes the minimum of the distances of points to their minimum locus. Then either p and m are conjugate to each other or there is a unique smooth closed geodesic through p and m such that both segments are minimal.

Let M be compact, even-dimensional, orientable, with positive curvature, and let p, m be as in corollary 1. Then p and m are conjugate.

Corollary 2.

Proof. Assume p, m are not conjugate, so by corollary 1, there is a unique smooth closed minimal geodesic loop y through p and m, y ( 0 ) = p. Using Synge's trick, we have a one-parameter family of smooth loops y u such that yo = y and 1 yu 1 < I y I for u # 0. T h e n the

Y FIG.45.

unique minimal segments from y J 0 ) to the other points of yu form all possible angles with yu at yu(0). Those which form a fixed angle 0 have a convergent subsequence, as a function of u, to a minimal segment from p to a point m' on y . By the uniqueness of rn as the minimum point of p on y , m' = m. This contradicts the fact that there can only be two minimal segments from p to m. Remark. This corollary shows that under these conditions there is a point at which the conjugate locus and cut locus intersect. This can be used to derive lower bounds on diameter from upper bounds on

246

11. SECONDVARIATION OF ARC LENGTH

curvature. Under the much stronger assumption that M is a simply connected Riemannian symmetric space the minimum and first conjugate locus coincide [28, 761. 11.8 Convex Neighborhoods [24; 33, p. 53; 931

A set B in a Riemannian manifold M is convex if for every m, n E B, there is a unique minimal segment from m to n and this segment is in B. T h e open ball B(m, ro) of radius ro about m is locally convex if each

FIG.46.

sphere S(m, r ) of radius r < r,, about m satisfies the convexity coildition: if y is a geodesic tangent to S(m, r ) at n = y(O), then for sufficiently small u, p(m, y(u)) 3 r . If B(m, ro) is locally convex then exp, must be one-to-one on B(0, y o ) c M n l ,for otherwise there would be in B(m, ro) points on the minimum locus of m ; if y were perpendicular to a geodesic T from m at a point ~ ( r beyond ) the minimum point, then p(m, y(u)) < r for small u, since p(m, ~ ( r )< ) r. T h e relation between the concepts of convexity and local convexity is not as simple as it is in Euclidean spaces. For example, on a flat cylinder a normal coordinate ball with diameter greater than half the circumference of the cylinder will be locally convex but will not be convex because it will contain opposite points, which have two minimal segments. On the other hand, if the convexity condition fails for S ( m , r ) , then B(m, r ) is not convex. T o show this let y be a geodesic tangent to S(m, r ) at n = y(0) and having p = y(u) near n inside S(m, r ) . Then a Jacobi field along y which points outward at p and vanishes at n will have a corresponding rectangle having longitudinal geodesics, of which only y is tangent to S ( m , r ) . T h u s there will be

11.8. Convex Neighborhoods

247

segments which start near p , pass outside S(m, r ) , and return inside S(m, r ) at n. However, it might happen that B(m, r’) is convex for some r’ > r . T h e example of a flat cylinder shows that the following is the best possible result of its kind. Proposition 1. Let B(m, 2r0) be locally convex. Then every minimal segment between a pair in B(m, ro) is entirely within B(m, ro). Proof. If p , q E B(m, yo), then a minimal segment between them cannot go outside B(m, 2r0). If p(m, y ) does not take on its maximum at an end of y , then the least parameter value of y for which p(m, y ) is maximum would give a point of tangency of y with a sphere S(m, r), r < 2r,, and one end of y from that point of tangency would be inside S(m, r ) , contradicting local convexity. Thus the maximum of p(m, y ) occurs at an end, so all of y lies inside B(m, ro).

If T is a geodesic from m to n = T ( Y ) , then N = exp,(T*(r)’) n U , where U is a neighborhood of n, is a submanifold containing all small geodesic segments tangent to S(m, r ) at n. Thus the index form I = I(m, N ) will determine largely whether S(m, r ) satisfies the convexity condition at n. If I has nonzero index, then there will be points on N which are closer to m than n is. Thus if the convexity condition on S(m, r ) is satisfied I will be positive-semidefinite; if I is positive-definite we need only the requirement that n be before the minimum point of m on T to obtain the convexity condition at n for S(m, r ) . I n turn, whether or not I is positive (semi-)definite is determined by the behavior of m-Jacobi fields along 7.For if V is an m-Jacobi field the end terms in I( V )are 0, because the second fundamental form of N at n is 0, so I ( V ) = (V’(r), V ( r ) ) , by corollary 3 to lemma 2. We summarize the result as follows: Let % be the space of m- Jacobi fields along geodesic from m, parametrized by arc length. Let b, be the quadratic form on X defined by b,(V) = ( V ’ ( r ) , V ( r ) ) . (a) If B(m, ro) is locally convex, then b, is positive-semidefinite for r E (0, ro). (b) If B(m, ro) is a normal coordinate ball and b, is positivedefinite for all such T and for all r E (0, ro), then B(m, r o ) is locally convex. Proposition 2.

T

248

11. SECOND VARIATION OF ARC LENGTH

When all such b, are positive definite we call B(m,r,,) strongly locally

convex, abbreviated SLC.

Problem 28. Let B(m,ro) be a normal coordinate ball, 7 a geodesic , r E (0, r o ) .Then the values of m-Jacobi from m,and z = ~ * ( r )where . fields { V ( r )I V E X } form the tangent space to S(m, r ) at ~ ( r )Show that:

(a) X is the space of S(m, r)-Jacobi fields along T , so m is a focal point of S(m, Y ) of order d - 1. (b) T h e second fundamental form H , of S(m, r ) is essentially the same as 6 , . Problem 29. Show by continuity considerations that B(m,r ) is SLC for sufficiently small r . If B(m,r ) is locally convex but not SLC, then B(m,rl) is not S L C for rl > r . Find examples to show that B(m,r ) can be normal for all r , SLC for Y E (0, a), locally convex for r E [a, b ] , and not locally convex for r > b, where a, b are arbitrary except for the 6 03. restriction a

<
d = dim M . Use the following theorem of Otsuki [68,69] to show that there is a plane section P such that K,(P) > K,(P). Let V be a real vector space of dimension n. Suppose Q1, ..., Qk , k < n, are symmetric bilinear real-valued forms on V such that

+

(Pi(%4 QkJ, 4 Qh, 4z) =

H 1 j 2sin ( H 1 % ) cos ( H 1 / 2 r() E , E ) .

Thus b, is positive definite if r Problem 33. Y

< ro , as desired.

Let C be a compact subset of M . Show that there is

> 0 such that for every m E C, B(m,r ) is convex and SLC.

Problem 34. If M is complete, simply connected, and has nonpositive curvature, show that every ball in M is convex. Problem 35. Let M have nonpositive curvature and let B be a convex ball containing a geodesic triangle with lengths of sides a, b, c and opposite angles a , /3, y . Show that u2

+ b2

-

< c2 +B +Y
2 f((Y’7 Y’>

(f’if)(7)

-

, Y )( Y , Y > )du

0

=

( Y , Y>’(r)

=

(g‘/g)(r)s

It now follows that f ( r ) 3 g ( r ) as long as there is no conjugate point of m on ~ ( ( 0r ,] ) .However, this was only needed to enable us to divide byf(r)li2, so the inequality and continuity now give that f ( r ) >, g ( r ) for r E (0, b]. Under the above hypothesis, the first conjugate point of n must occur before that of m. Corollary 1.

Corollary 2. (Bonnet). Let M have all curvatures of plane sections tangent to a geodesic y starting at m satisfy the inequalities 0 < L K ( P ) H , L and H constants. Then if s is the distance along y to the first conjugate point of m on y ,


0, then M is compact, the diameter of M is n / a , and the fundamental group of M is finite.