Treatise on Analysis

TREATISE ON ANALYSIS Volume IV This is Volume 10-IV in PURE AND APPLIED MATHEMATICS A series of Monographs and Textboo...

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TREATISE ON ANALYSIS Volume IV

This is Volume 10-IV in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.

Volume 10 TREATISE ON ANALYSIS 10-1. Chapters I-XI, Foundations of Modern Analysis, enlarged and corrected printing, 1969 10-11. Chapters XII-XV, enlarged and corrected printing, 1976 10-111. Chapters XVI-XVII, 1972 10-IV. Chapters XVIII-XX, 1974 10-V. Chapter XXI, 1977 10-VI. Chapters XXII, 1978

TREATISE ON

ANALYSIS J. DIElJDONNk Nice, France

Volume IV

Translated by

1. G. Macdonald University of Manchester Manchester, England

ACADEMIC PRESS

New York

San Francisco London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

1974

COPYRIGHT 0 1974, BY ACADEMIC PRESS, INC. ALL RIOHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York. Now’ York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 21/28 Oval Road, Loodon NWI

Libruy of Congress Cataloging in Publication Data Dieudonnk, Jean Alexandre, Date Foundations of modern analysis. (Pure and applied mathematics; a series of monographs and textbooks, 10) Vols. 2have title: Treatise on analysis. Vol. 1, “an enlarged and corrected printing” of the author’s Foundations of modern analysis, published in 1960. Includes bibliographies. 1. Mathematical analysis. 1. Title. 11. Title: Treatise on analysis. 111. Series. 73-1 0084 QA3.P8 vol. 10 1969 510’.8s 15151 ISBN 0-12-215504-1 (v. 4) PRINTED IN THE UNITED STATES OF AMERICA

79808182

9 8 7 6 5 4 3 2

“Treatise on Analysis,” Volume IV First published in the French Language under the title “Elements d’Analyse,”,tome 4 and copyrighted in 1971 by Gauthier-Villars, Editeur, Paris, France.

SCHEMATIC PLAN OF THE WORK

1

I. Elements of the theory of sets

m 1 1 . Real numbers

X X V . Nonlinear problems

This Page Intentionally Left Blank

CONTENTS

Notation..

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

ix

Chapter XVlll

DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD II. ELEMENTARY GLOBAL THEORY OF FIRST- AND SECONDORDER DIFFERENTIAL EQUATIONS. ELEMENTARY LOCAL THEORY OF DIFFERENTIAL SYSTEMS . . . . . . . . . . . . . .

..

1. First-order differential equations on a differential manifold. 2. Flow of a vector field. 3. Second-order differential equations on a manifold. 4. Sprays and isochronous second-order equations. 5 . Convexity properties of isochronous differential equations. 6. Geodesics of a connection. 7. Oneparameter families of geodesics and Jacobi fields. 8. Fields of g-directions,

1

Pfaffian systems, and systems of partial differential equations. 9. Differential systems. 10. Integral elements of a differential system. 11. Formulation of the problem of integration. 12. The Cauchy-Kowalewska theorem. 13. The Cartan-Kahler theorem. 14. Completely integrable Pfaffian systems. 15. Singular integral manifolds; characteristic manifolds. 16. Cauchy characteristics. 17. Examples: I. First-order partial differential equations. 18. Examples: 11. Second-order partial differential equations. Chapter XIX

LIE GROUPS AND LIE ALGEBRAS.

. . . . . . . . . . . . . . . . . . 127

1. Equivariant actions of Lie groups on fiber bundles. 2. Actions of a Lie group G on bundles over G. 3. The infinitesimal algebra and the Lie algebra of a Lie group. 4. Examples. 5. Taylor’s formula in a Lie group. 6. The enveloping algebra of the Lie algebra of a Lie group. 7. Immersed Lie groups

vii

viii

CONTENTS

and Lie subalgebras. 8. Invariant connections, one-parameter subgroups, and the exponential mapping. 9. Properties of the exponential mapping. 10. Closed subgroups of real Lie groups. 11. The adjoint representation. Normalizers and centralizers. 12. The Lie algebra of the commutator group. 13. Automorphism groups of Lie groups. 14. Semidirect products of Lie groups. 15. Differential of a mapping into a Lie group. 16. Invariant differential forms and Haar measure on a Lie group. 17. Complex Lie groups. Chapter X X

PRINCIPAL CONNECTIONS AND RIEMANNIAN GEOMETRY.

. . . 233

1. The bundle of frames of a vector bundle. 2. Principal connections on prin-

cipal bundles. 3. Covariant exterior differentiation attached to a principal connection. Curvature form of a principal connection. 4. Examples of principal connections. 5. Linear connections associated with a principal connection. 6. The method of moving frames. 7. G-structures. 8. Generalities on pseudoRiemannian manifolds. 9. The Levi-Civita connection. 10. The RiemannChristoffel tensor. 11. Examples of Riemannian and pseudo-Riemannian manifolds. 12. Riemannian structure induced on a submanifold. 13. Curves in Riemannian manifolds. 14. Hypersurfaces in Riemannian manifolds. 15. The immersion problem. 16. The metric space structure of a Riemannian manifold: localproperties. 17. Sirictlygeodesicallyconvexballs. 18. Themetric spacestructureofa Riemannian manifold: global properties. Complete Riemannian manifolds. 19. Periodic geodesics. 20. First and second variation of arc length. Jacobi fields on a Riemannian manifold. 21. Sectional curvature. 22. Manifolds with positive sectional curvature or negative sectional curvature. 23. Riemannian manifolds of constant curvature. Appendix

TENSOR PRODUCTS AND FORMAL POWER SERIES.

. . . . . . . 427

20. Tensor products of infinite-dimensional vector spaces. 21. Algebras of

formal power series.

References Index

.

.

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

432

.,......................

437

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

. . . . . ..

,

. .

NOTATION

In the following definitions, the first number indicates the chapter in which the notation occurs and the second number indicates the section within that chapter.

Q,,

R

FPZ' exp

unit vector field on R,such that r,(E(t)) = 1: 18.1 velocity vector (or derivative) T(u).E(t) of a mapping u of an interval of R into a manifold, at a point t E R: 18.1 interval of definition of the maximal integral curve u of X such that u(0) = x o , and its endpoints: 18.2 flow of a vector field X : 18.2 domain of the flow of X: 18.2 successive derivatives of a mapping u of an interval of R into a manifold: 18.3 domain of definition of the exponential of a spray 2 over M : 18.4 exponential mapping defined by a spray 2 : 18.4 canonical lifting to T(M) of a vector field X on M: 18.6, Problem 3 canonical vector fields on R2:18.7 partial derivatives of a mapping of an open subset of RZ into a manifold : 18.7 ix

NOTATION

x

v,

covariant derivatives in the directions of the vector fields E l , E 2 : 18.7 R-algebra of C" real-valued functions on M : 18.9 80 space of C" real differential p-forms on M : 18.9 8P d direct sum of the 8 p :18.9 so(x), (six, uI), ... , s&x, ul, . .., uq) integers defining an increasing sequence of integral elements: 18.10 rank of a k ) , where a is a differential 2-form: 18.16, rk,(a) Problem 2 f a section of a bundle E with base B, s an element Y (slf of a Lie group acting equivariantly on E and B: 19.1 P a differential operator from a vector bundle E to Y W avector bundleF, s an element of a Lie group acting equivariantly on E (resp. F) and B: 19.1 X a vector field on M, s an element of a Lie group Y(S)X acting on M: 19.1 actions on the sections defined by a right action: S(slf, S(4P 19.2 Int(s) inner automorphism XHSXS-' : 19.2 automorphism h e w s* he 3 - l of the tangent space Ad($) at the identity element: 19.2 8 infinitesimal algebra of a Lie group G: 19.3 and 19.17 Lie algebra of left-invariant vector fields on a Lie 9 group G : 19.3 and 19.17 algebra of distributions with support contained in @e { e } :19.3 and 19.17 Lie algebra of a Lie group G: 19.3 and 19.17 Se W G ) left-invariant vector field, equal to u at the point e : XU 19.3 image of a field of distributions P E G under a f*V) homomorphism f:G + G': 19.3 derived homomorphism f* : 6 -P (5' of a Lie group f* homomorphism f:G + G': 19.3 homomorphism Lie(G) -+ Lie(G') derived from a L i e u f* Lie group homomorphism f : G + G': 19.3 differential operator on M associated with a leftPU invariant differential operator P on G, where G acts on M on the right: 19.3 ZU, M Z" Killing field on M corresponding to u ~ L i e ( G ) , where G acts on M on the right: 19.3 gI(E), gl(n, R), gI(n, C),gI(n, H) Lie algebras of general linear groups: 19.4 v t 7

-

9

1 9

1

NOTATION

xi

3=C auXu

set of functions f such that f(x)/llx[lmf' is bounded in U: 19.5 formal Taylor series of$ 19.5

f

Taylor series off: 19.5

-

a,x" U

Z U

f

df

distribution with support {e} whose local expression for the chart rp is D U t O / a !19.5 : left-invariant differential operator taking the value Aa at the point e : 19.5 multi-index ( S i j ) , 5 j 5 n : 19.5 invariant vector field Zci:19.5 if a = (a1, .. ., an),Xu= X f * X i z X p : 19.6 enveloping algebra of the Lie algebra g: 19.6 Lie algebras of special linear groups: 19.7 exponential mapping of Lie(G) into G : 19.8 inverse of the exponential mapping: 19.8 normalizers in G and g, of a vector subspace tn of g,: 19.11 centralizers in G and ge of a vector subspace m of g:, 19.11 commutator group of H and K: 19.12 derived group of a group G: 19.12 derived ideal of a Lie algebra g: 19.12 nth derived group of a group G: 19.12 ideals of the descending and ascending central series of a Lie algebra 9: 19.12, Problem 3 automorphism group of a Lie group G: 19.13 automorphism group of the Lie algebra 9,: 19.13 group of inner automorphisms of a Lie group G : 19.13 image of a Lie group Gin Aut(g,) under the mapping st+Ad(s): 19.13 automorphism group of a group G: 19.14 semidirect product relative to a homomorphism CJ of L into J Q ( N ) : 19.14 semidirect product of Lie subalgebras n,, I, of g,, relative to a homomorphism cp: I, + Der(n,): 19.14 (left) differential of a mapping f of a manifold M into a Lie group G: 19.15 logarithmic differential of a mapping f:M + A of a manifold M into a finite-dimensional algebra A: 19.15

xii

NOTATION

canonical differential form on a Lie group G: 19.I6 real Lie group and Lie algebra obtained by restriction of scalars to R from a complex Lie group G and its Lie algebra ge: 19.17 complexification of a real Lie aglebra g : 19.17 S(C) frame bundle of a vector bundle E with base B: 20.1 Isom(B x F, E) Isom(F, Eb) set of isomorphisms of the vector space F onto the fiber Eb of E: 20.1 frame bundle of a differential manifold M : 20.1 affine group of R": 20.1, Problem 1 set of affine-linear bijections of R" onto the fiber E, of E: 20.1, Problem 1 P principal connection in a principal bundle : 20.2 value of a principal connection at a point b of the Pb base: 20.2 mapping u ~ r bu of the Lie algebra ge into the space of vertical tangent vectors G,, at the point r, of the principal bundle R with group G: 20.2 0 differential I-form with values in ge of a principal connection P : 20.2 Da,D,a covariant exterior differential of a vector-valued differential q-form a on a principal bundle endowed with a principal connection P : 20.3 n curvature form of a principal connection: 20.3 oApa exterior product of the connection form o with a vector-valued q-form a on the principal bundle R, with values in V, relative to an action p of G on V: 20.3, Problem 1 horizontal lifting at the point rx E R, of a tangent vector h, E T,(M): 20.5 canonical form on the frame bundle R(M): 20.6 torsion form of a principal connection in R(M): 20.6 canonical components of the connection form of a principal connection in R(M): 20.6 canonical components of the canonical form u on R(M): 20.6 moving frame : 20.6 connection form and canonical form on M, corresponding to the moving frame R: 20.6 canonical components of m:;),@), o i j ,ai (by abuse of notation) dR), dR): 20.6

NOTATION

1 oi ei 1

9

j

Quo,

R$:),

@IR),

wji

ej

abuse of notation for

C oi 0 ei i

and

xiii

1i w j i0 e j :

20.6 curvature and torsion forms of a linear connection on M, corresponding to a moving frame R : 20.6 R,,,Oi (by abuse of notation) canonical components of O'R':20.6 horizontal vector field on R(M): 20.6, Problem 1

(A

(A

T(M)*) 0 T(M)) : 20.6, Problem 2 direct sum of the Bg(M): 20.6, Problem 2 &(M)-module of C"-sections of B,P(M) (resp. B(M)): 20.6, Problem 2 differential operator in 99(M): 20.6, Problem 2 matrix(duij) for a matrix U = (uij): 20.6, Problem 2 canonical lifting to R(M) of a vector field X on M: 20.6, Problem 5 operator Y H Ox Y - V x * Y : 20.6, Problem 6 Lie algebra of infinitesimal automorphisms of a principal connection P in R(M): 20.6, Problem 6 Lie algebra of the affine group A(n,R): 20.6, Problem 20 G-structure on a differential manifold M: 20.7 canonical image of H in GL(g,/t),) (G a Lie group, H a Lie subgroup of G): 20.7 pseudo-Riemannian or Riemannian metric tensor: 20.8 scalar product (resp. norm) in T,(M) for a pseudoRiemannian (resp. Riemannian) metric tensor g on M: 20.8 abuses of notation for scalar product and norm relative to a pseudo-Riemannian or Riemannian metric tensor: 20.8 components of a pseudo-Riemannian metric tensor g, relative to a moving frame: 20.8 isomorphism of T,(M) onto T,(M)* defined by a pseudo-Riemannian metric tensor: 20.8 M-isomorphism h,w G, h, of T(M) onto T(M)*: 20.8 isomorphism of lowering the ith contravariant index to the jth place: 20.8 gradient of a real-valued function on a pseudoRiemannian manifold : 20.8

-

NOTATION

xiv

symmetric covariant tensor fields on r

A T(M) and r

A T(M)*, corresponding to a pseudo-Riemannian

11 h,

A

h,

A

vg vol, , u, vol ug, v *tL

K Khijk

r“( X )

K‘ KJk

Ric(h,)

A

h,ll

metric tensor g on M: 20.8 r-dimensional area of a tangent r-vector: 20.8 Riemannian volume: 20.8 canonical volume form an on oriented Riemannian manifold: 20.8 adjoint of a differential r-form a on a Riemannian manifold: 20.8 differential I-form ‘ G ( K ~on ) T(M), for a pseudoRiemannian manifold M : 20.8, Problem 3 (hl h) for a pseudo-Riemannian manifold M and h E T(M): 20.8, Problem 3 geodesic field C,(h, , h,): 20.9, Problem 2 pseudo-Riemannian metric tensor on T(M) induced by a pseudo-Riemannian metric tensor g on M : 20.9, Problem 3 Lie algebra of infinitesimal isometries of a pseudoRiemannian manifold M: 20.9, Problem 7 Lie algebra of the projective group PGL(n + 1 , R): 20.9, Problem 12 Riemann-Christoffel tensor of a pseudo-Riemannian manifold : 20. I0 components of the Riemann-Christoffel tensor relative to a moving frame: 20.10 M-morphism Z H ( r ( Z A X ) ) * Y of T(M) into itself: 20.10 r’(X, X ) : 20.10 Ricci tensor, defined by ( K , X@ Y ) = Tr(r’(X, Y)): 20.10 components of the Ricci tensor relative to a moving frame : 20.10 Ricci curvature in the direction of h, ET,(M): 20.10 endomorphism of T,(M) defined by (Fx(hx)lk,) = (K’(x), h,@ k,): 20.10 scalar curvature at x: 20.10 Gaussian curvature of a surface: 20.10 divergence of a vector field X : 20.10, Problem 3 Lie algebra of SO(n + 1): 20.11 hyperbolic n-space : 20.11

NOTATION

ds I Pj(x)

f(g1,***,gn)

xv

complex quadric of dimension n - 1: 20.1I , Problem 5 first fundamental form of a submanifold of a Riemannian manifold: 20.12 second fundamental forms, vector-valued second fundamental form of a submanifold of a Riemannian manifold : 20.12 total curvature of a submanifold of RN in the normal direction n,: 20.12, Problem 4 integral curvature of a submanifold of RN:20.12, Problem 4 abuse of notation for g: 20.13 second fundamental form of a hypersurface: 20.14 principal curvatures of a hypersurface at a point x : 20.14 mean curvature and total curvature of a hypersurface at a point x : 20.14 scalar differential (n - p)-form, constructed from p vector-valued functions and n - p vectorvalued differential 1-forms, with values in Rn: 20.14, Problem 9 length of a piecewise-C' path: 20.16 Riemannian distance from x to y : 20.16 length and energy of a path C, : t w f ( t , 5 ) : 20.20 index form of two liftings of v : 20.20 sectional curvature: 20.21 tensor product of two vectors: A.20.1 tensor product of two K-vector spaces: A.20.1 indeterminates, elements of K"': A.21.2 formal power series with coefficients in K: A.21.2 algebra of formal power series in n indeterminates over a field K: A.21.2 formal power series obtained by substitution of formal power series g j without constant terms in a formal power seriesf: A.21.3


CHAPTER X V l l l

DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD II. ELEMENTARY GLOBAL THEORY OF FIRSTAND SECOND-ORDER DIFFERENTIAL EQUATIONS. ELEMENTARY LOCAL THEORY OF DIFFERENTIAL SYSTEMS

Once the concept of a tangent vector to a differential manifold M has been established, it is easy to generalize the notion of a solution of a first-order differential equation (10.4.2) to functions defined on an interval of the real line, with values in M: the derivative u’(t) is replaced by the image of the unit tangent vector to R at the point t , and the right-hand side of the equation must therefore be a tangent vector to M at the point u(t). Locally, by means of a chart, we can reduce such an equation to a differential equation in the usual sense (10.4.1). However, since we now have an intrinsic formulation of the notion of a differential equation, problems of a global nature present themselves: the existence of a maximal integral curve and its behavior as a function of the “ initial condition ” which defines it or of “ parameters ’’ on which the equation depends. Here we shall indicate only the first rudiments of this extremely difficult subject, which involves subtle considerations of topology and integration theory (the reader will see some examples in Chapter XXV, and in [21] and [51]). The notion of a differential equation of the second order on a manifold M is less obvious, because it requires the concept of the “second derivative” of a function defined on an interval of R with values in M. Since the values of the “first derivative” are tangent vectors to M, it is to be regarded as a function with values in the tangent bundle T(M), and a second-order equation on M is therefore a first-order equation on the manifold T(M). The local and global questions which we shall consider for such equations (Sections 18.3 1

2

XVlll DIFFERENTIAL EQUATIONS

to 18.7) are largely concerned with a special type of second-order equation, namely, those which give rise to geodesics of connections, and which will be studied in more detail in Chapter XX in the context of Riemannian manifolds. The curves defined by a differential equation are characterized geometrically by the requirement that at each point they should touch a given line in the tangent space to the manifold at that point. Replacing the line by a vector subspace of arbitrary dimension, we have the general notion of a " Pfaffian system" on a manifold, which is the intrinsic counterpart of the " partial differential equations " of classical mathematics. Here, the global theory is practically nonexistent, except for completely integrable systems (see [69]) and the linear partial differential equations (and allied types), which we shall encounter in chapters XXIII and XXV. In this chapter, we shall beconcerned exclusively with local problems of existence and uniqueness, so that it would be possible to work in RN throughout. But the language of manifolds and especially the exterior differential calculus are extremely useful even in this local situation, as E. Cartan showed, in order to comprehend the intrinsic nature of the problems independently of any adventitious coordinate system. We have hardly skimmed the surface of the immense work of E. Cartan in this field, to which we urge the reader to refer ([54] and [59]).

1, FIRST-ORDER DIFFERENTIAL E Q U A T I O N S ON A DIFFERENTIAL MANIFOLD

(18.1.1) Given an open subset U' of R", a (real) autononious y'stent of differential eqirations in U is a vector differential equation (18.1.1.1)

DX = f(x),

where f : U + R" is a continuous mapping. A solution of this equation is therefore a continuously differentiable mapping ir of an open interval I c R into U such that Du(r) = f ( u ( t ) ) for all t E I (10.4). An autonomous system of differential equations is therefore a vector differential equation in which the variable t "does not appear" on the right-hand side. If u is a solution of (18.1.1.1) defined on I c R, it is clear that for each a E R, the function t H u ( t + a) is also a solution of (18.1.1.1), defined on the interval I ( - a ) . If we identify the tangent bundle T(U) with U x R" by means of the canonical trivialization (16.15.5), then the mapping x ~ ( xf(x)) , is the local expression of a continuous cectorfield X on U, defined by X ( x ) = t;'(f(x)) (16.5.2). Let E denote the vector field on R (called the iiriit vector field)

+

1 FIRST-ORDER DIFFERENTIAL EQUATIONS

3

defined by the condition that t , ( E ( t ) )is the vector 1 E R. Then, if u is a C' mapping of I into U, we may write Du(t) = ~ ~ ( t ) ( T t (.uE)( t ) )

and then the relation D u ( t ) = f ( u ( t ) ) is equivalent to

T(u) * E ( t ) = X ( u ( f ) )

(18.1.1.2)

for all t

E

I.

(18.1.2) The form (18.1.1.2) in which we have expressed that u is a solution of (18.1.1. l ) no longer involves the trivialization of T(U), and therefore allows us to generalize the notion of a n autonomous system of differential equations to an arbitrary differential manifold M. Given a vector field X of class C' ( r 2 0) on M, the (first-order) diferential equation on M defined by X is the relation T(u) E

(1 8.1.2.1)

0

=X

ou

and a C' mapping 1' of an open interval I c R into M is said to be a solution of this equation if for each t E I we have

T(13) . E( 1) = X(P( 1 ) ) .

(18.1.2.2)

For such a mapping

11,

we shall write? d(f) = T(r) * E ( f )E TD(,)(M)

(18.1.2.3)

for all t E I (ii'(t)is the deriratiile or idocity uector of u at the point t ) , and the equation (18.1.2.2) now takes the form L"(t) = X(P(t))

( 18.1.2.4)

for all

t E

1.

t We are therefore now assigning two different meanings to the symbols L.' and Du when I' is a mapping of an interval I into a rrc/or space R" (although in (8.1) we regarded these two symbols as synonymous). In this particular case we have

D d / )= T~,,,(I-'(/)).

4


I f f i s any real-valued function of class C' on M, it follows immediately from the definitions ((16.5.4) and (17.14.1)) that, by virtue of (18.1.2,4), we have (18.1.2.5)

for all t E I. A solution of (18.1.2.1) is also called an integral curre of the Lvctorjield X . A differential equation (18.1.2.1) is also called a dynamical systmi on M. If (U,cp, n) is a chart on M and if ZH(Z, f(z)) is the corresponding local expression of X (so that

the function (cp 0 u) I u - ' ( U ) is a solution in u - ' ( U ) of the vector differential equation (18.1.1.1), which is called the local expression of the equation (18.1.2.1) corresponding to the chart (U, cp, n).

Remarks (18.1-3) Consider an arbitrary vector differential equation DX = f(x, t ) ,

(18.1.3.1)

where f is defined and continuous on an open subset H of Rn x R and takes values in R". This equation may be reduced to an autonomous system as follows. Consider the continuous mapping Z H F(z) of H into R"" defined by F(x, t ) = (f(x, t ) , l), and the autonomous system Z' =

(18.1.3.2)

F(z).

If u = (u, cp) is a solution of this equation defined on an open interval I c R, where u ( t ) E R" and cp(t) E R, then we have cp'(t) = 1, so that cp(t) = t + c(, and u'(t) = f(u(t), t r ) for all t E I. Putting w ( t ) = v ( t - r ) , we have M"(t) = f(w(t), t ) in I + r , and w is therefore a solution of (18.1.3.1) in this interval. Conversely, if u is a solution of (18.1.3.1) in I, then it is clear that t w ( c ( t ) , t ) is a solution of (18.1.3.2) in I. Hence the problem of solving (18.1.3.1) is equivalent to that of solving (18.1.3.2). The counterpart of (18.1.3.1) on a differential manifold M is therefore a differential equation defined by a vector field Y on M x R such that Y ( x , t ) = ( X ( X , t ) , E ( t ) ) , when T(M x R) is identified with T(M) x T(R).

+

2 FLOW OF A VECTOR FIELD

5

2. F L O W O F A VECTOR FIELD

Let X be a i7ectorjeld of class C ' ( r 2 I) on M. I f ' v l and v2 are two integrul citriws of X , dejned respectiiwlv on open intervals I,, I, in R, and such that v,(to) = Li2(tO),for some point to E I n I , , tlim i l l and v, coincide in the interval I , n I, . It is enough to show that the set A of points t E I , n I, such that o , ( t ) = i i , ( t ) is both open and closed in I , n I, (3.19.1), since by hypothesis A is not empty. Now A i s closed, because r , and v 2 are continuous (3.15.1). On the other hand, if t , E A, there exists ;I neighborhood J c I , n I, of t , , and a chart ( U , cp, n ) of M at the point v , ( t , ) = u 2 ( t l ) for which cp 0 ( 1 1 , IJ) and cp o (u,l J ) are solutions of the same equation (18.1.1.1), where f is of class C' with r 2 I , and these two solutions take the same value at the point t , . (18.2.1)

,

I t follows, therefore, from (10.5.2) that v 1 and u2 coincide in J and the proof is complete. (18.2.2) Let X be a vector field of class C ' ( r 2 I ) on M . Then for each xo E M, there exists an open neighborhood J of0 in R and an integral curve v of X defined in J and such that 140) = xo . For by considering a chart ( U , cp, n) of M at the point xo such that the differential equation (18.1.2.1) has a local expression of the form (18.1.1.1), with f of class C ' ( r 2 I), the existence of a solution of this equation, taking the value cp(x,) at the point 0 and defined in a

neighborhood J o f0 in R, is guaranteed by (10.4.5), which proves the assertion. Now let J(x,) be the open interval which is the union of all the open intervals J containing 0 and in which there exists an integral curve of X which is equal to xo at t = 0. Since, by virtue of (18.2.1), any two of these functions agree in the intersection of their intervals of definition, it follows that there exists a unique function u defined on J(xo),such that each of the integral curves is a restriction of 1 1 , and it is clear that 11 itself is an integral curve of X . This function t' is said to be the maxinial intqval curve of X such that u ( 0 ) = x,, (or with origin x,,). We denote by t-(x,) and t + ( x , ) the left- and right-hand endpoints of the interval J(x,), so that t-(x,,) < 0 < tt(xo);it can happen that t - ( x , , ) = -co or t f ( x o )= + co.Also, we denote by F x ( x o , t ) the value at t E J(xo) of the maximal integral curve of x which takes the value xo at t = O . The function F, is defined on the set dom(F,) c M x R, consisting of the points (x, t ) such that /-(x) < t < t + ( x ) (the union of the sets {x} x J(x)). The function F, is called the flow of the vector field X , and dom(Fx) its domain. (18.2.3)

With the notation of (18.2.2), for each point t,, E J(x), we have

6

XVlll

DIFFERENTIAL EQUATIONS

and, for each t

E

J(x)

+ (-to),

For it is clear that the function t w F x ( x ,to + t ) is an integral curve of X defined on J(x) + ( - t o ) and taking the value Fx(x, to) at t = 0: and this function cannot be the restriction of an integral curve 11 defined on a strictly larger interval J' and taking the same value at t = 0, otherwise the function t ~ v ( -t to) would be an integral curve defined on an interval strictly larger than J(x) and taking at to the same value as the function t H F,(x, 1 ) ; which, in view of (18.2.1), would contradict the definition of J(x). The formula (18.2.3.2) is a consequence of this argument and the definition of F,. (18.2.4) Let M be a d?fSerential manijdd, N a closed submani/bld of M , and X a vector field on M of class C'(r 2 I ) . Supposi> that X(.u) E TJN) for all s E N (in other words, that the field X is tangent to N at all points of N). Then every inttyral curve of X which meets N is containid in N .

Let t t + u ( t ) be an integral curve of X defined in I , and suppose that N for some to E 1. I f the open set (3.11.4) of points t E I, such that t > to and v ( t ) 4 N is not empty, then it will have a greatest lower bound t , 2 t o , and we shall have v ( f , )E N . But if Y is the restriction to N of the vector field X , there exists an integral curve w of Y, defined on an open interval J c I containing t , and such that w ( t , )= ~ ( t , (18.2.2), ) and it is clear that w is also an integral curve of X . But then we have w ( t ) = v ( t ) for all t E J, by (18.2.1), hence c ( t )E N for all t E J, contradicting the definition of t , . Hence the set of points t E I such that t > to and u ( t ) 4 N is empty; likewise is the set of t E I , such that t < to and u ( t ) 4 N.

u ( t 0 )E

(18.2.5) Let X be a vectorfield of class C' on M, with r 2 I (possibly r =a). Then dom( F,) is an open subset of M x R and F, is a C' rnapping of dom(F,)

into M. Let xo E M and let t o E J(xo). Then we have to show that there exists an open interval ]a, b [ c J(xo)containing the points 0 and t o , and a neighborhood V of xo in M, such that for each x E V. the interval ]a, b[ is contained in J(x) and that F, is of class C' in V x ]a, b [ . Clearly, we may assume that to 2 0. Let [c, d ] be a compact interval contained in J(xo)= ] t - ( x o ) , t + ( x o ) [ , such that c < 0 5 to < d. Let L be the compact subset of M which is the image of [c, 4 under Fx(xo, .). and let W be a relatively compact open neighborhood of L in M . Then by (16.25.1) there exists an embedding of

2 FLOW OF A VECTOR FIELD

7

W in RN, for sufficiently large N , and we may therefore assume that M = W c RN.By (16.12.11) and (16.4.3) there exists an extension of X to a vector field Y of class C', defined on a neighborhood U of M in RN.Since the tangent bundle T(U) may be canonically identified with U x RN, we may write Y ( y ) = (,I*, f(y)) for all y E U, where f is a mapping of class C' of U into RN. Since M is /oca//y closed in RN (16.8.3) we may, by restricting U if necessary, assume that M is closed in U (12.2.3); and, replacing U again by a smaller open set containing L, that Df is bounded in U, say IIDf(y)ll 5 k for ally E U. Let 6 > 0 be chosen sufficiently small that, for each t E ]c, d [ , the open ball with center F,(x,, t ) , and radius6 is contained in U (3.17.11), and then choose I: > 0 such that E O ~ " - " ) < 6. Then it follows from (10.5.6) that for each point z E U such that llz - xoII 5 E , there exists a solution t H u ( z , t ) of the differential equation DJJ = f(y), defined on the i n t e n d ]c, d [ and such that u(z, 0) = z and ilu(z, t ) - Fx(xo,t)II 5 E for all t E ]c,d [ . By (18.2.4) applied to U and M, for each x E M satistjling IIx - xoII 5 E , the function t H u ( x , t ) is an integral curve of X in the interval ]c, d [ that takes the value x at t = 0. Consequently, we have ]c, d [ c J(x) and u(x, t ) = F,(x, t ) for all t E ]c, d [ . Moreover, it follows from (10.7.4) that by replacing the interval Jc, d [ , if necessary, by a smaller interval ]a, h [ such that a < 0 5 t o < b, we may suppose that the function (x, t ) H " ( x , t ) is of class C' in the product of a neighborhood of of L in M and the interval ]a, b[. This completes the proof.

This proposition leads directly to the following corollaries: (18.2.6) For each t E R, thc sct of points x E M such that ( x , t ) E dom(F,)

is open in M. This follows from (3.20.12). X H t+(.r)is lower semicontinuous, and the function is upper semicontinuous on M .

(18.2.7) The ,function XH~-(X)

For the set of points x such that t ' ( x ) > x > 0 is equal to the set of points 2 ) E dom(F,). hence is open by (18.2.6). The first assertion therefore follows from (12.7.2) and the second assertion is proved similarly.

x such that (x,

(18.2.8) Let U be an open set in M arid a a real number > 0 such that

U x ] -a, a [ c dom(F,). Then, ,for each t E ] -a, a[, the mapping X H F,(x, t ) is a homeomorphism (of class C') of U onto an open subset U, of M, and the mapping (of class Cr) x H F,(x, - t ) is the inverse homeomorphism.

8

XVlll


This follows from (18.2.5) and (18.2.3.2). (18.2.9) For a C" vector field X , the numbers t + ( x ) and t - ( x ) may be finite: take, for example, M = R and X(x) = (x, x2). When this is the case, the " global " analog of (10.5.5) is the following proposition: (18.2.10) Let X be a vectorjeld on M of class c'( r 2 I ) , and x a point qf M such that t'(x) < + co. Then, .for each compact subset K of M. there exists E > 0 such that, for each t > t'(x) - E , the point F,(x, t ) does not lie in K . (In other words, the integral curve " ends outside " every conipact subset of M.) The proof is by contradiction. I f the assertion is false, there will exist an increasing sequence (t,) of real numbers strictly less than t + ( x ) , with t + ( x ) as limit and such that F,(x, 1,) E K for all n. Passing to a subsequence, we may assume that the sequence of points F,(.u, t,) converges to a point z E K. By virtue of (18.2.5), there exists an open neighborhood U of z in M and a real number a > 0 such that t + ( y )> a for all y E U. Now, if n is sufficiently large, we have t+(x)< t, + a and F,(x, 1,) E U, and therefore

t'(F,(x,

1,))

> a;

but by (18.2.3.1), t+(F,(x, 1,)) = t+(x)- t,, whence t + ( x ) > t, diction.

+ a, a contra-

There is an analogous result for t - , the statement of which we shall leave to the reader. In particular: (18.2.11) Let X be a vector j7eld qf class C' ( r 2 1) on M, \cith compact support (in particular, this condition will be automatically satisfied if the manifold M is compact). Then J(x) = R for all x E M. Let K be the support of X . For each x 4 K we have J(x) = R, and the integral curve t H F,(x, t ) is the constant function t w x . Hence, if x E K, the function F,(x, t ) takes no values outside K , and therefore by (18.2.10) we have J(x) = R in this case also. If X i s a vector field of class C" with compact support K, then for each t E R we have a diffeomorphism (18.2.11.l)

h, : X H F,(x, t )

of M onto M, such that

(18.2.11.2)

h , + , . = h,

0

11,. =

It,.

0

h,

2

FLOW OF A VECTOR FIELD

9

for all r, r ' E R and such that ho = I M . This follows from (18.2.8) (taking U = M and ] - a , a [ = R ) and (18.2.3.2). The 17, form a group, called the oiie-par.uiiic~/cvgroup of d~fSeotnorpltisms of M dLifi,ied by X . Notice that if .Y $ K , we have h , ( x ) = x for all t E R . Remarks (18.2.12) If M is a real-analytic manifold and X is an analytic vector field on M, then i t follows from the proof of (18.2.5) and from (10.7.5) that the flow F, is aiiul.y/ic in the open set dom(F,). (18.2.13) If I' is a C' solution of (18.1.1.1) in I and if X ( u ( t ) )# 0 at a point r E I, then 1' is an in~mersioiiat 1. But it can happen that D is an injective immersion of I in M but not a n embedding (16.9.9.3). (18.2.14) Suppose that the vector field X is of class C L , and put g,(x) = F,(x, 1 ) . Then y-,(g,(s)) = x for all sufficiently small / E R (18.2.3.2). If Y is any CLvector field on M, put (18.2.14.1)

YAX)

= T q c , J g - , ). Y(y,(.u)),

which is a tangent vector at s and is defined for all sufficiently small t E R. With this notation. we have the following interpretation of the Lie bracket [ X , Y]: (18.2.14.2)

cl

"K Yl(x) = Y,(x)l, = O d/

in the vector space T,(M), endowed with its canonical topology (12.13.2). To prove this. we may assume that M is a n open subset of R"; then the fields X , Ycan be written in the form j ' ~ ( yGO))) , and y ~ ( y H(y)), , where G and H are C' mappings of M into R". Hence, for a fixed x, we have

Consequently. for all sufficiently small (18.2.14.3)

g,(x) = x

r, we may write

+ t G ( x )+ / M I ) ,

where h(/) tends to 0 with / (8.6.2). On the other hand, if Dg,(y) denotes the deribative at J! of the function z ~ g , ( z )then , rt+Dg,(x) is the solution of the linear differential equation

U' = DG(g,(x)) U , (

10

XVlll


which reduces to the unit matrix I at t = 0 (10.7.3). Hence Dyl(x) = I + tDG(x)

(1 8.2.14.4)

+ tW(f),

where the matrix W ( t )tends to 0 with 1. I t follows that the right-hand side of (18.2.14.2) is of the form ( x , V(X)), where

V(X)= lim

I

rzo

r-0,

- (Dg- r(gi(X))

*

H(g,(x)) - H(.y)).

f

Now, for ( t , y ) close to (0, x), we have g-,(g,(y)) = y and therefore DY-,(Y,(X)) O Dy,(x) = 1 by differentiating. Hence V(x) = lim 1-0.1#0

I

- Dg-,(g,(x))* (H(yl (x))-

1

t

H(x)).

But, by virtue of (18.2.14.3) and (18.2.14.4), we have

(18.2.14.6)

+

Dg,(x) * H(x) = H(x) tDG(x). H(x)

+ fo,(t),

where o , ( t )and o,(t) tend to 0 with r. Since Dg-,(g,(x)), the inverse of Dy,(x), tends to I as t -+ 0, we obtain V(X) = D H ( x ) * G(x) - DG(x) H(x),

which proves our assertion (17.14.3.2). More generally, if Z is any C' tmsor.fir/dof type ( r , s) on M, and if we put

then we have the formula (1 8.2.14.8) i n the vector space (T:(M)), endowed with its canonical topology. This follows immediately from the uniqueness statement in (17.14.6), since the right-hand side of (18.2.14.8) evidently satisfies the conditions of (17.14.6) by virtue of ( 8.1.4).

2

FLOW OF A VECTOR FIELD

11

PROBLEMS

1.

Let F be a closed set in R" and

a frontier point of F . A vector u f 0 is said to be a n ( I if there exists a point b := a 4 pu. with p > -: 0, such that the (Euclidean) open ball with center h a n d radius p is contained in the complement of F. A vector v t T,,(R") is said to be tangent to F if (T,(V)~u) = 0 for all outward nornials u to F at a. A vector field X defined on an open neighborhood U of F is said to be rang~vit/o F a/ong F if, for each frontier point a of F, the vector X ( a ) is tangent to F. (1

oirtwrrrl normti/ t o F at the point

(a) Let I * s ( I ) he a C 1 curve, i.e., a C' mapping of an open interval I c R into R". For each I c I let 8(/)denote the (Euclidean) distance d(.v(t),F), and let y be a point of F such that i ~ . v ( t ) - y -c/(.r(t), F). Show that, if x ( t ) $ F and if u is a unit vector proportional to .Y - jf,then

( I f (/I,,) is a sequence of real numbers converging to 0 and if yc is a point of F such that d ( . t ( t t /iJ,J',,) c/(x(t 1 / I , , ) , F), observe that 6 ( t ) , i x ( t )- y , , ~ . ) (b) Let X be a Lipsc4iit:irrn vector ficld dcfincd o n an open neighborhood U of F (we identify T,(R") with R" by means of T.,).Suppose that X i s tangent to F along F. Show that there exists a constant h- ,.O such that, with the notation of (a),

-:

for each integral curve tp - x ( t ) of the vector field X. (Use (a), a n d the definition of a tangent vector to F at the point j,.) (c) Show that every integral curve of X which meets F is contained in F. (Argue by contradiction,and suppose that a n integral curve / - + x ( / ) satisfies x ( / , ) E F a n d x ( / ) 4 F for t n , / ( I )- cft sI)for sl t 5 s2.) ~

2.

Let X I , Xz be two C" vector fields on an open set U c R" (a) Let h , , hL be two real-valued functions of class C' o n U, a n d let Z(x) =h,(x)X,(x) I

A*

(xW2

(x),

Suppose that a n integral curve /-..u(t) of the vector field Z is defined for 0 5 t 5 1 and that .r(O) -~ . y o . For each positive integer n , consider the continuous function t + z n ( / )

12

XVlll


defined on [O. I ] by the following conditions: Z"

(0) xo 1

which is (0 5 k 2 n). Show that z. +x uniformly on [0, I]. (Consider the function affine-linear on each interval [ k / n , ( k t I ) / n ] and such that y n ( k / n ) z , ( k / n ) for 0 5 k 5 n, and use (10.5,1).) (b) Let Z - [ X I , X,]. Suppose that an integral curve t - - x ( t ) of the vector field Z is defined for 0 5 t 5 I and that ~ ( 0=) x, . For each positive integer I f , consider the continuous function t i t z , ( f ) defined on [0, I]by the following conditions: :

4k t I - = Vcp-') * ((cp

O

o>(t),

Wcp

O

W)).

If u is of class Cz, this local expression shows that u' is a mapping of class C' of I into T(M) and is a lifting of u ; hence we may define the vector u"(t), which belongs to the tangent space Tv,,,)(T(M)) to the manifold T(M), and thus U"

18

XVlll


is a continuous mapping of I into the differential manifold T(T(M)). In this way we can define successively the higher derivatives of u ; the derivative u(‘) is defined if u is of class c‘,and is a continuous mapping of I into T‘(M), where T‘(M) is the manifold defined inductively by the conditions T’(M) = T(M) and T‘(M) = T(T‘-’(M)) for r > 1.

(18.3.2) In order to define an (autonomous) second-order differential equation on M, we must therefore start with a vector field Z of class C ‘ ( r 2 0) on the tangent bundle T(M), and consider mappings u of class C2 of an open interval I c R into M which are such that the mapping t H u’(t) of I into T(M) is an integral curve of the vector field Z, or in other words satisfies the equation (18.3.2.1)

v”(t) = Z(u’(t))

for t E I. However, this is possible only if the vector field Z satisfies a supplementary condition. For u’ has to be a Ijfting of [;; that is to say, u ( t ) = o,(u’(t)); differentiating this relation, we obtain T(u) * E ( t ) = T(oM) . (T(u’) . € ( t ) ) , which may be written as u’(t) = T(oM) * u”(t), and by virtue of (18.3.2.1), this gives v ’ ( t ) = T(oM) +

Z(U’(t)).

Since we wish to have solutions satisfying arbitrary initial conditions, u’(t) must be able to take all values in T(M). Hence we must impose on the vector field Z the condition

f o r all h, E T(M). A vector field Z on T(M) satisfying this condition is called a uectorfielddejning a second-order diferentialequation. In particular, (18.3.2.2) implies that Z(h,) # OhX if h, # 0,. An (autonomous) second-order equation on M is therefore by definition a differential equation of the first order on T( M), defined by a vector field Z satisfying (18.3.2.2),and a solution of such an equation is a mapping u of an open interval I c R into M, of class C2, satisfying (18.3.2.1)for all t E I. In terms of a local chart (U, cp, n) of M, the tangent bundle T(M) is identified locally with q(U) x R”, and a vector field Z satisfying (18.3.2.2)has a local expression

wheref: cp(U) x R” +R”is a mappingofclassC‘;if~~isasolutionof(18.3.2.1),

3 SECOND-ORDER DIFFERENTIAL EQUATIONS

the function eq u ii t ion

ii

= ip

I'

:I

-+

(18.3.2.4)

ip(

19

U ) satisfies the second-order vector differential

D'u( I ) = f( i i ( t ), Du(t ) ) .

I t comes to the same thing to say that ;i function u of class C2 on an open iaterval J c R IS ;I solution of the second-order equation defined by Z , or to say that I' = oh., w . where w is an integral curve of the vector field Z , defined i n J . For we have r ' ( t ) = T(o,) . w ' ( t )= T(o,) . Z ( w ( r ) )= w ( t ) by (18.3.2.2). A niu.Yittiu1 solution of the second-order equation defined by Z is a solution which cannot be extended to a solution on a strictly larger interval; or, equivalently, a solulion of the form oM0 w , where w is a maxitnal integral curve of the vector field Z. DiKererential equations of higher orders on M are defined analogously.

(18.3.3) The results of (18.2)can of course be applied to second-order differential equations since what is involved is a particular case of integral curves of vector fields on T(M). Particular interest attaches to the set of solutions of a second-order differential equation for which a(0) is a giaen point U E M and o'(0) takes all possible values in the fibre 'I',(M). From (18.2) we deduce : (18.3.4) Lct Z bc N iwtorjickl of duss C' ( r 2 1) on T(M), dejtiing a secondordm clifliwntiul cqiiution.

( i ) For eaclr a E M, tliere exists a r u l number LY > 0 and a neighborhood U of'0, in T(M ) sucli that,,fbr each point h, E U, there is a solution t I+ y ( t , h,) of tlw stw)rid-ortior equation dejined by tlie vectorjeld Z which is dejned in the open iritcrrlal I-%, a [ c R and is sucli tliut y(0, h,) = x = oM(hx)and Y'(0,h,) = h,. Wc c'an nioreoiw choose a and U suck tliat, ,for each point x E U n M ( ii) (wlicro M is iclrntijiipd "itli the zero section ofT( M)) and each t o # 0 in ] - a , a [ , the t n a p p i q h,, Hy( t , , h,) is a liorncomorphisni of U n T,(M) onto an open neigliborliood V, i?f s in M , this lioi7it~oniorpliisr~i and its iniierse botli being c f l cluss C'. l f ' d is u di3uncc wliicli dejines the topology of M , we inuy assume also thut ( , f i ) rJisc~rlt o ) tlicrc cxisrs p > 0 such that V, contains the open ball with center x uiid rudiiis p,Jor cach x E U n M.

to the integral curves of the vector field Z : the num( i ) We apply (18.2.5) ber a and the neighborhood U are chosen so that U x ] - a , a [ is contained in the open set dom(F,).

20


(ii) Since the question is local as regards T(M), we may assume that M is an open set in R" and that the vector field Z has the local expression

(x,Y) H((x, Y), (Y1 f(X, Y))), where f is continuously differentiable in a neighborhood of ( a , 0) in M x R". Changing notation, let us write ( x o , yo) in place of h, and (u(t, s o Yo), v(t, xo 3 9

Yo))

in place of y ( r , hJ, so that the vector-valued function t H( u ( t ,xo Yo),

xo 7 Yo))

V(t,

9

is the solution of the system of two vector differential equations (18.3.4.1)

Y' = f(x, Y),

x' = y,

such that u(0, xo , yo) = x o , v(0, x o , yo) = y o , where u and v are of class C' in ] - a , a [ x U. Put &t, xo Yo) = D,f(u(t,xo Yo) v(t, xo Yo)), H i , x o , yo) = D, f(u(t, xo, yo), v(t, x o , yo)). 7

2

9

It follows from (10.7.3) that, for ( t , x o , yo) in a neighborhood of (0, a, 0) in R x R2", the functions (with values in -Y(R'')) a t , xo, yo) = D3 g ( 4 xo Yo), W ( t ,xo Yo) = D3 h(t, xo Yo). 7

7

9

where g(t, xo , yo) = (u(t, xo , yo) - xo) - tY0 h(t, xo , Yo) = V ( t , xo Yo) - Yo

3

1

form (as functions o f t ) the solution of the linear differential system

Z ' = w, (w'= A + B o w + ( t A + B ) , o

z

such that Z(0, x o , yo) = 0 and W(0,x o , yo) = 0. Application of Gronwall's lemma (10.5.1) to this system shows that we may choose go and

u = {(xo

9

Yo) : II xo

- a II < rr /I Yo I1 < r )

such that, for all I t I < a. and (x,, yo) E U, (18.3.4.2)

l l w ( ~ , ~ o ~ Y oS ) lbI I l I ,

I I Z ( ~ ~ ~ v o ~ YSbItI', o)II

where b > 0 is a constant. Now consider the mapping (18.3.4.3)

yo ~

~

( 7 xo 1 90YO)

- xo = 10

YO + g(to xo YO) 9

9

3

SECOND-ORDER DIFFERENTIAL EQUATIONS

for some t, # 0 in ] - Y, Y [ ? where value theorem, we have

Y

21

< a , ) .By virtue of (18.3.4.2) and the mean-

for l/yl11 < r and l/y211 < r , provided that Y < clo has been chosen sufficiently small $0 that brr < 4. Hence we see that (18.3.4.3) is a n injectire continuous mapping of the compact ball B’ : llyoll 5 + r into R”, hence is a honieoriiorphisrn of B’ o n t o its image (3.17.12). It remains to show that this image contains a ball with center 0 and radius i/i&>pcnriwit o / x o (for Ilx0 - a ( / < r ) . For this, we shall show that (10.1.1) can be applied t o the function ( X , Y ) H x - g(fo, xo, fi-IY)

defined in the product of the balls

Now. by virtue of (18.3.4.1), there exists a constant c such that

Take Y < Y, sufficiently small so that we have

CY

< &r. Then, for Y , and Y, in B”,

l I g ( ~ , , - v o . f i ‘ Y , ) -g ( ~ o 3 ~ o 3 ~ i5~3 Y l I Y2 l~-I lY,Il

by virtue of (18.3.4.4), and for X

E

B,

l I x - g ( ~ o , . ~ o , o ) 5l lA r l ~ o l< ? r l t o I .

Hence there exists a continuous mapping X H F( X) of B into B” such that

x = F ( X ) + g(to,xo, tOIF(X)) and this proves that the image of B’ under the mapping (18.3.4.3) contains B. Hence we obtain the assertion (ii) of (18.3.4) by taking p = & r l r o \ , a n d observing that in a nietrizable compact space all distances are uniformly equivalent (3.16.5).

22

XVlll


4. S P R A Y S A N D I S O C H R O N O U S SECOND-ORDER E Q U A T I O N S

(18.4.1) In the theory of second-order differential equations, one is often interested less in the solutions tw u ( t ) (which are iinending p a t h in M (16.27)) than in the iniagcJ of these solutions. These images are called the frujcctorics of the equation or of the vector field Z that delines the equation. The image under 1 1 of a compact interval [a. p] contained in the open interval of definition of 1%is called an arc ojtrujcctor~:and ( ' ( a ) and o ( P ) are called respectively the origiti and endpoint of the arc. The tangent vectors ~ ' ( 2and ) a'(P) at the origin and endpoint of the arc are well-defined. I f z l is a solution defined in an interval 1-3, r [ , then for each positive real number c the function 'c1~: tt-+c(cf), defined on ] - c - ' a , c-'Y[, has the samt iniagt as but is not, in general, a solution of the same second-order equation. In the case where M is an open set in R", t i is a solution of (18.3.2.4), and we have 11,

d ( t ) = CU'(Ct),

so that

11%is

w"(r)

= c2P"(ct),

a solution of the vector differential equation IV"(t)

= C2f(ll.(t), c-'ll*'(t))

for - c - ' a < t < F l u . We are therefore led, in this case, t o consider mappings f satisfying the condition (18.4.1.1)

f(x, ('y) = c2f(x. y)

for all c E R . T o express this condition in an intrinsic form for a n arbitrary differential manifold M, we introduce the mapping ni, : h , H c . h, of T ( M ) into T ( M ) : for a vector field Z on T ( M ) defining a second-order differential equation, the condition corresponding to (18.4.1. I ) is (18.4.1.2)

Z(m,(h,)) = c . (T(m,) . Z ( h , ) .

The vector field Z is said to be isodironoim, or to be a s p r q ' over M. if it satisfies this relation for all h, E T(M) and all c E R. The corresponding differential equation is also said to be isochronous (if the variable t represents time, the equation "does not depend on the unit of tncasiirC of time"). We remark that if Z is an isochronous field, then Z(0,) = OOx for all .YE M and Z ( h , ) # OhX whenever h, # 0,; if z' is a solution of the corresponding equation, then (>it/irr II is constant or v ' ( t ) # O,,,,) throughout ez'ery open interval I of definition of v (and if Z is of class C", then 1' is an itntncrsiorz of I in M). If u is a solution defined in ] - a , a [ , then the function t w r ( - f )

4

SPRAYS AND ISOCHRONOUS SECOND-ORDER EQUATIONS

23

is also a solution defined in the same interval. Since for each c E R the function t H~ ( +t 0 ) is also a solution, it follows that an arc of trajectory with origin a and endpoint b i n M is also an arc of trajectory with origin b and endpoint a, and hence a and h are also called the endpoints of the arc. (18.4.2) Let Z bc N spray o w r M of class C ' ( r 2 0). With the notation of (18.2.3), j i i r [ w l i rral number L' # 0 and cnch i>ectorh, E T( M), we have

(18.4.2.2)

for all t

E

F,(ch,, t ) = c F,( h, , ct) 9

J(ch,)

This follows directly from the definitions. The trajectories,which are the images of the solutions t ~ o ~ ( F ~ ( c th) ), , (where t E J(ch,)), are therefore independiwt ofthe choice of c # 0 for h, # O ; they are called the niuxinial trajcctorics of the second-order equation defined by Z (or of the vector field Z ) passing through the point x and tangent at this point to tlic (lircction defincct bv the cector h, (or any of the vectors ch,, c # 0). (18.4.3) I n what follows, we shall denote by R,, R,(M), or simply R, the set of points h, E T(M) such that the open interval J(h,) contains the closed unit interval [0, I]. I n the notation of (18.2.3), this is equivalent to tf(h,) > 1. Since t + is a lower semicontinuous function on T(M) (18.2.7), the set R is open i n T(M), and it follows from (18.4.2.1) that the relation h, E R implies ch, E R for all L' E [0, I ] (in other words, each of the sets R nT,(M) is star-shapcrl in the fiber T,(M); but it is not necessarily symmetrical with respect to 0, (18.4.9)). The mapping h, HoM(F,( h, , I)) is called the exponential t?iappin~j&fined bj, Z and is written h, Hexp,( h,), or simply h, H exp(h,); its value at h, is the value at t = 1 of the solution u of the second-order differential equation defined by Z which satisfies ~ ( 0=) x, u'(0) = h, E R. (18.4.4) For c w h h, E T(M), the fiinction 1 1 : tHexp(th,), defined in the intiwul J( h,), is thc niu.uinin1 solution of the second-order diflerential equation clefincd b)! Z. nhicli sutisfirs tlic initial conditions c(0) = x, ~ ' ( 0=) h,. For ea(li t # 0 iri J(h,), we h a w (18.4.4.1)

u'(t) = t-'F,(th,,

1).

24


For it follows from (18.4.2.2) that the function which takes the value h, at t = 0 and the value t-’F,(rh,, 1) at t # 0 in J(h,) is equal to FZ(h,, t ) for all t E J( h,). We recall also (18.2.3.1) that for to E J(h,) we have (18.4.4.2)

J(L”(t0)) = J(h,)

+ (-to).

(18.4.5) Let Z be a spray oiler M of class C’, nhcre r 2 1. Then the exponential map is a C’ mapping of R, into M. For each x E M, the tangent linear tnapping Tox(exp) at the point 0, of the zero section OM of T(M), when restricted to the tangent space Tox(T,(M)) of vertical cectors (identijed with T,(M)), is the identity mapping, and when restricted to the tangent space Tox(OM) to the zero section is the canonical bijection of this space onto T,(M).

The first assertion follows from (18.2.5), and the others follow from the facts that the restriction of exp to the zero section of T ( M ) is by definition the canonical bijection OM -+ M (the restriction of the projection OM

: T(M) -+ M),

and that, considering the linear mapping r : t H th, of R into the vector space T,(M), we have h, = Z,,,,(r’(t)) and in particular h, = r’(O),so thatTOx(exp)* h, (being equal to v’(O), where u(t) = exp(th,)) is equal to h, (18.4.4). Hence we obtain, by virtue of (10.2.5): (18.4.6) For each a E M and each neighborhood Uo of 0, in T(M), there exists an open neighborhood U c Uo of 0, in T(M) such that the mapping

h,

c-,( x , exp( h,))

is a homeomorphisni of class C‘, nith inverse of class C‘, of U onto an open neighborhood of ( a , a ) in M x M, and such that the mapping ha H exp( ha) is a homeomorphism of U nT,(M) onto an open neighborhood V of a in M. (18.4.7) Let a be a point of M and let U be a neighborhood of 0, it1 T ( M ) such that U nT.(M) is star-shaped and contained in R, and such that the mapping h,wexp(h,) is a homeomorphism of U n T , ( M ) onto an open neighborhood V of a in M. Then, f o r each y # a in V, the arc of trajectory which is the image of [0, I ] under the mappity t ~ e x p ( t h , ) ,bchere ha is the unique solution of exp(h,) = y in U n T,(M), is the unique arc of trajectory with origin a and endpoint y which is contained in V.

4

SPRAYS AND ISOCHRONOUS SECOND-ORDER EQUATIONS

25

An arc of trajectory L with origin a and endpoint is necessarily the image of an interval [0, 33 under a mapping t w e x p ( r h i ) , where h: E T,(M). We shall show that if L c V, then th: E U for 0 5 t 5 a and therefore ah; = h,, which will establish the uniqueness of the arc in question. Now, it is clear that rh:, E U for all sufficiently small t ; if the closed set of t E [0, a ] such that th; 4 U were not empty, it would havea least element /I> 0. But, by hypothesis, we have z = exp(/lh;); if $ : V U nT,,(M) is the inverse of exp, then th; = $(exp(th:)) tends to $ ( z ) E U as t +/l, contrary to the hypothesis that Phi 4 u. --+

It should nevertheless be remarked that in general there will exist other arcs of trajectories with origin a and endpoint a p o i n t y E V, but not contained ~ ~ / ? o /it1/ yV. In other words, the exponential mapping is not necessarily itljectirc on the whole of R. Likewise, we shall meet, in Chapter XIX, examples where it is not surjective; and finally, even if M is compact, the open set l2 is riot tiecessarilj, (lie M~holeofT( M).

Exanip Ies

+

(18.4.8) Take M to be the open interval R r = 10, co[of R, and identify the tangent bundle T ( M ) with M x R. Consider the spray a x , Y ) = ((x, J9, ( Y , J,2/x))

corresponding to the differential equation x" = x"/x in M. Here we have R = T(M), and for each x E M the exponential mapping restricted to TJM) = {x} x R is the diffeomorphism (s.j ' ) c t x e y of T,(M) onto M . This example will be generalized in Chapter XIX, and will justify the nomenclature of exponential mapping. (18.4.9) Let M

=

R and identify T ( M ) with R x R. Consider the spray

Z(x. .I%)= ( ( x ,j,),( 1 9 , .I,')), corresponding to the differential equation x" = .Y" in R. It is easily seen that in this case the set R is the set of points ( x ,y ) with j x < 1 , and that the exponential mapping, restricted to 2 ! n T,(R), is the mapping (.Y, j-)H.Y - log( I which is a diffeomorphism of R n T,r(R)onto R . 13).

(18.4.10) I n the preceding example, the vector field Z was translationinvariant. Consider then the quotient group T = R/Z, for which R is a covering, and let n : R T denote the canonical homomorphism. There exists a unique vector field Z , on the tangent bundle T(T) for which Z is a lifting (relative t o the morphism T ( n ) : T(R) --+T(T)); Z , is isochronous, a n d if I' is any solution of the differential equation defined by Z , then t w r ( r ( t ) ) is a solution of the dilyerential equation defined by Z,. We see therefore that although T is compact, R,, is not the whole of T. --+

26

XVlll


5. C O N V E X I T Y PROPERTIES O F I S O C H R O N O U S DIFFERENTIAL E Q UAT10NS

(18.5.1) Let M be a differential manifold and let Z be a spray of class C' >= 1) over M . The manifold M is said to be conwx relatice to Z if there exists a C' mapping s of M x M into the open set R, in T(M) satisfying the following conditions :

(r

In other words, given any two points x,, x2 of M, the mapping (18.5.1.2) of the interval [0, I ] into M is a path with origin xI and endpoint x2 and is the restriction to [0, I ] of a solution of the differential equation defined by 2 (18.4.4). A manifold which is convex relative to a spray is therefore necessarily connrcted (3.19.3). An open set U in M is said to be coni-ex relutii>eto Z if it is convex relative to the restriction of Z to T( U), or equivalently, if there exists a C' mapping s : U x U + Q z satisfying (18.5.1.1) and in addition such that (1 8.5.1.3) for all xl, x2 E U and all

tE

[0, I].

(18.5.2) Suppose that t k c open set U c M is concex relatire to Z. Then, with t h e notation of (18.5.1), f;)r m c h x E U, tile imuge of U under the niupping J' HS(X, J-) is t h e star-shaped open set T,( M)n Qzu( U), where Z, denotcJs the rc)striction of tlie i w t o r Jield Z to T(U). Tlic niupping y H S ( X , J,) is a homoniorpliisni of U onto this open s e t . und thc inuerse honieomorphisni is h, H exp( h,) ; both homc~oi,rorj~liisiiis ure qf' class C'. In purtirdur, ,for euch J, E U, tlw i i n u p of tlie interi:ul [0, I ] wider t h mapping t H exp( ts(x, J,)) is tlic oiilj>arc of'trujectorji with origin x and endpoint y contuined in U.

Since only the vector field Z , features in the statement of the proposition, we may assume that U = M. The relation exp(s(x, J')) = J! shows that the composition of the tangent linear mappings of h,Hexp(h,) and .YHS(X, J!) is the identity, and therefore each of them is bijectiiv (since the tangent spaces ThX(T, 0

for all

Z E

L.

Indeed, we have s(a, z ) E U, hence t * s(a, z) E U for 0 2 t 2 I , and therefore the function u ( t ) = exp(t . s(a, z)) is defined in an open interval I containing [0, 11. By the definition of L, we have z # a, hence s(a, z ) # 0, and therefore ~ ' ( 0#)0; and because f(x, 0) = 0, we have also v ' ( t ) # 0 for all f E I , since otherwise c would be constant in I . I t follows then from (18.5.3.2) that the function ft-+Q,(u(t)) has its second derivative everywhere > O in I . Since Q,(u(O)) = Q,(a) = 0 and D(Q, ' v ) ( O ) = DQ,(a) . ~ ' ( 0=) 0, we conclude (8.14.2) that Q,(c(r)) > 0 for 0 < t 2 1 , and in particular, that Q,(z) = Q,(u(l)) > 0. Since L is compact, there exists r > 0 and a neighborhood W" c W' of a such that, for all b E W", (18.5.3.4)

Q b ( c )2 r

>0

for all c E L.

5

CONVEXITY OF ISOCHRONOUS DIFFERENTIAL EQUATIONS

29

Now let p be such that 0 < p < r , and let W be the open neighborhood of .Y E W " such that Q(x, x) < p. For each b E W, let C, be the connected component of b in the open set defined by the inequality Q(h, x) < p. It follows from (18.5.3.4) that cb cannot intersect the frontier L of W', and hence (3.19.9) we have C, c W'. Now let xl,x2 be two points of C,. Since C, c W', we have s(x,, x 2 ) E U and therefore t . s(xl, x2)E U for 0 5 t 5 I . The function t H q ( t ) = Q,(exp(t. s ( x , , x2))) is therefore defined in the interval [0, I ] , and its second derivative cp"(t) is 2 0 for all t E [O, I ] by virtue of (18.5.3.2). Hence we have the inequality a consisting of the points

(18.5.3.5)

s

for 0 5 t I . (For the function $(t) = ~ ( t -) ( ( I - t)q(O) + tcp(l)) has second derivative $ " ( t ) = cp"(t) 2 0 in [0, I], and i,b(O) = I)( 1) = 0 ; consequently the derivative $'(t) is increasing, and if we had $(%) > 0 for some x E 30, 1 [, we should have $'(/I) > 0 for s a n e p E [O, X I , whence $'(t) > 0 for t E [p, I ] and so $ ( I ) > $(/I) > 0, a contradiction.) From (18.5.3.5) we deduce that

(18.5.3.6) for all t

E

[0, I ] . by the definition of the mapping s. This implies that

exp(t s ( x , , x2))E

cb

for 0 5 t 5 I . by (3.19.7) and the definition of a connected component, and proves that C, is convex relative to Z . Q.E.D. Propo4tion (18.5.3) has the following corollary: (18.5.4) U i i c l w t k c hypotl1esc.r. qf (18.5.3), if K is an)! conipact subset of M, tlwrr csists u Jiiitc cmwiiiy o/' K bjs open sc'ts C i ( 1 5 i 5 ni) which, together w i t h ull thcir fiiiitr iiitcrscctioiis, ure c o i i w x rclatiw to Z.

Let d be a distance defining the topology of M . For each a E K , i t follows fro01 (18.3.4(ii))and (18.4.7) that there exists r, > 0 such that, for each point .Y satisfying ( / ( ( I , .u) < t r , and each x' satisfying d(x, x') < f r o , there is only one arc of tra.jcctory with origin x and endpoint x' contained in the open ball with center x and radius j r c , , Cover K by a finite number of open balls B, ( 1 5 h 5 i l l ) w i t h centers a, and radii t r a k . Then there exists r' > 0 such that r' < J r o kfor I 5 k 5 111, and such that for each x E K the open ball B(x) with center s and radius r ' is contained in one of the balls B, (3.16.6). By definition, for each point x' E B(x), there is only one arc of trajectory with

30

XVlll


origin x and endpoint x' contained in B(x). This being so, it follows from (18.5.3) that for each b E K there exists an open set Cbcontaining b which is convex relative to Z and of diameter < r'; moreover, if E is the intersection of a finite number of these open sets C,, ( 1 5 k 5 / I ) , then for each k and each pair of points x, x' of E, there exists one and only one arc of trajectory L, with origin ?c and endpoint x' contained in Cbr,(18.5.2); but since all these arcs L, ( 1 2 k _I p ) have diameter < r', they are all the same, and therefore E is convex (18.5.1.3). Finally, K can be covered by a finite number of the open sets Cbh. We shall see in (20.17.5) that there exist functions Q with the properties of (18.5.3) on any differential manifold M.

6. GEODESICS

O F A CONNECTION

(18.6.1) Let C be a linear connection on a differential manifold M (17.18.1). For each x E M and each h, E T,( M), the vector is such that T(o,) . G(h,)

=

h,; furthermore, for each c E R , we have

G(c . h,) = C,(C . h, , c . h,)

=c

*

C,(h,, c . h,)

by virtue of (17.16.3.3), and T(m,.) . G( h,)

= C,(

h, , c . h,)

by (17.16.3.5). I t follows from these formulas that h,-G(h,) is a sprajs over M . This spray is called the geodesic spray of the linear connection C; the sohrioirs of the second-order differential equation on M defined by G are called the geodcsics of the connection C; and the trajectories (resp. arcs qf trajectories) of this equation are called the geodesic trajectories (resp. geodesic urcs) of C. The differential equation of the geodesics of C is therefore (18.6.1.1)

u"(1 ) = C",,,(!If(1 ) ,

tl'( I ) ) ,

which, by virtue of (17.17.2.l), can also be written in the form (18.6.1.2) with the notation of (18.1.1).

V E ' v'

=0

6

GEODESICS OF A CONNECTION

31

Relative to a chart (U, cp, 11) of M for which the local expression of the connection C is given by (17.16.4.1),this equation is equivalent to the following system of (nonlihear) second-order differential equations:

d2ui

(18.6.1.3) where

11 =

--

(It

cp

I'

dLll + C r;&)- =o clt clt dll"

( I 2 i 5 n),

11.1

has components

21' =

r ( I 5 i 2 n).

cp'

(18.6.2) (i) In order that tM'o connections C and C' on M should haile the

sanie geodesic sprq', it is necessary and siificient that the bilinear nrorphisni

B : T(M) 0T(M) + T(M), the d@wnce of C and C' (17.16.6), should be antisjwmctric. (ii) Giren an)' connection C on M, there exists a Liiiique connection C' on M i4hiclt has zero torsion (17.20.6) and the satne geodesic spray as C. From (17.16.6.3) we have BJh, h,) = ~hx(C,(h,, h,) 1

-

C.L(h, h,)) 9

and to say that B is antisymmetric means that B,,(h,, h,) = 0 for all h, E T(M), whence (i) follows. Furthermore, if t and t ' are the torsions of C and C', respectively, then (17.20.7.2) we have t' - t = 2B if B is antisymmetric; hence the only connection C' having the same geodesic spray as C and zero torsion is obtained by taking B = - i t . (18.6.3) The geodesics of a connection C can be interpreted in a different tramport relative to C. Consider a way by introducing the notion of parull~~l path I' : [I,/I] -+ M of class C' in M (resp. an unending path c : 1 + M of class C', where I is an opin interval in R). Then a parallel transport along the path (resp. along t h e unending path L.) is by definition a C' mapping w of [ci, B] (resp. I ) into T(M) such that (18.6.3.1)

oM(~ ( t )=)It([)

and

V E. w

=0

in

[a, /I] (resp.

in I),

which by definition (17.17.2.1) is equivalent to

(18.6.3.2)

o M ( W ( t ) )= Z ) ( f )

and

W'(f) =

Cv(,)(L"(f),W ( f ) ) .

(18.6.4) For cacti path I' of class C' in M, &fined on [a, 83, and each tangent M), there exists a unique parallel transport w along the w c f o r h,,(,, E Tu(a,( path L., such that w(ci) = h,,,, .

32

XVlll


Let U be a relatively compact open neighborhood in M of the compact set L = v ( [ s ( , PI). By virtue of (16.25.1) there exists a n embedding of U in RN for N sufficiently large, and we may assume therefore that M = U c RN. Then by (17.18.5) there exists a n open set V in RN such that M is a closed submanifold of V, and a connection on V which extends the given connection on M . In view of (18.2.4), we are reduced to proving (18.6.4) in the case where M is an open set in R". But then, in the notation of (17.16.2), we may write w ( t ) = ( r ( t ) ,u(t)), where u(t) E R" and u is a solution of the vector differential equation

(the local expression of (18.6.3.2)) which is homogeneous linear, by the definition of a connection. The result therefore follows from (10.6.3). The mapping w satisfying the conditions of (18.6.4) is called the parallrl transport of tke cector h,,,, along tire path and w ( t ) is said to be obtained by parallel transport of the vector h,(,, f r o m z'(r)to ~ ( t along ) 11. The same method proves: L?,

(18.6.5) Zf(ej)ljjjn is a basis of the tangent space TL,(a,iM)and & f o r each index j , w j is the parallel transport of the i'ector ej along the path z', then f o r each t E [ r , 81 the w j ( t ) (1 2 j 2 n ) f o r m a basis ofTUc,,(M).

This is a consequence of (10.8.4). We may therefore say that the wj form a basis of the vector space of parallel transports along 21. (18.6.6) If now we compare equations (18.6.3.2) and (18.6.1.2), we see that every geodesic of the connection C may be defined as a C2 mapping I' of a n open interval I c R into M such that z'' : I + T(M)is a parallel transport along the unending path and that for any two points s < t in I the tangent vector I?'( 1) is obtained by parallel transport of the tangent vector r'(s) along 1'. 11,

Remarks

Let f : M, + M be a local difeonlorphistn and let C , be the connection on M, which is the inverse imageof C under f (17.18.6). Then it follows immediately from the definitions that, if is a n unending path in M I , and wI a parallel transport along v1 (relative to C , ) , then f w , is a parallel transport along f 0 If u I is a geodesic relative to C,, thenf'u 1 9 , is a geodesic relative t o C. (18.6.7)

ill

0

19,.

GEODESICS OF A CONNECTION

6

33

(18.6.8) More generally, we may define a rensor parallel rransport of type s) along I' as a / f l i n g Z of class C' of to T:(M), such that V, * Z = 0 (17.18.3). With the notation of (18.6.5), it follows from (17.18.2) that if, for each f E [ Y , p]. (wT(t))( I I j g n ) is the basis of T,(,,(M)* dual to the basis ( w j ( t ) )of T,,,,,(M), then the liftings (Y,

17

WT,@ W T L @ . . ' @

wj*,@ W i I @

Wil@...@

wi,

form a basis of the space of tensor parallel transports of type ( r , s) along u.

PROBLEMS 1.

Show that. i f j is the canonical involution of T(T(M)) (Section 16.20, Problem 2(b)), then j ( u " ) I,'' for every C' mapping I' : 1 + M (where I is an open interval in R). ~

2.

~

Let E be a vector bundle with base B and projection x . A real-valued functionfon E is said to be Iro/myr,ieorrs o/c/egwe I' (where I' is an integer -10) if the restriction off' to each fiber EDis a lio,iroge/ieorcs/ioic/io,rofrkcyree r , i.e., i f f ( c . U b ) = crj'(ub)for all uh E Eh and all c E R. A homogeneous function of degree 0 is therefore a function of the form y x . where g is a real-valued function on B. A vector field Z o n E is said to be /iouiogemou.s of'dqree r if Z ( C f Uh)

~

cr-

(TO)lc) ' z ( u h ) )

I

for all uhE Ehand all c E R. (a) When B is a n open set in R", and E = B

%

R", in order that

a x , Y) =: (g(.u, Y), f(.w, Y))

should be homogeneous of degree g(x, cy)

~~

r,

it is necessary and sufficient that

c'- 'g(.t-, y)

and

f(.u,

CY)

7

c'f(.r, y).

( b ) I f Z is a vector tield o n E, homogencous of degree I' and of class C o , and i f f is a rcal-valued function on E, homogeneous of degree s and of class C ' , then ./is a honiogeneous function of degree I' ~t s -- 1. Conversely, let Z be a Co vector field o n E such that ( i ) for every C' homogeneous function /of degree 0 on E, the function f& . / is homogeneous o f degree I' - 1 ; (ii) for each C 1 section s* of the dual bundle E*, if we define f'by f(UJ

,~S*(O),Uh"

(so that/is of class C ' and homogeneous of degree I on E), then 8, . f i s homogeneous of degree r . Undcr these conditions. show that L is homogeneous of degree r . A homogeneous vector field of degree 0 consists o f vertical vectors i n T(E). ( c ) I f % , , Z r are c" homogeneous vector fields o n E, of degrees I ' ~ r. 2 , respectively, then [ Z , ,Z L ]is homogeneous of degree I ' ' I r 2 - 1. ( d ) ILet y be a B-niorphism of E"' into C. With the notation o f Section 16.19, Probleni 11, show that the mapping

'.'

Ub'~-~(Uh,g(Uhl:'Uh~I7

u,))

34

XVlll


is a field o f vertical tangent vectors on E, homogeneous o f degree i s denoted by g". (e) For each u, c E,, , let

I'.

This vector field

H i s ;I C" field o f vertical vectors on E, honiogeneous of degree I and i s sometimes called the Liorruil/e fie/t/. I n order that a red-valued function / o f class C' on E should he hoiiiogeneous of degree r , it i s necessary and sutficicnt that H,r . f = r/'(Eu/er's i t / w / i / . v ) . For ;I vector field Z o f class C ' on E to be homogeneous o f degree I', it i s necessary and sufficient that [ H , Z] ( r I)Z. (f) Suppose that E T(B). I f 6 : h,- - ( h , , h,) is the diagonal B-niorphisni o f T(B) into T(B) . ,T(B), then we have H h - 6 . With the notiition and definitions o f Section 17.19, I~rohlems2. 3 and 4, show that ~

~

~~

[i,,

3.

H,11

=

i,,

H7iJ [J, N ] li,ll (/,I

J K I L 0. - [ H , J ] J, y-

i,,

[(IJ,

( I J 1 [ i J , i1lI

HI,]

O

Let X he a C' vector field on M. Show that there exists u unique vector field f o n T ( M ) such that Ohx

(0.1 .I")

(o,?. (if)(h.,)

for every function fofclass C 2 on M (the notation is th:il of Section 16.20, Problem 2). ,? is a Iioniogeneous vector lield of degree I . and we havc f j T ( X ) and T(o,,) f X ohl . The held f is called the ctrrrwrical //'//i//go f A'. ~

4.

Let G be the geodesic spray of a linear conncction C on M . L.ct g be ;I covarinnt tensor licld of order r and class C ' on M, and for each h, L T ( M ) put g(h.,)

g(s), h,

h,

...

h1

h.,

..'

h,

show that

(O,.g)(h.O

Lg,h,s

(whcrc thcrt are I ' 1 I f x l o r s h, in the tensor product). (Ci)nsidcr t h e lxirticti1:ir case where g w , ~ 1 )".~. . 'sw,.. the w , being ditferential I-forms on M . ) 8 .

5.

Let M be ii ditfercntial manifold and y an M-morphisni o f TI;(M) ( T ( M ) ) Q ' into T ( M ) . Let G be the geodesic spray of :I torsion-free connection C on M . I n Problem 2(d) we detined the field y v o f vertical vectors on T( M), which i s homogeneous of degree I'. Define also g'Yh.,)

C.,(h.,,g(h./

h.,'

. . . ' . h,))

for all h , v i - T ( M ) ;then g" i s a homogeneous vector field on T(M), o f degree Finally, let r 'g denote the M-morphism o f Tb+2(M)into T ( M ) defined by (r.g)(klc

Show that

k2i_ . . . # .k r 1 2 ) ( r . ( g . ( k l

'"(.

k,)

'

k,+,)).k,,Z

I'

I 1.

7 ONE-PARAMETER FAMILIES OF GEODESICS A N D JACOB1 FIELDS

35

X be a C ' vector field on a differential manifold M , and let g,(x) F x ( x , I ) , in the notation of (18.2.2). L.et C be a linear connection on M and let Z be a C ' tensor field on M . For each I J(.t-). let Z, be the parallel transport along the curves -.c/,(s) which takes the value Z(,q,(.I-)) at .\ I . Show that

6 . Let

~

t -

(T, 7.

'

ZKr)

tl

- Z,(-Y)l,,O

tlt

Let M he ;I dill'crcntial manifold, C ;I lincar connection on M ; let

LI

be a point of M and

let U be an open neighborhood of O,, i n T,(M) such that the restriction ofexp, to U is a diffcomorphism o f CI onto an open neighborhood V of t i i n M (18.4.6). For cach vector

h t U. let %h denote the C" vector field on V for which Zh(exp u) i s the tangent vector obtained by parallel transport of h from I / to exp(u) along the path / i--pexp,(iu). For each rcal-valiicil function/of class C1 on M , w'e have

(cf. (18.1.2.5)). Dedticc that, i f bl i s ii real-analytic manifold and i f f is analytic i n a neighborhood of(/, then in a neighborhood of I 0 we have

8.

On T ( R 2 ) (idcntilicd with H 2 . K ' ) consider the spray Z ( x , y ) ((x, y ) , ( y , O)), the ( 0 ) ) .of which trajectories o f which are the lines i n R'. Let M be thc cylinder R'/(Z H' is :L covering. xiid let Z,, be the unique spray over M which lifts to Z. Give an example 01' two open sets i n M which arc convex relatibe t o Zo but whose intersection is not connected ,j

9.

L e t G be ;I spray of cI;iss C " OKI' ii differential manifold M, and let B be an antisymmetric hilincar M-morphism o f T(M) 1 T(M) into T(M). Show that there exists on M ii unique linear connection C with torsion equal to 13, and geodesic spray C . ( I f C' i s ;I linear connection on M with torsion equal to B (17.20.7), show that there chists :I syninietric bilincar M-morphism A : T(M) ', T(M) --T(M) such that O'(h,) C . { ( h , . h , ) h ( h , , A ( h , . h,)) ( i n the notation o f Section 16.19, Problem 11).

7. ONE-PARAMETER FAMILIES O F GEODESICS A N D JACOB1 FIELDS

(18.7.1) Let El and E , denote the vector fields (called cunonical) on R2 delined by t(

I ] , ,

L ) ( E , ( f (,2.) )

= (1,

01,

t( , , , I

J W f , ,( 2 ) ) = (0, 1 )

Given two open intcrvals I , J in R, a mapping f ' : I x J --t M of class C' ( r 2 I ) , with values in a differential manifold M , is often called a oiw/ m m i i c t w / w i i / j ~of ci/rw\. (Consider for each 5 E J the mapping / H f ( t ,5 )

36

XVlll


of I into M ; the function f may be regarded as describing the “variation” of this family of “curves.”) When there is no risk of confusion, we shall write, by abuse of notation (generalizing the notation of (18.1.2.3)) (18.7.1.1)

,f;(t,5 ) = T ( f

.

t),

f i ( t , 5 ) = T(f)E2(t, 0,

so that the mappings ( f , t ) H f : ( t ,5 ) and (I, t ) w f ; ( t5,) are C - ’ mappings of I x J into T(M) which lift5 (18.7.2) Let C be a connection on M , and for each lifting w : I x J + T ( M ) o f f , of class C‘ ( r 2 3), w i f e (by abuse of notation)

v,

v,.

w=

v,,

V , . (V, . w ) - V , . ( V , . w ) = ( v ( f

A

w = V E ,* w ,

*

.w

(1 7.17.3). Then we haue

(18.7.2.1)

,f;)) . w,

where r is the curvature of C .

This is a particular case of (17.20.4.1) since [E,, E,] = 0. (18.7.3) With the notation of (18.7.1), and assuming M endowed with a connection C , the mappingf is said to be a one-paratnetcr family of geode.sic.r of M iffis of class C‘ with r 2 3 and if, for each 5: E J, t h e mapping t hf ( r , 5 ) is a geodesic. (18.7.4) If the connection C is torsion-free and i f f is a one-parat~ieterfamil~ of geodesics in M, then

v, . (v,.f ; ) = ( r . ( f ; A fi))

(18.7.4.1) for all ( t ,

5)E I

x

Since [ E l , E,]

.f ;

J. = 0,

the formula (17.20.6.4) gives us V ; f ; = v,*.f;

(1 8.7.4.2)

since the torsion is zero. Using (18.7.2.1), we obtain

v, (v,‘ f ; , = v, (v, f i ) = v, (v,‘f;)-k ’

‘

’

‘

( V .

(f;A\:))

‘f;.

But since f is a family of geodesics, we have V , .f;= 0 (18.6.1.2), whence the result.


37

(18.7.5) Throughout the remainder of this section we shall assume that the connection C is /orsioii~/ic,c. Under the hypotheses of (18.7.4), fix a valuc r E J and put ~ ( t =) / ' ( t , 9). so that I ' is ii geodesic. If we then put w ( t ) = , / J t , r ) , the mapping w of I into T( M) is of class C2 iind satisfies the lioniogeiic~or/sliiieur equation (18.7.5.1)

v, (v, '

'

W ) = (I" (1"

A W ) ) ' 1".

For each geodesic of C. defined on the interval 1, a Cz mapping w of I into T ( M ) which satisfies (18.7.5.1) is called ;I Jucohi,ficlddong thegeodc.sic 1 1 . I f P , = I ' ; q) is a geodesic obtained from 1' by an attine change of parameter ' p : t - i t + 11. then it is immediately verified that w (p is a Jacobi field along r , , for each Jacobi field w along P. 11

(18.7.6) For c d i iiiiircoiistuiit gooticsic 11 o/ C , tlefriictl o i l an opeii ititrriwl I c R, c d i poitit x E I urid ouch puir 04rc'ctors h, k E T",,,(M), there exists u iiiiiqiio Jucohi j c l d w uloiig 1' which is dcjiiird oil I uiid satisfies W(Y)

= h,

(V, . w ) ( Y ) = k.

Let ( e J lz i z r i be ;I basis of the veclor space T I , , = ) ( M and ) , let u idenote the parallel transport o f ei along the path (18.6.4).Then for each t E I the vectors u , ( t ) ~,,,,T, ( M ) form a h i . s of T,,,,)(M)(18.6.5), and every C2 mapping 11

w : I --+ T ( M) which lifts

I'

can be written uniquely in the form w

c wi u i , n

=

i= I

where the M ' , are real-valued functions of class C2 defined on I , by virtue of Cramer's rule. By definition, we have V, . u i= 0 (18.6.3.1), and therefore it follows from the rule (17.17.2.5) for calculating covariant derivatives that " V,' w = c w ; u , , V;(V; w)=cw:)uI. r=o

r=O

N o w we may a s u m e that the basis ( e l )ha5 been chosen so that en = D'(s(), since 1' is not constant. Since I' is a geodesic, it follows from (18.6.6) that u,,(t)= i " ( t ) for all t E 1. If we put

then the equation (18.7.5.1) for w is equivalent to the system of n secondorder differential equations n

(18.7.6.1)

38

XVlll


We have here a system of homogeneous linear equations of the second order with C’ coefficients; hence the result follows from (10.6.3). Furthermore ((10.7.4) and (10.8.4)): (18.7.7) Suppose that M is conrrcctcd. Thrn the Jacobi ficlds along a nonconstant geodesic of C dejintd on I are of‘class C“ arid fbrni a ivctor space of diimvisioti 2 dim( M ) . Those diich ranish at a g i w i point u E I form a ccctor spacc OJ dimension rqual to dim( M ) .

(18.7.8) We have derived the notion of a Jacobi field by considering a oneparameter family of geodesics, and the Jacobi field derived from this family along one of the geodesics appears as the “derivative with respect to the parameter.’’ I n fact, we shall see that e z w y Jacobi field which vanishes at one point at least can be obtained i n this way, and in a canonical fashion (cf. Problem 3). (18.7.9) Let a bc n point of’ M , Ict h,l be a nonzero ivctor in T,( M), arid lct o bt the geodesic t H exp(th,) of C, cle$ncd in an opcn intmml I o j R containing the point 0. For each opiw ititercal I , of R containing 0 and such that 1, c I , and euch vector k, E T,( M), therc exists an open interoal J c R containing 0 such that, ,for each 4 E J, the geodesic t I+ exp(t(h, + (k,)) = . j ( t , 5 ) is defined on I , . The Jucobi,field t ~ . f :t , (0 ) is then the irniipe Jacobijtdd w along 11 in I , sLIcIi that ~ ( 0=)0, and (V, . w ) ( O ) = k,. The existence of J follows immediately from (18.2.5). and it is clear thatf is a one-parameter family of geodesics such that ,f(O, 4 ) = a. Consequently, if we put w ( t ) = f ; ( t ?O), then we have w ( 0 ) = 0,. On the other hand, if we put g ( t , 0= t(h,

+ 5k,JE T,(W

and if we denote by exp,l the restriction of exp to R n T,(M), then

f ; V ? 5 ) = T,,,,&XP,) and therefore, for (18.7.9.1)

. Ti/,&k,)

t = 0,

w ( t ) = T,h,(eXP,)

*

T,a(tk,)

= t(Tfha(exp,)‘Tz:(k,)).

It remains to verify that (V, . w)(O) = k,. Using the formula (17.17.2.5), this reduces to the fact that TOa(exp,)is the identity mapping (18.4.5). (18.7.10) The properties of Jacobi fields which vanish at a point a of a geodesic u : t wexp(th,) are closely related to the properties of the restriction exp, of the exponential mapping at a point h, E R n T,(M):

7 ONE-PARAMETER FAMILIES OF GEODESICS AND JACOB1 FIELDS

39

(18.7.11) Fbr eucli point h, E R n T,l(M), the rank (16.5.3) of the niapping exp, at tlic point h, is q u a 1 to dim,(M ) - s, where s is the dinirnsion of the i'cctor spuw o j Jucohijelds along t k e ycodcsic i'h, : t Hexp(th,) nqhichwnish at t = 0 unrl t = 1.

For it follows from (18.7.9.1) that as w runs through the n-dimensional vector space J of Jacobi fields along I,,,', which vanish at t = 0, the mapping w H w( I ) is of rank .s, because the mapping w H (V, . w ) ( 0 )is an isomorphism of J onto T,( M), by virtue of (18.7.6). (18.7.12) The points h, E T,(M) n 0 at which Tha(exp,)is not invertible are r of exp,, and the images exp(h,) of these points are called the s i t ~ p / u poitits called the conjii{]atc'sof a on the geodesic rh,, . The set of nonsingular points of exp, is open i n TJM) and contains the origin 0, (18.4.5), and the restriction of exp, to this open set is a local rliflkmorpliisni onto an open set in M. But it should be observed that, for each h, E R n T,(M), there may exist no cotljugatc of a on i'h, (in other words, all the points of R n TL,( M) may be nonsingular) even though the mapping exp,, is not injective.

(18.7.13) I n general, if M is a coivring of M,, with projection n : M + M,, and if C, is :I connection on M, and C its i n i w x i ininye on M (17.18.6), then the geodesics for C, are the mappings 7r I , , where is a geodesic for C (16.28.2), and every Jacobi field along n is of the form T ( n ) w , where w is a Jacobi field along If no pair of points is conjugate on P, the same is true for 7r 1'. Take, for example, M = R and M, =T, and identify T(M) with M x R = R 2 . Let C be the connection on M defined by 0

5

11

13

0

it.

0

C,((X, y), (x, 2 ) ) = (b,Y), (Y, 0));

then the geodesic field is a x , Y) = ( ( x .Y)? (Y, 0))

and the geodesics are therefore the solutions of the differential equation D2.r = 0. Since C is translation-invariant, it is the inverse image of a connection C, on T. I t i 4 clear that for C we have R = T(M), and the exponential mapping is (s.y ) H s + y. so that exp, is a diffeomorphism for each a E M. Consequently the exponential for C, is also a local diffeomorphism, but evidently exp,,, is not injective, for any a, E M, .

40

XVlll


(18.7.14) Conversely, it can happen that exp, admits singular points, but is injective on T,(M) n R : in other words, a point conjugate to a on a geodesic ZJ is not necessarily a point through which there passes a second geodesic passing through a with a direction distinct from that of but is a point b a t which certain geodesics passing through a , with directions infinitesimally close” to that oft‘, have a distance from b which is “infinitesimally small to a higher order.” I % ,

“

(18.7.1 5) Let r be a geodesic for C , defined on an open interral I c R, and Ict [ a , /I] be a compact interral contained in I sircli that ci # /Iand a = t‘(z) and b = c(/I) are not conjugate points on r. Then, giceti an), tb.0 tangent rectors h E Tu(z)(M)and k E T,,B,(M), there exists a unique Jacobi field w along t‘ such that w ( a) = h and w(P) = k.

Clearly there is no loss of generality in assuming that a = 0 and /I = 1. Then we have ~ ( t=) exp(th’) for some h’ E Tu(o)(M),and since by hypothesis the rank of expo(,, is equal to ti at the point h’, there exists, by virtue of (18.7.9.1), a Jacobi field w1 along L’ such that wl(0) = 0, and wl( 1) = k. Likewise, we have P ( 1 - t ) = exp(( I - t ) k ’ ) for some k’ E To(l,( M), and since the rank of exp,,,) is equal to n at the point k’, there exists a Jacobi field w 2 along such that w,( I ) = 0, and w J 0 ) = h. The Jacobi field wI w 2 then has the desired properties, and is the only one because no Jacobi field # 0 can vanish simultaneously at the points 0 and 1.

+

13

PROBLEMS 1. Let M be a differential manifold, X a C’ vector field on M, and x’the canonical lifting of X to T(M) (Section 18.6, Problem 3).

(a) Let I‘ : I + M be an integral curve of Xand let w be a /i/fityof C ’ . Then we have T(o5,) ( w ’ ( f ) ?(w(f))) = 0

I‘ to

T(M), of class

~

for all t E I . Hence there exists a unique vector u(f) E TL(,,(M)such that X(w(r), u ( t ) )= w ‘ ( t ) - B(w(1))

for each IEI, in the notation of Section 16.19, Problem 11. The mapping f - u ( f ) (which is a lifting of I‘ to T(M)) is called the Lie clerirnfii~eo f w wiflr respect f o X , and is denoted by 0, w . For each C’ function rp : 1 -*R,we have

.

ex (9w)

9‘w

+ 9(ex . w ) .

For each C’ vector lield Y on M, we have

ex . ( Y . u ) [x,Y I

c.


41

(b) A C' lifting w 01 u to T(M) is said to be X-invariant if ex 0 w = 0. For each to E I and each tangent vector h E T,,,O)(M), there exists a unique X-invariant lifting w of u such that w(ro)= h. (c) I f J is a neighborhood of 0 in R and if 1':I x J + M is a C' mapping such that t - f ( t , 6) is an integral curve of Xfor each 6 E J, then t +-.f;(f, 0) is an X-invariant lifting of t + +f(t, 0) to T(M). 2.

Let M be a differential manifold, C a torsion-free linear connection on M, and G the geodesic spray of C. (a) Let u be a nonconstant geodesic of C. In order that w should be a Jacobi field along u , it is necessary and sufficient that w = T(ohl) 0 z, where z is a C-invariant lifting olu' to T(T(M)) (Problem 1). (b) Let X be a C' vector field on M whose canonical lifting XQSection18.6, Problem 3) satisties the condition [X,GI 0. Then, for each nonconstant geodesic v of C, the field X u u is a Jacobi field along u . (c) Let u be a nonconstant geodesic of C and let w be a C' lifting of u to T(M). Put ~

wv = h(u', w ) ,

WE' =

C(u', w ) ;

these are liftings of u' to T(T(M)). Show that every C' lifting I of u' to T(T(M)) can be expressed uniquely in the form z = w'; wy, where w , and w 2 are two liftings of u to T(M). Moreover, for each lifting w of u to T(M), of class C', we have

+

ec. wv = (0, . w ) v

-

W ~ ,

+ ((r . ( w A u'))

OG * wH = (V, . w)"

*

u')~.

3. Let M be a differential manifold, C a linear connection on M, and let H be the Liouville field on T(M) (Section 18.6, Problem 2(e)). (a) Consider the C" mapping c : T(exp) 0 H : !2 +T(M), which is a lifting of the mapping exp : R M. Show that O H . c = c. (b) Let L be a C*vector field on T(M), homogeneous of degree 1 (Section 18.6,Problem 2). Let s = T(exp) 0 L, which is a Cz mapping of R into T(M). Show that --f

VH ' (VH 's) = (r

(CA

s)) ' c

+ VH 's.

(c) Let a be a point of M, let h, be a nonzero vector in T.(M), and put u ( t ) = rh, for t E R. A lifting z of u to T(T(M)) is said to be homogeneous of degree 1 if z(ct) = T(m,) . z ( t ) for all c E R and all f E R. A necessary and sufficient condition for this is that z(r)

=W

t ) , h:)

+ G ( u ( t ) ,h:)

for two vectors h:, h," in T,,(M). An equivalent condition is that there should exist a vector field L on T(M), homogeneous of degree 1, and such that z = L u. (d) Let z be a lifting of u to T(T(M)) which is homogeneous of degree 1. Show that w = T(exp) z is a Jacobi field along the geodesic t -exp(th.) of C and that 0

0

z(t)

= h(th.,

t(Vsco, w ) ) -tC, (tho, N O ) ) .

In particular, the Jacobi fields along the geodesic t-exp(rh,) such that (8, * w)(O) = VEo . w = 0 are the fields of the form r r-f C,(th,, k,) for some k, E T.(M).

42

4.

XVlll

DIFFERENTIAL SYSTEMS

Let M be a real-analytic manifold, C a n analytic linear connection o n M ; let (/ be a point of M and let U be a neighborhood of 0, in T,(M) such that the restriction of exp, t o U is a difieomorphisni of U onto a neighborhood of t i i n M . Given two vectors h, k in T,(M), put f ( t , 5) - exp.(t(h [k)) a n d w ( 0 :&'(/, O), defined for sufficiently small f . Let exp,(z) denote the entire function ( e x p k ) I)/:. With the notation o f Section 18.6, Probleni 7, show that

+

~

W(t) -

(exP,(

~~

toz)

'

(tZk))(exp.(rh))

(Iniitat: the proof of (19.16.5). replacing !I, by End(E) and G by CUE), where F is a Banach space; apply this t o thc case where E is the Banach space of restrictions 0 1 C" vector fields t o a compact neighborhood of ( I , and consider the linear mapping : Y - t [ z h , Y ] of this space into itself.) Deduce that

oLh

T,h(eXp,) ' k ~- (eXp,(-loLh) 'Zk)(exp.(th)). 5. With the notation and hypotheses of(18.7.9) show that, for each by piirci//e/ trtimsport along r of the vector

tk;t where

0 3 ( / )is

$ , ( ( r ( t i ) . ( h'

such that lim

o.,(/)//~

Ii

I , , w ( r )is obtained

k,,)). h.)t3 i o , ( t ) t T , ( M ) , 0. (Use (18.7.6,1).)

r-0

8. FIELDS O F p-DIRECTIONS, PFAFFIAN SYSTEMS, A N D SYSTEMS O F

PARTIAL DIFFERENTIAL E Q U A T I O N S

Let M be a differential manifold. A p-clirrctiori (or a t u r i p i t 1)-dirwtioii) a point x E M is a vector subspace of T,( M) of dimension 17. that is to say (16.11.8), it is an element of the Grassiiiuiiiiiuii G,,(T,(M)). A j d c l qf' p-clirrctions on M is a mapping which assigns to each x E M a p-direction L, E G,(T,(M)). If we identify a sufficiently small neighborhood V of a point .yo E M with an open set in R" by means of a chart, then for each point x E V the tangent space T,(M) may be identified with {x) x R", and G,(T,(M)) with .[XI x C,(R"), and a field of p-directions is therefore a mapping . u w ( x , L~k), where L: E G,(R"). The field is said to be of'cluss C' ( r an integer >= 0, or + m ) if the mapping x w L: of V into the differential manifold G,(R") (16.1 1.8) is of class C'. This definition is independent of the choice of chart, by virtue of the following proposition : at

8 FIELDS OF p-DIRECTIONS

43

( F o r r = r.therefore, an equivalent condition is that the union L of the subspaces L, for all x E M be a i w f o r sirbbundlc (16.17.1) of T(M).) By choosing a suitable basis ((x,, ej))l j 5 n of Tx0(M),we may assume that L,, is the subspace spanned by the first p basis vectors ( x o , el),. . . , (x,, e p ) . By identifying iis before a neighborhood U of xu with a n open subset of R", we may assume (since Li is a continuous function of x) that for each x E U the subspiice LI of R" is spanned by p vectors e,(x), . . . , e,,(x) such that the projection of e j ( s ) o n RP (R'being identified with Rp x R"-", and R P with Lie) is ej for I 5.j 5 p ; this follows from the description of the atlas of a Grassmannian given in (16.11 .lo), and linear algebra (A.4.5). By the definition of the dilTcrential structure of G,(R") (16.11.10), to say that XWL,is of class C' then signifies that each of the mappings XH e j ( x ) of U into R" is of class C', and i t is now clcar that the p mappings X H ( X , e j ( x ) ) are uc'cfor fields satisfying the required conditions. Conversely, if there exist p vector fields X j ( x ) = ( x , vj(x)) satisfying these conditions, we may assume that (with the same notation) v,(x,) = e j . I f uj(x) is the projection of vj(x) on RP,the continuity of the X j implies the continuity of the uj, and therefore we may assume that the u,(x) form a busis of R P for all x E U. If P ( x ) is the 11 x p matrix whose columns are the ) if QCx) is the n x ti matrix vectors uj(x) (relative to the basis ( e j ) , 5 , 6 pand

then the vectors e,(x) = P ( x ) - ' * v,(x) ( I 5 j 5 p ) form a basis of L: such that the projection of e , ( x ) on RP is e, for I 5.j 5 p. If the X , are of class c',then so are the u, and Q - I , hence also the mappings x w e , ( x ) . Consequently, XH L i is of class C', and the proof is complete. Reriiurks

(18.8.2) If M is a pure manifold of dimension t i , there exists a unique field of +directions on M , for which L, = T,(M) for all x E M . On the other hand, there does not necessarily exist any C" field of I-directions on M , as we shall see in Chapter XXIV. Moreover, if x w L, is a c' field of p-directions o n M , therc need not cxist C"vector fields X j dejitwd on all ofM such that the vectors X , ( x ) are a basis of L, for each x E M , even in the case where M is a n open set in R" and 1) = I (Problem 3). We remark also that it follows from the proof of (18.8.1) that when the conditions of the proposition are satisfied, the set of vector fields X of class C' on U such t h a t X ( x ) E L, for all x E U is a free module over the ring A E ' ( U ) of real-valued functions of class C' on U, and has a basis the

xj ( 1 S j 5 / I I ) .

44

XVlll


(18.8.3) Let .YH L, be a field ofp-directions on M, of class C ' ( r 2 0). and for each x E M let L: be the anniliilator of L, in the dual T,(M)* of T,JM), so that dim(L:) = ti - p (A.9.5). (18.8.4) With the notation of (18.8.3),each point x,, E M adinits an open neighborhood U such that there exist n - p diferential 1-f o r m coj ( 1 _Ij 5 n - p ) of class C' on U w9ith the property that,.for each x E U , the w j ( x )f o r m a basis of L:. ConilerseIj; if w, ( I jj 5 ti - p ) are diflereiitial I - f o r m of c1a.w C' 011 U such that, f o r each x E U , the corectors ojj(x) are linearly indopendent, and i f L, is the p-diniensional subspace of T,( M) defined bj! the t i - p equations (h,, w j ( x ) ) = 0 ( 1 5 j 5 n - p ) , then X H L, is a C ' j r l d qf'p-directions on U .

To prove the first assertion, let

with the notation of the proof of (18.8.1), and take

where (e)), s i 6 n is the basis dual to the basis ( e i ) lI i s n of R". C.onversely, if the w j are such that the covectors w j ( x ) are linearly independent at each point x E U, then in a neighborhood of each point x , of U we may assume (after a permutation of the basis vectors of R") that the w j ( x ) are given by (18.8.4.1) by virtue of (A.4.5) and the continuity of w j ; then Cranier's rule shows that the i k j are of class C', and the vectors "-P

ej(x) = e j +

k= I

j.jL(x)eP+k

form a basis of L;, Hence the result, by virtue of (18.8.1). I t follows, as in (18.8.2), that the set of C' differential I-forms w on U , such that w(x) E L: for each x E U, is a free niodule over the ring d;'(U), having the w j ( I s j 2 n - p ) as a basis.

(18.8.5) Let X W L , be a C' field of p-directions on a differential manifold M. For each q = 1 , . . . , p , an iiitegral nianifold of diriiension q of the j e l d X H L , is defined to be a pair consisting of a pure differential manifold N of dimension q and a mappingj : N -+ M of class C' and (maximum) rank q at each point of N , such that for each point z E N the image of T Z ( N )under

8 FIELDS OF p-DIRECTIONS

45

T( j) is c o n t u i n d iti LjCz,. I f r 2 I , it follows from (16.7.8) that each z E N has an open neighborhood V in N such that the restriction o f j to V is inji>ctiw; and i f j is of class C ', t h e n j is a n it~uncrsiotiof N in M. Consequently, as far as the k o c d study of the integral manifolds of the field x t t L, is concerned, we niay always assume that N is an open subset of R". (18.8.6) When p = I , the notion of an integral manifold (necessarily 1dimensional) of a field of I-directions X H L , ~on M coincides locally with the notion of an intcgrul ciiri-e of a vector field uhiclr is w c r ) w h w nonzcro. To see this, we niay assume that N is an open interval 1 c R, and i f , j : I -+ M is an integral manifold, we may also assume that 1 is sufficiently small so that j ( 1 ) is contained i n an open set U of M on which there exists a vector field X for which, a t each .Y E U. the vector X ( x ) is nonzero and forms a basis of the line Lx(18.8.1). By hypothesis, for each t E I , we may write,j'(t) = p ( t ) X ( j ( t ) ) , where p ( t ) is a nonzero scalar and the mapping t t + p ( t ) is continuous in I . Siticep cannot change sign i n I , the mapping cp : t

h

H

p ( u ) dir is a C' homeo-

morphism of I onto an open interval I' c R (4.2.2), and the inverse mapping is also a C' homeomorphism; the m a p p i n g f = j o cp-' is an integral curve of the vector field X defined on 1'. The converse is clear. 'p-'

(18.8.7) Throughout the remainder of this chapter (except for (18.14.6)) we shall limit our considerations to the case of fields of y-directions oj'class C" defined on un o p 7 scv M i n R". and we shall be concerned only with the local existence and uniqueness of integral manifolds of this field. I n view of the remarks in (18.8.5), we may therefore always assume (by replacing M by a smaller open set if necessary) that the integral manifolds are closed subniani~blrlsof M . By virtue of (18.8.4), we may assume that there exist 11 - p differential I-forms (oj ( I 5.j 5 n - p ) of class C" on M such that the covectors w j ( x ) form a busis of L: for each point x E M . A submanifold N of M , of dimension q 5 p, i b an integral manifold of the field XH L, if and only if the restrictions to N of the n - p j i ) r n i s ujure zero, and this property is expressed by saying that N is an integral nianfold of the Pfaffian .r)?steni (18.8.7.1)

W,

=0

( I s j 5 I?

-

p).

It should be observed that this definition still makes sense when the ( o j ( x ) are not everywhere linearly independent. Conversely, if we are given n - p differential I-forms w j of class C" o n M, such that the w j ( x )are linearly independent at each point x E M, then the subspace L, of the tangent space TJM) on which all the covectors w j ( x ) vanish is of dimension p at each x E M, and .YH L, is a C" field ofp-directions

46

XVlll


on M . Every integral manifold of the Pfaffian system (18.8.7.1) is an integral manifold of this field. (18.8.8) If x i ( I 5 i 5 n ) are the canonical coordinates of a point x E R", then we may write

c n

(18.8.8.1)

olj(x)=

(1

a,,(x) dx'

i= I

S j 5 n -p )

for all x E M , where the aJiare C" functions on M. Let N be a q-dimensional submanifold of M, and let h : N -+ M be the canonical injection. Replacing M by an open subset, we may assume that there exists a chart cp of N in Rq. Let z i = pri / I cp-' ( I 5 i 5 n ) ; then, in order that N should be an integral manifold of the Pfaffian system (18.8.7.1), it is necessary and sufficient that, in the open set q ( N ) c Rq, the ti functions ( u ' , . . . , uq)t-+zi(u1,. . . , i f q ) should satisfy the system of q ( n - 1)) partial differential equations 0

(18.8.8.2)

azi

n

CU,;(Z', . . . , z") - = 0 i= I c?uL

( I sh-sq, 1 5 j S t i - p ) .

(18.8.9) Conversely, let us show how the determination of the solutions of a s)steni of'rst-ordcr partial rii@ential equations can usually be reduced to the determination of the i n t q r a l tmnijblds of a PfajEan system. Let q and n be two positive integers, let N = 9 + ti qn, and consider, on an open set U c RN, a certain number of functions F, ( I 5 /I 5 r ) of class C". A solution of the system of first-order partial differential equations

+

(18.8.9.1) FJ,(xl,. . . , xq, ZI, . . . , z", p 1 1 , . . . , p l y , . . . , p " ' ,

is by dcfitiiriori a system of (XI,

ti

. . . , pnq)= 0

functions

. . ., x ~ ) H z J ( x ,~. ,.. xq)

(I

s i 2 n)

of class C' on an open set V c R4 such that, for each point x=(xl,

. . . , x")EV,

the point of RN with coordinates XI,

. . . , xq, P'(.Y), . . . , ~"(x),D,tl'(x),. . . , D,u"(x)

belongs to U and satisfies the equations (18.8.9.1).

( I 5 /I 5 r )

8

FIELDS OF p-DIRECTIONS

47

Assume that the set of points of U satisfying (18.8.9.1) is a submanifold M of U. Then it is immediately clear that, if we consider the n differential 1-forms (18.8.9.2)

4

(1 s j

mj = dz'- x p " d x '

5 n)

i= 1

on U, and if we denote by oj the 1-form induced by mi on the submanifold M, then the submanifold of R4 x R" defined by the relations XEV,

z'=d(x',

..., x4)

(I

5jSn)

is the projection on R4 x R" of an n-dimensional integral manifold of the Pfaffian system (18.8.9.3)

oj

=o

(1

IjSn)

on M. Conversely, however, the projection on R4 x R" of an integral manifold of the system (18.8.9.3) cannot always be defined in a neighborhood of a point by equations of the form z j = d(xl, . . . , x4) (1 s j 5 n), where the ui form a system of solutions of (18.8.9.1). (18.8.10) Finally, given a system of equations which contain, besides the variables x i ( I 2 i 2 9) and the unknown functions z j (1 s j 5 n), the partial derivatives of each zi up to order mj ( 1 5 j 5 n), it is easy to reduce this system to one of type (18.8.9.1) containing only first-order partial derivatives. The method is the same as in the case of systems of ordinary differential equations, which is the case9 = 1 (cf. (10.6.6)): we take as new unknowns the derivatives of each zi of order < mj (because of the symmetry of partial derivatives (8.12.4), it is not necessary to take all these derivatives). For example, suppose that we have two variables x and y , one unknown function z, and a single second-order equation

where F is of class C" on an open set U c R'. Introduce two new unknown functions p , q with the additional equations

48

XVlll DIFFERENTIAL SYSTEMS

then the original equation is reduced to the system of type (18.8.9.1) consisting of these two equations and the equation x, y , z , p , q,

2,2, 3)= 0. ax ay ay

If the equation F(x, y , z, p , q, r, s, t ) = 0 defines a submanifold M of U, then the solution of this system of equation reduces to finding the solutions of the Pfaffian system (18.8.9.3) on M in which the left-hand sides are the 1-forms induced on M by the three differential forms dz - p d x - q d y ,

dp - r d x - s d y ,

d q - s d x - tdy.

PROBLEMS

1. Let M be a connected differential manifold which is a submanifold of RN, so that T(M) may be identified with a submanifold of T(RN) = RN x RN. Let x HL, be a C" field of I-directions on M. For each x E M, consider the two points on the line L, c T,(M) whose Euclidean distance from x is equal to 1. The set M' of these points is a twosheeted covering of M, whose projection on M is the restriction of oh(.Show that a necessary and sufficient condition that there should exist a C" vector field X on M such that X ( x ) # 0, and X ( x ) E L, for all x E M is that M' is not connected. In particular, such a vector field will exist if the fundamental group x , ( M ) contains no subgroups of index 2, for example, if M is simply connected. 2.

(a) On M = R2 - (01, let x ++L, be the C" field of 1-directions such that, for x = (r cos 8. r sin 8 ) with r > 0, L, is the line with gradient tan(+$). Show that there exists no continuous vector field X on M such that X ( x ) E L, and X ( x ) # 0 for all x E M. (b) Let p : R2 +T2 be the canonical mapping. For each point z = ( f , 7) E R2, let L: c T,(RZ) be the line with gradient tan in(l - 25). Show that, for each point x E T2, there exists a unique line L, C T,(T2) such that T(p)L: = L, (where x = p ( z ) ) and that x -L, is a C" field of I-directions on T 2with the property that there exists no continuous vector field X on T 2such that X ( x ) E L, and X ( x ) # 0 for all x E T2.

3. An analytic field ot p-directions on a real-analytic manifold is defined exactly as in (18.8.1). Let M, M'be two real-analytic manifoldsand let r - L x (resp. x'++L:,) be an analytic field of p-directions on M (resp. of p'-directions on M'). Let f :M + M' be an analytic mapping. If M is connected, and if there exists a nonempty open subset U of M such that T(f)(Lx) c L&) for all x E U, show that this relation holds for all x E M. (Use the principle of analytic continuation.)

9


49

9. DIFFERENTIAL SYSTEMS

(18.9.1) Let M be a pure differential manifold of dimension n. We shall denote by 6, the space of real C" differential p-forms on M (which was denoted by €,,. R(M)in (17.6.1)), and by €, the R-algebra of real-valued C" functions on M (which was denoted by €(M;R ) in (16.15.6)). Also we shall Put

d=f?(;5",0&, @...@d"

.

We recall that, if M is an open set in R", then d is the exterior algebra of the free &,-module &, (A.13.5), and each €, is a free &,-module with a basis consisting of the p-forms dx" A dxi2A . * * A dx'p (il i, < . . < i,). Consider now a Pfaffian system (18.8.7.1) on an open set M of R".If N is an integral manifold of this system, it is clear that the restrictions to N of the exterior differentials dw, are also zero (17.15.3.2). Hence the same is true of the restrictions to N of the differential p-forms (1 s p 5 n) which belong to the graded ideal a of d (A.18.1) generated by the o,and the d o j . For if a differential p-form a on M is such that its restriction to N is zero, then the same is true for each (p y)-form a A /3 ((16.20.15.4) and (16.20.9.5)); hence the set of elements a = a, + a2 a, E d (where a, E g P ) such that the restriction of each a,, to N is zero is a graded ideal b of d which contains the w j and the dw,, and hence contains a (in general b # a). Let a,, = a n &,,, the set of p-forms contained in a, so that we have a = a, @ a, 0 . .. @ a,, . Then a, is just the €,-module generated by the o,. We shall show that the relation a,, E a, implies da, E a,,+,. For this, we remark yn E d (with y p E &), such that that the set c of elements y1 + y 2 + * dy, E a,,++., for all p is a graded ideal of d,as follows from the formula giving the exterior differential of an exterior product (17.15.2.1); since c contains the w j , it contains a.

-=

+

+ +

+

(18.9.2) These results suggest the following generalization of the notion of a Pfaffian system. For brevity, let us define a differential ideal to be any graded ideal a of the algebra d such that, for each p-form a ~ a we , have also da E a. We remark that a = a, 0 a, 0 . .. @ a, (where a, = a n a,,) but that here we may have a, # (0).Such an ideal is said to define a diflerential system on M ; a submanifold N of M is said to be an integral manifold of the system if the restrictions to N of all the differential forms belonging to the different homogeneous components of the idea a are zero. A Pfaffian system corresponds therefore to a particular type of differential system, in which the differential ideal is generated by I-forms.

50


PROBLEM

Given any differential system on an open set U c R", and a p-dimensional integral manifold N of this system, show that there exists a nonempty open set V c U such that N n V is the graph of a solution of a system of first-order partial differential equations in N - p unknown functions ofp variables. Consequently N is also (in general) an integral manifold of a Pfaffian system.

10. INTEGRAL ELEMENTS O F A DIFFERENTIAL SYSTEM

We shall make repeated use of the following lemma, which generalizes the remark at the beginning of (10.3). (1 8.1 0.1) Let X be a topological space, and let (fA)l be a family of continuous mappings of X into the dual E* of ajnite-dimensional real vector space E. Then the rank of the system of linear forms (L(X))~€is a lower semicontinuous function of x (12.7), and the set of points x E X at which this rank is locally constant is a dense open set in X .

If ( e , ) l g i 6isr a basis of E, and (e;) the dual basis of E*, then we may write

where the cil are continuous real-valued functions on X. If the system of linear formsf,(x) has rank p , then there exist p indicesl, ( I 5 k 5 p ) such that is nonzero; at least one of the p x p minors of the matrix ( C ~ ~ J X 6) )i 4~ r , the corresponding minor of the matrix (cilk(x')) is then also #O for all x' in a suitably small neighborhood of x , and therefore for x' in this neighborhood the rank of the system ( h ( x ' ) ) is L p . This proves the first part of the lemma. Next, for each open set V in X, the maximum of the values taken by the rank of the system (fi(x)) for x E V is a finite number q. If xo E V is such that the rank of (h(xo))is equal toq, then it is equal to q at all points of a neighborhood W of xo contained in V, by lower semicontinuity. This proves the second part of the lemma. (18.10.2) With the notation of (18.9.1), let a be a differential ideal of d , and let ap denote its homogeneous component of degree p . Recall that, at any point x E M, the tangent space at x to M is identified canonically with {x} x R". For each x E M and each p > 0, a p-dimensional vector subspace

10 INTEGRAL ELEMENTS OF A DIFFERENTIAL SYSTEM

51

F, of {x} x R" is said to be a p-dimensional integral element of the differential system defined by a if: (i) all the functionsf€ a, vanish at x; (ii) for each integer q such that 1 5 q 5 p , and each q-form w, E a,, the restriction to F, of the q-covector w,(x) is zero, or equivalently, (18.10.2.1)

(UIA U1A

"'

A

u,,m,(x))=O

for any q vectors u,, . . . , u, in the space F,. From this definition it is immediately clear that if F, is an integral element of dimension p , every vector subspace G, of F,, of dimension q < p , is an integral element of dimension q. An integral element of a Pfaffian system (1 8.8.7.1) is by definition an integral element of the corresponding differential system (18.9.2). It is clear that if N is an integralmanifold(l8.9.2) of the system defined by a, containing the point x and of dimension q at x , then the subspace T,(N) of {x} x R" tangent to N at x is an integral element of dimension q. The existence and uniqueness theorem for differential systems will establish under certain condi/ions a converse of this proposition. (18.10.3) For this purpose we shall consider a point x E M at which all the functions,fE a, vanish, and then successively the integral elements of dimension I passing through x, and then for each of these the integral elements of dimension 2 containing them, and so on; and we shall seek to answer the following question: given an integral manifold N of dimension q, the tangent space T,(N) at a point x E N (which is an integral element of dimension q ) and an integral element F, of dimension q + 1 containing T,(N), does there exist (at any rate locally) an integral manifold N' of dimension q + 1 containing N and such that T,(N') = F,? More precisely, we shall first define a sequence of integers 2 0 as follows. For each point x E M (not necessarily a common zero of the functionsfe ao), let so be the rank of the system of linear forms ml(x) on { x } x R",where m, runs through the homogeneous component a, of a. Also let M,(x) denote the subspace of {x} x R" of dimension n - so(x) on which all the forms wl(x) vanish. Now consider a fixed nonzero vector u, E M,(x), and the vectors V E { X } x R" such that (18.10.3.1)

for all ml E a, and w2 E a,. The rank of the system (18.10.3.1) of linear equations in v is at least so(x), say so(x) sI(x, ul). Let M2(x, u,) denote the

+

52


subspace of all vectors V E ( X } x R" which satisfy (18.10.3.11, so that the dimension of M,(x, ul) is n - so(x)- sl(x, u,). Clearly ul E MJx, ul). Proceeding now by induction, let ul, ..., U,E{X} x R" be q linearly independent vectors such that, for each r = 1, . . . , q and each sequence of integers (i,, . . . , i,) satisfying 1 5 il < . < i, 5 q, we have (18.10.3.2)

(Ui,

A Uil A

* ' *

A

Ui,,m,(X))

=o

for allm, E a,. Consider the vectors v E {x} x R" such that, for r = I , .

we have (18.10.3.3)

(Uil A Ui2 A

" *

.., q + I ,

=0

A Ui,-l A V, m,(X))

for all sequences (i,, . . . , i r - , ) satisfying 1 4 i, .c . . < i r - l 5 4 . (When r = 1, (18.10.3.3) is replaced by (v, ml(x)) = 0.) Denote by s,(x)

+ s1(x, u1) + + *.*

S&X,

u,,

. . , u,) *

the rank of this system of linear equations in v, and by M,+,(x. u,, . . , , u,) the subspace of {x} x R" defined by the equations (18.10.3.3). It is clear that this subspace contains u,, . . . , u,, and therefore (18.10.3.4)

so(x)+ sl(x, ul)

+ + s,(x, u l , . . . , u,) 5 n -4.

If we have strict inequality in (18.10.3.4) then we can choose a vector u , + ~E M,+,(x, ul, . . . , uq)linearly independent of u,, . . . , u,, and the induction can continue. It follows immediately from these definitions that if x is a common zero of all the functionsf€ a,, and if I, denotes the q-dimensional vector subspace of {x} x R" spanned by the vectors ul, . . . , u,, then the I, for I 5 r 5 q form an increasing sequence of integral elements. (18.10.4) From now on we shall assume that the set M, of points X E M at which all f E a, vanish is a submanifold of M, of dimension p 2 1, and that a, is generated as an &,-module by n - p functions fk (1 5 k 5 n - p ) such that, at each x E M,, the rank of the system of differential 1-forms & is n - p (16.8.9). Now let MI be the set of pairs (x, ul) E M x R" such that x E M,, u, E M,(x), and ul # 0. Proceeding by induction, suppose that Mr

has been defined, and define M,+ to be the set of points such that

((xi ~

u , + ~E M,+l(x,ul,.

l * *r *

. . , u,)

~ r ) rur+I) E

and

Mr

uI

x

A

R" uz A

-.*

A

u , + ~# 0.


The projection of M, so(x)

+I

53

on M, is the set of points (x,u,, . . . , u,) such that

+ sl(x. ul) + . . . + s,(x. ul,

. . . , u,) < n - r.

Since the dfk belong to we have s,,(x) 2 n - p and therefore the set M,+l will certainly be eniptj' when r 2 p . By virtue of (18.10.1). the set V, of points .YE M, such that the rank s o ( x ) is lo call^^ con.stai~tat x is a dense open set in M,. By induction again, suppose that the open subset V, of M, has been defined. Then the set Ur+] of points (x. ul. . . . , u,+,) E M, + I such that qJx)

+ . ' . + S,+](X..

UI,

.

. .. ur+l)

is locallj~c*onstant at this point is a dense open set in M,+] c M, x R",and therefore V r + l = U r + lnpr;'(V,) is also open in M,+l. A n integral element of dimension r contained in {x} x R" is said to be regular if it is spanned by r vectors uI. . . , , u, such that (x, ul, . . . , u,) E V,. We have then, by definition. (x. ul. . .., u,) E V,s for s < r , and therefore the integral element of dimension s spanned by ul, . . . , usis also regular. Moreover, the definition shows that, for (x',ui, . . . , u:) suflciently near to (x, u l . . . . . u,) in M,, the integral element spanned by ui, . .., u: is also regular, and we have s,,(x, u;, . . . , u;, = sy(x. ul, . . . , uq) E Vr-] and if for 1 2 q 5 r . Observe that if (x. uI.. . . ,

the integral element generated by (x, uI,. . . , u,) is regirlar by definition, a n d there exists no integral element of dimension r + 1 containing (x. uI,. . . , u,). An integral element which is n o t regular is said to be singular. (18.10.4.2) The vectors ul, . . . , u, will certainly generate a regular integral element if x. ul. . . . u, are successively chosen so that the values of ~

s,,(x). s,(x. u1). . *

* >

s,(x,

u1,

. . . , u,)

are as lar,ge as possiblt (18.1 0.1). (18.10.5) So far we have not made use of the fact that a is a n ideal in the exterior algebra d.This has the following supplementary consequences: (18.10.6) ( I ) I n order that a subspace F, o f { x } x R" (where x E M,) should he an integral element If dimension p . it I S necessary and suflcient that, for each m,,E n p , the restription to F, of the p-coi>ector m,(x) should be zero.

54


(ii)

If F, is an integral element of dimension p , the sum

(18.10.6.1)

so(x)

+

S'(X,

and the subspace M,,+,(x, ul, of F,.

u,)

+ + sp(x, . * *

U,,

. . . I

up)

. . . , up) are independent of the basis u,, . .. , up

Assertion (ii) is a consequence of (i), because the equations (18.10.3.3) for r < p + I are consequences of the equations(l8.10.3.3) for r = p + I by virtue of (i). Hence it is enough to prove (i), and for this it is enough to show that if each m,,E a,, is such that the restriction of m,,(x) to F, is zero, then the same is true for all m4 E a,, , q < p . Let u , , . . . , uqbe a sequence of q linearly independent vectors in F,; then we have to show that (ul A A uq, m,(x)) = 0. Extend the sequence ( u ~ ) , ~to~ a~ basis ,, (uj)lsjip of F,; let (ur) be the dual basis of F:, and consider the ( p - q)-covector u,*+ A * * * A u: = z*, say. There exists a C" differential ( p - 9)-form m P d qon M such that m,,-,,(x) = z* (16.4.3). The p-form w p= mq A mP-,, belongs to the ideal a, and m,(x) = m,(x) A z*. Now, by hypothesis we have (U1 A U2 A

A U p , m p ( x ) ) = 0,

* * .

and it follows immediately from the choice of z* and the rules of calculation i n the exterior algebra (A.14.2.1) that the relation above is equivalent to (U1 A

*

.-A

Uqrm,(x))

= 0.

Example

(18.10.7) Consider the case in which the differential system corresponds to a Pfaffian system on an open set M in R4, consisting of a single equation w = dx4 - A(x', x2, x3)dx'

- B(x', x2,x3) dx2 - C(X', x2, x3) dx3 = 0,

where A, B, C are of class C". The corresponding ideal a is generated by w and the differential 2-form dx3 + B'(x', + C'(xl, x2, x3) dx' A dx2, aB ac ac - -a A A'=--B' = ax3 ax2' axi ax3'

dw

where

= A ( x l , x2, x3) dx2 A

x2,x3) dx3 A dx'

aA C'=---

ax2

aB ax1'

It is immediately verified that we have so(x) = 1 for all x E M, and that M,(x) is the hyperplane H, with equation u4

- A(x)ul - B(x)u2 - C(x)tr3 = 0.


55

Every vector v E R3 is therefore the projection of a unique vector u E H, . In order that two nonproportional vectors ui,u2 E H, should define a 2-dimensional integral element, it is necessary and sufficient that their projections vl, v2 on R3 should satisfy the relation t(x) A v1 A v2 = 0, where t(x) = (A'(x), B'(x), C'(X)).

If t(x) # 0, then sl(x, ul) = I for each vector ul E H, such that t(x) A v1 # 0, and therefore 'M,(x, u,) is a plane whose projection on R3 is the plane containing t(x) and vl ; there is no integral element of dimension > 2 containing ul. If t(x) and v1 are proportional, then sI(x, u,) = 0 and M,(x, ul) is equal to H,; but if we choose u2 E H, not proportional to ul, we see again that there is no integral element of dimension > 2 containing u1 and u 2 . If on the other hand t(x) = 0, then H, is an integral element of dimension 3. In order that s1 should be locally constant near the point (x, ul), it is therefore necessary and sufficient that either t(x) # 0 and v1 be not proportional to t(x), or t(x') = 0 for all x' sufficiently close to x. Consequently, the singular integral elements of dimension I are the lines in {x} x R4 spanned by a vector u1E H, , such that either x is a frontier point of the closed set t-'(O), or else t(x) # 0 and the projection vl of ul is proportional to t(x). The singular integral elements of dimension 2 are the planes contained in H,,where x is a frontier point oft-'(0). We see therefore that a regular integral element of dimension 2 may contain singular integral elements of dimension 1. (18.10.8) Suppose that the differential ideal a is generated by a set S of q-forms, where q 5 r. Then the condition (i) of (18.10.6) is met by requiring that for each m, E S the restriction of nr,(x) to F , is zero. For, by hypothesis, every element of ap is a sum of exterior products a1 A . . A a,, where a j is a pj-form, with pl + p2 + * . . + ps = p , and at least one of the forms belongs to S. Now, in order that the restriction to F, of a q-covector should be identically zero, it is sufficient that the same should be true of the exterior product of this q-covector by an arbitrary covector (A.14.2.1). This remark will be particularly useful in the case where the differential system is a Pfafian system (18.8.7.1); we may then take S to consist of the ojand the dwj.

PROBLEMS

1. Assume that the differential ideal

(I is

generated by a, 0 a2 (so that a,

= (0)).

(a) With the notation of (18.10.4), let F, be an integral element of n, of dimension r 2, generated by r + 2 vectors u,, . . , , u,+*, and suppose that the integral element

+

56


.

generated by ul, . . . , u,+ is regular. Let G , be the subspace of F, generated by UI, . . , u , - ~ ,and let H, be a supplement of G , in {x} x R", containing u,, u , + ~ and , u,+z. Then M,(x, ul, . , u,is the direct sum of G, and V, = H, n M,(x, ul,.. . , u,of dimension

..

m =n - r

+1

- (so(x)

+ sl(x, uI) + + s,-I(x.

~

1

. .., ~ , - d ) . ,

Let ( w l , . . . , w,) be a basis of V, such that w1 = u,, wz = w3 = u,+z. For j = 1,2, 3, let W, denote the (m- 2)-dimensional subspace of V, generated by W, and

w4,. . . ,w,. Every integral element Fi of dimension r + 2 containing G , and sufficiently close to F, is determined by its intersections Rv, with W , ( j = 1, 2, 3), where the VJ are subject to the relations (v, A vk, m2 (x)) = 0 for all forms w z E (11 and j # k. Let f be the mapping (VI,

of W Ix Wzx W3 into

(

Vz,

V3) ++(VIA Vz I Vz A

V,)'.

V3

9

V3

A VI)

Show that f i s an immersion in a neighborhood of 1

(wI, w z , w3). Next, let L be the vector subspace of 7 V, on which all the linear forms

mz(x) ( m 2E nz) vanish. Then L is of codimension s, = s,(x, ul, . . . , u,) andf-'(L3) is the set of all triples (vl, vz, v3) E Wl x W 2 x W3 such that = 0,

0 and a system of power series

convergent for (xil < p (1 5 i 6 p xi, the point

(

XI,

. . * ,xp+ 1, cl(s),. .

*

+ 1) and such that, for these values of the

dv SL., , cr(s).7 ., . . , -, OX

dXP

i?Vr . . . ,, . . . , 3) ZxP is

belongs to B, the equations (18.11.4.5) are satisfied, and uj(xl,.. . , xp, 0) = 0

t Cf. J.

(1 $ j 5 r).

Dieudonne, '* Calcul Infinitksimal," pp. 260-261, Hermann, Paris, 1968.

12 THE CAUCHY-KOWALEWSKA THEOREM

65

Then for l x i l < p (1 S i S p + I), we have

where &k is the multi-index ( 8 k 1 ) 1 s [ s p + l (Kronecker delta) and the series on the right-hand side of (18.12.1.3) are convergent. Moreover, the constant terms of the series (18.12.1.2) and (18.12.1.3) for k S p are zero by hypothesis. It follows, therefore, from the substitution theorem for power series (9.2.2) that there exists a positive real number p' < p such that for \ x i /< p' ( 1 5 i 5 p + I), the series (18.12.1.2) may be substituted for the uj ( I S j j r ) and the series (18.12.1.3) for the w j k (1 S j s r , 1 S k $ p ) in the power series Hj. Let us now express that each of the power series (18.12.1.3) fork = p + 1 and 1 5 j 5 r is equal to the power series obtained from Hi by these substitutions; then, for 1 s j S r and each multi-index 2 (9.1.61, we have (1 8.1 2.1.4)

where p m h and vnhk are multi-indices (so that ( V & ) k is the kth component of vnhk)and for given 1 the summation is over the finire set of multi-indices satisfying the relation

(which expresses the equality of the coefficients of X ' - ~ P + in ~ the two series of the jth equation (18.11.4.5), by considering products of terms in the same way as in (9.2)). In particular, looking at the components of indexp + 1 in (18.12.1.5), we obtain (18.12.1.6)

iP+ I - 1 = CYp+ I -k

9

h = l m=O

(\l,~,h)p+ 1

+

f

(Vnhk)p+ I '

h = l k = l n=O

From this we shall deduce that the aji are ~iniqiielydetermined by the relations (18.12.1.4). When i.p+l= 0, all the aji are zero, by virtue of the hypothesis ~ ~ ( x. .l .,, x p , 0 ) = 0 (9.1.6). Now argue by induction on A,,+!: by virtue of (18,12.1.6), the multi-indices plll,land Vnhk which feature on the right-hand side

66


of (18.12.1.4) are such that (&h),+l < A,+1 and ( v , , , ~ ) , + ~< A,+l, for a given multi-index A. Hence, when A,+1 2 1, if the ahofor o,,+~< are known, the relations (18.12.1.4) determine the ajI uniquely. This already establishes the uniqueness assertion in (18,12.1), in view of (9.1.6) and the principle of analytic continuation (9.4.2). It remains to show that, conversely, if we dejne by induction on A,,+l the real numbers ail so that aj, = 0 for A,+1 = 0 and so that ajI is given by (18.12.1.4) when A,+1 2 1, then the power series (18.2.1.2) with these numbers as coefficients will converge in some neighborhood of 0. For if this is established, then since the constant terms of the series (1 8.12.1.2) and their derivatives (18.12.1.3) for 1 5 k S p are zero, these series may be substituted for the uj and the w j k , respectively, in the series (18.12.1.1) for [ x i [< P'~,where p" is a sufficiently small positive real number (9.2.2), and the analytic functions v j defined by (18.12.1.2) for lxil < p" ( I 5 i 5 p + 1) will then satisfy the required conditions, by virtue of the relations (18.12.1.4) satisfied by the aj, * To establish this convergence we shall use Cauchy's " method of majorants." The idea is to find power series (18.12.1.7)

. ., u r , w I 1 , .. ., w,,)

Gj(x', ... , x"+', ul,.

= a. P, Y

convergent in some neighborhood of 0 and having the following two properties : (i) for all values of the indices, C$\ is a real number 2 0 and

(ii) the system of partial differential equations (18.12.1.9)

admits a solution consisting of functions analytic in a neighborhood of the origin, (18.12.1.10)

.

v j ( x l ,. . , x P + I )=

where the A , are real numbers 20.

1A j , x A I

(1 5 j 5 r ) ,


67

Suppose that the C$,, and the A , have been found to satisfy these conditions. Then the first part of the proof shows that (18.12.1.11)

It is enough to show that for all choices of the indices, these relations imply

.

laj.il S A j L .

(18.1 2.1 12)

But this is obvious when A,+, = 0, because then ail = 0; and for >0 it follows immediately by induction on A,+], from comparison of the relations (18.12.1.4) and (18.12.1.11), and the inequalities (18.12.1.8). It remains, therefore, to construct power series G, with the properties (i), (ii) listed above. We shall first show that, as a matter of convenience, we may assume that the constant terms c, = ch$o of the series (18.12.1.1) are all zero. For if u, ( I S j r ) are solutions of the system of equations (1 8.11.4.5) which reduce to 0 when xP+' = 0, then the functions Cj(X',

. . . ,X P + l ) = v j ( x1,..., x P + l ) - c j x P + '

are solutions, vanishing for xP+' = 0, of the analogous system of equations obtained from (1 8.11.4.5) by replacing each Hi by the function

-

H j ( x l ,. . . ,x P + ' ,u l , . . . , u, , w11, . . . , w,,) = Hj(xl, . . . , x"', ~1 + C 1 X p + ' , . . . , U ,

+ c,x'+',

. . , w,.,,)- ~

~11,.

j

.

Since the series f i j have zero constant terms, we have achieved the desired reduction. Next we remark that the power series on the right-hand side of (18.12.1.1) converge also for complex values of the x i , uj , and wjk less than Ro in absolute value (9.1.2). Denote the sums of these series again by H j , so that each H, is now an analytic function of the complex variables xi,u j , and wjk. Let R be a positive real number Cool, the relation A, 2 0 is a consequence of the existence of a solution (18.12.1.17) of (18.12.1.16) such that Y'(0) = 0. For, just as at the beginning of the proof, the A,, satisfy the relations


69

where now 3,/I, y are integers 2 0 such that cy + /I + y > 0, the triple (0, 0, 1) being esclircled, and /is 2 0, vr 2 I are integers satisfying

(1 8.12.1 .I 9)

a

a + Clis+ s=O

I

C(v,-l)=n-l. r=O

When 17 = I . this relation is satisfied only by cy = /I = 0, y 2 2 and otherwise arbitrary, and v, = 1 for 0 5 t 5 y ; hence for A, we have the equation

(18.12.1.20) which admits A, = 0 as solution. Suppose now that n 2 2 and that A,,, 2 0 for in 5 n - I , and observe that the relation A, = 0 implies that the only nonzero terms on the right-hand side of (1 8.12.1.18)correspond to the case v, 2 2 when y # 0, and ps 2 2 when p # 0. Now it follows from (18.12.1.19) that if cy # O , we have ps I n - 2 and v, S n - 1 ; if a = 0 and p # O , we have n - 2 for all t ; and if ct = = 0, we must 2 5 ps 5 n - I for all s, hence v, I have y 2 2, hence v, 5 n - 1 for all t . It follows that the right-hand side of (1 8.12.1.18),for n 2 2, is a polynomial in A 2 , . . . , A,- with coeficients 2 0 , and the inductive hypothesis (together with the relation 8 - Cool > 0) therefore implies that A, 2 0. We have, therefore, finally to show that with this choice of 8, the differential equation (18.12.1.16)admits a solution Y ( x ) which is analytic in a neighborhood of 0, and such that Y(0) = Y'(0) = 0. This equation is not of the usual type (10.4.1),but may be reduced to it as follows: if we put q x , u, w ) = ~ ( xu,, w) - ew,

we have O(O,O,0) = 0 and

a@

- (o,o, 0) = cool - e # 0; aw

hence, by the implicit function theorem (10.2.4),there exists a function Y(x, u ) analytic in a neighborhood of (0, 0 ) , such that "(0, 0) = 0 and @(x,u, Y(x, u)) = 0 identically. It is clear that every solution of the differential equation

(18.12.1.21)

Y' = Y(x, Y )

will also be a solution of (18.12.1.16).But now the existence theorem (10.5.3) applies to (18.12.1.21), and a solution Y(x) of this equation satisfying

70


Y(0) = 0 will automatically also satisfy the condition Y'(0) = 0. The proof of the Cauchy-Kowalewska theorem is now complete. (18.12.2) It is clear that we may replace R by C throughout in the statement of (18.12.1): the proof is unchanged (except, of course, that the extension of

the Hito CP+'+ r + r p is now superfluous). (1 8.12.3) Consider now a system of partial differential equations depending on " parameters " : (18.12.3.1)

where the functions Hj are analytic in a neighborhood of 0 in Rp+l +r+rp+q

(resp. c p + ' + r + r p + q 1.

Then there exists a neighborhood T of 0 in Rq (resp. Cq)and a connected open neighborhood V of 0 in RP+' (resp. CP") such that in V x T there exists a unique system of analytic functions u j ( x l , . . . , x P + l , zl, . .. , ZJ satisfying (18.12.3.1) and such that Uj(X',

. . . ,x p , 0, z,, . . . , ZJ

=0

in (V n Rp) x T (resp. ( V n C p ) x T). The proof follows exactly the same pattern as before: the right-hand side of (18.12.1.1) has to be replaced by

C ('$;i:yd xau~wyzd

a. /J, Y . i:yd

and the right-hand side of (18.12.1.2) by

C ajlPx?zP. 1. P

We leave it to the reader to write down the relations corresponding to (18.12.1.4). In the expression of the " majorant " function (18.12.1.14), the second term on the right-hand side must be multiplied by

fi

I = 1 (I -;)-I.

The function (18.12.1.15) is then replaced by a function F(x, U, W , Z) = C Cntns(~)~"'~"Ws,


71

analytic in a neighborhood ofO. where the Cmns(z) are power series in zl,...,zq which all converge in the same polydisk T and have coefficients 20. By a homothety on z. we may assume that the point (1, 1 , . . . , 1) belongs to T. If we put

COOl(Z)=

c Coola 6

Zd>

the coefficients C o o , ,(which by construction are 20) are bounded by a number N independent of 6 (9.9.5); we then choose the number 0 such that 0 > N, and complete the proof as before. (18.12.4) For systems of equations (18.11.4.5) in which the functions Hj are analytic, there is no general result analogous to (10.5.6) expressing that, for a “ small variation of the H i , the solutions taking the same initial values are also subjected only to “ small ” variations (cf. Problem 5). ”

PROBLEMS

2 1. Show that there exists a real number C > 0, depending only on c, such that for each pair of integers tn 2 0, n 2 0 we have

1. (a) Let r be a real number

for 0 5 p 5 m and 0 _I v 5 n. (First establish the inequality for c = I and then show that when c > 1 and p > l / ( c - I), m - p > I/(c - l), there exists a constant C o , depending only on c, such that

r(cp

+ v + 1)

(pi-v)!

r ( c ( m - p) *

+

n -v (m-p+n-v)!

For this purpose, use the fact that

r(a)r(b)

+ +

+ 1) “0

r ( c m n 1) (m+n)! ’

c,r(a+ b - 1)

whenever a or b is sufficiently large, by considering the integral expression for Euler’s beta function B(a, b).) (b) Deduce from (a) that there exists a constant C‘, depending only on c, such that

(Observe that the sum is unchanged by replacing p by m - p and v by n - v, and hence that we may assume that cp v 2 t(cm n). Then majorize the sum by replacing r ( c ( m - p) n - v - 3 ) by r(c(n7 - p) n - v - l), and remark that the sum C ( p q)-4 is finite ( p , q each running through the set of positive integers).) PI II

+

+

+

+ +

(c) Let w(x, t) be a real-valued function of class Cwin a neighborhood of (0,O)in R2.

72


Assume that there exist real numbers c 2 1, M > 0, N > 0 and an integer n 2 1 such that, for all integers m 2 0,

Show that there exists a real number K > 0, depending only on c (and not on M or N) such that, for all integers j 2 1,

for 1 5 k 2.

5 n. (Use (b) and Leibniz's formula, and induction on j . )

Let cl, c2, c3 be three numbers in the interval [l, +a[.A functionf(x, t , y) of class C" in an open subset D of R3 is said to be of type (cl, c2 , c3) if for each compact subset L of D, there exist M > 0 and N > 0 such that altJtk

+

f ( x , t , y) 5 MN' tJ+kr(cli i)r(c, j

+ l)r(c3 k + 1)

for all ( x , t, y) E L and all integers i, j , k 2 0. An equivalent condition is

or that

+

for three numbers a , , a2 ,a3 (with the convention that r(u 1) is replaced by 0 when u 5 0). If cI = cz = c3 = 1, thenfis analytic in D (Section 9.9, Problem 7). Consider the partial differential equation

+

where r < p and q r 5 p , the function j'is of class Cmin a neighborhood of 0 in R3, andfis of type (c, 1, l), where 1 5 c 5 (q - r)/q.In a neighborhood of 0 in R3 we may therefore write

where the series converges for all (x, y, t) sufficientlynear 0, and the u'(x., t ) are of type (c, I), so that m

aJ(x, f ) =hzobJh ( X ) f h

for ( x , t ) sufficiently near 0, the series being convergent and the bJhof type c.


73

(a) Show, by induction on k 2 p , that a sequence (uk) of functions of class Cm in a neighborhood of 0 in R may be determined such that

is an identity between formal power series in T (A.21.2). (b) Show by induction on k > p that there exist two constants M > 0, N > 0 such that, for all k 2 p and all m such that m 2k 2 q 2r 1 , we have

+

+ +

for all x in a neighborhood of 0. (Start with the inequalities

and proceed by induction, using Problem l(c) and the inequalities

cm

m + 2(k - p ) 6 nt + 2k - q - 2r - 1, + (k p ) - 3 5 c(m - q) + k - r - 3.) -

(c) Deduce from (b) that the equation (*) has a unique solution m

of class C" in a neighborhood of (0, 0), analytic in t in this neighborhood and such that

at u(x, 0) = 0 at'

for 0 5 k 5 p - 1

in a neighborhood of 0; furthermore, this function is of type (c, 1).

3. Generalize the results of Problem 2 by replacing the equation (*) by a system of equations

(1 s i i m ) , wherethefunctionsfi(x, ,..., x n , f , i i ,,.... ~ i " ~ , ( w i k J ) l ~ k ~ m are, lof ~,~,) class C'O and of type (c, 1, I ) with respect to the three vectors x = (xl, . . .,xn) E R",I E R, and u = ( u , , .. ., i i m , (wIkj))E Rm+ms; the initial conditions are replaced by ak

~ u i ( x,..., l .x.,O)=O atk

for O ~ k k p l - l ,

+

and the p i are assumed to satisfy p i 2 q , k J rill and p i > I ' i k j for all (i, k , j ) . Hence deduce another proof of the Cauchy-Kowalewska theorem.

74

4.

XVlll DIFFERENTIAL SYSTEMS Consider the partial differential equation (the heat equation)

m

Suppose that u ( x , y ) = c u k ( y ) x kis a C" solution in a neighborhood of (0. 0) and is analytic in x .

k=O

(a) Show that there exists a nonempty open interval 1 c R in which the vk are defined, and a number M > 0, such that IVk ( y ) 1 5 M kfor all k 2 0 and d / y E 1. (Observe that in a neighborhood V of 0 in R,the radius of convergence of the series

m

k=o

vk(y)xk is > O by

hypothesis, for all y E V. Then use Problem 9 of Section 12.7, and (12.16.2).) (b) Show that the functions v k are of class C" in a neighborhood of 0, and that u;(Y) = (k

+ 2)(k + I)'k+Z(Y).

Deduce that there exists a neighborhood J of 0 in R such that the function u is of fype (2, 1) (Problem 2) in J x I. Hence the condition c 5 ( p - r)/q in Problem 2 cannot be improved.

5. Consider the system of partial differential equations (+) in the problem of Section 18.1 1, and replace f' by a function g analytic in a neighborhood of 0. Let (u?), up') denote the unique analytic solution of this system of equations such that ~ ~ (x02 , ,x 3 ) == ~ ~ (x z0, x3) , = 0. Show that it is not possible that there should exist a compact neighborhood 1 of 0 in R,a neighborhood V of 0 in R3,an integer k 10 and a number A > 0 such that, for each function g analytic in a neighborhood of 1, the functions vF), are defined in V and satisfy in this neighborhood the inequalities

13. T H E CARTAN-KAHLER THEOREM

(18.13.1) We shall take up again the problem posed in Section (18.11), but with the following additional hypotheses: M is a real-analytic manifold (in fact, it will again be an open subset of R")and the differential forms in the ideal a defining the differential system are analytic; furthermore, the only manifolds we shall consider will be analytic submanifolds of M. Under these conditions, the functions fk (1 8.11 . I .I), the functions gk in (18.11.2.1) and the preliminary diffeomorphisms effected in (1 8.11.2) and (1 8.11.3) are analytic, and so are the functions C j and D in (18.11.3.4). Hence we obtain, after the reductions performed in (18.1I),a system of equations (1 8.11.4.5) in which the right-hand sides are analytic, and we seek analytic solutions.

THE CARTAN-KAHLER THEOREM

13

75

(18.13.2) The Cauchy-Kowalewska theorem (1 8.12.1) therefore applies, and shows that (restricting M if necessary) the system of equations (18.11.4.3) admits a unique (analytic) solution ( u l , . . . , u,) satisfying the initial conditions (18.11.3.2). In order to resolve the problem posed in (18.11.2), it is necessary to show that the vj satisfy not only the equations (18.11.4.3) but all the equations (18.11.3.4) corresponding to all ( p + l)-forms mp+l E ap+l. For this purpose, we shall need to make use of the hypothesis of regularity of the integral element F,, from which we started (18.11.I). (18.13.3) We recall (18.10.4) that we are assuming that the subset M, of M on which all the functionsf€ a, vanish is a submanifold of M. In applications it is most often the case that M, # M (cf. (18.17) and (18.18)); but we can always reduce to the case M, = M (i.e., a. = {0}), because in a neighborhood of the point x, we may assume that M, is defined by equations x ~ + ~ = g k ( x l , . . . , xP) (1 k n - p ) , where the gk are analytic, and then as in (18.11.2) we reduce to the case where all the gk are zero, i.e., Mo = M n RP. But then we may operate in the space RP,by considering the differential forms induced on RP by the forms belonging to the ideal a. Hence, we shall assume from now on that M, = M.

s s

(1 8.13.4) Let q be any integer between 0 and p , and let rq = so

Let m$', ( I 5 p Srq) be ( q + 1)-forms belonging to that the rq linear forms (18.13.4.1)

WH(U1O

A

' * *

A U,O A W,

+ - + sq.

with the property

mi?p!1(Xo))

are linearly independent. By definition (18.10.3) and by (18.10.6) there exists such a system of (q + 1)-forms, and the hyporhesis ofregularity implies that, for ( x , ul, .. ., uq)sufficiently close to (x,, ul0, . . . , u,,) in M,, the rq linear forms (18.13.4.2)

W H ( U 1 A ' " A Uq A

W, I u $ ? 1 ( X ) )

are still linearly independent, and that for any (q + 1)-form, wq+lE a q + l ,there exist rq scalars A:? l(x, ul, , , . , ue) such that (18.13.4.3)

(ul

A

... A

uqA w, m,+l(x))

for all vectors w E R" and all (x, ul, (xo I

. . , , uq)E M,

"10,

*

* 9

u,o>.

sufficiently close to

76


(1 8.13.5) For q

5 p , put

d q + l = ( q + 1 ) S o + q s 1 + ' . . + 2 S q - 1+ s q = r 0 + r 1 + . * . f r , .

+

We shall show that there exist d,, (q 1)-forms m$' E a,+ such that, in a neighborhood of the point ( x o , ulo , . . . , in M x R("')", the (analytic) mapping

of this neighborhood into R d q + *has a (total) derivative of constant rank dq+,.This will imply that, in a neighborhood of the point (xo,

u10,

' . a 7

U,+l,O),

the set M,+ coincides with an analytic submanifold of dimension ( 4 + 2)n - d,+l

in M x R(qfl)n(16.8.8). The proof is by induction on q. For q = 0, we have d, (1 8.13.4) the mapping (1 8.13.5.1), which in this case is (x,

= so = ro

and by

Ul)H(
=

h

+

(aklh2' - akJh2') cljk

and 6 1 is generated by the 1-forms & (1 6 k

9

5 r), the dHIJkand

Let f :P Q be an analytic mapping such that the graph offis an integral manifold of are the components of frelative to the last the Pfaffian system defined by a. n - p - r vectors of the canonical basis of R", define scalar-valued functions fhlanalytic in a dense open subset PC1)of P by the formulas --f

Then the graph of the mapping

(f,( f h l ) l 5 , S p . I 6 h b n - p - r) of P(l) into Q ( I ) = Q x Rp("-p-') is an integral manifold of the differential system defined by 6. The differential system defined by b is called thefivsr-orderprolongationof the Pfaffian system defined by a. Replacing PC1)if necessary by a dense open subset, we may assume also that the set ML1)defined in (18.10.4) for the ideal b is an analytic submanifold of and that its projection onto Pcl)is a surjective submersion. Pcl)x 0")

84


Themthorderprolongationof thesystemdefined byaisthendefined by inductionas the first-order prolongation of the ( m - 1)th order prolongation. Show that the study of the numbers s; corresponding to the prolongations (of all orders) of the given system reduces to that of the following algebraic situation: Consider a field K of characteristic 0 (for example, the field of rational functions in infinitely many indeterminates over KO),three vector spaces Fp, S, T and a linear mapping 6 : FPOKT S. Define Fh, I h , Th, p h + Nh , and V = I, as in Problem 2. The quadruple (FP, S,T, 6) is called a Cartun quadruple. Define the first-order prolongation (FP,SC1),T(l), to be the Cartan quadruple in which S") = T, T")= V, and : FP8 V -+T is defined by 8 ( ' ) ( u O g ) = g(u). The mth order prolongation is then defined by induction to be the Cartan quadruple (FP,S("'),T("'), 8")), which is the firstorder prolongation of the ( m - 1)th order prolongation. Let I:'"), $'), pr+ I , N:"'], and Vcm)denote the objects defined for the mth order prolongation as above for the Cartan quadrupole (Fp, S,T, 6). -+

4. The hypotheses and notation are as in Problems 2 and 3.

(a) Show that TCm) = V(m-l) may be identified with the vector space of symmetric m-linear mappings of F; into T such that

S(u0 Og'm'(u1, u2, .. ., urn))= S(U1 O g ' T u o , u2, ... , u,))

for all choices of uo , ul, . . . , u, E FP. The mapping S ( m ) : FP OKT(m)+. S ( m ) = T ( m - 1) is defined by (S(YU @g'")))(u1, ...,urn-,)= g(,)(u1, ..., urn-1 , u).

+

The space Ip)(ul, . . . , uh) may be identified with the space of ( m 1)-linear mappings F; x Fh(uI,, . . , uh) into T which satisfy the conditions:

girn+l) of

(i) ~ A ' " + " ( U ~..., ( I ~u) ,~ ~ ~ m t l ~ ) = 8 ~ m t 1 ) (uf,+J U~l~...,

where i l , ...,i, are any integers between 1 and p, and i, tation of {l, 2, ...,p } such that u(imt 5 h ;

5 h, and u is any permu-

(ii) S ( U , ~ S P + ~ ..., ~ ( UU k~ m ~ ,, u I ) = ~ ( u I @ ~ i m t + ~ ~ ~ Uk,,r t l , . .~. 1 r )) for all indices k , 6 p and i 5 h, j 5 h. The kernel Nimlof the homomorphism pk":l is canonically isomorphic to the space Qi") c Ten) of mappings 8"") such that g ( y v l ,v*,

. . . , v,)

=0

whenever at least one of the vectors v, belongs to Fh(ul,.. , , u,,). The isomorphism transforms g&$') E Nlm)into the mapping

. ., Vm)H8E$1)(V~9 ..., vm, uhtl).

(VIP *

Define QLm)to be T("). (b) For each u E FP, let #'"(u)

be the linear mapping g'"'*S'"'(u Og'"))

of T("')into T("-'). The restriction h('")(u)of p("')(u) to QP) maps Qim) into Qim-'), and the kernel of h('")(u,,,,) is Q~'$I. (c) Deduce from (a), (b) and Problem 2 that the quadruple (Fp, S("'), T('"), 8'"))is

14 COMPLETELY INTEGRABLE PFAFFIAN SYSTEMS

85

involutory relative to P if and only if P+ I ) is isomorphic to the direct sum of the Qim) (for a suitable choice of the basis U ) . Deduce that an equivalent condition is that each of the mappings A'"' ' ) ( U h + ,) (1 5 k g p - 1) is surjective. 5. If dim(T) = q, the space L,,, of symmetric m-linear mappings of FF into T may be identified (after choosing bases of FPand T) with the dual of the subspace A,,,, of the space A = K[Z,, . . . , Z,, Z:, . . . , Zi] of polynomials in p q indeterminates, consisting of the polynomials which are homogeneous of degree m in the Z, and homogeneous of degree I in the Z j . In this identification, FP is identified with the subspace A l . o of A, and Z,, . . . , Z, with the elements u l r . .. , up of the chosen basis of Fp.

+

(a) The mapping 8("')(u)( m 2 I ) is defined everywhere on L, , If U is the element of corresponding to u, show that the transposed mapping r8cm)(u)may be identified with the multiplication mapping Wm-,, -UW,,,- ,,, , which is a linear mapping of A,- ,,,into A,,,, I . Let Bmnh be the annihilator of Qim) in A,,,, then we have UBm-,,hc Bmh,

and in order that h('")(u) should be surjective it is necessary and sufficient that the relation UW t Bnh for some w E L,- should imply W E B,- h . (b) Let bh be the ideal of A generated by the B,,, for m 2 1 and fixed h. Then

,

bh

nAm- I

Bmh.

(c) Prove the following algebraic lemma: let c be an ideal of A and let c = q 1 n q2

n q,

be a reduced primary decomposition of c,t where each primary ideal q, corresponds to a prime ideal p J ; also let m be an integer such that p 7 - I c q, for 1 5 j 5 r . Then there existsanelementUEAl,osuchthattherelationUWEcnA,,,, implies W E C ~ A , , , - , , ~ . (Consider each index j , and distinguish two cases according to whether p , =I Al,o or not; in the first case, we have qJ 3 A,,,- I , o . Deduce that if U does not belong t o any of the intersections u, n A l , o # Al,o, then U has the desired property.) (d) Deduce from (a) and (c) that, under the hypotheses of Problem 2, there exists an integer m o such that the mth order extensions of the given Pfaffian system are involutory relative to P for all m 2 mo (Cartan-Kuranishi prolongation theorem). (Choose the elements u,, . . ., up of the basis of FP successively so that the A'm)(uh+l)are surjective and the 7im)(ul,. . . , uh) take their minimum values; at each stage we have to choose uh+ outside a certain algebraic subvariety of FP, distinct from FP, by (c) and Problem 2.)

14. COMPLETELY INTEGRABLE PFAFFIAN SYSTEMS

(18.14.1) We shall now take up again the problem of the existence of integral manifolds of afield of p-directions (18.8.5), which locally is equivalent to that of the existence of integral manifolds of a Pfuflun system (18.8.7.1) (18.14.1.1)

t See, for example, N. Bourbaki, "Commutative Algebra," Chapter IV, Section 2. Addison-Wesley, Reading, Massachusetts, 1972.

86


where the wj are C" differential 1-forms such that the oj(x) are linearly independent at each point x E M. These integral manifolds are also the integral manifolds of the differential system defined by the differential ideal a generated by the oj (1 8.9.1). (18.14.2) In particular, we propose to investigate under what conditions there exists, in a neighborhood of each point of M, an integral manifold of maximum dimension p passing through the point. When this is the case, the Pfaffian system (18.14.1.1) (or the field of p-directions XHL,) is said to be completely integrable. We remark first of all that the hypothesis of linear independence of the wj(x) at each point implies that so(x) = n - p at each x E M, in the notation of (18.10.3). Secondly, if L, is ap-dimensional integral element, we must have (u, A u2,dwj(x)) = 0 for all vectors ul, u2 in L,, and by definition (18.10.3) this implies that sI(x, u) = 0 for all vectors u # 0 in L, . It follows that we have also s,(x, ul, . . . , u,) = 0 for all sequences of linearly independent vectors ulr .. ., u, in L,, for 2 5 q S p - 1 : for each q-form m, E a4 is a linear combination of exterior products of forms, each of which contains at least one of the ojor doj, and the assertion is therefore a consequence of the rules of calculation in exterior algebra (A.14.2.1). Hence the condition imposed on the Pfaffian system (18.14.1.1) already implies that every vector subspace L, is an integral element containing a regular integral element of dimension p - 1. We could therefore apply the Cartan-Kahler theorem, provided that the manifold M and the differential forms oj are assumed to be analytic; but, as we shall see, it is possible to improve on this, and to obtain an existence and uniqueness theorem under the weaker hypotheses that M is merely a differential manifold and the wj are qf class C" : (18.14.3) In order that the Pfafian system (18.14.1.1) should be completely integrable, it is necessary and suficient that for each x E M there should exist an open neighborhood W of x and C" differential 1-forms cijk defined in W, such that in W we have

(18.14.3.1)

n-p

doj=

1 wk A u j k

k=

1

(1 S j

S n -p).

When this condition is satisfied, for each xo E M there exists an open neighborhood W of xo in M, a product U x V of a connected open set U c RPand an and a difleomorphism cp of U x V onto W such that, in open set V c Rn-p, U x V, the p-dimensional connected integral manifolds of the Pfaflan system

14 COMPLETELY INTEGRABLE PFAFFIAN SYSTEMS

87

S j 5 n - p ) are exactly the submanifolds of the form T x ( y o } , where T is a connected open subset of U, and y o a point of V.

'cp(wj) = 0 (1

Since the question is local, we may assume that M is an open subset of R". Linear algebra shows that for each point x , E M there exists an open neighborhood W, of x , in M and p indices ik (1 S k s p ) such that the oj and the dxikform a basis of the €,-module of C" differential I-forms on W, (A.4.5). By performing a linear transformation on R", we may assume that ik = k ( 1 S k g p ) . Hence, for p + 1 s j 5 n, we may write (18.14.3.2)

dx' =

n-p

1Ajkwk + 11

k= 1

P

h=

Bjh

dxh,

where the Ajk and Bjh, being uniquely determined by Cramer's formulas, are functions of class C" on W,, and the determinant det(Ajk) is # 0 in W, . On the other hand, the exterior products,

wjAdxh UjAwk

dxh A dx'

(1 S j s n - p , 1 S h s p ) , (1 $ j < k g n - p ) , (I 5 h < I g p ) ,

form a basis of the &,-module of C" differential 2-forms on W, , and hence we may write (18.14.3.3)

where the coefficients are again uniquely determined by Cramer's formulas, and are therefore C" functions on W, . For each x E V, the space L, is the set of vectors u such that ( u , w j ( x ) ) = 0 for 1 S j 5 n - p , and therefore there exists a basis (uh(x))I6 h s p of L, such that (uh(x), d x ' ) = 6,,(Kroneckerdelta). Expressing that the restriction of doj(x)to L, is zero, we obtain the relations Ejhl(x)= 0 for x E W,, whence (18.14.3.1) follows immediately; the converse is obvious. We now remark that the projection of L, onto the subspace of Rnspanned by the first p vectors of the canonical basis is surjective. Hence (16.8.3.2) if there exists a p-dimensional integral manifold of the system (18.14.1 .I) passing through x , , then it is defined in a neighborhood of x, by n - p equations (18.14.3.4)

X P f k = uk(xl,

... ,X p )

(1 2 k

5 Iz - p ) ,

88


where the vk are of class C".Substituting into (18.14.3.2)and bearing in mind that the restrictions of the oj to this integral manifold must be zero, we obtain the system of partial differential equations

(18.14.3.5)

.. . , xp, Ul(x1 , . . ., x p ) ,. . . , fJ,-p(x1, . .. ,x'))

-=Bkh(X1,

axh

and conversely, if the

vk

satisfy this system of equations and are such that

uk(xA,...,xg)=Xg+k

(1 < = k $ n - p ) ,

then it follows from (18.14.3.2) and the nonvanishing of det(Ajk(X)) in W, that the restrictions of the oj to the manifold defined by (18.14.3.4)are zero. This being so, let us show that the system (1 8.14.3.5)is completely infegrable in the sense of(10.9.6). For this it is enough to take the exterior derivatives of the 1-forms in the equations (18.14.3.2);we obtain

(18.14.3.6)

n-p

(dA,

A

wk -k Ajk

A

do,) -k

k=l

P h= 1

dBjh A d X h = 0.

Replace dBjhin this relation by

and then each of dxP+', . . . ,dx" by itsexpressiongiven by(18.14.3.2).Expressing that the coefficient of dxh A dx' (1 (['

dt2 +f(([')'

-f((t3)' + (t4Y

-

=0

dt2 - 5' dt4 - 5" dt3),

&'I

of class C". Show that the Pfaffian system w = 0 on M is completely integrable; all the leaves are proper and exactly one (contained in the closure of each of the others) is compact and diffeomorphic to T2 (Rvebfdiution). 12. Let M be a differential manifold which is the product of a compact manifold S of

dimension n and a Euclidean space R"; endow S with a distance compatible with its

96


topology, and Rm with the Euclidean distance llz - 21 ' 1, and M with the distance derived from these by the procedure of (3.20). Let x++LZbe a completely integrable Cm field of n-directions on M, such that S x {O} is an integral manifold of the field. (a) Let y : [0, 11 +S be a path in S , with origin a and endpoint b. Show that there exists a neighborhood B of 0 in Rm and a unique continuous function ( t , z)

-

(y(r), f0,2))

from [0, I ] x B to M such that, for each z E B, the path t - ( y ( t ) , f(t, z)) defined on [0, 11 is contained in the leaf (Problem 4) passing through the point (a, z). (Cover S by a finite number of open sets which are domains of charts, such that for each of these open sets U there exists a neighborhood V of 0 in R" such that, for each point ( y , z) E U x V, the connected component of the intersection of U x R" and the leaf passing through (y, z) project bijectively onto U. For this, use the integration method of (10.9.4) and the majoration of (10.5.6).) Show that z-f(l, z) is of class C" on B. (Use (10.7.31.) (b) Let : [0,1] x [a,B] + S be a Homotopy of the path y = cp(., a)onto y 1 = v(.,8) leaving the endpoints a, b fixed. For each f E [a, 81 there exists a neighborhood B; of 0 in Rm and a function ( 1 , z)++(v(t, f(r, z, 5)) defined on [0, 11 x B e , having the properties stated in part (a). Show that we may assume that there exists a neighborhood W of 0 in Rm contained in all the Bc and such that f(1, z, f ) is independent o f f for z E W. (Follow the proof of Cauchy's theorem (9.6.3).) (c) Suppose that S is simply-connected. Show by using (b) that for each multi-index v = ( v l , , , .,v,) E Nm there exists a C" function gy defined on S with values in R", such that at each point b E S the vector g,(b) is the derivative of order Y of the function z++f(l, z) (defined in (a) above) at the point 0. (d) Suppose moreover that S is a real-analytic manifold. Then, with the notation of (a), the function zHf(1, Z) is analytic in a neighborhood of 0, and the functions gv defined in (c) are analytic in S.Deduce that there exists a neighborhood V of 0 in R" such that for each z E V the leaf F, through the point (a, I) is compact and projects bijectively onto S.(Observe that there exists a neighborhood of 0 in Rm in which the Taylor series

t),

converge, for all 6 E S.) (e) Extend the result of (d) to the situation where S is a compact real-analytic manifold with finite fundamental group. (Consider the universal covering S' of S, which is compact, and lift the fieid of n-directions x-Lx to S' x R",)Compare with Problem 10.

13. (a) Let XI, .. . , X , be C" vector fields on a differential manifold M, and suppose that at some point xo E M the p tangent vectors Xl(xo),. .. , X p ( x o ) are linearly independent. With the notation of (18.2.2), show that there exists an open neighborhood U of 0 in RPsuch that the mapping

(*I

(tl,

.. . , rp)-

FtIx ...

rpXp

( x o , 1)

is defined and of class C" in U, and is an embedding of U in M. (Reduce to the case M = Rnand use (10.7.4)) (b) With the same notation, let N be the image of U under the mapping (*). Show that if the vector fields [X,, X,](1 5 i <j 5 p ) vanish at all points of N, then the fields

15 SINGULAR INTEGRAL MANIFOLDS; CHARACTERISTIC MANIFOLDS

97

XI, . .. , X , are tangent to N at the points of N. (If X , Yare two linear combinations of the X , with constant coefficients, then for all sufficiently small real numbers s, t, the point u(s, t ) = FrcSx+ y ) (xo, 1) belongs to N . Reduce to the case M = R" and form the differential equation satisfied by the function t - ul(0, I )(in the notation of (18.7,1), i.e., the tangent vector at the point s = 0 to the curve s'-fu(s, f)), using (10.7.3).) (c) Let M be a real-analytic manifold, S ( M ) the ring of real-analytic functions on M, and let ? be a submodule of the S(M)-module of analytic vector fields on M. For each x E M, let v ( x ) denote the dimension of the subspace of T,(M) spanned by the tangent vectors X ( x ) with X E ?. We shall not assume that v(x) is constant. We define an integral manifold of 2 to be an analytic manifold N together with an analytic immersion j : N + M such that v(j(z))= dim, N for all z E N and such that for each X E C the vector X ( j ( z ) )belongs to the image of T,(N) under Tz(j). We shall assume that ? is a Lie algebra (in general infinite-dimensional over R). Let xo E M and put p = v(xo),so that there exist p vector fields X I , . . , X , E g such that the vectors Xi(xo) (1 ii 5 p ) are linearly independent. Also let C 0 be the set of all X E L' such that X ( x o ) = 0. Show that ?, is a Lie subalgebra of ?, and that the X , can be chosen so that [ X , , X,] E Po for all pairs of indices i,j . With this choice of XI,. . ,X , , form thep-dimensional submanifold N of M which is the image of U imder the mapping (*) in the notation of part (a). Show that the vector fields belonging to ?, vanish a t each point of N (use Problem 4 of Section 18.2). Deduce, with the help of (b), that N is an integral manifold of 2. (d) With the hypotheses of (c), show that tnaxirnal integral nianifolds of 52 can be defined as in (18.14.6), and that they form a partition of M.

.

.

14. Let f be a C" function on R such that f a n d all its derivatives vanish at the point 0, and such thatf(x) # 0 for all x # 0 (16.4.1). On RZ,consider the Lie algebra 2 over the ring Q R 2 ) of C" functions, generated by the two vector fields (tl,

t J + + E I ( t l .t J

and

( t i , 1 J + + f ( t l E01,t J

(in the notation of (16.7.1)). Show that for this Lie algebra the conclusion of Problem 13(c) does not hold.

15. SINGULAR INTEGRAL MANIFOLDS; CHARACTERISTIC MANIFOLDS

(1 8.1 5.1) Given an integral manifold N of a differential system on adifferential manifold M (18.9.2), the tangent space TJN) to N at a point x may be a regular integral element, in which case the same is true of the tangent spaces to N at all points of some neighborhood of .Y in N (18.10.4); or it may be singular (18.10.4), but such that every neighborhood of x in N contains points at which the tangent space is regular; or finally it may happen that the tangent space T,.(N) at every point s' of some neighborhood of x in N is a singular integral element. As far as the local theory is concerned we may assume, in this case, that TJN) is singular for all points x E N, and then N is said to be a singular integral manifold.

98


Example: Singular Integrals of a First-Order Scalar Differential Equation (18.1 5.2) If we take q = n = 1 in the general definition (1 8.8.9.1) of a system of first-order partial differential equations, we have a first-order scalar diferential equation

F(x, z, PI

(18.15.2.1)

=0

(where F is a C" function on an open set U c R3),and a solution of this equation is a function v, defined on an interval I of R, such that for each x E I the point (x, v(x), v ' ( x ) ) belongs to U and satisfies the relation F(x, ~ ( x ) ~, ' ( x ) )= 0. Scalar equations of type (10.4.1) are therefore particular cases of this general notion. A differential equation (18.15.2.1) is often written F(x, z , z') = 0. In order to apply the general theory, we shall assume that the set M, of points of U satisfying (18.15.2.1) is a submanifold of dimension 2, at all points of which we have dF # 0. Then we have to consider the differential system on U defined by the differential ideal a generated by the 0-form F and the 1-form dz - p dx, and the solutions of (1 8.15.2.1) will be the projections on RZof the integral manifolds of dimension 1 of this differential system (provided that these projections are manifolds of dimension I). It is clear that in this case a, is generated by the two 1-forms (I8.15.2.2)

d z - p dx,

aF dF = - dx

aF dF + -d z + -d p ; ax aZ aP

hence (18.10.3) we have so = 2, except at points where simultaneously (1 8.15.2.3)

aF

-=o, aP

dF aF -+p-=o,

ax

aZ

in which case so = I . In general, the set S of singular points (xo, z o , p o ) satisfying all three relations (18.15.2.1) and (18.1 5.2.3) will be empty or will consist entirely of isolated points: an example is the equation x - zz' = 0, where S consists of the points (0, 0, i-1). In this example, through each point of S there passes exactly one integral manifold of the differential system defined by n. However, it may also happen that there exists a curve C (i.e., a I-dimensional manifold) contained in S . The hypothesis that the three partial derivatives of F do not simultaneously vanish at any point of M implies that C

15 SINGULAR INTEGRAL MANIFOLDS; CHARACTERISTIC MANIFOLDS

99

cannot be contained in any plane x = x o . For the restriction of dF to C must be zero, which (in view of (1 8.15.2.3)) implies that the restriction of aF +d~ az

dF -dx

ax

to C must be zero; and we cannot have dF/dz = 0, otherwise it would follow from (18.15.2.3) that dF/dx = 0 also. Hence we may assume that locally C is given by z = v(x), p = w(x), and therefore we must have dF ax

aF

- + - v‘(x)= 0,

aZ

dF ax

-

aF + w(x) aZ = 0,

whence v‘(.u) = w ( x ) , because dF/dz # 0. The curve C is therefore a singular integral manifbld of the differential system, and its projection z = u(x) on RZ is called a singular integral (or singular solution) of the differential equation F(x, Z, z ’ ) = 0. An example of such an integral is given by Clairaut’s equation,

z

(18.15.2.4)

= xz’

+ g(z’),

where the derivative g ’ ( p ) is not a constant. The second equation (18.15.2.3) is satisfied automatically in this case, and the set S is given by the equations (1 8.15.2.5)

x = -g’(p),

z = g ( p ) - pg‘(p).

+

The module a, is generated by dz - p dx and (x g’(p)) dp. Through each point (xo, z o , p o ) E Mo not belonging to S, there passes a unique integral curve of the differential system, namely the line (1 8.15.2.6)

P =Po

9

z = XPO

+ g(p0);

whereas if (xo, zo , p o ) E S and if g”(po) # 0, there pass through this point two integral curves of the differential system, namely S itself and the line (18.15.2.6). The projection of S on R2 is the envelope of the projections of the lines (18.15.2.6): at each point it has the same tangent space as the line z = -up, + g ( p o )passing through the point. We shall not attempt a general investigation of the singular integral manifolds of a differential system (cf. [54]and [57]). Given a regular integral manifold of dimension p of a differential system, i.e., such that all the tangent spaces T,(N) are regular integral elements, it may happen that in each of these tangent spaces there exist singular

(18.15.3)


100

integral elements (18.10.7) of dimension q < p . If there exists a q-dimensional integral submanifold P of N whose tangent space at each point is a singular integral element (so that P is a singular integral manifold of the differential system), then P is said to be a characteristic submanifold of N. For such a manifold, the uniqueness property of the Cartan-Kahler theorem (18.1 3.2) is in general no longer valid (cf. (18.17)).

16. CAUCHY CHARACTERISTICS

Let M be a pure differential manifold of dimension n, and let a be a differential ideal of the algebra d (18.9.2). We shall assume that no = {0}, which is always possible in the local study of differential systems (18.13.3). For each point x E M, a tangent vector h, E T,(M) is a Cuucliy ckaracteristic vector of the ideal a (or of the differential system defined by a) if, for each p = 1, . . . , n and each p-form m,,E a, the ( p - 1)-covector i(h,) * m,(x) (16.18.4) belongs to the vector space n,-l(x) spanned by the ( p - I)-covectors wP- l(x), where wp- E ap(18.16.1)

By reason of the hypothesis no = {0}, for p = 1 this condition reduces to (h,, m , ( x ) ) = 0 for all 1-forms m, E a,, or equivalently h, E M,(x) in the notation of (18.10.3). In other words, the lines containing a Cauchy characteristic vector h, # 0 are the integral elements of dimension 1 at the point x. More generally, let F, be an integral element of dimension p , and let ( u ~s )j s~p be a basis of this vector space. If a Cauchy characteristic vector h, does not belong to F, (i-e., if uI A u2 A . . . A up A h, # 0), then we have (18.16.2)

(u,

A " . A

U,,A

h , , ~ ~ + ~ ( x ) ) = (Au* ~- * A up,i(h,)~mP+,(x))=O

by virtue ofthe definition (18.16.1). Furtherforall(p + 1)-forms mPtl E more, the same argument shows that if v is a vector such that (U1 A

.*.

A Up A V, W,,+l(X))

=0

for all ( p + I)-forms m p + l E a p t l , then also (ul

A

. - .A up A h,

A

v,mPt2(x)) = O

for all w p + 2E a,+2; or, in other words, s,,+,(x, uI, . . . , up, h,) = 0 (in view of (18.10.6)). Consequently we also have sq+ , ( x , uil ,. . . , ui,, h,) = 0 for 1 Sqspaandi, < . * * < i , . Nonzero Cauchy characteristic vectors do not always exist (Problems 2 and 3). If h, is a nonzero Cauchy characteristic vector that belongs to a regular integral element G,of dimension r (18.10.4), then an integral element

101

16 CAUCHY CHARACTERISTICS

of dimension q such that I < q < r, which is contained in G , and contains h,, is necessarily singular, except when s,, = 0. This justifies the name “characteristic vector” (18.15.3). (18.16.3) Let p p ( s ) denote the dimension of the subspace ap(x) of the space

A (T,(M))* of p-covectors. P

We shall show that the set of points x E X at which pp is locally constant is a dense open set in M. We may assume that M is P

an open subset of R”.and hence that A(Tx(M)) is identified with M x R‘;)’ and ap(s)with a vector space of linear forms on{.v} x R‘:’. Let U be an open neighborhood of x, and let y be a point of U such that pp(y) = mp is equal to the least upper bound of pp in U; then there exist inp differential p-forms mF’E (1, (I I 01 S nip) such that the ( y , mjP’(y)) form a basis of the vector space ap(y). I t follows now from (18.10.1) that there exists a neighborhood V c U of y such that, for each z E V, the covectors (z, wF)(z)) for 1 5 01 m p form a basis of ( l P ( z ) .This proves our assertion.

s

We may moreover assume (A.4.5) that there exist p-forms

ebb)

( 2 p 5 (;) 1

(3

-

m P differential

- nip), which we may take to be of the type devil A

such that for each z E V the

*

A

Jx‘P,

p-covectors Obb)(z)and mF’(z) form a basis

of A(T*(M))*. (18.1 6.4) We shall deduce from the foregoing that if, for each x E M, we denote by v(x) the dimension of the space N, of Caucliy characteristic vectors of the ideal a at the point x, then v is locally constant in a dense open subset of M . For by restricting to an open subset of M we may assume that the numbers pp(x) = mpare constant in M, and also that for each p we have chosen p-forms m:’ and OY’ having the properties stated in (18.1 6.3). We have then to express that, for I S p 5 n and 1 5 a S m,,, the ( p - I)-covector i(h,) mF’(x)

belongs to oP-,(,v), or equivalently that its ( p

- m p - , components

relative to the Obb2,(x) are zero. But it is clear that each of these components is a linear form in the coordinates of h, (relative to the canonical basis of {x} x R“),whose coefficients are C”-functions of x. Hence the Cauchy characteristic vectors h, are determined by a system of linear equations to which (18.10.1) applies, and this proves our assertion.

102


I t follows from this and from (18.8.4) that there exists a nowhere dense closed subset F of M such that, in each connected component Uj of M - F, the function XI+ N, is a C"fie1dof vj-directions, where v j denotes the constant value of U(X) for x E Uj . (18.16.5) On every open subset of M in which V ( X ) is constant, t h e j e l d of directions X M N , (where N, is the space of Cauchy characteristic vectors at x for the ideal a)) is completely integrable.

In view of (18.14.5), it is enough to show that if X, Yare two vector fields (on an open subset of M ) such that X(X) E N, and Y(x) E N, for all x, then also [X,Y](x) E N,, Now, for each p-form w E a,,, it follows from the formula (1 7.15.3.4)

8, * m

= i,

dm + d(ix * w),

*

and from the facts that a is a differential ideal and X(x) E N,, that we have i , dw E ap and d(i, * w ) E a,, , and therefore 8, * w E a, . Next, if X ( X )E N, and Y(x) E N, for all x, then for each p-form w E a,, we have Ox * m E a,, and i, * (0, m) E ; also i , m E ap- I and 0, * ( i , . m) E from above. Hence iLx, w E a,,-I by virtue of (17.14.10.2), and the proof is complete. (18.16.6) With the same notation, if F, is an integral element of dimension p and if F, n N, = {0}, then F, N, is an integral element of dimension p v. For if (ui)ljigpis a basis of F, and ( v j ) l d j savbasis of N,, then for each ( p v)-form w,, ,E ap+,we have

+

+

+

j ) . Let us show that this is possible in general. Without loss of generality, we may limit ourselves to the case where x i = t i for 1 6 i 5 n - I ; then the equations (1 8.18.3), apart from the last one, become

(1 8.18.5)

(1 5 k

5rI

-

1,

1 SiSn).

The equations (1 8.18.4) determine the p i for i 5 n - 1, once p" is known. Equations (1 8.18.5), when i = n, give (1 8.18.6)

124


and then, for i < n, by substituting the value of p'" =psi obtained from (18.18.6) into (1 8.1 8.5), (18.18.7)

ik

- api

p -

apn agn i 5 li2 5 ii - 1). axk - -+ ~ " "( 1 5~ axi axk agn

agn

Hence all the p i k are known once p"" is known. This being so, consider a point of the manifold Mo, YO

1

= (xo

1

* * * 9

x:

3

~0

1

3

PO

9

* * * 3

.. ( p i ) , L i d jsn)

, 1) = ,Y:, the values i k of the We can determine g" so that g"(xA, . . . , yderivatives dg"/dxk at the point wo = (x: , . . . , x:-') being for the moment arbitrary. The equations (18.18.4) then give the values of the d/r/dxk at this point, and also we have h(xA, . . . , x2,-') = zo . Clearly we can construct (in infinitely many ways) functions g", h, ' p k (I 5 k 5 n) of class C" whose values, and the values of their first derivatives, at the point w o , are the numbers above; and it now remains to show that there exists a function p r defined in a neighborhood W of wo which takes the value p y at this point and is such that @(XI,

..*,

xn-1,

p"") = 0

identically in W, where O is the function obtained from F by replacing x", z, p k (1 6 k 5 n) by the functions g", 17, cpk, and the p i k other than p"" by the functions determined by (1 8.18.6) and (1 8.1 8.7). The implicit function theorem then guarantees the existence and uniqueness of p"" under the hypothesis c?O/ap""# 0

at the point w,; in view of (18.18.6) and (18.18.7), this is equivalent to

the derivatives of F being evaluated at the point y o . It will always be possible to satisfy this inequality provided that these derivatives are not all zero, and this establishes our assertion. We shall now show that, under the assumptions we have made, there is a unique integral element of dimension n containing the tangent space T,,(T') at the point y o . Let (ej)l j s n be the canonical basis of R",and for 1 s j I n - 1 let uj be the vector in T,,(T') whose projection on R" is ej + Cje,,.These n - 1 vectors form a basis of TJT'), and it is sufficient to show that there is

18 EXAMPLES: II. SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

125

only one vector u in the integral element in question whose projection on R" is en.Let u = (0, . . . , 0, 1 u , q I

The equation (u, cu(y,))

= 0 gives

u

1 9

.. q", ( Y i j ) , s i s j s n ) .

=p:,

(u, w i ( y 0 ) ) = 0

3

and the equations ( I 5 i 5 n)

give q i = &. Next consider the n(n - I ) relations (uj~u,hi(yo))=O

(I S j g n - I ,

1Sisn);

for 1 2.j 5 n - I , we have for k 5 n - I

(d.vh, uj) = O (x) = s Sf(s-1

*

x)

for all s E G and all x E E. The B-morphism f is said to be G-invariant if s .f=ffor all s E G , in other words if (1 9.1.2.1)

f ( s * x) = s - f ( x )

for all x E E and all s E G. In particular, G acts on the set T(B, E) of C" global sections of E. In this case we write y(s)f in place of s .f. More particularly, if X and Y are two manifolds on which G acts, then G also acts on the set of C" mappings of

1 EQUIVARIANT ACTIONS OF LIE GROUPS ON FIBER BUNDLES

129

X into Y, which may be considered as the sections of the trivial bundle X x Y over X; if G acts trivially on Y, the formula (19.1.2) becomes

and to say that f is G-invariant means that f ( s x ) = f ( x ) for all s E G and all x E X. If H is another bundle over B and if G acts equivariantly on H and B (the action on B being the same as before), and if g is a B-morphism of F into H, then we have

(19.1.3) Suppose now that E is a vector bundle over B. Then G is said to act equilinearly on E and B if it acts equivariantly on E and B, and if moreover, for each b E B, the (bijective) mapping U b S H U b of E, onto E s . b is linear.

Example (19.1.4) Let M be a differential manifold on which G acts (differentiably); then G acts equilinearly on T(M) and M by the action (s, k,)t+s k, defined in (1 6.10). For, by virtue of (1 6.1O.l), G acts on T(M), and it is enough to show that the action is differentiable. If so E G and a E M, and if s i 6 n (resp. ($i)lsisn)is a local coordinate system at the point a (resp. the point so a), then the functions ~ ) ~ ( s* x ) = gi(s,tl, . . . , t"), where t j = &x), are of class C" in a neighborhood of (so, cp(a)), and the mapping SHS * k, has as local expression (1 6.3.1)

-

which proves the assertion. (19.1.5) If G acts equilinearly on E and B, and also on F and B, where F is

another vector bundle over B (the action of G on B being the same in each P

case), then G also acts equilinearly on E 0 F, E €3F, A E and Hom(E, F), by virtue of (16.16.6). To verify, for example, that we have a differentiable action of G on E 0 F, we may assume that E and F are trivial, say E = B x R" and F = B x R"; and then, if the actions of G on E and F are defined by (s, (b, u))

(s * b, 4 s , b) u),

(s, (b, v)) H (s * b, N s , b) * v),

130

XIX LIE GROUPS A N D LIE ALGEBRAS

where A and B are n x n and m x m matrices, respectively, whose elements are C" functions on G x B, then the action of G on B x R"" is given by (s, (6, w))

(s * 6, (A(#, 6) 0 &, b)) * w), P

and the result follows. The proof is analogous for EO F and A E. In the case of Hom(E, F), we may again assume that E and F are trivial in order to verify that the action is differentiable; then Hom(E, F) may be identified with B x Hom(R", R"), and with the notation introduced above the action in question is given by (s, (b, U ) ) H ( S b, B(s, b)UA(s-', s * b)),

from which the result again follows. It is easily verified that the canonical morphisms defined in (16.18) are invariant under the action of G . (19.1.6) We have seen (19.1.2) that if G acts equilinearly on E and B, then it acts on the set T(B, E); likewise it acts on I"')(B, E) for each integer r > 0, and moreover for each s E G the mappingf-s .fof T(B, E) (resp. T(')(B, E)) into itself is linear and continuous for the topology defined in (17.2). For, by virtue of (3.13.14) and (17.2), we may assume that E = B x R,and then it is enough to show that if a sequence cfk) of functions in d(B) (resp. &')(B)) tends uniformly to 0 together with all its derivatives (resp. all its derivatives of order 5 r ) in a compact subset K of B, then the sequence (s .A) has the same property relative to the compact set s K ; but this is immediate, by virtue of the formula (19.1.2) and Leibniz's formula. (19.1.7) Consider again the situation of (19.1.5). Then G acts by transport of structure on the set of C" differential operators from E to F (17.13.1). For such an operator P and all for s E G , y(s)P is defined by

for all f E T(B, E). The operator P is said to be G-invariant if y(s)P = P for all s E G. If Q is a C" differential operator of F into a third vector bundle H over B on which G acts equilinearly (with the same action of G on B as before), then we have

1

EQUIVARIANT ACTIONS OF LIE GROUPS O N FIBER BUNDLES

131

for all h, ,E I , ,( bl ) by \. I I I tic o f (1 9 1 5) I o r each (xalar) difrerenttal I-form ('1 on bl. I I follow\ ili'it y ( ~ ) c o( 0 1 . ( I ) ) I\ delitied b y the relation

(19.1.9.2)

( ( A . (rj)(.v).

h,)

=

((I)(.s-' . s),

. h,)

by v i r ~ t i cof (19.1.9.1) m c l (19.1.5) By ttan\port of structure (17.15.3.2) we Ii,ive fot ;I p-f?)riii Y of c h s s C'

(19.1.9.5)

(19.1.9.6)

d(y(.s)u)= y(.s)(dcl)

(y(\)'Y)(\) = s . X ( S

'

'

.\).

132

XIX LIE GROUPS AND LIE ALGEBRAS

Consider the corresponding differential operator O x ; by transport of structure (17.14.8), we have

r(sPx = Qr)X

((9.1.9.7)

9

and, for two C" vector fields X, Yon M, (19.1.9.8)

Remarks (19.1.10) (i) Let E be a vector bundle over B, and suppose that G acts equilinearly on E and B. Then G acts equilinearly on T(E) and E (19.1.4), and it follows immediately that if T(E) is regarded as a fiber bundle over B, then G acts equivariantly on T(E) and B. Since however there is no canonical vector bundle structure on T(E) as a bundle over B, we cannot say that G acts equilinearly on T(E) and B. (ii) The definitions and results are analogous when G acts on the right.

PROBLEMS

1. Let G be a Lie group, H a Lie subgroup of G, and let X

= G/H be the homogeneous space. Let (E, X, T ) be a fibration over X,and let xo denote the point eH = H E G/H = X,and Eo = n - ' ( x o ) the fiber of E at x o .

(a) Suppose that G acts equivariantly on E and X (the action of G on X being the canonical action, by left translations). Then the mapping (1, y ) - r . y defines a left action of H on Eo Show that if G is regarded as a principal bundle with base X and group H (16.14.2), there exists an X-isomorphism f of G :r ' Eo onto E, such that f(s * I) = s .f(r) for all s E G and all I E G x " Eo (the left action of G on G x 'Eo being defined by s ( r y ) = (sr) y). (Consider the mapping (s, y)-s * y of G x Eo into E.) Consider the converse, and the case where E is a vector bundle over X,and G acts equilinearly (b) Let ( E , X, T ' ) be another fibration over X on which G acts equivariantly, and let EA = 7'-' ( x 0 ) be the fiber of E' at xo . Let u : Eo -+ EA be a C" mapping such that u ( r . y ) = r . u(y) for all t E H. Show that there exists a unique X-morphism P : E + E' extending u such that P(s * z) = s * F(I) for all s E G and all z E E. (c) Let I be the set of points y E Eo that are invariant under H. For each y E I, let a, : X E be the mapping defined by oy(s.x o ) = s . y for all s E G. Show that every G-invariant section of E over X is of class Ca,and that the mapping y- o, is a bijection of I onto the set of these sections.

.

-

.

--f

-

1 EQUIVARIANT ACTIONS OF LIE GROUPS ON FIBER BUNDLES 2.

133

Let G, G' be two Lie groups, p : G --z G' a Lie group homomorphism, X (resp. X ) a differential manifold on which G (resp. G') acts differentiably, and f:X + X ' a Cm mapping such that f ( s .x ) = p(s) - f ( x )for all x E X and all s E G. Let E be a fiber bundle over X', and suppose that G' acts equivariantly on E and X'. Show that there exists a unique differentiable action of G on the inverse image E = X x x' E' of E under f such that if (Ly) is the canonical morphism of E onto E (16.12.8), then g(s . z) = p(s) g(z) for all z E E and all s E G. 3

3. (a) Let G be a compacf Lie group, X a compacf differential manifold on which G acts differentiably, and E a vector bundle over X such that G acts equilinearly on E and X. The group G then acts continuously and linearly on the Frkchet space r(X,E) (resp. the Banach space P ( X , E), for each integer r 2 0; cf. (17.2.2)). For each section u E P ( X , E) ( r an integer 2 0 or a ) ,the set of sections s . u (19.1.2), where s E G, has a compact closed convex hull in I"')(X, E) (Section 12.14, Problem 13). Deduce

+

that the integral

where P is a Haar measure on G , has a meaning (Section 13.10, Problem 2) and is a

G-invarian/ section of class C'.

(b) Let A be a submanifold of X which is stable under the action of G. Show that each C' section ( r an integer 2 0 or a)of E over A that is invariant under G can be extended to a G-invariant global C' section of E. (Use (16.12.11) and part (a).) (c) Deduce from (b) that if F is another vector bundle over X on which G acts equilinearly, and if there exists a G-i.somorphism f of E IA onto F [A (that is, an A-isomorphism such that f(s * z) ==s * f ( z )for all z E E and s E G), then there exists an open neighborhood U of A in X which is stable under G, and a G-isomorphism of E JU onto F IU which extendsf. (Apply (b) to the vector bundles Horn (E,F)and Hom(F, E), and then use (8.3.2.1) applied to the Banach spaces

+

rCo)(X, End(E))

and

rto)(X,End(F)).)

(d) Let Y be a differential manifold on which G acts differentiably, and let

v : Y x J+X be a C" mapping (where J is an open interval in R,containing [0, 11) such that *

Y , 1 ) = s * P(Y, 1 )

for s E G, y E Y, and f E J. If Eo, El are the inverse images of E under fo and fl v(., I), show that Eo and E, are G-isomorphic. (Use (c).) 4.

=

v(., 0)

Let G be a Lie group and X a principal bundle with base G and group G . (a) Let G act on the base of X by left translations. Show that if there exists a differentiable action of G on X such that G acts equivariantly on the bundle X and the base G , then X is trivializable. (Observe that a G-orbit in X is a section of X.) (b) Give an example of a principal bundle with base G and group G which is not trivializable. (Consider the Klein bottle (1 6.14.10)J

134

XIX

LIE GROUPS AND LIE ALGEBRAS

5. Let G be a compact Lie group, M a compact differential manifold, J an open interval in R containing 0. Suppose that G acts differentiably on M x J in such a way that

.

s (x,

where for each

5) = (mc(s, x ) , 5 ) ,

6 E J, me is a differentiable action of G on M.

(a) Show that the vector field

on M x J, where is a Haar measure of total mass 1 on G and E is the unit vector field on R, is invariant under the action of G on M x J. (b) Deduce from (a) that there exists a diffeomorphism ( x , &-(h,(x), & of M x J onto itself (so that he is a diffeomorphism of M onto itself, for each 5 E J) for which

m e h x ) = hc(mo(s, h; ' ( x ) ) ) .

-

In other words, the actions me are isotopic. (Consider the flow of the vector field

(X,El.) 6. Let G be a compact Lie group, acting differentiably on a differential manifold M, and let xo E M be a point fixed by G. Then G acts linearly on the tangent space T,,(M) by (s, hZo)Hs* h,, (16.10.1). The point xo has arbitrarily small G-stable open neighborhoods (12.10.5). If V is one, let fo : V + T,,(M) be a C" mapping such that fo(xo)= Ox,

and such that T,,(f0) is the identity. If fi is a Haar measure on G, show that the mapping f : V +. T,,(M) defined by f(x) = JG t fo(t-'

*

x)

is of class C", that f(xo)= Ox,, and that Txo(f)is the identity; also show that f(s .x ) = s f(x) for all s E G. Deduce that there exists a G-stable open neighborhood W c V of x0 and a chart of W such that the local expressions of the diffeomorphisms x - s x of W onto itself for all s E G, relative to this chart, are linear transformations (Bochner's theorem).

-

-

2. ACTIONS OF A LIE GROUP G ON BUNDLES OVER G

If G is a Lie group, then G acts on itself by left translations x w s x . The definitions of (19.1) may therefore be applied to fiber bundles with base G, and we shall say that G acts left-equivariantly on a bundle E over G if G acts equivariantly on E and G (the action of G on G being left-translation). Likewise we define a right-equivariant action of G on E: the formula which replaces (19.1.2) in this case is cr'.s)(x)=f(x's-').s, and we shall write G(s-')fin place off s. Similarly, for differential operators, we shall put (G(s)P)* f = 6(s)(P * 6(s-'lf).

2 ACTIONS OF A LIE GROUP G ON BUNDLES OVER G

135

(1 9.2.1) For example, G acts left-equivariantly on the tangent bundle T(G) by (s,h,) HS * h, ,and right-equivariantly by (s, h,) Hh, s. Moreover, G acts equilinearly in both cases, and we have s (h, * t ) = (s * h,) * t (1 6.9.8). By virtue of (1 9.1.5) there are analogous statements for the tensor bundles T:(G) and the exterior powers of T(G) and T(G)*. For such bundles it is clear that

r ( W t )= W r ( d for all s, t in G, for the actions of G on the sections or on the differential operators. In particular, for each s E G, the mapping h,Hs

*

he* s-'

is an automorphism of the tangent space T,(G) at the neutral element e of G. It is denoted by Ad(s), and it is clear that (19.2.1.l)

Ad(s)

= T,(lnt(s)),

where Inth) is the inner automorphism X I + S X S - ~ of G. By (19.1.4) the mapping s H Ad(s) is a Lie group homomorphism of G into GL(T,(G)), called the adjoint representation of G . (1 9.2.2) Let E be a bundle over G, on which G acts left-equivariantly. Then every invariant section f of E over G is uniquely determined by its value f ( e ) E E, at the neutral element of G, because we must have f ( s ) = s * f ( e ) for all s E G . Conversely, it is clear that, for each element u, E E,, the mapping SHS . u, is a G-invariant section of E, of class C". In particular, if E is a vector bundle of rank n over G, and if G acts equilinearly on E, then the mapping which assigns to each u, E E, the invariant section S H S u, is an isomorphism of the vector space E, onto the vector subspace I of T(G, E) consisting of the G-invariant sections (and therefore I has dimension n). Furthermore, if ( f J I S i S , is a basis over R of this vector space, the sections f i form a frame of E over G. Hence every vector bundle over G on which G acts equilinearly is trivializable (16.15.3). (19.2.3) Let E, F be two bundles over G on which G acts left-equivariantly. Each invariant morphism g of E into F is uniquely determined by its restriction ge : E, -+ F, to the fiber E, at the neutral element; for it follows from (19.1.2) that g,(u,) = s * ge(s-' * us)for all s E G. Conversely, given any C" mapping g, : E, + F,, if we define g, for each s E G by this formula, it is clear that the mapping g : E + F which is equal to g, on each fiber E, is an invariant morphism.

136

XIX


(19.2.4) Finally, if E and F are two vector bundles over G , on which G acts left-equivariantly and equilinearly, then every G-invariant differential operator P from E to F is uniquely determined by the continuous linear mappingft+(P*f)(e) of r ( G , E) into F,. For by (19.1.8) and (19.1.2), we have ( P .f)(s) = s . ( ( P * y(s-')f)(e)) for all s E G. We shall consider more particularly the case where E = F = G x R, in other words the fieIds of real point-distributions on G (1 7.13.6) which are left-inuariant (i.e., invariant under left translations of G ) .

3. T H E INFINITESIMAL ALGEBRA A N D T H E LIE ALGEBRA O F A LIE GROUP

(19.3.1) Let G be a Lie group. The set of real differential operators P E Diff(G) (17.13.6) which are left-inuariant forms is, by virtue of (19.1.7), a sub-Ralgebra 8 of the (associative) algebra Diff(G), called the injinitesimal algebra of G . Its identity element is the identity mapping of &(G). We have seen (19.2.4) that the mapping Pt+P(e) is a bijection of 6 onto the set of real distributions with support contained in {e}; also we have the relation

which shows that PI+ P(e) is an isomorphisn? of the algebra Q onto the algebra of real distributions with support contained in {e} (the multiplication in this algebra being convolution), which we shall denote by 6,. To prove (19.3.2), consider an arbitrary functionfE &G). We have

and by virtue of the invariance of Q, we have (1 9.1.8) for each x E G ,

(Q * f ) ( x >= ( Q

*

y(x-'If)(e) = ,~ ( x - ' l f ) ;

putting Q(e) = S, this last expression may also be written in the form I f ( x y ) dS(y) (17.3.8.1), and therefore (P

(Q* n ) ( e )= p ( e ) ( Q .f>= *

5 s

d N x ) f(xr) d W ) ,

where R = P ( e ) . By virtue of (17.11.1) and (17.10.3), this proves (19.3.2).

3

INFINITESIMAL ALGEBRA; LIE ALGEBRA OF A LIE GROUP

137

(19.3.3) The set g c Q of left-invariant differential operators which are of order 5 1 and annihilate the constants may be identified (via the mapping X H ~ , ) with the set of left-invariant C" vectorjelds on G (in other words, the invariant sections of the tangent bundle T(G)) (17.14.2). It follows from (19.1.8) that g is a Lie subalgebra of the Lie algebra of all C" vector fields on G (17.14.3), in which the bracket is defined by [ X , Y ] = X o Y - Y o X. The mapping XI+X(e) is an isomorphism of the vector space g onto the tangent space T,(G) (1 9.2.2); the inverse isomorphism assigns to a tangent vector u E T,(G) the invariant vector field (19.3.3.1)

xu: s H s . u.

The distribution X,(e) is just the differentiation 0. in the direction of the tangent cector u at the point e (17.14.1). By reason of the above isomorphism, given any two tangent vectors u, veT,(G), there exists a unique tangent vector, denoted by [u, v], with the property that (19.3.3.2)

o[u,v] = 0, * 0, - 0, * 0.

I

so that we have (1 9.3.3.3)

The tangent vector [u, v] is called the Lie bracket of the two tangent vectors u, v. The tangent space T,(G), endowed with this law of composition, is clearly a Lie algebra ge, and the mapping XH X(e) is a Lie algebra isomorpliism of g onto g,. The algebra g, (or g) is called the Lie algebra of the Lie

group G , and will sometimes be denoted by Lie(G). A little further on we shall see that g, and the identity element generate the (associative) algebra 6, (19.6.2). (19.3.4) Let G, G' be two Lie groups, e, e' their respective neutral elements, f : G + G' a Lie group lronromorphism (16.9.7), and 6, 6' the infinitesimal algebras of G , G', respectively. For each field of point distributions P E 6, the irnagef(P(e))is a point distribution belonging to 6:.(17.7.1); hence there exists a unique invariant operator P' E 6' such that

138


The operator P' is called the image of P underf, and will sometimes be denoted byf,(P). By definition, we have for each s E G f*(P>cf(s>)= r(f(~)>f(P(e>) and hence, for each function u E 6 ( G ) ,

(u,f*(P)(f(s)))= ( 4 rCf(s>>f(P(e))>= (yCf(s>-')u7f(P(e)>). By definition, u = ycf(s)-')u is the function t ' H u ( f ( s ) t ' ) , and therefore the composition u ofis the function t H u(f(s)f(t)) = u(f(f(st)),

so that u 0 f = y(s-')(u

(y(f(s)-'~~,f(P(e)))

Hence we have

of).

= (r(s-')(u o f ) ,

P(4> = (u

of,

w> = = f ( ( P Q)(e)) =f(f'(e> * Q(e>>= f ( P ( e ) )* f ( Q ( e > > 0

0

by (17.11.10), and by definition this last distribution is equal to

(f*(P)(e'))* U*(Q>(e')>, whence (19.3.4.3) follows. In other words, f* is a homomorphism of the algebra 8 into the algebra Q', called the derived homomorphism off. If g : G' + G" is another Lie group homomorphism, then it is immediately verified that (19.3.4.4)

(9 of)* = 9* o f *

(19.3.5) Since the image underfof a distribution of order 5 1 is a distribution of order 5 1, it is clear that if g, g' are the Lie algebras of left-invariant C" vector fields on G, G', respectively, then we have f*(g) c g', and the restric-

3 INFINITESIMAL ALGEBRA; LIE ALGEBRA OF A LIE GROUP

139

tion off,, to g is a Lie algebra homomorphism of g into g', by the definition of the Lie bracket. Under the canonical identification of g with g, and g' with 9:. , the homomorphismf, is identified with the tangent linear mapping T,(f), which we shall denote also byf, or Lie(f). Hence, if u, v are any two vectors in g,, we have

In particular, for each s E G, we have (19.2.1.1) (19.3.5.2)

Ad(s) * [u, V]

=

[Ad(s) . U, Ad(s) * v].

(19.3.6) If we suppose merely that f is a local homomorphism (16.9.9.4), the mapping S Hf(S) is still a homomorphism of the algebra 8, into the algebra 8Lr(17.11.10.2), which is again denoted by f*; its restriction to ge (which may be identified with T,df)) is therefore again a Lie algebra homomorphism of ge into 9:. .

Remark (19.3.7) Let G be a Lie group acting differentiably on the right on a differential manifold M. We shall show that to each (lefl-invariant) differential operator P E 8 on G , there is canonically associated a differential operator PMon M, whose order is at most equal to that of P. For each x E M, let a, denote the mapping SHX * s of G into M, which is of class C", and put

for all f E d(M), or equivalently PM(x)= (a,)*(P(e)) (17.7.1). From the local expression (1 7.13.3) of P(e) i t is immediately verified that PMis a differential operator of order at most equal to the order of P. Moreover, for each s E G , we have

because the right-hand side is equal to (P(e), y(s-')(fo ox)), and y(s-')( f 0 ax) is equal to f 0 o x . s .From this we deduce that PHP, is a homomorphism of the algebra 8 into the algebra Diff(M) of differential operators on M (17.1 3.6); that is to say, (19.3.7.3)

( P o Q ) M = P M QM o

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XIX


for P, Q E 6.For if we put R = P ( e ) and S = Q(e), then for all f e &( M) (with the notation of (17.10.3.2)), we have

= jdR(s) s/((x

= ss/(x =

sf(.

*

*

s) t ) d S ( t )

( s t ) ) dR(s) d

0) d(R

w)

* S )(v )

= ((PC, Q)M ..f)(-r)

since (P Q ) ( e )= R * S. In particular, for U E ~ , ,the operator (ex,), is equal to OZ,.M. where 2u,M (also denoted by 2,) is the vector field on M 0

(1 9.3.7.4)

s++x * u

(cf. (16.10.1)). For by definition, we have

((e,J,

* ( f o a ,))@) = <de(f’c a x ) , u> (dxf, T,(ox) . u> = ,

. f ) ( x ) = (Ox, =

from which the assertion follows. The operator PM (resp. the vector field Zu,M) is called the trunsporr of P (resp. Xu)by the action of G on M. By virtue of (19.3.7.3) we have (1 9.3.7.5) so that the mapping UHZ, is a Lie ulgebru liotirotnorphism o f g, into the Lie algebra 3 A ( M ) of C” vector fields on M. The field Z , is also called the Killingfield on M corresponding to u. For each s E G we have (19.3.7.6)

Z,(x

. s)

’=ZAdlS), JX)

For the left-hand side is (x s) * u

= (x

. s.

(s * u s-l)) . s (16.10.1).

3 INFINITESIMAL ALGEBRA; LIE ALGEBRA OF A LIE GROUP

141

PROBLEMS

Let G be a Lie group and X a differential manifold. A partial right action of G on X is by definition a C" mapping (/I : R + X,where R is an open subset of G x X containing {e}x X, such that: +(e, x) = x for all x E X; (2) there exists a neighborhood R, of { e } x {e) x X in G x G x X such that, for all (s, I , x ) E R,, we have ( t , x) E Q, ( t s , x) E R, (s, $0,x)) E R, and (1)

+(s, $ ( t , x ) ) = $(ts, XI.

We write +(s, x) = x * s for (s, x) E R. Let (X.) be a locally finite denumerable open covering of X.For each n, let (/I, be a partial right action of G on X, , and suppose that for each pair of integers p , q and each X E X,nX,, the actions +p and $, coincide in some neighborhood of (e, x). Show that there exists a partial right action (/I of G on X such that, for each n and each x E X., the partial actions and (/In coincide on some neighborhood of (e, x). (Use Problem 5 of Section 18.14.) Let G be a Lie group, M a differential manifold, and let (s, x)-x s be a partial right action of G on M (Problem 1). Show that, for each differential operator P E (3, the definition (19.3.7.1) remains valid; likewise the formula (19.3.7.3), and the formula (19.3.7.2) for all pairs (s, x ) E Q. In particular, the mapping u-Z, is a Lie algebra homomorphism of oe into the Lie algebra Y & M ) of C" vector fields on M. Such a homomorphism is called an infinitesimal action of ge on M. Conversely, suppose we are given an infinitesimal action u- Y, of gc on M. Show that there exists a partial right action of G on M such that Z, = Y, for all u E R ~ , (For each point (s, x) E G x M, consider the n-direction L(a,x) in TdS, x,(G x M) =T,(G) x TJM) (where n = dim(G)) generated by the tangent vectors (Xu($),Y,(x)) as u runs through oe. Show that this field of n-directions defines a completely integrable Pfaffian system, and then use Problem 1J Give an example of a partial right action of G on a differential manifold M which cannot be extended to a differentiable action of G on M (16.10). (Observe that if U is an open subset of M, and p : (s, x)-x s an action of G on M, then there exists a neighborhood R of {e} x U in G x U such that p(R) C U.) (see Section 19.8, Problem 3). Let G be a separable metrizable connected topological group, and suppose that we are given, on a symmetric open neighborhood U of e, a structure of a differential (resp. analytic) manifold with the following property: if V is a symmetric open neighborhood of e such that V2 = U, then the mapping (s, t)-st-' of V x V into U is of class C" (resp. analytic). Show that there exists a unique structure of differential (resp. analytic) manifold on G, compatible with the group structureof G,whichinduces on each symmetric open neighborhood W of e such that W' = U the manifoldstructure induced by that given on U. (Use (16.2.5) to define the differential (resp. analytic) manifold structure on G. Then show that for each s E G there exists a neighborhood

142

XIX


Z, of e, contained in W, such that sZ,s-' c V and such that the mapping X - S X S - ~ of Z , into V is of class Cm(resp. analytic). For this purpose, use the fact that s can be written as a product slsz . * * s, of elements of W.) 5.

Let G be a connected Lie group and M a connected differential manifold on which G acts differentiably on the right. (a) For G to act transitively on M it is necessary and sufficient that, for each x E M, the mapping u-x u = Z,(x) of (1, into TJM) should be surjective. (To show that the condition is sufficient, observe that if it is satisfied every G-orbit is open in M.) (b) Suppose that G acts transitively on M, so that M may be identified with a homogeneous space H\G, where H is the stabilizer of some point xo E M. The kernel of the homomorphism u n Z , of ne into the Lie algebra YA(M) is then the Lie algebra of the largest normal subgroup K of G contained in H, namely, the intersection of the conjugates sHs-' of H (cf. 19.8.11); M may be identified with the homogeneous space (K\H)\(K\G). (c) With the hypotheses of (b), the action of G on M is said to be imprimifiue if there exists a closed submanifold V of M such that 0 < dim(V) < dim(M) and such that each transform V * s of V by an element of G is either equal to or disjoint from V. For this to be the case, it is necessary and sufficient that there should exist a closed subgroup L of G such that H c L c G and dim(H) < dim(L) < dim(G). If it is not the case, the action of G on M is said to beprimiriue; this will occur whenever there exists no Lie subalgebia of nr containing the Lie algebra be of H, other than ge and l i e . (d) With the hypotheses of (b), let I' be the image of oe under the homomorphism UHZ,. Let 8 be the algebra of C"-functions on M, and let III be the maximal ideal of E consisting of the functions vanishing at xo. For p = - I , 0, 1, 2, . . . , let i!, denote the set of vector fields X = Z , E ? such that Ox . / E 1 1 1 p + ~for all / E 8,so that ? - = ?, and 2, is the image of the Lie algebra b, of H. If c = (U, tp, I ) ) is a chart of Mat the point xo such that y4xo)= 0, and if(X,), b l b n are the vector fields associated with this chart (16.15.4.2), then the elements of ?, are the vector fields

n 1= I

al Xi, where

a, E n i p + ' (1 5 i 5 n). Show that [i!,, , ?.J c L',,, (with the convention that ? - 2 = i!), Furthermore, if there exists a vector field Y E ?, (for p 2 0) such that Y 4 ?,,+]. then there exists X E i!such that [ Y, XI 4 ?, (but we have [ Y, XI E 2,- ,). (Observe that, by (a), for each index i there exists X E ? such that

(Ox

. tpi)(xo)

#0

and

(8, . tp9(xo) = 0

for j # i . )

The 2, are Lie subalgebras of P and are stable under the mappings X-S(r)X forall t E H. For each p 2 0, Yp is an ideal in 2,. If, for each r E H, p ( t ) denotes the endomorphism h- h . t of T,,(M), then p is a linear representation of H on T,,(M), and if R is the image p(H) of H in GL(T,,(M)), the Lie algebra of fi is isomorphic to (e) Suppose further that i!, = {O}. Then there exists a largest index r such that

n P

2, # {O} and all the P, with j 5 r are distinct. For e a c h p >- 0, we have

(f) If G is a real (resp. complex) analytic group acting analytically on a real (resp. complex) analytic manifold M, the condition I?, = (0)is always satisfied.

n P

3 6.

INFINITESIMAL ALGEBRA; LIE ALGEBRA OF A LIE GROUP

143

Let u- Y . be an infinitesimal action (Problem 2) of the Lie algebra oe of a Lie group G on a differential manifold M. Given a point x o E M we may then define the 2, ( p 2 - I ) as in Problem 5(d). The action of (1, is said to be transitive at x o (or the Lie algebra ? is transitive at x o ) if dim(!!-l/?o) = dim,,(M). An infinitesimal action which is transitive at xo is said to be primitive at xo if there exists no Lie subalgebra of ? containing C 0 , other than 2 and F 0 . If the infinitesimal action u- , ' l is transitive (resp. primitive) at the point x o , what can be said about the corresponding partial action (Problem 2)?

7. (a) Let M be a differential manifold of dimension I . If the Lie subalgebra ? of YA(M) is transitive at a point xo (Problem 6) , then the condition 2, = {O} is satisfied.

n P

(Identifying a neighborhood of xo in M with an interval in R containing xo = 0, the restrictions to this neighborhood of the vector fields belonging to ? are of the form xi - f ( x ) E (in the notation of (18.1.1)) and at least one of the functions f d o e s not vanish at 0. By change of variable, we may assume that E E ?. Deduce that, if dim(?)= m, each of these functions f satisfies a homogeneous linear differential equation of order 5 m, with constant coefficients, and hence is analytic.) (b) With the hypotheses of (a), show that dim(!!) 5 3 and that the partial action corresponding to ? is necessarily of one of the following types: (I)

dim(?) = 1, G

= R,

and the partial action is ( 1 , x)-x

+t

(t E G , x

E

M).

(2) dim(?) = 2, G is the group defined in Example (19.5.11), and the partial action is ( ( I , , t r ) , x ) - t I x i t2.

(3) dim(?)

=

3, G

PGL(2, R), and the partial action is

( t l close to I , and t 2 , t 3 , x close to 0). (Observe that

[?r-l,

?,] # {0} and deduce that

r i I.)

If M is connected (and therefore diffeomorphic to either R or S , ) , which of the partial actions defined above can be extended to an action of G (or a connected group locally isomorphic to G ) on M? (c) Give examples in which ? is not transitive at a point and has arbitrarily large dimension. (Observe that there exist C"-functions .f, g on R, with nonempty compact support, such thatfg' - g f =f.)

8.

For all integers t

> 0, the vector fields (in the notation of (18.7.1)) €1,

Ez,x'Ez,(xl)ZE2, .. ., (x1YE2

form a basis of a Lie subalgebra of .F-b(R2) defining an infinitesimal action which is transitive and imprimitive at every point. 9. (a) With the notation of Problem 6, for each p 2 0 such that ?, # {O}, the vectors X ( x ) with X E ?, generate a vector subspace E,(x) of T,(M) at each point x # xo of a neighborhood V of x o . There exists a nonempty open subset U c V - {xo}

144


such that xo E 0 and such that dim(E,(x)) is constant and nonzero on U; the field of directions X- E,(x) in U is completely integrable. If this field of directions is invariant under the partial action on U corresponding to the given infinitesimal action, and if the latter is transitive at the point xo (and therefore in the neighborhood V, for sufficiently small V), show that it cannot be primitive at the points of U unless dim(E,(x))= dim,(M) a! these points. (b) If the infinitesimal action defined by I! is transitive at the point xo and if the Lie algebra I! is commutative, then go = {0}(replacing M if necessary by a neighborhood of xo). (c) Suppose that 2, # {0}and I!,+ = {0}for some integer r 2 0, and that the infinitesimal action defined by I! is transitive at the point xo and primitive at all points of a neighborhood of xo. Let n = dim,,(M). Show that, if r > n, there exists a nonempty open set U c V - {xo} such that xo E 0 and dim(E,-,,+&)) = n for all x E U. (If for some p 5 r we have dim(E,(x)) < n at all points of an open set whose closure contains xo , show that on another such open set we have dim(E,- l(x)) 2 dim(E,(x))

+ 1,

using (a) above and arguing by contradiction.) (d) Deduce from (c) that under the same hypotheses we have r hence that

6 2n + 1, and

+

(Observe that I!,-,, is commutative if r > 2n 1 and that by (c), there exists an open set, containing xo in its closure, on which the infinitesimal action defined by 2,-" is transitive; then obtain a contradiction by using (b) and a lower bound for dim(I!,-").) 10. With the notation of Problem 6, suppose that I! is transitive at the point xo, and that I!, # {O}, 2,+ = (0) for some integer r 2 0 ; then 2, is a commutative Lie algebra.

Show that if the infinitesimal action defined by 2, is transitive on some open subset of M - {xo}whose closure contains xo ,then we have r = 1 and dim(2,) = n. (Observe that if r > 1, the algebra 2,- would also be commutative, and show that this would contradict Problem 9(b).) Deduce that dim(2) 5 n(n 2).

+

11. Let G be a connected Lie group acting differentiably and transitively on a differential manifold M. For each integer k 2, the action of G on M is said to be k-ply transitive if there exists an open orbit for the action (s, (xl, .. ., xk))w(s * x,, . . ., s xk) of G on M'. If so, we have dim(G) 2 kn. If H is the stabilizer of a point of M,then for the action of G on M to be k-ply transitive it is necessary and sufficient that there should

. ..,Hh of H such that exist k conjugates HI,

dim(G) - dim(H1 n * * * nHk)= kn. When this condition is satisfied, there exists an orbit of the action of H on M, on which the action of H is (k - 1)-ply transitive. Moreover, the action of G on M is primitive (Problem 5). 12. (a) State the definitions and results corresponding to those of Problem 11 for partial

actions and infinitesimal actions. (b) With the notation of Problem 6, suppose that the infinitesimal action defined

4 EXAMPLES

145

by ? is k-ply transitive in a neighborhood of x o . Suppose also that there exists an integer r 2 0 such that ?, # {0} and Y r + = {O}. Show that k 9 n 2, where n = dim,,(M). (We may assume that k 2 3. The infinitesimal action defined by -Oo in an open subset of M - {xo} whose closure contains xo cannot be (k - ])-ply transitive unless, in the notation of Problem 9(a), we have dim(E,(x))=n in this open set. Deduce from Problem 10 that dim(?) 2 n(n 2))

+

+

4. EXAMPLES

(19.4.1) We shall begin by determining the infinitesimal algebra 0 of the commutative Lie group G = R". From (17.7.3), the space 8, of distributions with support contained in (0) is the set of all p(D)&,,where p(D) = c1 DAis

1

a polynomial in the partial differentiation operators Di = d/axi (1 Also, for any two polynomials p , q, we have (1 7.1 1. I 1.2)

1

4 i S n).

P(D)Ee * dDNe = (p(D)q(D))Ee

9

which proves that the algebra 8 , is isomorphic to the algebra R[X,, . . . , X,] of polynomials in n indeterminates over R. The invariant field of distributions P corresponding to p(D)&,is such that, for allf E &R") and all x E R", we have

(P* f

= (AD)&,3

Y(- x ) f ) = (@)f

)(XI,

so that P is just the diyerentiul operator p(D). The Lie algebra ge may be canonically identified with R"; it is obviously commutatioe, i.e., [u, v] = 0 for all vectors u, v. (19.4.2) Let A be a finite-dimensional (associative) R-algebra with an

identity element e, and let A* be the Lie group of invertible elements of A (16.9.3). Since A* is an open subset of the vector space A (15.2.4), the tangent space T,(A*) may be canonically identified with the tangent space T,(A) (1 6.8.6), and we have a canonical linear bijection (1 6.5.2) t, : T,(A) + A. We shall show that

which will allow us to identify the Lie algebra Lie(A*) with the vector-space A endowed with the bracket operation [x, y] = xy - yx. It is enough to prove that the values taken by an arbitrary linear form f on A at the vectors t,([u, v]) and t,(u)r,(v) - t,(v)t,(u) are the same. Since Df =f (8.1.3), we have

146

XIX


by (17.4.1). By virtue of (19.3.3.2), we have to calculate

(4* 0,) *f=

s

s s

d u x ) f ( x y )d w ) ;

but f ( x y ) de,(y) is by definition the derivative of the linear form y ~ f ( x y ) at the point e in the direction of the vector v, hence (8.1.3) is equal tof(xz,(v)). The same remark applies to f(xt,(v)) do&), and so we obtainf(z,(u)z,(v)) and hence the formula (1 9.4.2.1). In particular, if E is a real vector space of dimension n and if A = End(E), we have A* = GL(E) (16.9.3). The Lie algebra of this group is denoted by gI(E), and may be identified with End(E) endowed with the bracket operation. More particularly, if E = R", we denote by gI(n, R) the Lie algebra of the group GL(n, R); it has as a basis over R the canonical basis ( E i j )of the matrix algebra M,(R) (where E i j is the matrix of the endomorphism uij of R" defined by uij(ej)= e i , uij(ek)= 0 for k # j ) with the following multiplication table:

s

1

[Eij 3 E h k l

(I9.4.2.2)

[Eij 7

=

Ejrl = Eih

[ E i j , Ehi]

= - Ehj

[ EI J. '. EJ. I. ] = E,, - E .J .J .

if j # h and k # i, if k # i, if h # j ,

Likewise we denote by gI(n, C) and gI(n, H) the Lie algebras of the groups GL(n, C ) and GL(n, H) (16.9.3). The matrices Eij again form a basis of gI(n, C ) as a vector-space over C (resp. of gI(n, H) as a left-vector-space over

H). (19.4.3) Let G be a Lie group, H a Lie subgroup of G . Since the tangent space T,(H) may be identified with a subspace of T,(G), it follows that the Lie algebra lje of H is thereby identified with a Lie subalgebra of g,. Consider in particular a finite-dimensional R-algebra A with identity element, and suppose that we are given an involution XHX* on A. (The definition is the same as in (15.4), except that here (Ax)* = Ax* for A E R, i.e., the mapping XHX* is linear; an involution in a C-algebra is also an involution for the underlying R-algebra.) Let a be an invertible element; we shall show that the set H of elements x E A* such that (1 9.4.3.1)

x*ax = a

is a Lie subgroup of G = A*. Clearly H is a subgroup of G , so that it has to be shown that H is a submanifold of G . For this it is enough to show that the mapping x ~ - + x * a xis a subimmersion of A into itself (16.8.8). Since the

4 EXAMPLES

147

mapping XHS* is linear, and the mapping (x, y ) ~ y a xbilinear, it follows from (8.1.3), (8.1.4), and (8.2.1) that the derivative of the mapping x w x * a x at xo E A * is h ++ h*axo

+ $ah

= $((hx,')*a

+ a(hx~'))xo.

The assertion now follows, because h H hx;' is a bijective linear mapping and therefore the rank of the derivative at the point xo is equal to that of the linear mapping h w h*a + ah. which is independent of xo. The same calculation shows (16.8.8) that the Lie algebra of H may be identified with the vector subspace of A defined by the equation (1 9.4.3.2)

x*a

+ ax = 0.

This applies in particular to A = M,(R), M,(C), or M,(H), the involution being XH'X; in this way we obtain the Lie algebras of the groups O(n), U(n, C) and U(n, H) (16.11.2 and 16.11.3). More generally, consider a nondegenerate symmetric or alternating bilinear form 0 on R" ,and let S be its matrix relative to the canonical basis. The group of endomorphisms of R" which leave 0 invariant may be identified with the group of matrices X E M,(R) such that ' X S X = S ; the Lie algebra of this group is then identified with the Lie algebra of matrices X E M,(R) such that (19.4.3.3)

' X . S + s. X = O

(the bracket operation being [ X , Y ]= XY - YX). Remark (19.4.4) Consider GL(E) as acting on E by the canonical action (S,v) H S v, the product of the automorphism S and the vector v. Fix a vector vo E E and consider the mapping g : S w S . v 0 of GL(E) into E; it follows immediately from (19.4.2) that the differential of g at the neutral element I of GL(E) is given by (19.4.4.1)

d,g

*

U = (I* v0

for U E End(E) (identified with the Lie algebra of GL(E)). Consider now a Lie group G , a C" linear representation p : G + GL(E) and a differential manifold M on which G acts differentiably on the right. Consider G as acting on the right on E: (s, v) H p(s - 1) * v.

I

148

XIX


Let f : M + E be a G-invariant C" mapping, i.e. (1 9.1.2.1) such that (1 9.4.4.2)

f(x * s) = p(s - 1) * f(x).

Then for each vector u E g,, in the notation of (1 9.3.7), we have (1 9.4.4.3)

(eZu

a

-f

f ) ( ~= ) - P*W

~

,

pJu) being the image in gI(E) = End(E) of the vector u under the derived homomorphism of p. For by the definition of a Killing field and by (17.1 4.9) we have

(OZu * f)(x)

= d,f

*

( x * u).

On the other hand, if we evaluate at u the differentials at the point e of the two sides of (19.4.4.2), considered as mappings of G into E (for fixed x ) , we obtain on the left-hand side d,f (x * u) (16.5.8.5); and since T,(p) h = p,(h) by definition, it follows from (19.4.4.1), (16.5.8.5), and (16.9.9(i)) that on the right-hand side we obtain - p*(u) * f(x).

5. TAYLOR'S FORMULA IN A LIE GROUP

We shall first give some supplementary results on Taylor expansions (8.1 4.3). (19.5.1) Let U be an open neighborhood of 0 in R",and let llxll be a norm on R" compatible with the topology (e.g., the Euclidean norm). In the ring b,(U) of real-valued C" functions on U, we denote by o,(U) (or simply om), for each integer m 2 0, the set of allfe &,(U) such thatf(x)/llxll"+' remains bounded as x + 0 (and x # 0). It is clear that om(U) is an ideal in &,(U), and that oo is the set of C"-functions which vanish at the origin. We shall use the same notation o,(U) to denote the set of C"-functions f on U with values x~\m+l in a finite-dimensional real vector space F, such that ~ [ f ( x ) [ ~ / ~ ~remains bounded as x + 0 (and x # 0); or, equivalently, such that the components o f f relative to a basis of F are functions in &,(U) belonging to o,(U). (19.5.2) For each function f~ b,(U) and each integer m 2 0, the sum of the first rn + 1 terms of Taylor's formula (8.1 4.3) for f:

5 TAYLOR'S FORMULA

IN A LIE GROUP

149

is called the Taylor. polynomial of degree S m of the function f: Since

f','(O)

x'") = (x'D,

+

* *

+ x"D,)Pf(O)

by virtue of (8.13), we may also write (19.5.2.1)

with the notation of (17.1). (19.5.3) The Taylor. polynomial P, is the unique polynomial P of degree 5 m such that

f- p E 0 , m .

(19.5.3.1)

I t follows from Taylor's formula (8.14.3) that .f- P, E o,(U). Suppose that there exists a polynomial P # P, of degree 5 m satisfying (19.5.3.1). Then the polynomial Q = P - P,, belongs to o,,#(U).Writing Q = Qo Q, . . * Q,, where Qk is homogeneous of degree k (0 5 k 5 m), suppose that p is the smallest integer such that Q , # 0. Then, for x # 0 , putting x = llxllz, we shall have

+ + +

,

and since Q, # 0, there exists z E S,,- such that Q,(z) # 0; but then, in the formula above, when z is fixed and x + 0, the expression in brackets tends to Qp(z) # 0, and the absolute value of the left-hand side would tend to 00, which is absurd.

+

(19.5.4)

For each function f E 8,(U), put ](XI,

. . . , X,) = "

a for/nulpoiwr series belonging to R[[X,, LX = (zI, . . . , a,,) (A.21.2).

1

- D"f(O)X", a!

. . . , X,]],

where X" = X;' * . * :X if

(19.5.5) Let V be an open neighborhood of 0 in RP, and let g = ( g l , . . ., g,) be a C" mapping of V into U such that g(0) = 0. Then for each function f E BR(U), if11 - f o g, we have

150

XIX


By virtue of (19.5.3) it has to be shown that, for each integer rn 2 0, if S, is the sum of the terms of degree 6rn in the formal power series on the righthand side of (19.5.5.1), then h - s,,,E o,(V). Let P,, Qlm, . . . , Q,, be the o f f , gl, .. . , g,, respectively. Since the Taylor polynomials of degree formal power series s" I , . . . , 8, have constant terms equal to 0, it follows from the properties of formal power series (A.21.3) that S, is the sum of the terms of degree 6 rn in the polynomial

srn

Rm(Y1, . *

1 ,

Yp) = P m ( Q l m ( Y I

7

. . ., Yp), . .

* 9

Qnm(Y1,

.. Y p ) ) . *

9

Since the function y~ R,(y) - S,(y) belongs to o,(V), it is enough to show that zI - R, E o,(V). We may write gjcY) = Qjrn(Y)

+ rj(Y),

where r j E o,(V), and therefore the function Y H Pm(S1 (Y), * . * Sn(Y)) - Pm(Q Im(Y), = Pm(gl(Y), . . gn(Y>) - RAY) 9

* * *

9

Qnm(Y))

. 7

belongs to o,(V), since the latter is an ideal in d,(V). Hence it is enough to show that the function Y ++f(S,(Y),

..

* 9

S,(Y))

- P,,(9l(Y>9

. . .19,(Y))

belongs to o,(V). Now, the hypothesis g(0) = 0 implies that there exists a neighborhood W, c V of 0 and a number k 2 0 such that Ilg(y)ll 6 kllyll for all y E W,. On the other hand, by the definition of P,, , there exists a neighborhood W c Wo of 0 and a constant A > 0 such that, for all y E W, we have

This completes the proof. (19.5.6) We shall use the following notation to express that the formal power series 1a,X" is equal to the formal power seriesf(X,, . . ., X,): U

(1 9.5.6.1)

(The use of this notation does no! imply that the series on the right-hand side converges for any x # 0, nor that if it does converge its sum is equal to f ( x ) . )The right-hand side of (19.5.6.1) is called the (infinite) Taylor expansion offat the point 0.

5 TAYLOR'S FORMULA IN A LIE GROUP

151

After these preliminaries, let G be a Lie group and let c = (U, cp, n) be a chart of G at the neutral element e , such that q ( e ) = 0. Let V be a symmetric open neighborhood of e in G such that V3 c U. Then the function

(19.5.7)

is defined and of class C" on the open set cp(V) x q ( V ) c R'"; it is called the local expression of the multiplication law in G , relative to the chart c. As functions of (x, y), for 1 5 i S n, we may write

in order to express in a concise form the Taylor expansions of the functions m i . The fact that e is the neutral element of G is expressed by the conditions m(x, 0) = x

m(0, Y) = Y,

for x, y E V. Consequently, the formal power series on the right-hand side of (19.5.7.2) are of the form (19.5.7.3)

For each multi-index y = (yl, . . . , y,,), let

and write its Taylor expansion in the form (19.5.7.5)

so that, by virtue of (19.5.7.3) and (19.5.5), we have

This shows immediately that (1 9.5.7.6)

caBy=o

if

1011 + IPI < 171

152


and that the coefficients cusy for which la1 + IpI polynomial (X, YJY' * * ' (X" Y,)Yn;

+

=

Iy( are those of the

+

in other words, the only nonzero coefficients cagYwith l a ( + those for which a + j3 = y, and

IpI

= IyI

are

(1 9.5.7.7) (19.5.8) Consider now, for each multi-index a = (a1, . . . , an), the real distribution A, with support { e } on G defined by (19.5.8.1)

for all f

E gPR(U). In

A, * f

1

= 7 D"(f0

a.

other words, A,

= cp-'

~p-')(0)

c.

1

-i D'E,, . Since the distributions

D'~Oform a basis over R of the space of real distributions on R" with support contained in (0) (17.7.3), it follows that the Au form a basis of the algebra 6,. If Z , E Diff(G) is the left-invariant differential operator on G which reduces to Au at the point e (19.2.4), the Z , form a basis of the injinitesimal algebra 6,and we have

We have now the following result : (19.5.9) For each s E V, the function y w f ( s q - ' ( y ) ) , dejined on q ( V ) , has the Taylor expanison (19.5.9.1)

and the multiplication table for the basis (Z,) of 6 is given by (19.5.9.2)

where the coe@cients cuByare defined by (19.5.7.5).

Let us write the Taylor expansion of the function ywf(sCp-'(y)) in the form

5 TAYLOR'S FORMULA IN A

LIE GROUP

153

where the P, are differentia1 operators; we have 1

(P,* f ) ( e )= a . D"(J0 cp-')(O)

= A,

*f.

Next, we may write f(scp-'(y)) = (y(s-')f)(ecp-'(y)), and so by replacing f by y(s-')fin the formula above we obtain another Taylor expansion:

Comparing the two expansions and bearing in mind (1 9.5.8.2), we obtain

(P,*f)(s)= (2, * f ) ( s > for all s E V. If we replacefby ZB. f i n (19.5.9.1), we get

The TayIor expansion of the function ( x , Y>Hf(Scp-l(X)cp-l(Y))

on q ( V ) x cp(V), where s E V, is obtained by replacing s by s q - ' ( x ) in the right-hand side of (19.5.9.1) and is therefore, by (1 9.5.9.5),

On the other hand, ~ p - ' ( x ) ~ p - ' ( y= ) cp-'(m(x, y ) ) and therefore, by virtue of (19.5.5), we obtain the Taylor expansion of the function (x, Y)-f(Scp-'(m(x,

Y)))

by substituting, for each y, the formal power series

(the Taylor expansion of ( m ( x , Y ) ) ~ for ) Yy in the formal power series (Zu.f)(s)Y Y . This gives rise to the series Y

(19.5.9.7)

and now a comparison of this series with (19.5.9.6) gives the relation (19.5.9.2) for each pair of multi-indices (a, p).

154

XIX

LIE GROUPS A N D LIE ALGEBRAS

The formula (19.5.9.1) is called Taylor’s formula at the point e in G, relative to the chart c = (U, rp, n). (19.5.10) Let E , denote the multi-index ( ~ 3 , ~ s) ~j s n , where aij is the Kronecker delta. Then the invariant vector fields Z,, = X i ( I 5 i S n) form a basis of the Lie algebra g of invariant vector fields. Moreover, by (19.5.7.6) and (19.5.7.7), we have cEiejy= 0 for I y I > 2, CEi,e,.eitel ‘Ei,

el.

m

=1

- b (e ki e)i

for i + j ,

(which we shall write as b$) for simplicity). Hence it follows from (19.5.9.2) that, for i # j, we have (1 9.5.10.1)

and therefore the multiplication table for the basis (Xi)lgisn of the Lie algebra g is given by

Since Xi(e) = ui = (derp)-’ e, (16.571, the basis ( u i ) of the Lie algebra

ge of the group G has the same multiplication table: n

(19.5.10.3)

[Ui, Uj] =

1 (bi$’- b : j ’ ) U k k= I

(1

i,j

n).

Example (19.5.11) Consider the Lie group G whose underlying manifold is R* x R, and multiplication given by (19.5.11.1)

(s,’ s 2 ) ( t ’ ,

t 2 ) = (s’t’, s’t2

+ SZ),

so that the neutral element is e = (I, 0). As chart we take (G, rp, 2), where cp is the translation in R2 which takes (1, 0) to (0, 0). Then, with the notation of (19.5.7), we have in this case (19.5.11.2)

6 THE ENVELOPING ALGEBRA

155

and by virtue of (19.5.10.3) the Lie algebra of G has the multiplication table (19.5.11.3)

[UI, U z l = u2 *

PROBLEMS

1. Let G be the Lie subgroup of CL(3, R) consisting of the matrices

3.

Show that the Lie algebra 0, has a basis (u, v, w) for which the multiplication table is [u, v] = w,

[u, w] = 0,

[v, w l = 0.

2. Show that the Lie algebra of the group SL(2, R) has a basis (uI, u2, us) for which the multiplication table is [UI,

uz1= 2u2,

tu,, u3l= -2u,,

[Uz,

u J = u1.

6. T H E ENVELOPING ALGEBRA O F T H E LIE ALGEBRA OF A LIE GROUP

We retain the hypotheses and notation introduced in (1 9.5.7), (19.5.8), and (19.5.10), and we denote the composition Xo Y in 8 by XY. For each multi-index u = (ctl, . . . , u,) we shall write

with the convention that X , = 1. It should be remarked that it is not legitimate to permute the X iin this product, because in general the algebra 8 is not commutative. (19.6.2) The operators Xu form a basis of the infinitesimal algebra 8 of the group G .

For each integer rn > 0, let 8, denote the vector subspace of 8 formed by the invariant operators of order Sm.By virtue of (19.5.8.1) and (19.5.9.2), the space 8,,,has as a basis the set of the 2, such that [ a [ 5 m. Since the X , and the Z, have the same set of indices, it is enough to show that the X,

156

XIX


with 1. g rn span 6, (A.4.8). For m = 1, this is clear from the definition of the X i , and therefore it will be enough to prove, by induction on m, that for ICII= rn, we have (19.6.2.i )

with coefficientsqal E R.Now, if CI = ( M , , . . . , an),let i be the first index such that cti > 0; then by definition we have Xu= X i X a - 6 i , and the inductive hypothesis gives (19.6.2.2)

From (19.5.9.2), (19.5.7.6), and (19.5.7.7), we obtain

and likewise, for

< m - 1,

(19.6.2.4)

Now substitute these expressions into the right-hand side of (1 9.6.2.2) multiplied on the left by X i , and we obtain (19.6.2.1). (19.6.3) We shall show that the associative algebra 6 is the "enveloping" algebra of the Lie algebra g, in the following sense: (19.6.4) For each (associative) R-algebra B with identity element, and each linear mapping f :g -P B satiflying the relation

for all X, Y E9, there exists a unique homomorphism h of the algebra 6 into the algebra B which extends f and is such that h(1) = 1.

The uniqueness of h is clear, because g and the identity element generate 6,by (19.6.2). As to the existence of h, we remark that for each multi-index a we must have

6 THE ENVELOPING ALGEBRA

157

with the notation introduced above. There exists a unique linear mapping k : Cfi + B satisfying (19.6.4.2) for each a, and to prove that h is an algebra homomorphism it is enough to verify that for all LY and 8, we have

/?(XuXo)= l?(Xu)A(Xp).

(19.6.4.3)

Consider first the case where a = E ; and /3 = t i . If i 5 j , we have X i X j = XEi+E,( and the relation (19.6.4.3) follows from the definition (19.6.4.2); whereas if i >,j, we write

X i X j = [ X ; ,Xi]

+ xjx;,

and then the relation (19.6.4.3) follows from the previous case and the hypothesis (1 9.6.4.1). Consider next the case where LY = E~ and /3 is arbitrary. We shall proceed by induction on //?I = m, and by induction on i. There exists an index j such that X , = X j X 7 with y = /3 - E ~ If. i S j (which in particular will be the case when i = I), we have XiX,= and the relation (19.6.4.3) again follows from the definition (19.6.4.2); whereas if i > j , we have

h(Xi X j X,) = l7([Xi, X j ] X , )

+ h(Xj X ; X,),

and since [Xi, X i ] is a linear combination of the X,, and 171 = rn - I , it follows from the inductive hypothesis that

/ ? ( [ X i ,XjlX,) = l7([X; XjI)l?(X,); 7

also /?([ X i , X j ] ) = 17(Xj,17(Xj)- l?(Xj)17(Xj),

/?(Xj)17(XJ= &Yo), and therefore

/ ? ( [ X,i X j ] X , ) = l?(X;)/?(Xo) - l?(Xj)h(Xi)h(X,). On the other hand, we may write X i X ,

I satisfy I I 1 5 tn, and therefore /?(XX j ;X , ) =

1

=

2 r AX,, A

rA /7(XjX,)=

where the multi-indices

1rA 11(Xj)17(XA) i.

= h(Xj)h(XiX,)

since ,j < i. Since 17 I = /n - 1, we have / ( X i X,) = /t(Xi)h(Xy)by the inductive hypothesis. Hence we obtain /?(XiX,) = h(Xi)ll(X,) for all i and all p, as desired.

158

XIX


Consider finally the general case, by induction on I LY I = m.We may again write X , = X i Xy, where y = LY - .ci, for some index i. We have then Il(X,

and since X,, X

,-

Xp) = h(X, XyX,),

r1 X , , it follows from above that

--

1

/ ( X i XyX,)

=

C1 r1 h(Xi X,) = 1 r, h(Xi)h(X1) = /?(Xi)/?(X, X,).

Finally, since (71 = tn - I , the inductive hypothesis implies that

h(X, X p ) = h(Xy)h(X,). This completes the proof, because h(Xi)h(X,) = h(X,). The associative algebra 8 is also denoted by U(g).

PROBLEM

Let T"(n)denote the real vector space @"o, the nth tensor power of the Lie algebra ii of a Lie group G, and let T({i) be the tensor algebra of o, i.e., the direct sum of the T"(o)for all n 2 0. (a) Show that the enveloping algebra U(n) is isomorphic to the quotient of T(o) by the two-sided ideal 3 generated by elements of the form X 0 Y - Y @ X- [ X . Y]in T(o), for all X , Y E o. (b) Let T,, be the direct sum of the T'(11) for 0 5 k 5 n. The canonical image U. of T, in U(0) is the vector subspace spanned by products of n elements which are either scalars or elements of 11. Let P. = U,,/U,- I for n 2 O(with the convention that U- I = {O}), and let P(1i) be the vector space direct sum of the P.. The multiplication in U(o) induces bilinear mappings P, x P.+P,,,+, for all pairs of integers m, n 2 0, and in this way P(o) becomes a graded associative R-algebra. If dim(n) = ni, show that P(o) is isomorphic to the polynomial algebra R [T,. . . . ,T,] in m independent indeterminates. (c) Deduce from (b) that the algebra U(ki) is left-and right-Noetherian. (If a is a left ideal of U(a), consider the graded ideal of P(o) generated by the canonical images of a n U. in P., and lift a finite system of generators of this ideal.) (d) Let a,, ..., a, be left ideals of finite codimension in U(n). Show that the left ideal a, az * . . a, is of finite codimension. (Induction on p. using (c).) (e) Show that the algebra U(n) has no zero-divisors (use a method analogous to (c)). and that the only invertible elements in U(o) are the scalars. (f) Let A , B be two nonzero elements of Up. Show that there exist two nonzero elements C, D of U(o) such that CA = DB. (Compare the dimensions of U. A , U. B, and U,,+,.) Deduce that U(ci) is isomorphic to a subalgebra U' of a division ring K (which is an R-algebra), such that every element of K is of the form 6 - l ~and of the

7 IMMERSED LIE GROUPS A N D LIE SUBALGEBRAS

159

form 7’F-I. where f , 6 , ~?’are . elements of U’. (Imitate the construction of the field of fractions of a commutative integral domain, by considering in the product U(a) x (U(d - {OH

the following relation between (A, A’) and (B, B’): for each pair (C, D ) of nonzero elements of U(o) such that CA‘ = DB’, we have CA = DB.)

7. IMMERSED LIE G R O U P S A N D LIE SUBALGEBRAS

In the remainder of this chapter we shall see how the theory of Lie groups may in large measure be reduced to the theory of (finite-dimensional) Lie algebras over R. (19.7.1) Let G, G’ be two Lie groups with neutral elements e, e’, respectively, and Lie alqebras ge = Lie(G ) ,g.: = Lie(G‘).Let f :G + G’ be a Lie group homomorphism and let f, = Lie(f ) = T,( f ) : g, -+ 9:. , which is a Lie algebra homomorphism, whose image f*(g,) is a Lie subalgebra of 9:. , and whose kernel it, is an ideal of the Lie algebra g,, the quotient ge/nebeing isomorphic to f*(ge)* (i) Iff (G) is a Lie subgroup of G’, then its Lie algebra may be ident$ed with f,(9,) ( c f . (19.7.5)). (ii) The kernel N off is a Lie subgroup of G whose Lie algebra may be identifed with It,. (iii) In order that f should be an immersion (resp. a submersion, resp. a local diffeomorphism), it is necessary and sufJicient that f* should be injective (resp. surjective, resp. bijective). Iff is a submersion, then f ( G ) is an open subgroup of G’, and G/N is locally isomorphic to G‘.

The assertions (ii) and (iii) are immediate consequences of (16.9.9(iii)) and (16.7.5), and (i) follows from (iii) by considering f as a submersion of G onto f(G). Example (19.7.1.l) Consider the Lie group homomorphismf: XHdet(X) of GL(n, R) into R*.By virtue of the identifications of (19.4.2), the derived homomorphism ,f* is identified with the derivative Dfof f at the point I. The expansion of a determinant shows immediately that det(Z + Z ) = 1

+ T r ( Z )+ r,(Z)

160


with r 2 ( Z )E 0 2 , in the notation of (19.5.1). It follows that .f* is the trace mapping Z H T r ( Z ) of M,(R) = gI(n, R) into R = Lie(R*). Since the kernel of f is the special linear group SL(n, R), it follows from (19.7.1) that the Lie algbra sI(n, R) of SL(n, R) is the Lie subalgebra of gl(n, R) consisting of the matrices of trace 0. (19.7.2) Let G,, G, be two Lie groups and G = G, x G 2 their product (16.9.4). Then the mapping u-(T,(pr,) . u, Te(pr,) . u) is an isomorphism of Lie(G) onto the product Lie algebra Lie(G,) x Lie(G,).

For by (16.6.2) this mapping is bijective, and evidently it is a Lie algebra homomorphism. We shall usually identify the invariant vector fields on G, x G, with pairs ( X ' , X"), where X'(resp. X " ) is an invariant vector field on G, (resp. G2), the vector field X ' being identified with ( X ' , 0) and X " with (0, A'"), so that ( X ' , X " ) may also be identified with X ' + X". Let ( A ' ~ ) l s i s m (resp. (X:)lsjs,,) be a basis of the Lie algebra g1 (resp. g,) of invariant vector fields on G I (resp. G2). Since A'; and X ; commute, it follows that the infinitesimal algebra of G , x G2 has a basis consisting of all products X i X i with r E N" and /lE N" (in the notation of (19.6.1)), and that we have

(Xix;)(x;x;,= (x; x;)(x;X;;); in other words, the infinitesimal algebra of G, x G 2 may be identified with the tensor product 8,O 8, of the infinitesimal algebras of G, and G, (A.20.4). It should be remarked that if two connected Lie subgroups HI, H, of a connected Lie group G are such that the Lie algebra ge of G is the direct sum of the Lie algebras of H, and H,, it does not necessarily follow that G is isomorphic to the product H, x H,; all that can be asserted is that G is locally isomorphic to HIx H, (cf. (19.7.6) and Problem 1). (19.7.3) We have seen (19.4.3) that the Lie algebra of a Lie subgroup of a Lie group G may be identified with a Lie subalgebra of Lie(G) = g,. But, conversely, an arbitrary Lie subalgebra 6, of ge is not necessarily the Lie algebra of a Lie subgroup of G (which is necessarily closed). However, there is the following proposition: (19.7.4) Let G be a Lie group, ge its Lie algebra. For each Lie subalgebra

be of ge, there exists a connected Lie group H and an injective Lie group homo-


161

morphismj : H -+ G suck that j , is an isomorphism of Lie(H) onto Ij, . Moreover, H and j are determitied up to isomorphism by these conditions: ifj' : H' + G is an iiijectii*ehomomorphism of a connected Lie group H' into G, such that j ; is an isomorphism of Lie(H') onto be, then there exists a unique isomorphism u of H' onto H such that j ' =j 0 u. Let (ui)16ismbe a basis of 6,. and for each index i let X , be the leftinvariant vector field on G such that X,(e) = ui . Since be is a Lie subalgebra of g, the brackets [ui, uj] are linear combinations of the uk with real coefficients, hence (19.3.3) the brackets [ X i , X j ] are linear combinations of the X , with constant coefficients. A Jbrtiori (18.14.5), if L, denotes the subspace of TJG) spanned by the vectors X j ( x ) (so that L, = x . be), the field of m-directions X H L , is completely integruble. Consider then the set '331 of maximal integral manifolds of this field (18.14.6) (we recall that they are not in general submanifolds of G ) .Since the field of directions X H X ~ ) , is invariant under left translations, it follows that the left translations by elements of G are homeomorphisms of G onto itself for the topology F defined in (18.14.6) relative to this field of directions. Hence, for each maximal integral manifold C E 911 and each s E G, the translate sC is a connected integral manifold for the topology .T,hence contained in some C' E 91i; and conversely s-'C' is a connected integral manifold which intersects C , hence is contained in C; in other words, sC is a maximal integral manifold. Now let H be the maximal integral manifold containing the neutral element e. We shall first show that H is a subgroup of G. For if s, t are any two points of H, then st-'H is a maximal integral manifold containing the points, hence s t - ' H = H ; since e E H, this shows that s t - ' E H and therefore H is a subgroup of G. We now give H the topology induced by F and the structure of a differential manifold defined in (18.4.6), and we shall show that H endowed with these structures (and its group structure) is a Lie group. In order to prove that ( s , t ) H s t - l is a C' mapping of H x H into H, we may assume that s (resp. t ) is in an arbitrarily small neighborhood of a point so (resp. to). Now there exist such open neighborhoods U, V on which the manifold structure induced by that of H is the structure of a submanifold of G, and such that the image of U x V under ( s , t ) H . s t - ' is contained in a neighborhood W of sot;' in H having the same properties. The assertion is now obvious. and it is also clear that j is an immersion and thatj,( Lie(H ) ) = 6,. Finally. the uniqueness: by hypothesis, H' has the same dimension as H, and therefore (in view of (16.9.9)) the niappingj' factorizes as j u, where u is a diffeomorphism of H' onto an open subset of H (for the topology F ) (18.14.6). SinceJ' is given to be a group homomorphism, the same is true of u, and therefore u(H') is an open subgroup of H (for the topology F ) .But H is connected for the topology Y, hence 14H') = H . Q.E.D. 0

162

XIX


The Lie group H defined in the proof of (19.7.4) is called the connected Lie group immersed in G corresponding to be, when H is identified with the subgroupj(H) of G. In general it is not closed in G , even if G is simplyconnected (cf. Problem 2 ) . Whenever we consider H as a topological group (and a fortiori as a Lie group) it is always the topology F defined here (called the proper topology of H) that is to be understood, and not the topology induced by that of G, unless it should happen that the two topologies coincide, in which case, H is a submanvold of G (16.8.4), and hence a Lie subgroup of G and closed in G (16.9.6). The uniqueness assertion of (19.7.4) shows immediately that if lj: is a Lie subalgebra of be, then the connected Lie group immersed in H, corresponding to $:, is the same as the connected Lie group immersed in G, corresponding to 5:. (19.7.5) Let G, G' be two Lie groups, f :G + G' a Lie group homomorphism. If G is connected, then f ( G ) is a Lie group immersed in G', corresponding to the Lie subalgebra f , ( g e ) of g:, . If N

=f

-'(e') is the kernel off, then f factorizes into

G 5 G/N

G',

where p is the canonical homomorphism and u is an injective Lie group homomorphism (16.10.4). Hence we may restrict our attention to the case where,f is injective, and therefore an immersion (16.9.9). Because f * is a Lie algebra homomorphism, ,f*(ge) is a Lie subalgebra of 9.: , to which there corresponds a connected Lie group H immersed in G'. Since, moreover, ,f* is injective, it is an isomorphism of ge ontof,(g,); now apply the uniqueness assertion of (19.7.4), and the proof is complete. (19.7.6) Let G, G' be two Lie groups, and ge, 9:. their respective Lie algebras. For each Lie algebra homomorphism u : ge -+ gb., there exists a C" local homomorphism h from G to G' (16.9.9.4), such that Te(h)= u. Moreover, any C" local homomorphism h, from G to G' such that Te(h,) = u coincides with h on some neighborhood of the neutral element e of G. In particular, ifG is connected, the mapping h-h, of the set of homomorphisms of G into G', into the set of hornomorphisnis of g, into gb., is injective; and if moreover G is simply-connected, this mapping is bijrctice. Consider the Lie group G x G', whose Lie algebra may be identified with (19.7.2). It is immediately verified that the graph r. of u is a Lie subalgebra of ge x 9:. . Let H be the connected Lie group immersed in G x G', 9, x g;


163

corresponding to I-, , and let j : H + G x G‘ be the canonical injection. We have (pr, o j ) * = (pr& o j , ; but (prl)* is the first projection ge x 9:. + ge, and j,(Lie(H)) = I-,. so that (pr, o.j)* is an isomorphism of Lie(H) onto Lie(G) = $1,. I t follows (19.7.1) that the restriction v of pr, 0 j to a sufficiently small open neighborhood U of ( e , e’) in H is a local isomorphism of H with G (16.9.9.4). Consequently (loc. cit.) there exists an open neighborhood V c v ( U ) of e in G such that v - ’ I V is a local isomorphism of G with H, andj, 0 ( v - ’ ) , is the mapping X H ( x , u(x))of ge onto T,,. I t is clear that h = pr2 o j o ( v - l I V) is a local homomorphism from G to G’ such that h, = u. Conversely, if h, is a local homomorphism from G to G’ such that h,, = u, then g : xt+(x, h,(x)) is a local homomorphism from G to G x G’ such that g,(g,) is the Lie subalgebra T,,.We deduce that, in a neighborhood of ( e , e’) in G x G’, the graph of h, is an integral manifold of the field of directions formed by the translates of I-,,; since H is a maximal integral manifold of this field and contains ( e , e’), it follows that the graph of h, is contained in H, and therefore h, and h coincide in a neighborhood of e. If G is connected, any two homomorphisms of G into G’ which coincide in a neighborhood of e are equal, because each neighborhood of e generates G (12.8.8). Finally, if G is connected and simply-connected, then every local homomorphism from G to G’ extends to a Lie group homomorphism of G into G’ (16.30.7). (19.7.7) In order that two Lie groups G , G ‘ should be locally isomorphic, it is necessary and sujicient that their Lie algebras should be isomorphic. If G , G’ are connected, simply-connected Lie groups whose Lie algebras are isomorphic, then for each isomorphism u : ge + 9.: of the Lie algebras there exists a unique Lie group isomorphism f : G + G’ such that f , = U . This is an immediate consequence of (19.7.6). (19.7.8) We shall see later (Chapter XXI) that (’very finite-dimensional Lie algebra over R is the Lie algebra of some Lie group. There is, therefore, by virtue of (19.7.7), a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of connected and simply-connected Lie groups. Furthermore, we obtain all the connected Lie groups with a given Lie algebra g by taking the connected and simply connected group G (which is determined up to isomorphism) whose Lie algebra is g, and forming the quotients G/D of G by the discrete subgroups D contained in the center of G (16.30.2).

Example: Connected Commutative Lie Groups (19.7.9) If a Lie group G is commutative, then so is its infinitesimal algebra

164


6 (17.11.8) and therefore, if dim(G) = n, the Lie algebra of G is isomorphic to R" (19.4.1). Every connected commutative Lie group of dimension n is therefore isomorphic to a quotient group R"/D, where D is a discrete subgroup (hence closed (12.8.7)) of R" (16.30.2). We shall determine all these subgroups D, up to isomorphism. (19.7.9.1) For each closed subgroup F of R", there exists an autonzorphism u of the vector space R" such that u(F) = Zp x R', where p >= 0, r >= 0, and p r S n, Zp is the discrete Z-module with basis el, . . . , ep, and R' is the vector subspace spanned by e p + , ,. . . , ep+' ((ei)ls i s n being the canonical basis of R").

+

We shall begin by showing that if F is not discrete, it contains at least one line Rb (with b # 0). Let llxll be the Euclidean norm on R"; by hypothesis, there exists a sequence (a,) of points of F such that 0 < Ila,,II < I/m for each m > 0. Let b, = a,/~~a,Il, so that 11 b,II = 1. Since the sphere is compact, the sequence (b,) has a subsequence (bmk)which converges to a limit b E S,,We shall see that Rb c F. For this purpose let t be any real number, and for each mk let t k be the unique integer such that tkllamkII

5 (H’ is the derived homomorphism of the diagonal homomorphism A ’ : G ’ + G ’ x G’. In order that a subalgebra 4 of B should be the infinitesimal algebra of a Lie group H immersed in G , it i s necessary and sufficient that A*(@) c 6 06. When this condition is satisfied, the basis (X.) of (il (19.6.2.1) may be chosen so that the X, for which a, = 0 for j > p (where p = dim(H)) form a basis of 6.

8. INVARIANT CONNECTIONS, ONE-PARAMETER SUBGROUPS. A N D T H E EXPONENTIAL MAPPING

Let G be a Lie group and let u be a nonzero vector in the Lie algebra g, of G . Since the additive group R has R as (commutative) Lie algebra, the mapping w : ? H tu is a Lie algebra homomorphism of Lie(R) into Lie(G). Since the group R is simply-connected (16.27.7), there exists a unique Lie group homomorphism v : R -+ G such that v* = w (19.7.6). Let Xu denote the left-invariant vector field on G such that Xu(e) = u (19.3.3). We shall show that v is the integral curve of t h e j e l d Xu which passes through the point e ; in other words (18.1.2.4), that (1 9.8.1)

-

v ‘ ( t ) = X,(V(t)) = v ( r ) u

for all t E R. For, by (19.7.6), the graph T v is the image of the integral curve through (0, e) of the left-invariant vector-field on R x G which takes the , This value ( I , u) at the point (0, e), i.e., the vector field ( t , x ) ~ ( E ( t )XJx)). establishes our assertion. The image of the integral curve t H v ( t ) is called the one-parameter subgroup of G corresponding to the vector u E g,. (19.8.2) We shall show that the one-parameter subgroups and their lefttranslates (i.e., the left cosets of the one-parameter subgroups) are the geodesic trajectories of certain linear connections on G (18.6.1). Since a connection C on G is a morphism of T(G) 0 T(G) into T(T(G)), we shall say that C is invariant if this morphism is invariant (19.1.2). In other words, for all elements s, x in G, and all vectors h, and k, in T,(G), we must have

c, . A s

*

h,, s k,) = s * C,(h,,

k,),

170

XIX


which determines C completely once the mapping (u, v)HC,(U, v) of ge x ge into (T(T(G))), is known: it must of course satisfy the conditions (17.16.3.2)-(17.16.3.4) for x = e. Conversely, given such a mapping (which is equivalent, once a chart at the point e of G has been chosen, to being given n arbitrary bilinear forms l-j on ge x g, by virtue of (17.16.4)), we may define an invariant linear connection by the formula

by virtue of the fact that G acts equilinearly on T(G) and G , and also on T(T(G)) and T(G). (19.8.3) Relative to an invariant linear connection C on G, if X , Yare two invariant vector fields, then the field V , Y is also invariant. This follows from the formu!a (17.17.2.1) which defines V , * Y, and the relation

TS,(s * Y ) ( s h,) = s * (T,( Y ) . h,) for

S E G.

Furthermore, it is clear that the mapping ( X , Y ) H V ,

*

Y of

g x g into g is R-bilinear (17.18.1). Conversely, if we are given an arbitrary R-bilinear mapping p : g x g + g, there exists a unique invariant connection C for which V x Y = p ( X , Y). This follows from the formula (17.17.2.1) and the fact that, for all x E G , there exists an open neighborhood U of x such that the restrictions to U of a basis of g over R form a basis of T(U, T(G)) over b(U) (19.2.2).

-

(19.8.4) In order that an invariant linear connection C on G should be such that, for each u E g, , the integral curves of the vector field Xu are geodesics of C, it is necessary and suflcient that V , * X = 0 for all invariant vector fields X on G (in other words, the R-bilinear mapping ( X , Y ) w V , * Y of g x g into g must be alternating). Ifmoreover the torsion of C is zero, then.for any two invariant vectorfields X , Y, we have (19.8.4.1)

v, . Y = f [ X , Y]

and for any three invariant vectorfields X , Y, Z , the curvature morphism of C is given by (19.8.4.2)

( r . ( X h Y ) ) * Z = -)[[A’,

Y],Z].

To say that a mapping t ~ v ( tof) K into G is a geodesic signifies that Vv.(,)* v ’ ( t ) = 0 for all t E R (18.16.1.2). By virtue of (19.8.1), to say that every

8 INVARIANT CONNECTIONS AND THE EXPONENTIAL MAPPING

171

integral curve of Xu is a geodesic therefore signifies that the field VXu * Xu is zero at all points of the one-parameter subgroup corresponding to u. But since this vector field is invariant, it is enough that it should be zero at one point to be zero everywhere on G. This establishes the first assertion. Since we now have V y . X = -V, Y, the formula (19.8.4.1) (when the torsion of C is zero) follows from the definition of the torsion (17.20.6.1). Then the definition of the curvature (17.20.4.1) shows that, by virtue of (19.8.4.1), (r * ( X

A

Y ) )* 2 = t[X [ Y, 211 - t [Y, [X, 211 - N X , YI, 21,

which is equal to -$[[A',

Y], 21 by Jacobi's identity.

(19.8.5) The solution u of (19.8.1) which takes the value e at t = 0 may therefore be written o ( t ) = exp(tu)

(19.8.5.1)

by virtue of (18.4.4) since X J e ) = u. The mapping uHexp(u) (also denoted by exp,) is called the exponential mapping of the Lie algebra ge into the group G . We have (1 9.8.5.2) (19.8.5.3)

exp(s(tu)) = exp((st)u). exp((s

+ t)u) = exp(su) exp(tu),

for all s, t E R, which justifies the notation; but in general for u, v E ge.

exp(u

+ v) # exp(u) exp(v)

(19.8.6) There exists an open neighborhood U of 0 in ge such that the exponential mapping is a diffeomorphism o j U onto an open neighborhood of e in G.

In view of the remarks above, this is a particular case of (18.4.6). The inverse diffeomorphism, of exp(U) onto U, is denoted by x Hlog x (or log, x). Given a basis of ge, the composition of log, and the bijection of ge onto R" determined by the basis defines a chart of G at the point e, called the canonical chart relative to U and the chosen basis of g e eThe local coordinates corresponding to this chart are called canonical coordinates (or canonical coordinates of the first kind) in U, relative to the chosen basis of g e .

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XIX


It follows from (18.4.5) that at the point 0, E ge we have

Examples (19.8.7.1) If G = R", it is immediately seen that if we take as chart the identity mapping, the local expression of the equation (19.8.1) is Dx = u. Hence exp(tu) = tu, i.e., the exponential mapping is in this case the identity mapping 1,. (1 9.8.7.2) Consider next the multiplicative group A* of a finite-dimensional R-algebra A with identity element. For each S E A * , the mapping XHSX of A into A is linear, hence equal to its derivative (8.1.3), and therefore, for each tangent vector u E g, the local expression of the invariant vector field Xu is X H ( X , xu), where u = z,(u) E A (19.4.2). Consequently the local expression of the differential equation (19.8.1) is

DX= X U . The solution of this equation, which takes the value e at t = 0, is the exponential series ? H e + -t u l!

+ -t u2 2 + 2!

+ ?1" u " + n. which is normally convergent in every bounded interval of R, relative to any norm for which A is a Banach algebra (15.1.3). The notation (19.8.5.1) for this series is therefore consistent with the usual notation. *

*

a

,

Remark (19.8.8) Take for example A = M,(R), so that A * = GL(2, R). Regarding A as a subalgebra of the complex matrix algebra M2(C),if Y EA is any real matrix there exists an invertible complex matrix P such that X = P Y P - ' is either a real triangular matrix

or a diagonal matrix

(" "p

a

_O jp)

with complex numbers on the diagonal (i.e., p # 0). In the first case, exp(tX) is a triangular matrix with diagonal entries e"', evf,and in the second case exp(tX) is the diagonal matrix


173

In either case, the matrix f - ' exp(tX)f = exp(tP-'XP) = exp t Y

(where f is real) cannot be equal to the matrix

(-;

-1")

for 1 > 0 and 1 # I , which belongs to the identity component CL'(2, R) of GL(2, R). This example therefore shows that, for a connected Lie group, the exponential mapping is not neewarily sirrjective. We have already seen (18.7.13) that it is not necessarily injective (cf. Section 19.14, Problem 4). It can be shown (Problem 2) that in this case the matrices in GL'(2, R) which are of the form exp(t Y ) are the matrices

with A = ad - be > 0 and a

+ d > -2JA,

(-t

and the matrices

-3

with 1. > 0. This set of matrices is therefore neither open nor closed in

GL'(2, R). (19.8.9) Let G , G' be two Lie groups, and let .f: G --t G' be a Lie group homomorphism. Then for each i'ector u E ge and each t E R we have

The mappings 11 : t -f(exp( ru)) and w : t t-,exp(tf,( u)) are homomorphisms of R into G' such that ~ ' ( 0 = ) w'(0) =f*(u),by virtue of (16.5.4). The result therefore follows from the uniqueness assertion of (19.7.6). (19.8.10) Let G be a Lie group, H a connected Lie group immersed in G, and 5, the Lie algebra of H , identified with a Lie subalgebra of ge. In order that a vector u E ge should belong to be, it is necessary and suflcient that exp(tu) E H for all t E R.

The necessity of the condition follows from (19.8.9) applied to the canonical injectionj : H + G. Conversely, if the image of u : t Nexp(tu) is contained in H , then since this mapping is of class C"' as a mapping of R into G , it is also of class C" when regarded as a mapping of R into H, for the manifold structure of H (18.14.7). Hence we have u = u'(0) E Te(H) = be.

174

XIX

LIE GROUPS AND LIE

ALGEBRAS

Remark (19.8.11) Let M be a differential manifold on which G acts differentiably on the right, and consider for each vector u E ge the Killing vector field Z, on M, which is defined by Z , ( x ) = x * u (19.3.7.4). Then the maximal integral curve of this vector field with origin x o E M (18.2.2) is defined on the whole of R and is given by

t ++xo . exp( tu).

(19.8.11.I)

For if we put o ( t ) = exp( tu) and (19.8.11.2)

-

w ( t ) = xo v ( t ) ,

T(w) E(t) = X O (T(v) * E(r)) = x0 *

then by (16.10.1) we have *

(y(t)

U) = ~ ( t* u)

by virtue of the definition of exp(tu) (19.8.1). From this we shall deduce that, in order that a tensor field SEFL(M) should be G-invariant, it is necessary that

ez, s = o

(19.8.11.3)

a

for all u E ge , and moreover this condition is sufficient if G is connected. We have seen (18.2.14.8) that the value of Oz, S at a point x E M is the limit as t + 0 of the tensor

which may also be written in the form

I

- ( S ( x . exp(ru))

t

- S ( x ) exp(tu)) - exp( -tu).

Since the question is local, we may identify T(M) with M x R" in a neighborhood of x ; by considering the vector part (16.15.1.3) of the above tensor, it follows that if we put (19.8.11.4)

F ( t ) = S(x

exp(tu)) - S(x) * exp(tu)

(which is an element of (T~(M)),.,,,,,,,), the derivative F'(0) = T(F) E(0) is given by the expression (19.8.11.5)

F'(0) = G(f,((ez, * S)(X)).


175

If S is G-invariant, that is to say if S(x s) = S(x) * s for all s E G and all x E M, then we have F ( t ) = O for all t E R and hence the relation (19.8.11.3). Conversely, if this relation is satisfied, let xo E M, and put (19.8.11.6)

Fo(t) = S(xo * exp(tu))

- S(xo) exp(tu),

so that tt+Fo(t) is a lifting to T:(M) of the mapping (19.8.11.1). Writing x = xo * exp(ru), we have Fo(t

+ t’) = Fo(t)

*

exp(t’u)

+ (S(x

*

exp(t’u)) - S(x) exp(t’u)),

from which it follows, by use of (19.8.11.2) and (19.8.11.5) and the hypothesis (19.8.11.3), that (19.8.11.7)

Fh(t) = Fo(t) * U,

the right-hand side of which corresponds (16.10.1) to the action of G on T;(M) induced from the action of G on M ((19.1.3) and (19.1.5)). Since Fo(0) = Ox,, the only solution of (19.8.11.7) is evidently 0 (18.2.2). By virtue of (1 9.8.6), we have, therefore, S(xo * s) = S(xo) s for all s in some neighborhood of e in G. The hypothesis that G is connected now implies that this relation is valid for all J E G , by virtue of (12.8.8). 9

PROBLEMS 1. With the hypotheses and notation of (19.8.7.2), give the series expansion of log(e (for x in a sufficiently small neighborhood of 0 in A) in powers of x . 2.

+ x)

Determine the image of the exponential mapping for the groups GL(2, R), SL(2, R), GL(2, C), and SL(2, C) (use the reduction of a matrix to Jordan form). Deduce that the exponential mapping for GL(n, C) is surjective, and hence also for PCL(n, C), the quotient of GL(n, C) by its center. On the other hand, the exponential mapping for SL(n, C) is not surjective, although the quotient PSL(n, C) of SL(n, C) by its center is isomorphic to PGL(n, C).

3. Let G be a simply-connected Lie group, and M a compact differential manifold. Show that every infinitesimal action of 0. on M arises, by the formula (19.3.7.4), from a differentiable action of G on M. (With the notation of Section 19.3, Problem 2, consider a maximal integral manifold of the field of n-directions (s, x ) ++ L(s,x ) on G x M; then use Problem 5 of Section 16.29, together with (18.2.11).) Does the above result remain true if G is connected but not simply-connected? (Consider the case where M is a compact Lie group G’, and G = G’/D is the quotient of G’ by a finite subgroup D # { e }of the center of G’.) (Cf. Section 19.9, Problem 9.)

176

XIX


4. Show by use of the exponential mapping that a Lie group has no arbitrarily small subgroups (Section 12.9, Problem 6). 5. Let G be a separable, metrizable, locally compact group with no arbitrarily small subgroups. We shall use the notation and terminology of Section 14.11, Problems 9-14. Let LG be the set of one-parameter subgroups of G (Section 14.11, Problem 12).

Let V be a symmetric compact neighborhood of e in G which contains no subgroup # {e} of G, and such that for all x , y E V the relation x 2 = y 2 implies x = y . Let K denote the set of all X E LO such that X(r) E V for (rI 5 1. Denote by exp the mapping X - X ( l ) of L G into G, and put KI = exp(K). The mapping exp is injective on K.

(a) Show that K1 is closed (hence compact) in G. (If a sequence ( X , ) in K is such that the sequence (X,(I)) tends to a limit a, remark that the sequence (Xi(l/i)f)has limit a, and use Section 12.9, Problem 7.) (b) Let g be the function defined in Section 14.11, Problem 14(b). Let XI, ..., X,, be one-parameter subgroups of G such that

Show that for each functionf'that is Xk-differentiable(Section 14.11, Problem 12) for each k, we have

Xm(l/i)EG, and show that the (Consider the sequence of elements bf = X,(l/i) functions i(y(bf)g - g) tend uniformly to 0 as i -+ m, by using the formula

+

(*I

y(sflg - B = Y ( ~ ) ( Y ( ~-BS)

+ (y(s)g - 9).

Use Problem 13(a) of Section 14.11 to show that a subsequence of the sequence (by'])(independent of r E R) converges to X(r), where Xis a one-parameter subgroup, and show that X = 0 by using Problem 12 of Section 14.11. Conclude that the sequence (byi1)converges to e in G.) (c) With the notation of Section 14.11, Problem 14, let (a,) be a sequence in G converging to e and let (mJ) be a sequence of integers tending to cc. Show that if the sequence of functions mJ(y(aJ)gj- 9,) converges uniformly to a function h, then the sequence (a:'"']) converges to X(r), where X is a one-parameter subgroup, and that Dxg = h. (Use Problem 13(a) of Section 14.1 1 and Problem 7 of Section 12.9 to show that there exists a subsequence of the sequence (aymJ1)which converges to a oneparameter subgroup X ( r ) ;then use part (b) to show that this one-parameter subgroup is independent of the choice of subsequence.) (d) Given two one-parameter subgroups X, Y, show that there exists a unique oneparameter subgroup such that each function f which is both X-differentiable and Ydifferentiable is 2-differentiable, and

+

Dzf= Dxf+ DYJ (With the same notation as before, consider the sequence of functions and use the formula (*). Then use part (c) to show that the limit of the sequence ( ( X ( l / i ) Y(l/i))c"]) exists and is the one-parameter subgroup Z(r)required.) We have therefore equipped LO with a structure of a real vector space.


177

6. The hypotheses and notation are as in Problem 5 . For each X E LG, put llXll= IIDxg 11.

(a) Show that llXll is a norm on LG ,and that the set K (Problem 5) is a neighborhood of 0 relative to this norm. (Argue as in Section 14.11, Problem 13(a), and use Section 14.11, Problem ll(c).) (b) Let (A',)be a sequence in K. Show that if the sequence (Xi(l)) converges to X(1), then the sequence (Dx,g)converges uniformly to Dxg.(Remark that, by virtue of Section 14.11, Problem 14(b), the set of functions Dyg, where Yruns through K, is uniformly equicontinuous and uniformly bounded. Hence there exists a sequence of integers hi tending to co such that some subsequence of the sequence

+

converges uniformly. By using Problem 5(c), show that the limit of this subsequence is necessarily Dxg. (c) Prove that LGis afinile-dimensionalvector space. (Use (a) and (b), the compactness of K,,and F. Riesz's theorem (5.9.4).) (d) If G is a Lie group, show that LG may be canonically identified with Lie(G), and exp with the exponential mapping defined in (19.8.5). (Use (19.10.2)) , 7. (a) Let G, H be two separable, metrizable, locally compact groups with no arbitrarily small subgroups. For each continuous homomorphism f : G +. H, and each one-parameter subgroup X E LG, the groupf,(X) = Xofis a one-parameter subgroup of H. Deduce from Problem 5(d) thatf, is a linear mapping of LG into LH (b) In particular, an inner automorphism Int(w) of G induces in this way an automorphism Ad(w) = Int(w), of the vector space LG , for each element w of G. Show that iv-Ad(w) is a continuous homomorphism of G into GL(L0). (It is enough to prove that w-Ad(w). X i s a continuous mapping of G into LG, for each X E LG; bearing in mind the definition of the norm in LG (Problem 6), use Section 14.11, Problem 1l(c).) (c) Put ad = Ad*, which is a linear mapping of LG into oL(L0) = End&). Show that for each X E LG the linear mapping ad(X) of Lo into LG is the limit in End(LG) of the sequence i(Ad(X(I/i)) - I)as i --f m. The convergence is uniform on compact subsets of La. (d) Show that for all X, Y E LG and all r E R,we have

.

+

(

)

(ad(Y)*X)(r)= lim Jim (Y(l/j)X(l/i)Y(-l/j)X(-l/i))c'*J1 . J-a

1-a

(Use part (c) and Problem 5(d).) Deduce that ad(X). X = 0, and hence that ( X , Y)-ad(X).

Y

is an alternating bilinear mapping of LG x LG into LG. (e) With the hypothesis of (a), show that f*(Ad(w) r)= Ad(f(w))

.f*(Y)

for all Y E LO and w E G, and deduce that fXad(X>. Y)= ad(f+(x))*f*(Y)

178


for all X , Y ELo. Hence show that if we define [ X , Yl = ad(X)

Y,

the vector spa- LG together with this bracket operation is a real Lie algebra. (Tab H = GL(LG) and f; = ad(Z) in the formula above.) (f) Show that, for all X E Lo and r E R,we have Ad(X(1)) = exp(ad(rX)) in GL(LG), and (Ad(X(r)) * Y)(r)= A'(t)Y(r)X(-r) in G for all r E R. 8. The hypotheses and notation remain the same.

Let ( Yl)be a sequence of elements of Lo tending to Y, and let X integer m, let

(a)

bml =

E

LG. For each

Yd- 1)( Yl(l/m)X(l/mi))m.

show that for each r E R,we have

(

)

lirn lirn bCr1l = lirn lirn b"l1

1-m

m-m

.+m(l+m

)

(all the limits exist in G). (Use Problem 5(c) and Section 14.11, Problem 14(b), together with (7.5.6), in order to reduce the problem to the existence of the repeated limit

Observe that if we put

X L ~=( Yd-t/m)X(r) ~ Yl(t/m),

we may write Use tbe relation (*) of Problem Yb) to show that

where Xl(r) = Y(-t)X(r) Y(t).) (b) Deduce from (a) that if Y = 0 we have

+

(Use Problem 5(d), which defines Yl (l/i)X.) (c) With the notation of Section 14.11, Probkm 11, let (m,)be a sequence of integers tending to m, and for each j let a,, b, be two elements of U,, Suppose that the

+

.

sequence (arm'])converges to X(r), and the sequence (brmJ1)to Y(r),where X and Y


179

belong to LO. Show that, for all sufficiently small r E R, the sequence ((a,b,)['"'') converges to (X+ Y)(r).(Use Section 14.11, Problems 14(a) and 14(b).) (d) Show that K1 = exp(K) (Problem 5 ) is a neighborhoodofe in G. (For each x E G, let v ( x ) be the least integer n 2 0 such that x"+' 4 V. Suppose that there exists a sequence (a,) in G, converging to e, such that a, 4 K, for all j. Let Y, E K be such that v( Yl(- 1)aJ takes the greatest value of all the v ( x i 'al)as x I runs through K1, and let v1 be this greatest value; we have vf co as i 00. We may assume, by passing to a subsequence if necessary, that the sequence (( Yl(- I)a,)r"il) converges to X(r), where X # 0 in LO (Section 12.9, Problem 7). The sequence (Y,) then converges to 0 in LG, and therefore we have --f

--f

1

Yi+-X€K VI

for all sufficiently large i. Put

3

(

bl=exp -Y,--X

al=ci'(Y,(-l)al).

Deduce from (b) that cIIylJconverges to X(r), and from (c) that the sequence (bj"ll) converges to e. This contradicts the definition of v(b,) and v I . ) 9. The hypotheses and notation remain the same as in Problems 5-8.

(a) Let Y E L O , so that (Problem 7(f)) we have Ad( Y(r)) =

Cm

k=o

l (ad( Y))'t' k.

in GL(L0). Put

in End(Lo); Syis invertible whenever Y is sufficientlyclose to 0. Show that the sequence

in G tends to (Sy . X)(r), uniformly in Xfor Xnear 0. (Use Problem 8(b).) (b) Use Problem 8(d) to show that in a sufficiently small neighborhood K of 0 in Lo, a law of composition ( X , Y) H X . Y may be. defined by exp(X. Y)= exp(X)exp( Y). We have X . 0 = 0 . X = X, and X * (-X) = (-X) X = 0. Moreover, there exists a neighborhood K" c K' of 0 in LG such that the products X * ( Y * Z) and (X . Y).Z are defined and equal for all X, Y, Z in K", and we have X

' ((f

+ t? Y)

= (X'

(t Y)).(t' Y)

for all sufficiently small t , t ' E R. (c) With the above notation, show that in LO we have 1

lim - ((- Y). ( Y + t(S;' 1-0

f

*

Z)))= Z

180


for Yfixed and sufficiently small, uniformly in Z for Z sufficiently close to 0. Deduce that 1 lim - (( Y.rz)- Y) = Syl * Z . 1-0

t

Hence show that the function t H W(t)= Y .tZ satisfies the differential equation dW = SG&,.z. dt

(d) Deduce from (c) and from the expression for Sy given in (a) that there exists a Lie group structure on the neutral component of G, for which the underlying topological group structure is the given one (Gleason- Yamabe theorern).t (Use Section 19.3, Problem 4.) 10.

Let G be a metrizable topological group and N a closed normal subgroup of G. (a) If N and G/N are locally compact, then G is locally compact. (Let Vo be a symmetric closed neighborhood of e in G such that Vo n N is compact. If V, is a symmetric closed neighborhood of e in G such that V: c V 0 , then V, n xN is compact for each x E V1. Let n : G -+G/N be the canonical homomorphism, and let C be a compact neighborhood of x(e) in G/N, contained in d V 1 ) . If V2is a symmetric closed neighborhood of e in G such that V: cV,, show that W = V t n n - , ( C ) is compact in G. For this purpose, if 3' 3 is a covering of W by open subsets of G, then for each y E C there is a finite number of sets of % which cover V1 n n - Y y ) ; if T, is their union, show that there exists a neighborhood S, of y in G/N such that

V2 n n-'(C n S,) cT,, and finally use the compactness of C.) (b) Deduce from (a) and Problem 9 that if N and G/N are the underlying topological groups of Lie groups, then the same is true of G.

9. PROPERTIES O F THE EXPONENTIAL MAPPING

(19.9.1) Let G be a Lie group and let u E ge. Let f E b(G), and put g ( t ) = f(exp(tu)) for t E R. Then (wiring X * ffor Ox -f)we have g'")(t) = (X;.f)(exp(tu)).

Clearly it is enough to prove the formula when m = 1. Putting s = exp(tu), we have by virtue of (16.5.4) and the definition of a differential (16.5.7),

g'(0

=(4f, ~ ' ( 0=)( d S L X&)>

=

(xu. f ) ( s )

by (17.14.1.1) and (19.8.1).

t The method of proof is taken from unpublished lecture notes of the late Yamabe.

9 PROPERTIES

OF THE EXPONENTIAL MAPPING

181

Hence we obtain the Taylor expansion

Choose a basis ( u ~ ) ~of~g,. ~ Then, ~ , , in an open neighborhood exp(U) of e in G on which the mapping log, is defined, the canonical coordinates of

,corresponding to the given choice of basis of ge, are t , , . . . , tn. If we put X i = X U i( I S i S n), the formula (19.9.2) gives

on the right-hand side of which the coefficient oft" (for an arbitrary multiindex a) comes only from the power in this operator is

(iIlC

ti Xi )'"I

. Now the coefficient of t"

s, = C x,,x,,... xilal, the sum being over all sequences ( i l , . . . , ilal) of I tc. I integers between 1 and n, (19.9.4)

in which the number of terms equal to k is ak,for 1 5 k S n. Hence, with the notation of (19.5.8), we have the following explicit expression relatiue to canonical coordinates (of the first kind) at the point e :

(19.9.5)

For example, in the notation of (19.5.10), we have (19.9.6)

zei+E,= $(Xi X j + xj Xi)

for all i , j . By comparing this formula with (19.5.10.1) we see that, relarive to canonical coordinates (of the first kind) at the point e, we have

for i # j . Hence the multiplication table for the basis (ui) of the Lie algebra can be written in the form (19.9.8)

[ui, uj] = 2

n

k= 1

brj'q

(i Z j ) .

182


Relative to a system of canonical coordinates (of the first kind), the function m(x, y) of (19.5.7.1) is log(exp(x) exp(y)). Comparison of the formulas (19.9.8) and (19.5.7.2) therefore shows that

where r E o,(U x U) (19.5.1). Equivalently, (19.9.10)

exp(x) exp(y) = exp(x + Y + HX,

Yl + rz(X

Y)).

It follows by induction that, for each integer k > 2,

where r2 E 02(Uk).In particular, since exp(-x) = exp(x)-',

with r; , r;' E o2 (U x U). (19.9.14) Suppose that g, is the direct sum of k vector subspaces ol, . . . , v k . Then the C" mapping

-

(xl, x2, . . ., xk) I-+ exp(xl) exp(x2) * * exp(xk) of g, = D~ x ... x Dk into G is a direomorphism of a neighborhood in 0 in g, onto a neighborhood of e in G .

It follows from (19.9.11) that the tangent linear mapping at the point 0 is the identity mapping of g, (identified with o1 x * . . x Pk), and the result therefore follows from (16.5.6). (19.9.15) Consider in particular a basis (ui)lsi16nof g,, and take vj = Ru, in (19.9.14) for 1 s j n. Then the mapping

s

(tl,

.. . , 1,)

exp(tlul) e x ~ ( t z~

2 *) *

exp(t,, un)

9 PROPERTIES OF THE EXPONENTIAL MAPPING

183

is a diffeomorphism of an open neighborhood of 0 in R" onto an open neighborhood U of e in G . The inverse mapping rp is therefore a chart of G at the point e, and the coordinates t,, . . . , t, of ~ ( s for ) a point s E U are called the canonical coordinates of the second kind of the point s, relative to the basis (ui)of 9,.

PROBLEMS

1. Let G be a Lie group, u a vector in oe, and x a point of G. Let f~ I ( G ) and put g ( i ) = f ( x * exp(tu)) for t E R.Then we have g("')(t)= (.%.'"

. f ) ( x .exp(tu)).

Deduce that, relative to canonical coordinates of the second kind (19.9.15), we have 1 z=-x, cc! in the notation of (19.6.1). 2.

Let G be a Lie group. Show that there exists a neighbourhood U of e in G having the following property: for each sequence (x.) of elements of U, if we define inductively y , = xl,yn = (x., Y , , - ~ ) ,the sequence of commutators ( y n )converges to e (cf. Problem 6).

3. (a) Show that in the unitary group U(n) there exists a compact neighborhood V of P which has the property of Problem 2, is stable under all inner automorphisms, and is such that if x, y E V do not commute, then x and the commutator ( x , y) do not cornmute. (For the third property, see Section 16.1 1, Problem 1. For the second property, remark that for each neighborhood W of e in a compact group G, and each s E G, there exists a neighborhood V, c W of e and a neighborhood T, of s such that the relation t E T, implies t V , t - ' c W.) (b) Let by the normalized Haar measure (14.3) on U(n), and let V, be a compact symmetric neighborhood of e in U(n) such that V: C V. Letf(n) be the smallest integer such that p(V,) > l/f(n). Show that, for everyfinite subgroup F of U(n), there exists a commutative normal subgroup A(F) of F such that

(F : A(F)) sf(n) (Jordan's theorem). (Consider the subgroup A(F) generated by F n V, and use (a).) 4.

Let G be a connected Lie group and let u, v E oe. If [u, v] = 0, show that exp(t'u) and exp(t"v) commute for all t', t" E R,and that exp(t'u) exp(t"v) = exp(t'u

+ t"v).

184


5. Let G be a Lie group and let U be a neighborhood of e in G of the form exp(V), where V is an open ball in 0, with center 0 (relative to some norm on fie), such that the exponential mapping is a diffeomorphism of V onto exp(V). Let W c V be an open ball ) ~U. Show that if u, v E W are such that exp(u) and with center 0 such that ( e ~ p ( W ) C exp(v) commute, then [u, v] = 0. (Consider the image of the one-parameter subgroup corresponding to u under the inner automorphism x-exp(v) .x .exp(-v).) 6. Let G be a Lie group. Show that for each numberp E ]0,1[ thereexists a neighborhood U of e on which the log function is defined and such that, for a given norm on Be. we have for all x, y E U

7. Let G be a connected Lie group and Z a connected Lie subgroup of G contained in the center of G. Suppose that the exponential mappings exp, and exp,,, are diffeomorphisms. Show that exp, is a diffeomorphism. (Let n : G G/Z be the canonical homomorphism, n,,: 0, -+ e,/a, the derived homomorphism, and let u be a linear mapping of gb/& into 0, such that n, u = lge,lr. Consider the mapping a : x - ~ ( l o g ~ / ~ ( n ( x ) ) ) of G into 0:, show that expc(-a(x))x E Z, by showing that its image under n is the neutral element of G / Z . Then put P(x) = a(x3 logz(expo(-a(x))x), and show that expo (P(x)) = x and P(expG(u)) = u for all x E G and u E oe . --f

0

+

8. Let G be a Lie group and 0, i t s Lie algebra. Let of G x g, into T(G).

denote the C" mapping (s, U)HS . u

(a) Show that if T(G) is trivialized over an open set U on which log, is defined, by means of the corresponding canonical chart (so that U is identified with an open subset of oe, and T(U) with U x oe), then for all u, v, w E 0, we have T ( w ( P ) * ((e,

4, (v, w))

= ((e, 4

, (v, w

+ Hv, ~ 1 ) ) .

(Use(19.9.1O)J

(b) Deduce that if M is a differential manifold and iff: M mappings, then we have

-+

G, u : M

-+

op are C1

relative to an invariant linear connection C on G satisfying ( I 9.8.4.1). 9. Let G be a simply-connected Lie group of dimension n, let 0. be its Lie algebra, and let UY, be an isomorphism o f e . onto a Lie subalgebra of the Lie algebra FA(M) of C" vector fields on a differential manifold M.For each point (s, x) E G x M, let

L(s,x)

To, JO (G x M)

be the n-direction spanned by the vectors (Xu(s), Yu(x)) for all u E gc (Section 19.3, Problem 21, and for each point x E M let N, be the maximal integral manifold of this completely integrable field which contains the point (e, x ) (18.14.6). If G acts on G x M by left-translation on G, the image s * N of any maximal integral manifold N is another such.

10 CLOSED SUBGROUPS OF REAL LIE GROUPS

185

(a) The field Y,, is said to be complete if every maximal integral curve of the field (18.2.2) is defined on the whole of R. When this is so, we put fr.,,(x) = FYU(x,1) for x E M and t E R, in the notation of (18.2.3). We have A,,, 0 tl.,,, =fi+,,.,, . Show that if the fields Y,,,, . . . , Y,,, are complete, the mapping (ti,.

. .,r,)~(exp(tlul)...exp(trnurn), (ft,,.,,

0

OA,,~~)(X))

of R"' into G x M is of class C", and its image is contained in N, (cf. (18.14.7)). Show that if u, v are such that Y,,and Y, are complete, then so also is Yru.vl. (Consider the mapping t+z(exp(u) exp(tv) exp(-u),

(f-l.,, ~~.vofl.u)(x))

and use (19.11.2.2) and (19.11.2.3).) (b) Suppose that there exists a system of generators of the Lie algebra gc such that the fields Y,, corresponding to the generators are complete. Then it follows from (a) that there exists a basis (u,), d J s n of ne having this property. Let W be a connected neighborhood of 0 in R" such that the mapping w : ( t , , .. . , t.)Hexp(f,u,)exp(tzu,)...exp(t.u.)

is a diffeomorphism of W onto a symmetric neighborhood V of e in G. For each s E v, put hs = A n . u n

..

0

A1.ui

7

where ( t l , .. . , 1). = w - ' ( s ) . Show that the mapping @ : (s, X)H(S, h,(x)) is a diffeomorphism of V x M onto itself and that, for each x E M,the image of V x {x} under @ is the connected component of (e, x ) in .N,n ( V x M).(Show that this set is both open and closed in N, n( V x M) with respect to the proper topology of N, .) (c) Deduce from (b) that the restriction of the projection pr, to each N, is a diffeomorphism of N, onto G. (Prove that this mapping makes N, a covering of G, when N, is endowed with its proper topology.) Hence show that CP is a differentiable action of G on M, such that for this action we have Z,, = Y,, for all u E ne. 10.

Let G be a Lie group, fle its Lie algebra, and let u be a mapping of 0, into G such that u(nx) = (u(x))" for all I I E Z and all x E 0,. Suppose also that rtiscontinuousatthepoint 0 and that if U is a neighborhood of 0 in (ie such that u(U) is contained in a neighborhood of e on which logG is defined, then the function x-log(u(x)) = u(x) is differentiable at the point 0 and has as derivative the identity mapping. Show that u = exp,. (Observe that u(x/n) = u(x)/n for all integers n > 0, and deduce that u(x) = x.)

10. CLOSED SUBGROUPS O F REAL LIE GROUPS

(19.10.1) (E. Cartan's theorem) Every closed subgroup H ef a real Lie group G is a Lie subgroup of G (in other words, the subspace H of G is the underlying space of a submanifold of G). Let denote the subset of the Lie algebra ge of G consisting of all vectors u E ge such that exp(/u) E H for all t E R. We shall begin by proving that

186

XIX LIE GROUPS A N D LIE ALGERBAS

(19.10.1.1) $, is a Lie subalgebra of g,.

It is clear that if u E be, then fu E be for all t E R.Let us show that if u, v are two vectors in $, then u + v E 8,; this will prove that $, is a vector subspace of g,. Now, if t E R and if n is an integer >O, then by virtue of (19.9.10) we may write (u

t2 1 + v) + 2n 7[u, v] + -5 w,(t, u, n

where, for fixed t , u, v, the llw,(t, u, v)ll form a bounded set. It follows that (19.10.1.2)

(exp(t u)exp(i v)), = exp(l(u

+ v) + n

where the sequence of numbers (la,(l is bounded (for fixed t, u, v). By definition, the left-hand side of (19.10.1.2) belongs to H, and as n + 00 it tends to the limit exp(t(u + v)); since H is closed, it follows that u + v E 6,. Next, let us show that [u, v] E be. The formula (19.9.1 3) gives

where the sequence of numbers (1b,(1is bounded (for fixed t, u, v). We deduce that (1 9.10.1.3)

f

f

(exp(f u) exp(f v) exp( - u) exp( - v)),* = exp(t'[u, v]

+n

and the same argument as before shows that [u, v] E 9,. We may therefore consider the connected Lie group K immersed in G (19.7.4) which corresponds to the Lie subalgebra 6, of ge, Since exp($=) is a neighborhood of e h K (for the proper topology of K)(19.8.6), and since K is senerated by any neighborhood of e (12.8.8), it follows that K c H by the definition of 6,. It will therefore be enough to prove that (1 9.10.1.4) The subgroup K is open in H (for the topology induced by that of G), and the topology induced on K by that of G is the proper topology of K.

For it will then follow from (16.8.4) that K is a submanifold of G; moreover, K is the identity component of H, and therefore by translation H is a submanifold of G, and hence is a Lie subgroup of G.


187

To prove (19.10.1.4), it is enough to show that every neighborhood N of e in K (for the proper topology of K) is a neighborhood of e in H (for the topology induced by that of G): for by translation, the same will then be true for every neighborhood of any point of K. Suppose therefore that there exists a neighborhood N of e in K that is not a neighborhood of e in H. Then there exists a sequence (a,,) of points of H - N which tends to e in G. Let us decompose the Lie algebra g, as a direct sum be@ m, where in is a vector subspace of ge. There exists a bounded neighborhood V of 0 in 9, and a bounded neighborhood W of 0 in m such that the mapping (x, y ) ~ e x p ( xexp(y) ) is a homeomorphism of V x W onto a neighborhood U of e in G (19.9.14). We may assume that a,, E U for all n, and hence for each n there exist well-defined vectors x,, E V and y,, E W such that a,, = exp(x,,)exp(y,,).Moreover, by replacingV by a smaller neighborhood, we may assume that exp(x,,) E N, since N is a neighborhood of e in K (for the proper topology of K). Since a,, E H - N, we must have y,, # 0 and lim y,, = n-r m

0. As W is bounded and y,, # 0, there exists an integer r,, > 0 such that r,, y,, E W and (r,, + l)y, 4 W. Furthermore, since W is relatively compact in in, we may, by passing to a subsequence of (a,,), assume that the sequence (my,,) has a limit y E W. Since lim y,, = 0, the sequence ((r,, + l)y,,) also

+

n-1 m

tends to y; but because (r,, I)y,,4 W, this shows that y belongs to the frontier Fr(W) of W in in, and therefore y # 0, so that y 4 9,. We shall now show that exp(ty) E H for all t E R; by the definition of $, , this will imply that y E be and will give the desired contradiction. Since H is

(; +1

closed, it is enough to show that exp - y E H for all rationalintegersp,q(where q > 0). Now, we can write pr,, = qs,, u,,, where s, and u,, are integers and 0 s u,, < q. This implies that Un lim y,, = 0 4

n+m

and hence, in G,

(: )

r:

exp - y = lim exp - y, n*m(

))

= lim(exp(y,,))”l n+m

But since exp(x,,) E N c H and a,, E H, we have exp(y,,) E H. Consequently, as H is closed in G, we have

and the proof of (19.10.1) is complete.

188


(19.10.2) Let G, G‘ be two Lie groups. Every continuous homomorphism

f :G -,G’ is a homomorphism of Lie groups (i.e., is of class CL).

For the graph rs o f f is a closed subgroup of the Lie group G x G‘ (12.3.5), hence is a subatanifoldof G x G’ by (19.10.1). The result now follows from (16.9.10). In particular: (1 9.10.3) Two structures of drflerential manifold which are compatible (1 6.9.1) with the same structure of topological group are identical.

This follows from (19.10.2), applied to the identity mapping of the group in question. Remark (19.10.4) Let G be a Lie group and let ( H J 2 . L be any family of Lie subgroups of G. Since H = () HAis closed in G, it is a Lie subgroup of G, and it IEL

follows immediately from (19.8,lO) that the Lie algebra of H is the intersection of the Lie algebras of the H A .

PROBLEMS

Show that if a one-parameter subgroup of a Lie group G is not closed, then its closure in G is compact (hence a torus). (Use (19.10.1) to reduce to the case where G is commutative and connected, and then use (19.7.9.2).) (a) Let H be a closed subgroup of a Lie group G, and let L be a connected Lie group immersed in G . If the intersection of the Lie algebras of H and L is zero, show that H n L is discrete in L (for the proper topology of L). Give an example in which H n L is dense in H. (b) Give an example of two connected Lie groups L, L’ immersed in G, such that the Lie algebras of L, L‘ have zero intersection and L n L’ is dense in G. (a) Let G be a connected Lie group, or its Lie algebra. If u E 0, is such that the oneparameter subgroup exp(Ru) of G is not closed in G, show that there exists a vector v E Be arbitrarily close to u (for the canonical topology of the vector space 0,) such that exp(Rv) is closed in G. (Use Problem 1 to reduce to the case where G is a torus.) ) such ~ that ~ each ~ of~the ~one(b) Deduce from (a) that there exists a basis ( u ~ of ciP parameter subgroups exp(Ru,) is closed in G.


4.

189

Let G, G' be two Lie groups and / I : G -,G' a homomorphism (of abstract groups). Suppose that, for each continuous homomorphism u : R + G , the composite homomorphism 11 : L! : R G' is continuous. Prove that I I is a homomorphism of Lie groups. (Use (19.9.15).) +

5. In this problem, assume the theorem that every finite-dimensional Lie algebra over R is the Lie algebra of some Lie group (cf. Chapter XXI). Let M be a connected differential manifold and 1' a group of diffeomorphisms of M. Let S be the set of all vector fields Y E . F b ( M ) that are complete (Section 19.9, Problem 9) and such that for each r E R the diffeomorphism x -Fy(s, t ) of M onto itself (18.2.8) belongs to Assume that the Lie subalgebra n of ,FA(M) generated by S is finite-dimensional.

r.

(a) Let G be a simply connected Lie group whose Lie algebra is isomorphic to n. Show that there exists a homomorphism (of abstract groups) h : G + r such that, for each vector u in the Lie algebra iieof G . if V,, E 0 is the vector field corresponding to U, we have h(exp(ru)) .x - FYu(.r, t ) for all t E R and all x E M. Furthermore, there exists a neighborhood V of the identity element of G such that the restriction of h to V is injective. (Use Section 19.9, Problem 9.) (b) Show that there exists on Go = h(G) c I' a unique structure of Lie group such that h is a surjective Lie group homomorphism of G onto G o . The group Go is normal in r, and for each w t I', the mapping II -> w' II w - ' is an automorphism of the Lie group G o . (Use Problem 4.) Deduce that there exists on r a unique topology 3 which is compatible with the group structure of I'. induces on G o the topology defined above, and for which G o is open in I' (and therefore the identity component of r). (c) Show that the topology .F has a basis of sets W(U, K), where U (resp. K) runs through the open (resp. compact) subsets of M, and W(U, K) is the set of all w E such that w(K)c U. (Reduce to proving that the topology induced by 9 on h ( V ) can be defined in this way.) Deduce that I', endowed with the topology ,7,is metrizable and separable, and hence that I' is a Lie group acting differentiably on M (Palais' thcarem). r

6.

Let M be a parallelizablc connected differential manifold (Section 16.15, Problem I ) of dimension n, SO that there exist / I vector fields XJ E .Fb(M) ( I z j < n ) such that at each point the n vectors X,(X) form a basis of TJM). Let r denote the group of diffeomorphisms of M which leave i/ruariant each o f the fields ,'A . (a) With the notation of Problem 5, show that S is contained in the set n of vector fields Y E .Fb(M) such that [ Y, X J ] 0 for I i j n. (b) For each point u = ( 1 1 ~ . . . . , 11.) E R", put X(u)= X ~ 4 j X j .For each x E M, J

there exists a neighborhood V of 0 in R" such that Fntu,(x.1) is defined for all u E V and such that u--. Fx,,,,(x, I) is a diffeomorphism of V onto a neighborhood of x in M. Deduce that if Y E a. the set of points .Y E M such that Y ( x ) = 0 is both open and closed in M, and hence that for each x E M the mapping Y-+ Y ( x ) of a into TAM) is injective. (Observe that [ Y, M u ) ] = 0 for all u E V.) (c) Deduce that, for the topology .7 described in Problem 5(c), r can be endowed with a manifold structure which makes it a Lie group acting differentiably on M, with 3- as underlying topology. (Use Problem 5 . )

190


11. T H E A D J O I N T REPRESENTATION. NORMALIZERS A N D CENTRALIZERS

(19.11.1) Recall (16.9) that a linear representation of a Lie group G is a Lie group homomorphism f :G -,GL(E), where E is a finite-dimensional real vector space. The tangent linear mapping to f at the point e is therefore a Lie algebra homomorphism

(1 9.3.5); it follows in particular, by virtue of (1 9.8.9) and (19.9.1) that for all x E ge, we have

(recall that f,(x) E gI(E) may be canonically identified with an element of the ring End(E) (19.4.2)). We shall consider in particular the aa'joint representation sHAd(s) of a Lie group G in its Lie algebra ge (19.2.1). (1 9.11.2) The tangent linear mapping at the point e to the aa'joint representation s ~ A d ( s is) the homomorphism x ~ a d ( x o)f g, into gI(g,) (we recall that ad(x) * y = [x, y]; the homomorphism x w a d ( x ) is called the adjoint representation of gej.

-

For each x E g, the mapping ywAd(exp(x)) y = exp(x) y exp(x)-' of g, into itself is linear. For fixed y, the derivative of the mapping 9

x H Ad(exp(x)) * y a t the point 0, is therefore ((8.1.3) and (8.2.1)) hH((Te(Ad) 0 T,,,(exp))

h) * Y.

Since To,(exp) is the identity mapping of ge, we obtain hw(T,(Ad). h) * y. But y~ Ad(exp(x)) * y is the derivative at the point 0, of the mapping

11 THE ADJOINT REPRESENTATION

191

of ge into itself, by virtue of the definition of Ad(s) (19.2.1.1) and of the fact that T,(log) is the identity of g,. Now it follows from Taylor's formula (19.9.12) that the derivative of (1 9.11.2.1) at the point 0, is the linear mapping

Y H Y + [x, Yl

+ r1(x)

* Yl

where r l ( x )E End(g,) is such that rl(x)/llxll tends to 0 as llxll + O (16.8.9.1). For each fixed y, the derivative at 0, of the mapping (of ge into itself) X H Y

+ [x, Yl + r1W Y

is therefore h H[h, y], which proves (1 9.1 1.2). Hence, by virtue of (19.11.I.I), we have for each vector x E ge (19.11.2.2)

Ad(exp(x)) = exp(ad(x)) =

" 1 (ad(x))" ,,,=om.

in the algebra End(g,). Likewise, by applying the formula (19.8.9.1) to the case where f is the inner automorphism Int(s) : f-sfs-', we obtain, for x E go: (1 9.11.2.3)

Int(s)(exp(x))

= s(exp(x))s-

= exp(Ad(s)

. x).

Remark (19.1 1.2.4) If A is a finite-dimensional R-algebra with identity element, it follows immediately from (8.1.3) that in the Lie group A* the adjoint repre-

sentation is given by (19.11.2.5)

Ad(s) : U H S U S - '

(where A, endowed with the bracket operation, is identified with the Lie algebra of A* (19.4.2)). (19.11.3) Let G be a Lie group and let algebra g, of G .

in

be a vector subspace of the Lie

(i) The set of all s E G such that Ad(s) * in c in (or equivalently such that Ad(s) * in = in, since Ad(s) is an automorphism of 9,) is a closed subgroup H of G , whose Lie algebra Ij is the set of all u E ge such that ad(u) in c in. 9

192


(ii) The set of all s E G such that the restriction of Ad@) to m is the identity mapping 1, is a closed normal subgroup K of H, whose Lie algebra f is the set of all u E 9, such that the restriction of ad(u) to m is zero. (i) Choose a basis of ge containing a basis of in. Then to say that Ad(s) m c m is to say that certain of the entries in the matrix of Ad(s) relative to this basis are zero. From this it is clear that H is closed in G; and since Ad(s) m = m is equivalent to m = (Ad(s))-’ * m = Ad@-’) in it follows that H is a subgroup of G. Now let $ = Lie(H). For each y E m, the derivative at the point 0, of the mapping x HAd(exp(x)) . y of 9 into in must be an element of Hom(4, in); but this derivative is the restriction to 9 of the mapping U H [u, y] (19.11.2), hence we must have [u, y] E in for all y E in and all u E $. Conversely, let $‘ be the Lie subalgebra of ge consisting of all u E 9, such that ad(u) m c m.For each x E Ij’, the restriction of ad(x) to in is an endomorphism of this vector space, hence the same is true of exp(ad(x)) (the exponential being taken in GL(m)). By virtue of (19.11.2.2), we have therefore Ad(exp(x)) y E in for all y E m; in other words, exp(9’) c H. Since $’ 3 $, it follows that exp(9‘) is a neighborhood of e in H, which implies that $’ c Q (19.8.10). Hence 9’ = 9. (ii) It is immediate that K is a closed normal subgroup of € Let = fI =. Lie(K). For each y E m, the mapping x ~ A d ( e x p ( x ) ) y off into in is constant (equal to y), hence its derivative is zero, which as above gives [x, y] = 0,. Conversely, let f’ be the Lie subalgebra of ge consisting of the vectors u such that the restriction of ad(u) to m is zero. For each x E f’, the restriction of ad(x) to m is zero, hence the restriction of Ad(exp(x)) = exp(ad(x)) to 111 is the identity mapping (19.11.2.2). Consequently, we have exp(f’) c K, and since f’ I> f it follows as before that f’ = f. The group H is called the normalizer in G of the vector space in c 9, , and is denoted by N ( m ) . Its Lie algebra 9 is called the normalizer of m in g, and is denoted by %(m). The group K is called the centralizer in G of the vector space m, and is denoted by O(m). Its Lie algebra f is called the centralizer of m in ger and is denoted by 3(m). The connections between these notions and those of the normalizer and centralizer in G of a subset of G (12.8.6) are brought out in the following results. (19.11.4) Let G be a Lie group, H a connected Lie group immersed in G (19.7.4), corresponding to a Lie subalgebra $, of 9,. Then the normalizer M(H) of H in G is the closed subgroup N(b,),whasse Lie algebra is ‘iR(f)e).


193

It is clear that, for each s E G, the group sHs-’ is a connected Lie group immersed in G, with Lie algebra Ad@) * $, (19.2.1). Hence (19.7.4) we have s E N ( H ) if and only if s E A”($,). The proposition is therefore a consequence of (19.11.3(i)). (19.1 1.5) Let H be a connected Lie group immersed in a connected Lie group G. Then H is normal in G if and only i f its Lie algebra be is an ideal in ge. To say that H is normal signifies that M(H) = G. This is equivalent to A”($,) = G, and implies that %(be)= g, i.e., that $, is an ideal in g;, but = G. since G is connected, the relation ge = %($,) also implies that N($,) (19.11.6) Let G be a Lie group, H a connected Lie group immersed in G, corresponding to a Lie subalgebra $, of ge , Then the centralizer 9 ( H ) of H in G is the closed sirbroup 9($,), whose Lie algebra is 3($,). To say that s E 2(H) signifies that the restriction of Int(s): tt-+sts-’ to H is the identity automorphism of H. Since H is connected, this is equivalent to saying that the restriction of Ad@) to be is the identity automorphism of 6, (19.7.6). The proposition is therefore a consequence of (1 9.1 1.3(ii)). In particular: (19.1 1.7) The Lie algebra of the center C = 9 ( G ) of a connected Lie group G is the center c, of the Lie algebra ge of G (i.e., is the set of all x E ge such that [x, y] = 0, for all y E ge). (19.11.8) We shall prove in Chapter XXI that if g is any finite-dimensional Lie algebra over R, there exists a Lie group G such that Lie(G) is isomorphic to g. Here we shall prove a particular case of this theorem: (19.1 1.9) r f g is any jnite-dinzensional Lie algebra over R and if c is the center of g, then there exists a Lie group H such that Lie(H) is isomorphic to the Lie algebra g/c. Consider the adjoint representation x H a d ( x ) of g: its image is a Lie subalgebra $ of gI(g), isomorphic to g/c. Ifn = dim(g), then gl(g) is isomorphic to the Lie algebra of the Lie group GL(n, R). Hence (19.7.4) there exists a connected Lie group H immersed in GL(n,R), whose Lie algebra is isomorphic to $, hence to g/c.

194

XIX


PROBLEMS 1. Let G be a connected Lie group, 8 the infinitesimal algebra of G, formed by the leftinvariant differential operators (1 9.3.1).

(a) For each left-invariant vector field X E 0, consider the derivation ad(X) : Z - Xo Z -Z

0

X

of 6. With the notation of (19.6.2), show that ad(X) is an endomorphism of each of the vector subspaces 0, of 0, and hence that exp(ad(X)) is an automorphism of each vector space a, and therefore an automorphism of the vector space W. Using the fact that ad(X) is a derivation, together with Leibniz's formula, show that exp(ad(X)) is an automorphism of the algebra (*. (b) For each differential operator P E Diff(G) and each s E G, define Ad(s). P = y ( s ) l ( ~' )-P .

If P E 0, we have Ad(s) * P = S(s- ')P,and Ad(s) . P is the differential operator whose value at the point e is the image under Int(s) of the point-distribution P(e).Show that for each u E oc we have Ad(exp(u)) * Z = exp(ad(Xu)) * Z for all Z E U. (Observe that the two sides agree when Z E $1, and that $1generates the associative algebra W.) (c) Deduce from (b) that the center of the algebra 0 consists of the operators

P E Diff(G) which are both leff- and right-invarianr. (Observe that in each space W,", we have 1

lim - (exp(ad(X)).Z-Z) I-0

t

=

ad(X)*Z

for X E0.) If c is the center of the Lie algebra ,I, it is therefore contained in the center of 0. 2. (a) For the group G considered in (19.5.11), show that the center of the infinitesimal algebra @ consists only of scalars. (b) Let G be the Lie group SL(2, R), and take as basis for the Lie algebra i 4 2 , R) the basis (u,, u2, u3) of Section 19.5, Problem 2. Show that the center of this Lie algebra is zero, but that the element u: 2(u, u3 -tu3u2) belongs to the center of

the infinitesimal algebra W,

.

+

3. Let G be a connected Lie group, H a connected Lie group immersed in G ; let H = H' be the closure of H in G and let tie, 11,. 1: be the Lie algebras of G, H, H', respectively.

(a) Show that if a connected Lie group K immersed in H is normal in H, then it is also normal in H'. (If f, is the Lie algebra of K, observe that Ad(s). I, c I, for all s E H,and use (1 9.11.4)J


195

(b) Show that if a connected Lie group L is immersed in G and if H c L c H = R, then L is normal in H . (c) Suppose that H'= G (Le., that H is dense in G). Show that the Lie algebra o,/b, is commutative. (Observe that every Lie subalgebra 1, such that be c I, c 0, is an ideal of nr .) (Cf. Section 19.16, Problem 11.) (d) Under the hypotheses of (c), show that oc is the direct sum of an ideal f, contained in the center c, of ar, and an ideal I. 2 S. such that I, n c. c 1). . (Decompose 0./6, as the direct sum of (re IIJII. and &/fie, and c, as the direct sum of c, nb, and L.) Furthermore, every element u E 1, such that [u, v] = 0 for all v E 0, belongs to c, n he. (Remark that the hypothesis implies that Ad(s) u = u for all s E H, and deduce that the same is true for all s E G . )

+

-

4.

Let G be a connected Lie group, H a connected Lie group immersed in G and dense in G. For each s E G, the image of lie under Ad(s) is contained in be (Problem 3(a)). Let Ad&) denote the restriction of Ad(s) to he, which is an automorphism of the vector space be. (a) Suppose that the image of H in GL(b,) under the adjoint representation is a closed subgroup Ad(H) of CL(b,). Show that Ad(H) is equal to the image AdH(G) of G under snAdH(s). Deduce (in the notation of Problem 3(d)) that I, = 8. and hence that oe is the direct sum of 11, and an ideal f. c c, . (b) Deduce from (a) that the center Z(G) of G is the closure in G of the center %(H) of H. (Let s E %(G) be the limit of a sequence (8.) of elements of H. Show that there exist elements t,, E I ( H ) such that x. = s, 1;' tends to e in H; for this purpose, observe that Ad(H) is isomorphic to H/T(H), where H is endowed with its proper topology.) Show that G = H I ( G ) and that Ad(G) = Ad(H) is closed in GL(&). (c) Under the hypotheses of (a), deduce from (b) that for H to be equal to G, it is necessary and sufficient that T(H) be closed in G.

5.

Let G be a connected Lie group, u : G --+ G' a homomorphism of Lie groups. Assume that Ad(G) is a closed subgroup of GL(0.). Then u(G) is closed in G' if and only if u ( T ( G ) )is closed in G'. (Use Problem 4.)

6.

Let G be the closed subgroup of GL(3, C) consisting of the matrices

I 3,

where s, t take all values in R and x , y all values in C. Show that 9 ( G ) consists of the identity element. Let o! be a given irrational number and let H be the subgroup of all elements of G such that t = as. Show that Ad(H) is not closed in GL(b,). (Use Problem 4.) 7. Let G be a Lie group, H any subgroup of G. For each x

E H, let 11, denote the set of vectors u, E T,(G) for which there exists an open interval I c R containing 0 and a C" mapping f : I G such that f(0) = x , f(1) c H, and T d f ) * E(0) = u, . --f

(a) Show that for each x E H we have 11, = x be = 11, * x , and that Ad(x) . b e c 11,. (b) Show that 11, is a Lie subalgebra of oe = Lie(G). (Use (16.9.9(ii)) to prove that be is a vector subspace of ae. Then observe that if u, v are two vectors in 6, and if

196

XIX


f : I -+G is a C" mapping of a neighborhood I of 0 in R such thatf(0) = e,f(l) c H, and To(f) . E(0) = u, then we have Ad(f(r)) v E [I, for all r E I. Finally use (1 9.11.2).) (c) Let Ho be the connected Lie group immersed in G, such that Lie(Ho) = I),. Let M be a differential manifold, f : M -+ G a C" mapping such that f(zo) = e for some zo E M, andf(M) c H. Show that there exists an open neighborhood U of zo in (Use (18.14.7)J In particular, H, is a normal subgroup of H. M such thatf(U) c Hoe (Use the definition of be and (19,9.15),) Deduce that there exists a unique topology on H which is compatible with its group structure, which induces on Ho its proper topology (19.7.4), and for which Hois the identity component of H. (Cf. Section 12.8, Problem 1.) 8. Let M be a differential manifold. A subset A of M is said to be C"-connected if, for all x, y E A, there exists a sequence of points of A such that zo = x, zn = y, and for 1 5 j 5 II, a C" mappingfi of an open interval I c R into M such that f,(l) contains z,-~ and z, and is contained in A.

(a) Show that if a subgroup H of a Lie group G is C=-connected, then H is a connected Lie group immersed in G. (With the notation of Problem 7. show that H, = H.) (b) Deduce from (a) that if A, B are two connected Lie groups immersed in a Lie group G, then the subgroup H of G generated by A u B is a connected Lie group immersed in G, whose Lie algebra is generated by the union of the Lie algebras of A and B. (To establish the latter point, show that if u E Lie(A) and v E Lie(B), then [u, v] E Lie(H), by using Problem 7 and (19.9.1 3). 9. Let A and B be two connected Lie groups immersed in a Lie group G, and let a,, 6, be their Lie algebras.

(a) Show that the mapping (x. y ) - x y of A x B into G (where A, B are endowed with their proper topologies and their Lie group structures) is a subimmersion of constant rank dim@, 6,) = dim a, dim be - dim (a, n be). Moreover, if AB is a lotally closed subspace of G, then AB is a submanifold of G. (Consider the left action of A x B on G defined by ((x, y), s) = xsy-' and use (16.10.2) and (16.10,7).) Give an example where AB is open and dense in G but is not closed. Section 19.7, Problem 3. (b) Suppose that A nB is closed in G and normal in B, and that AB is a submanifold of G. The group A nB acts on the manifold AB by right translations, and the orbit manifold AE%/(AnB) exists. Show that it is canonically diffeomorphic to

+

+

( M A n B)) x (BAA n B)).

(Use(16.10.4)J Deduce that A and B are Lie subgroups of G. (c) Show that if AB = BA, a, 6, is a Lie subalgebra of oe = Lie(G). Use (a) and Problem 8(b); show that the immersed Lie group AB cannot have dimension > dim(a, be) by using (18.14.7) and Baire's theorem (12.16,1).) (d) Suppose that B is a normal Lie subgroup of G. Show that when A and AB are endowed with their proper topologies, the quotient groups A/(A n B) and AB/B are canonically isomorphic.

+

+

10. The automorphism group L of T is a discrete group of two elements, the identity and the automorphism x--x (Stetion 19.7, Roblern 6). Consider the nonconnected

12 THE LIE ALGEBRA OF THE COMMUTATOR GROUP

197

Lie group G = L x, T, the semidirect product of L and T relative to the identity automorphism u : L + Aut(T) (19.4.5). Show that the center o f G is discrete. (Compare with (19.11.7).)

12. T H E LIE ALGEBRA O F T H E C O M M U T A T OR e R O U P

Let G be a group and let H, K be two normal subgroups of G. The subgroup of G generated by the commutators h k h - l k - ' , where h E H and k E K, is called the comrizutator group of H and K and is denoted by (H. K). It is clear that ( H , K) is a nornial subgroup of G and is contained in H n K. The coninzutator group of G is the group (G, G), which is also denoted by 9 ( G ) and is also called the deriuedgroup of G. It is the smallest normal subgroup N of G such that G/N is commutative. We recall also that iff), f are two ideals in a Lie algebra g, the vector subspace generated by the elements [x, y]. where x E lj and y E f , is denoted by [f), €1.I t follows immediately from the Jacobi identity that [b, €1is an ideal of g contained in b n t. In particular, the ideal [{I, g] is denoted by D(g)and is called the derived ideal of g. For Lie groups, these notions are connected by the following theorem: (19.1 2.1 ) Let G be a connected Lie group, gc its Lie algebra. Then the derived group 9(G ) is the underlying group of the connected Lie group immersed in G tr4icli corresponds to the Lie subalgehra TYy,). Let G' denote the connected Lie group immersed in G which corresponds to B(g,);then G' is normal in G (19.11.3). We shall first show that 9 ( G ) c G'. The quotient Lie algebra b = y,/B(g,) is commutative, hence is the Lie algebra of a group H isomorphic to R" for somem 2 0. I f o : g, + f) = ge/'D(g,) is the canonical homomorphism, there exists a local homomorphism w from G to H of class C" such that T,(w) = o (19.7.6). Let U be a symmetric neighborhood of e in G on which the function log, and the local homomorphism MI are defined, and let V be a symmetric neighborhood of e such that V4 c U . Since H is commutative, the relations S E V, t E V imply w ( s t ) = w ( t s ) , hence w ( s t s - ' t - ' ) = 0. But since the function log, is defined on U, we have s t s - ' t - ' = exp(log(sts-'t-')),andtherefore(l9.8.9) w ( s t s - ' t - ' ) = exp(w(log(.sts-'t-'))) = 0. By definition, thissignifiesthat log(sts-'t-') E D(g,) and hence implies that s t s - ' t - ' E G'. Since G is connected, the neighborhood V generates G (12.8.8); hence the formula

' '

( s t ) U ( s t ) - ' u - 1 = s(tut - ' u - 1)s- '(sus- u - )

198

XIX

LLE GROUPS AND LIE ALGEBRAS

and the normality of G' (19.11.3) show that every commutator sts-'t-' belongs to G', i.e., 9 ( G )c G'. Conversely, we shall show that G' c 9(G). Since G' is connected, it will be enough to show that there exists a neighborhood of e in G (for the Lie group topology of G') contained in 9 ( G ) (12.8.8). Let n = dim ge, r = dim %(g,) S n. Then there exist r pairs (aj, bj) of elements of ge such that the vectors cj = [aj, bj] (1 S j S r) form a basis of D(g,). Complete this basis, by adjoining n - r elements c,, . .., c, , to a basis of ge . Let V be a symmetric neighborhood of e in G, which is such that the function log, is defined on V4. Let (u,), sjsnbe the system of canonical local coordinates of the first kind at e corresponding to the basis ( c j ) , so that uj(x) is the jth coordinate of log(s). Let I be an open neighborhood of 0 in R such that, for A, p E I, the points sj(A)= exp(Aaj) and f j ( p ) = exp(pbj) belong to V for 1 g j 5 r. Then the point gj(L P) = sj(A)tj(p)~j(-A)tj(-p) belongs to V4, and it follows from (19.9.13) that we may write

where Aj and Bj are bounded in I x I. Taylor's formula, applied to the functions of one variable p w u i ( g j ( A ,p)), now gives (19.12.1.2)

ui(gj(L PI) = puij(A) + p2wij(A P),

where u i j , w i j are C" functions on I x I. By virtue of (19.12.1.1) and the definition of the c j (I S j S r), we have u&) = 6,1

(19.12.1.3)

+ A2Ci,(IZ)

for 1 i, j 6 r ( S i j being the Kronecker delta), where the C, are bounded in I. Hence there exists 1, # 0 in I such that det(uij(&)) # 0 (I 6 i, j 5 r). Now define, for pl, p 2 , ..., pr in I, (19.12.1-4) g(P1, ~ 1 2 ,

m m . 9

Pr)

=~I(Ao, Pl)g2(&,

P Z ) ' * * A ( bPr). ,

It follows from (1 9.9.1 1) and (19.12.1.2) that the mapping ( ~ 1 ,*

* * ?

pr)Hlog(&1,

- - * ,

Pr))

of I' into R' has Jacobian equal to det(uij(Ao)) at the origin; hence g is a diffeomorphism of a neighborhood W of 0 in R' onto a neighborhood ofe in G' (for the proper topology of G');but, by definition, g ( p l , p2 , . ..,pr) E 9(G). Q.E.D.

12 THE LIE ALGEBRA OF THE COMMUTATOR GROUP

199

(19.12.2) In particular, we obtain the fact that the commutator group of a connected Lie group G is connected (cf. Section 12.8, Problem 4); on the other hand, it is not necessarily closed in G (Section 12.8, Problem 7). (19.12.3) For any group G and any integer n 2 1, the nth deriued group 9"(G)of G is defined inductively by the conditions

9'(G) = 9 ( G ) = (G, G ) and, for n 2 2, 9"(G) = 9(9"-'(G)) = (Q"-'(G),9"-'(G)).

It is clear that the sequence (9"(G)) is a decreasing sequence of normal subgroups of G. The group G is said to be solvable if there exists an integer n 2 I such that 9 " ( G )= {e}. If G is a connected Lie group, it follows from (19.12.1) that 9 " ( G ) is the underlying group of a connected Lie group immersed in G , corresponding to the Lie subalgebra B"(g,) of ge. For G to be solvable, it is necessary and sufficient that B"(g,)= (0) for some integer n 2 I , i.e., that the Lie algebra g, should be soloable.

PROBLEMS 1.

Let 11 be a Lie algebra of finite dimension over R or C. (a) If (1 is solvable, then so is every Lie subalgebra of a. (b) Let (1 be an ideal in 0. Then 0 is solvable if and only if the Lie algebras n and g/a are solvable.

2.

Let n be a Lie algebra of finite dimension over R or C. Show that the following properties are equivalent: (i) b i is solvable. 3 a. = {0} of ideals in (ii) There exists a decreasing sequence 0 = no 3 a, 3 11 suck that the algebras at-,/ai are commutative ( I 2 i 5 n). (iii) There exists a decreasing sequence o = 11, 3 6 , 3 * . . 3 lip = {O} of subalgebras of $1 suck that Oi is an ideal in 0 1 - , and the algebra hr-l/bl is commutative for I iisp. (iv) There exists a decreasing sequence (1 = ino3 in, 3 *..3 litq = {O} of Lie subaIEbras of (1 such that iiil is an ideal in iiii-l and i n f - l / i ~ l is of dimension I for 1 5 i 5 q. (To prove that (iv) implies (i), use Problem 1.)

3. For any Lie algebra 0, the descending central series is the decreasing sequence of ideals ((5p(n))p defined by 6 Y U ) = n,

csP+'(0) =

[o,

&P(n)I.

The s s c e d w cmfral series is the increasing sequence of ideals (G,(d),2~, defined by 61do)

= {O},

6,+l(o)/@,(ii) = center

of nP,(o).


200

Show that, for a finite-dimensional Lie algebra over R or C, the following conditions are equivalent : 3 ap = {O) of ideals of (i) There exists a finite decreasing sequence 0 = a, 3 a, =I such that [a, a,-,] c a, for 1 5 i z p . (ii) &'(a) = {O] for sufficiently large k . (iii) at(@) = 0 for sufficiently large k. (iv) There exists an integer n such that, for all sequences (xJ), dJBn of 11 elements of 0, ad(x,) = 0 in End(a). we have ad(x,) 0 ad(x,) 0 (v) There exists a decreasing sequence of ideals a = a. 3 a1 3 . . 3 a, = {O] of o such that [e, a,-,] C a, and ar-,/ar is of dimension 1 for 1 5 i 5 q.

0,

-.

.

(Observe that if (i) is satisfied, then a, 3 Qf+'(t$ and a,,-, c &(g).) A Lie algebra g satisfying these conditions is said to be nilpotent. If m (resp. n ) is the smallest integer such that 6,(0) = a (resp. V ( g ) = {0}), show that n = m 1 and that &,(a) 3 Q"-'(a) for 0 5 i 5 n - 1.

+

Let 0 be a nilpotent Lie algebra of dimension n whose center CI,(o) has dimension 1. Show that there exists a basis of 8 consisting of three elements a, b, c and a basis of a subspace tu of dimension n - 3, such that [a, b ] = c and [b, u ] = 0 for all u E tu. (Take for c a basis of the center a,(e), for b any element of &(a) not in 6,(g), and consider the subspace of 0 which commutes with b.) In a solvable but not nilpotent Lie algebra 0 over R,there may exist a decreasing sequence 3 a. = {0} such that a,-,/a, is of dimension I for 1 5 i 5 n. of ideals 0 = a, 3 a, 3 This is so, for example, for the Lie algebra of (19.5.11). Show, on the other hand, that there exists a solvable Lie algebra 0 over R having a basis of three elements u, v, w such that [u, v] = w, [u, w ] = -v, and [v, w ] = 0, and that in this Lie algebra there exists no decreasing chain of ideals with the property stated above. Let A, B be two connected Lie groups immersed in a Lie group G. Show that the group C = (A, B) generated by the commutators aba-lb-', wherea E A and b E B, is a connected Lie group immersed in G. (Use Problem 8(a) of Section 19.11.) If a,, 6 , are the Lie algebras of A, B respectively, show that the Lie algebra of C contains the vector subspace [a,, be] of ge = Lie(G) generated by all [u, v ] with u E a, and v E b, (Section 19.11, Problem 7). Given an example where A and B are closed and of dimension 1 in G = GL(2, R), and the Lie subalgebra generated by [a,, 6,] is distinct from Lie(C). Let G be a Lie group, K a one-parameter normal subgroup of G . Show that the commutator group S(G)i s contained in the centralizer of K. 13. AUTOMORPHISM GROUPS O F

LIE GROUPS

(19.13.1) We recall that an automorphism of a Lie group is by definition an automorphism of the group structure of G which is also a diffeomorphism. These automorphisms form a group, denoted by Aut(G). If Go is the identity component of G, then it is clear that u(Go)= Go for every u E Aut(G), and the restriction uo of u to Go is an automorphism of the Lie group G o . We have, therefore, a homomorphism U H uo of Aut(G) into Aut(G,). This homomorphism is not necessarily injective (consider for example the case

13 AUTOMORPHISM GROUPS OF LIE GROUPS

201

where G is discrete, so that Go and Aut(Go) are reduced to the identity); nor is it necessarily surjective-in other words, an automorphism of Go cannot necessarily be extended to an automorphism of G (Problem 1). (19.13.2) In what follows .we shall restrict our attention to Aut(G,), that is to say, we shall assume that the group G is connected. For each automorphism u of G, the tangent linear mapping u* is then an automorphism of the Lie algebra g,; for if u is the inverse of the automorphism u, we have u* 0 u* = (u o u)* = I,*, and likewise t'* 0 u* = I,*. (19.13.3) Let G be a connected Lie group.

(i) The mapping u w u * is an injective homomorphism of Aut(G) into the automorphism group Aut(g,) of the Lie algebra of G . (ii) /f G is simply connected, then u Hu* is an isomorphism of the group Aut(G) onto the group Aut(g,). (iii) In general, if G = e / D , where is the simply connected miversa1 covering group of G (16.30.1) and D is a discrete subgroup of the center of G, then Aut(G) may be identged with the subgroup of Aut(e) cwsisting of automorphisms 17 suck thar u"(D) = D (or, equivalently, u'(D) c D).

e

The injectivity of U H u* in general, and the surjectivity when G is simplyconnected, both follow from (19.7.6). Let p : + G be the canonical homomorphism. For every (Lie group) homomorphism u of G into G, the mapping u o p is a Lie group homomorphism of into G, and therefore (16.30.3) there exists a unique homomorphism u" : -, such that p 0 17 = u 0 p. Moreover, if 2) : G + G is another homomorphism, we have ( u 0 u)" = 0" o 17, because p 0 (i; ti) = ( u 0 p ) 0 17 = u 0 ( u p ) . Consequently, if u is an automorphism of G, then u" is an automorphism of and UHU" is an injective homomorphism of Aut(G) into Aut(c). Furthermore, the relation p o u" = u o p shows that we must have G(D) = D. Conversely, if u" has this property, there exists a homomorphism u of G/D = G into itself that p 0 u" = u o p , and this homomorphism is of class C" (16.10.4). Likewise there is a Lie group homomorphism v : G + G such that p 0 6 - l = v op. From this we deduce immediately that v 0 u = u u = l G , and the proof is complete.

e

e e

0

0

e,

0

(19.13.4) In the notation of (19.13.3), the group Aut(g,) is a closedsubgroup of GL(g,), hence a Lie subgroup of GL(g,) (19.10.1). For if (aj)lsjsn is a

basis of gc, an automorphism v of the vector space ge is also an automorphism of the Lie algebra g, if and only if u satisfies the conditions

u([aj akl) = [u(aj),~(ak)l for all pairs of indices ( j , k) such that I 5 j < k S n. The coordinates (relative to the basis (aj)) of the two sides of this equation are polynomials in the 9

202

XIX


elements of the matrix of u relative to the basis (aj), and therefore (3.15.1) Aut(g,) is closed in GL(g,). Now assume that G is simply-connected. Then (19.13.3) we have an isomorphism of groups cp : U H U , of Aut(G) onto Aut(g,). By transporting via cp-' the differential manifold structure of Aut(g,) to Aut(G) (16.2.6), we obtain canonically a Lie group structure on Aut(G). In future, whenever we speak of Aut(G) as a Lie group, it is always this structure that is meant. (19.13.5) Lei G be a connected Lie group, covering group of G.

e the simply-connected

universal

(i) The group Aut(G) is closed in the Lie group Aut(G) (hence is a Lie group, by (1 9.10.1)). (ii) The mapping (u, X)HU(X) ofAut(G) x G into G is of class C". We shall begin by proving (ii) when G is simply connected, in which case is an isomorphism of Aut(G) onto Aut(g,). Let uo E Aut(G), and let U be a symmetric open neighborhood of e such that the function log, is defined on an open set W containing U and uo(U). Then there exists a neighborhood V, of (uo)* in Aut(g,) such that, whenever u* E V, and z E U, we have exp(u,(log z)) E W. If V is the inverse image of V, in Aut(G), this shows (19.8.9) that u(z) E W whenever u E V and z E U.Bearing in mind the definition of the Lie group structure of Aut(G), this proves that (u, Z ) H U ( Z ) is of class C" on V x U. Now let xo be any point of G; then there exists a finite of points of U such that xo = alu2 . a, (1 2.8.8). For each sequence (aj)l x E xo U we may write u(x) = u(al)u(a2)* * * u(u,)u(x; 'x). Now, each of the mappings U H u(aj) is of class C" on V, and the mapping (u, x ) H u(x; 'x) is of class C" on V x U. Hence (u, X ) H U ( X ) is of class C" on Aut(G) x G. If we now drop the assumption that G is simply-connected, so that G = G/D, then (19.13.3) Aut(G) may be identified with the subgroup of Aut(G) consisting of the automorphisms u of T; such that u(D) c D. For each z E D, the set F, of automorphisms u of T; such that u(z) E D is closed, because D is closed and U H U ( Z ) is continuous (3.11.4). Since Aut(G) is the intersection of the sets F, (z E D), this proves (i). The mapping UHU* is therefore an isomorphism of the Lie subgroup Aut(G) of Aut(e) onto a Lie subgroup of Aut(g,). The argument of the previous paragraph can now be used without any changes to prove (ii) in the general case. UHU,

-

Example (19.13.6) Let G = T , so that G = R" and D = Z". The Lie algebra ge is commutative, and therefore Aut(G) = Aut(g,) is the general linear group GL(n, R). Now an automorphism of the vector space R" maps Z" into itself if

13 AUTOMORPHISM GROUPS OF LIE GROUPS

203

and only if its matrix, relative to the canonical basis, is a matrix of integers; it follows therefore that Aut(G) is the discrete subgroup GL(n, Z) of GL(n,R) = Aut (G). (19.13.7) With the same notation, let us now determine the Lie algebra of the Lie group Aut(ge). Since this group is a closed subgroup of the linear group GL(g,), its Lie algebra a may be characterized as the set of endomorphisms U of the vector space ge such that exp(tU)EAut(g,) for all r 6 R (19.8.10), i.e., such that

for all t E R and all x, y E ge (19.13.8) The Lie algebra a of the Lie group Aut(ge) is the Lie algebra Der(g,) of deriuations of ge .

Since the two sides of (19.13.7.1) are equal at t = 0, it is sufficient to express that their derivatives are equal. Since the derivative of ti+exp(tU) is Uexp(tU) = exp(tU) U, we obtain the equation

which for t = 0 reduces to (19.1 3.8.2)

U ' [x, Y l = [ U . x, Yl

+ [x, U . Yl.

This shows that U must be a derivation of ge . Conversely, if this is the case, then the derivative of the right-hand side of (19.13.7.1) is equal to U [exp(tU) * x, exp(tU) * y]. If v ( t ) denotes the difference between the two sides of (19.13.7.1), we have therefore v'(t) = U * v ( t ) ; and since v(0) = O,, it follows from (10.8.4) that V ( t ) = 0, for all r. (19.13.9) Let G be a connected Lie group, ge its Lie algebra. Recall that for each s E G, the inner automorphism t ~ s t s - lof G is denoted by Int(s). Clearly s ~ I n t ( sis) a homomorphism (of abstract groups) of G onto a subgroup of Aut(G). This subgroup is denoted by Int(G) and is (algebraically) isomorphic to G/C, where C is the center of G. If now we endow Aut(G) with

204


its Lie group structure (19.13.5), then the homomorphism sHInt(s) is a Lie group homomorphism of G into Aut(G). For by composing this homomorphism with the isomorphism U H U * of Aut(G) onto a Lie subgroup of Aut(ge), we obtain the homomorphism s ~ A d ( s ) which , is of class C" (19.2.1). Hence (19.7.5) Int(G) is a connected Lie group immersed in Aut(G), and normal in Aut(G) (because u 0 Int(s) u-l = Int(u(s)) for any u E Aut(G)). Identifying Aut(G) with a Lie subgroup of Aut(ge) via U H U * , the group Int(G) is identified with the connected Lie group Ad(G) immersed in Aut(g,). As a Lie group, it is isomorphic to G/C. The subalgebra Lie(Int(G)) of the Lie algebra Der(ge) of Aut(g,) is, by virtue of (19.11.2), the image ad(ge) of ge under its adjoint representation, and is isomorphic to the quotient ge/ceof ge by its center. 0

PROBLEMS 1. Let A be a connected commutative Lie group (written additively) which contains an element a # 0 of order 2 (for example, a torus T",where n 2 1) Show that the manifold G = A x (0, 1) (which has two connected components) becomes a solvable Lie group if

the multiplication is defined by ( x , ONY, 0 ) = ( x

+

Y , 0)( x , W Y , 1) = ( x +Y, l), ( x , I X n 0 ) = ( x - Y , I), ( x , l)(Y, 1) = (x - Y a, 0).

+

If there exists an automorphism u of A such that u(a) # a (which will be the case when A = T",n 2 2), show that u cannot be extended to an automorphism of G (the group A being identified with the identity component Go of G ) . 2. Let G be a connected Lie group, T a normal subgroup of G. If T is isomorphic to a torus, show that T is contained in the center of G. (Observe that the group Aut(T) is discrete, and consider the homomorphism S- Int(s) 1 T of G into Aut(T).)

3. Give an example of a connected Lie group G such that Int(G) is not closed in Aut(G). (See Section 19.1 I,Problem 6.) 4. Let G be a connected Lie group. For each compact subset K of G and each neighborhood V of e in G, let W(K, V) denote the set of all u E Aut(G) such that u(x)x-' E V for all x E K . Show that the W(K, V) form a fundamental system of neighborhoods of the identity automorphism lG in Aut(G). (Use (12.8.8).)

*

5. Let G be a Lie group, K a one-parameter subgroup of G which is closed and normal in G. Let a be an dement of G such that K Z ( a )

14 SEMIDIRECT PRODUCTS OF LIE

GROUPS

205

(a) If K R, then S ( a ) n K = {e). If K g T, then S ( a ) n K has two elements. (Consider the restriction of Int(a) to K and use Section 19.7, Problem 6.) The second possibility is excluded if G is connected (Problem 2). (b) If in addition G/K is commutative, show that G = LZ(a)K and that, for each closed subgroup A of S ( a ) , the group AK is closed in G. (Observe that the mapping x ~ x - I a - ~ of . ~Ka into K is surjective.) 6.

Let G be a Lie group, K a one-parameter subgroup of G which is normal in G (but not necessarily closed in G). Let A be a closed subgroup of G. Show that, if AK is not closed in G , then K is contained in the identity component of the centei of E.(Reduce to the case where G and G is connected. Then B = Z ( K ) n A is a closed normal subgroup of G. Replacing A, K by their images i n G/B, we reduce to the situation B = { e } . Using Section 19.12, Problem 7, show that A is then commutative. I f K were closed in G, then AK would also be closed in G, by virtue of Problem 5. Hence K is not closed in G ; now use Problem 2, and Section 19-10, Problem 1.) Give an example of two one-parameter subgroups A, K of the commutative group R x T2such that A is closed and the product AK is not closed.

AK

14. SEMIDIRECT P R O D U C T S O F L I E G R O U P S

(19.14.1) Let G be a group (not necessarily topologized), N a normal subgroup of G. Then for any subgroup L of G, we have LN = NL (recall that if A, B are subsets of G, the notation AB denotes the set of all products xy, where x E A and y E B). G is said to be the semidirect product of N and L if every z E G is uniquely expressible as z = xy with x E N and y E L; or, equivalently, if G = NL and N n L = {e}. (For if N n L # {e}, an element z # e in N n L can be written as z = ez = ze; hence in two ways as a product of an element of N and an element of L. Conversely, if x’y’ = xy, where x’, x E N and y’, y E L, then we have x-’x’ = yy’-’ E N n L, and hence if N n L = {e} we must have x’ = x and y’ = y.) If IC : G + G/N is the canonical homomorphism, the restriction of n to L is an isomorphism of L onto G/N; for the relation G = LN shows that R 1 L is surjective, and the relation N n L = {e} shows that it is injective. It should be remarked that, for a given normal subgroup N of a group G, there need not exist a subgroup L of G such that G is the semidirect product of N and L (Problem 1).

Suppose that G is the semidirect product of N and L. For each L, the mapping o,, : x ~ y x y - ’ is an automorphism of the group N. Moreover, for any two elements u, v of L we have a,, = ou a,, so that y~ a,, is a homomorphism of L into the group d ( N ) of automorphisms of N. Furthermore, this homomorphism and the laws of composition in N and L (19.14.2)

y

E

0

206


determine the law of composition in G, because for x, x‘ in N and y, y‘ in L we have

Conversely, let N and L be any two groups and y ~ o a, homomorphism of L into d ( N ) . Then we may define a group structure on the set S = N x L by the rule (19.14.3)

Associativity follows from the formulas ((x, Y W l y’))(x”,Y”) = ( x ~ , ( x ’ ) ~ , , ~ (YY’Y”), x”), ( x ,Y W , Y X X ” , Y”)) = (xa,(x’a,~(x”)), VY’Y”)~ and the relation a, uy, = a,.,,. If e‘ (resp. e”) is the neutral element of N (resp. L), it is clear that (e’, e”) is the neutral element of S. Finally, we have 0

( x , Y ) ( ~ y - l ( x - l ) , Y -=l )( o y - l ( x - l ) , Y - l ) ( x > Y=)(e‘, e”),

which completes the proof of our assertion. If N’ (resp. L’) is the set of all ( x , e“) for x E N (resp. (e‘,y ) for y E L), it is immediately seen that the group S is the semidirect product of N’ and L’, which are isomorphic respectively to N and L. We shall sometimes use the notation S = N x , L (the direct product of N and L corresponds to the trivial homomorphism a : y~ 1, of L into d ( N ) ) . If now G is the semidirect product of subgroups N and L, and if y~ oy is the corresponding homomorphism of L into d(N), then it is clear that the mapping ( x , y ) H xy of S = N x L into G is an isomorphism of S onto G, by virtue of (19.14.2.1) and (19.14.3.1). ~

(19.14.4) Suppose now that G is a connected Lie group, N a connected closed normal subgroup of G and L a connected Lie group immersed in G , and suppose that G is the semidirect product of N and L. The restriction to L of the canonical homomorphism II : G + G/N is continuous for the topology induced on L by G, and afortiori for the proper topology of L (since the latter is finer than the induced topology). It is therefore a Lie group homomorphism (19.10.2), and since it is bijective, it is an isomorphism of Lie groups (16.9.9). For each y E L, it is clear that oy : x ~ y x y - ’is a Lie group automorphism of N, i.e., is an element of Aut(N) c d ( N ) . Furthermore, y w o y is a Lie group homomorphism of L into Aut(N). For, by virtue of the definition of the Lie group structure of Aut(N) (19.13.5), if n, is the Lie algebra of N, the tangent linear mapping Te(oy) is the restriction to n, of the automorphism

14 SEMIDIRECT PRODUCTS OF LIE GROUPS

207

Ad(y) of 9,; and since ywAd(y) is a Lie group homomorphism of G into Aut(gJ, it follows that ywAd(y)In, is a Lie group homomorphism of L iat0 Aut(n,), which proves the assertion. Conversely, let N and L be two connected Lie groups and let a Lie group homomorphism of L into Aut(N). It follows from (1 9.1 3.5) that the mapping (x, y) I+ ay(x) is of class C“. The group structure defined on the product N x L by (19.14.3.1) is then compatible with the product manifold structure on N x L, and we denote by N x a L the Lie group so defined. (19.14.5)

y w u y be

(1 9.14.6) Every connected Lie group G which is the semidirect product of a connected closed normal subgroup N and a connected Lie group L immersed in G, is isomorphic (as a Lie group) to a Lie group of the form N x a L, where y w a y is a Lie group homomorphism of L into Aut(N). I n particular, L is necessarily closed in G .

For if cYis defined as in (19.14.4), we may construct the Lie group N x a L defined in (19.14.5) (L being taken with its Lie group structure). It is then clear that the mapping (x, y ) ~ + x of y N x a L into G is a bijective Lie group homomorphism, hence is an isomorphism (1 6.9.9). (19.14.7) In the Lie algebra g, of the semidirect product G = N x u L defined in (19.14.5), the Lie algebra it, of N is an ideal, and the Lie algebra I, of L is a subalgebra which (as vector space) is a supplement of n, . We have seen (19.14.4) that, for each y E L, the mapping T,(a,) = (ay)* is an automorphism of n,, i.e., is an element of Aut(n,), and the mapping y-(ay)* is the same as y ~ A d ( y ) ) n , Its . tangent linear mapping is therefore a Lie algebra homomorphism rp : I, + Der(n,), which is the restriction to I, of the homomorphism V H ad(v) 1 n, . We remark that knowledge of the homomorphism cp and of the Lie algebra structures of n, and I, completely determines the Lie algebra structure of 9,; for each w E g, can be uniquely written as u + v with u E n, and v E I,, and for two such elements w = u + v, w‘ = u’ + v’, we have (19.14.7.1)

[U

+ V, U’ + v’] = [u, u’] + [v, u’] + [u, v’] + [v, v’] = [u, u’] + rp(v) - u’ - rp(v’) u + [v, v’]. *

The algebra g, is called the semidirect product of 9,and I, corresponding to Q, and is denoted by n, x ,,I,.

208


(19.14.8) The construction of semidirect products will allow us to prove in Chapter XXI that every finite-dimensional Lie algebra over R is the Lie algebra of a Lie group, by virtue of the following result: (19.14.9) Let g be ajinite-dimensional Lie algebra over R, let n be an ideal in g, and let 1be a subalgebra of g supplementary to n. Suppose that there exists a simply-connected Lie group N (resp. L) such that Lie(N) (resp. Lie(L)) is isomorphic to n (resp. 1). Then there exists a simply-connected Lie group G whose Lie algebra is isomorphic to g and such that the manifold underlying G is difeomorphic to N x L.

For each v E I, the restriction of ad(v) to n is a derivation cp(v) of n, and the mapping V H ~ ( V is) a Lie algebra homomorphism of I into Der(n). Since L is simply-connected, there exists a unique Lie group homomorphism t,b : L + Aut(n) such that JI* = cp ((19.7.6) and (19.1 3.8)). Furthermore, because N is simply-connected, the mappingfwf, of Aut(N) into Aut(n) is an isomorphism of Lie groups (19.13.3); hence there exists a Lie group homomorphism y w a , of L into Aut(N) such that (ay)* = $ ( y ) for all y E L. If we now consider the semidirect product G = N x, L, it follows from (19.14.7) and the definition of y ~ a that , the Lie algebra ge of G is isomorphic to g. In view of (16.27.10), the proof is complete. This result gives in particular a partial answer to the problem raised in (1 9.14.8) : (19.14.10) Every solvable Lie algebra of dimension n over R is isomorphic to the Lie algebra of a simply-connected solvable Lie group, which is diffeomorphic

to R". The proof is by induction on n : the result is trivial for n = 1, by (19.4.1). If g is a solvable Lie algebra of dimension n > 1, then by definition the derived algebra g' = [g, g] is not equal to g, hence is of dimension = -Ad(f (x))

O

+ dxg, dxf.

It follows from (1 6.9.9) and the composite function theorem (1 6.5.4) that for all h, E T,(M) we have T,(f.)

-

- h,

=( u f )

. h,) - g(x) +m) - (T,(g).

-

hxj,

and since dx(fg) h, = (f(x)g(x))-’ * (Tx(fg) hx), the formula (19.1 5.4.1) follows, by virtue of the definition of Ad(s) (19.2.1). We then derive (19.15.4.2) by replacing g by f in (19.15.4.1) and using the fact that swAd(s) is a homomorphism. In particular, if we take f or g to be the constant mapping X H S of M into G, we obtain (19.15.4.3)

dX(S9) = dxg,

(19.15.4.4)

d , ( g ~ )= Ad(s-’)

0

d,g.

16 I N V A R I A N T DIFFERENTIAL FORMS A N D H A A R MEASURE

217

(19.15.5) Let J g be two mappings of class C' of M into a Lie group G. Then fg-' is locally constant ifand only i f d f = dg.

For to say that ,fg-' is locally constant is equivalent to saying that T,( f g - ' ) = 0 for all x E M (16.5.5), which in turn is equivalent to dx(f g - ' ) = 0. But, by virtue of (19.15.4), we have d J f g - 9 = Ad(g(x)) O (dxf - dxg), whence the result follows. (19.15.6) Let f be a mapping of class C' of M into a Lie group G, and let u : G -,G' be a homomorphism of Lie groups. Then for s E G and x E M we have

(d,u) * k, = u*(s-'

(19.15.6.1)

*

ks),

dx(u 0 f ) = U* 0 d,J

(1 9.15.6.2)

The formula (19.15.6.1) follows directly from the definitions and from (16.9.9). As to (19.15.6.2), we have, again from (16.9.9),

d,(u

o f

1

*

'

h, = ( d f(4))- * ( T f ( , ) ( 4 * ( T x ( f ) * h,)) = T A U )* ( f ( 4 - l * ( T J f * h,)) = u*(d,f (hx)).

16. INVARIANT DIFFERENTIAL FORMS A N D HAAR MEASURE ON A LIE GROUP

On a Lie group G, the canonical differential form o or 0, is by definition the (left) differential d( IG) of the identity mapping 1, : G + G: it is therefore a vector-cahred l-form with values in the Lie algebra ge. For each x E G we have, by (19.15.6.1), (19.16.1)

(19.1 6.1.1)

O(X)

*

h,

= X-'

*

h,.

Notice that the knowledge of this form determines the differential of any

C" mapping f : M + G of a differential manifold M into G. For the definition (19.15.2.1) can be rewritten in the form

dxf = N f (4) T x ( f 1, O

218

XIX


or equivalently (1 6.20.15.3), df

(19.16.1.2)

=

tf(~)).

(19.16.2) For each s E G , we have (19.16.2.1)

y(s)o = 0 ,

(1 9.1 6.2.2)

~ ( s ) w= Ad(s-')

0

B).

For by definition ((19.1.2) and (19.2)) we have, for each x E G, (6(s-')o)(x) = o(xs-1) * s.

(y(s)o)(x) = s ' o(s-lx),

But also, from the same definitions. (S

o(s-~x)) * h, = B)(s-'x) * ( s - ~ * h,) = (x-'s) = X-' h, = W(X) h,

-

*

( s - ~* h,)

because G acts here trivially on ge; and likewise

-

(~(xs-')

S)

*

h, = B)(xs-~)* (h, = s (o(x)

(19.16.3)

*

8-l)

= ( s x - ~ )* (h,

*

s-l)

h,) . S - ' = Ad(s) * ( ~ ( x* )h,).

Let (eJISiSnbe a basis of the Lie algebra ge. Then we can write

(19.16.3.1)

i=l

and since m(x) is a bijection of T,(G) onto ge = T,(G), the (scalar-valued) differential 1-formswi are linearly independent. Since they are left-invariant by (1 9.16.2), they form a basis over R of the vector space of left-invariant di$ ferential I-forms on G , since T,(G)* is of dimension n (19.2.2). It follows (19.2.2) that, for each p such that 1 5 p S n, the differential p-forms on G (19.16.3.2)

mil A oil A

A

mi,

(il < i,

0 such that, whenever llSll< r < r, the family of matrices and

where n takes all integer values 2 1, and for each n the pairs ( p c , qc)(1 I ,i 5 n) take all values in N x N except (0,O), is absolurely summable (5.3.3). Likewise the series

- 5 -nI (1 "=I

is absolutely convergent, and its sum is equal to that of the family (*), and to the matrix log((exp S)(exp T)). (Reduce to (9.2.1) by majorizing the norms of the elements of the family (*).) 5. Let G be a Lie group, g, its Lie algebra, endowed with a norm compatible with its

topology. The function

W ,u, v) = log(Ad(expo(u))Ad(exPG(tV))) with values in the Banach algebra End(n,) (the logarithm being that of GL(0,)) is defined for t E [0, I ] and for u, v in a sufficiently small neighborhood U of 0 in 0. Show that for such values of 1, u, v, the family of elements of oe (**)

(-I)"+' -. n(n

+ 1)

+

((ad u)''(ad rvIq1(adu P ( a d w ) ' ~- - - ( a du)'"). v p , ! ".p.!yI!

(where n 2 1, pi qc > 0 for 1 that its sum is equal to

**.qn-l!

i 6n - 1, and pn > 0) is absolutely summable, and

(Use (19.16.5) and Problem 4.) Deduce that for u, v E U, the family

(***I

((ad u)''(ad v)ql(ad u)P2(adv ) " ~ * * (ad u)'~)* v (-I)"+ I -. n(n+ 1) pl!...pn!ql!...qn-l!(ql + . * * + q . - 1 + 1)

is absolutely summable and that its sum is log&xpG(4 exp&))

- (u + v)

(Campbell-Hausdorffformula). Hence give another proof of the result of Problem I(b). 6. (a) Let A, B, C be three connected Lie groups immersed in a Lie group G , and let a,, be, c, be their Lie algebras, which are subalgebras of op= Lie(G). Assume that [a,, c,] c re and that [be. c.] c c,. Show that if [a,, be] c c,, then (A, B) c C. (By considering the Lie subalgebra a. +be c, of n., reduce to the case where c, is an ideal in a,, and then use the Campbell-Hausdorff formula (Problem 3.)If [a,, 6,] = c,, show that (A, B)= C (cf. Section 19.12, Problem 6).

+

16 INVARIANT DIFFERENTIAL FORMS AflD HAAR MEAWE

223

(b) Deduce from (a) that if G is a connected Lie group, oe its Lie algebra, then the groups W ( G )of the descending central series of G, defined by W(G) = G, W(G) = (G, ‘eP-’(G)), are connected Lie groups immersed in G, whose Lie algebras are the Q’(Oe).

7. (a) With the hypotheses of Problem 8(b) of Section 19.14, show that the Lie subalgebra generated by the u k such that p 2 k 5 n has a multiplication table relative to this basis in which the structure constants are rutional numbers. (Show that this is the case for the subalgebra in, generated by the Uk such that k z j , by descending induction on j ; use Problems 8(a) and 6(a) of Sectionl9.14, togetherwiththeCampbel1Hausdorff formula (Problem 9.) (b) Conversely, suppose that the Lie algebra oe of a simply-connected nilpotent Lie group G has a basis relative to which the structure constants are rational numbers. Show that there exists a discrete subgroup D of G such that the homogeneous space G/D is compact. (Show first that there exists a basis ( u ~ of )oe such ~ that, ~ ~if ~ a, is the subspace of oe spanned by the ukfor which k 2 j , then the a, satisfy condition (v) of Section 19.12, Problem 3, and the structure constants relative to the basis (uk) are rational. Next observe that if

exp(tlu, + 5 1 u z

+ * * a +

5n~.)=exp(S~U~)exp(S2uz)...exp(bu3,

then the SJ are polynomials in f , , . ..,f n with rational coefficients. Complete the proof by induction on n.) (c) With the hypotheses of (b) above, let Do be a discrete subgroup of G such that, in the description of Do given in Section 19.14, Problem 8(b), the structure constaRts with respect to the basis (u,) of oe are rational. Show that there exists an increasing sequence (D,) of discrete subgroups of G containing D o , such that D, is dense in G.

u m

(With the notation of Section 19.14, Problem 8(b), show that there exist arbitrarily large integers N with the property that the center Z of G contains an element b, such that bN, = an;then proceed by induction on dim(G), by considering G/Z.) 8. (a) Let Q he a vector space of dimension 7 over R, and let (elll sll, Show that the formulas

[el, e,] = al,el+, [e,, ell = 0

(1

he a basis of 0.

5 i < j 6 7, i + j $ 7 ) ,

(i+j>7)

define a Lie algebra structure on 0, provided that the real numbers conditions a 2 3 als

alz a34 - a24 a16

all

satisfy the

- a13a 2 4 = 0,

+ a14

a25 =

0.

The Lie algebra 0 so defined is nilpotent, and QJ(0) is spanned by the el such that i z j + 1 for j = 2, . . ., 6; also, the centralizer b of Q5(o) is spanned by ez, . . ., s,. Deduce that if another basis ( e h 1 5 7 of o is such that QJ(o)is spanned by thee: such that i j 1 for j = 2, , . . ,6, and such that 5 is spanned by e: , . . . ,e; ,then we must have

+

Ie;, e;l

= ate;+,

+ uiJ,

~

224

XIX


where u;, is a linear combination of the e; with indices k 2 i + j we have

+ 1. Furthermore,

cl;4a:5a:~1cl~:1 = a14a25cli~a;~.

(b) Deduce from (a) that for a suitable choice of the a l , E R there exists no basis

of a for which the structure constants are rational. If G is the simply-connected nilpotent group having a as Lie algebra, there exists no discrete subgroup D of G such that the homogeneous space G/D is compact. (c) Let a be a real vector space of dimension 8, (e,), I a basis of 0. Show that the formulas = e3 [el,

9

[el, e31 = e 4 ,

[el, e71 =

[el,

[ez, % I = 2e~1, [ % , %I = [ e l ,e,l= 0 for i + j > 8

e4I = eS

= es,

[ez,

-F e8 ,

eSl = e 6 r

9

[ez, e4l=

e6,

[ez, e s l =

e7,

[e3,e5l=-ea3

define a Lie algebra structure on 0. The Lie algebra a so defined is nilpotent. By considering the the ideal [a2((]),(s2(0)] and the transporter of cZ2(0) into a4(n), show that if (6,) is any other basis of 0 for which the structure constants are rational, then there exists a third basis (el) with the same property which is derived from ( G I ) by a transition matrix with rational entries, and is such that el = pi el u;, where u; is a linear combination of the ek for which k > i, and the constants pl are rational. (d) Let be a simply-connected nilpotent Lie group whose Lie algebra is isomorphic to the algebra described in (c) above. Show that there exists a discrete subgroup N of the center of e such that the connected nilpotent group G = e / N contains no discrete subgroup D such that G/D is compact.

e

D’)be a discrete subgroup of G (resp. G ’ ) ; suppose that G / D is compact. Show that every homomorphism f:D .+ D’ has a unique extension to a homomorphism g : G -+ G’. (Use the description of D given in Section 19.14, Problem 8 in terms of a basis (uk)ljks. of 0.; we may writef(expc(uk)) = exp,.(u;) for each k, where u; E a:. is well determined. To show that the linear mapping F : (ie 0:. defined by

9. Let G, G’ be two simply-connected nilpotent Lie groups, and let D (resp.

-+

F(Uk) = U;

(I

5 k 6 /I),

is a homomorphism of Lie algebras, proceed by induction on the dimension of G, by introducing the subalgebras 111, of pe (Section 19.14, Problem 8). Use the CampbellHausdorff formula, Section 19.14, Problem 6(a), and the formulas (19.1 1.2.2) and (19.11.2.3)J If f is surjective (resp. bijective) and if G’/D’ is compact, then g is surjective (resp. bijective). 10. Let D be a countable discrete nilpotent group which admits a sequence

D=D1 IDz=’D3

> * *D *nI 2 Dn+l={P}

of normal subgroups such that D,/D,+ E Z for 1 5 j 5 n, and (D, D,) c D,+ Show that there exists a simply-connected nilpotent Lie group G such that D c G and G/D is compact. (Induction on n: if Gz is a simply-connected nilpotent Lie

16 INVARIANT DIFFERENTIAL FORMS AND HAAR MEASURE

225

group containing D2and such that GJD2is compact, and if d e D is such that the image of d in D/Dzgenerates D/D2, consider the automorphism z-dzd-' of Dz, and use Problem 9.) 11. Let G be a connected Lie group, H a connected Lie group immersed in G , and let oe. 11, be the Lie algebras of G, H respectively.

(a) Let s E G. Then sts- It-

E

9 ( H ) for all t E H if and only if

Ad(s) * u - u E T(be) = [be, 61, for all u E 6,. (Use the Campbell-Hausdorff formula t o prove that the condition is sufficient.) (b) Assume that H is dense in G . Show that g ( H ) = g ( G ) . (Use (a) to show that ( G , H ) c O(H), which implies that [oo, li,] c [[I,, be]; then show analogously that (G, G ) c ./f?(H).)

+

12. Show that if o is the canonical differential on a Lie group G, then d o [w, w] (Muurcr-Cur/uii c4ynarion). (For each left-invariant scalar I-form w , evaluate (w,

=0

X" 1, X">

for any two elements u, v E 0 9 , by using (17.15.3.6).) Consequently, if (el)lsljn is lie,and if we put

a basis of

then we have (1)

13. Let a be a left-invariant C' diflerentialp-form on a connected Lie group G. Show that tc is right-invariant as well if and only if du = 0. (Observe that the condition of right-

invariance, which is A ('Ad(s)) . a(e) = a(e)for all s E G , is equivalent to the condition

for all choices of p + I vectors u, u l r . . . , up in 11,; then use (17.15.3.5) for left-invariant vector fields.) In particular, the dimension of the vector space of I-forms on G which are both left- and right-invariant is equal to dim o, - dim[ii,, bieI. 14. For a connected Lie group G to be unimodular, it is necessary and sufficient that Tr(ad(u)) = 0 for all u en,. In particular. every connected nilpotent Lie group is

unimodular 15. Let G be a connected Lie group and H a connected closed subgroup of G : let I],, be be the Lie algebras of G and H, and let $1: be the Lie algebra of the closed subgroup 9 ( G ) of G. Show that there exists on G/H a nonzero relatively G-invariant measure (Section 14.4, Problem 2) if and only if the kernel of the linear form u->Tr(ad,(u)) on 11, contains 0: n1 1 , .

226

XIX

LIE GROUPS AND LtE ALGEBRAS

16. Let G be a connected Lie group of dimension n, let H be a connected closed subgroup of G of dimension n - p . and let ( w I ) ,S I S n be a basis of the space of left-invariant

differential I-forms on G (19.16.3.1). Assume that the basis ( e l ) l S l Sofn ge has been chosen so that the inverse images ' j ( w k )of the wk,k 6 p, under the canonical injection j : H + G , are zero. If there exists a C" differential q-form fl on G/H, invariant under the action of G, then the q-form a = 'dfl on G (where : G -+ G/H is the canonical submersion) is a linear combination with constant coefficients of exterior products of P

the I-forms w , , . . . , u p q, at a time, and furthermore we must have A('Ad(s)) a@)= a(e) for all s E H. Show that this condition is equivalent to R",E) be the disjoint union of the sets Aff(R", Eb)for all b E B and let a : A+B be the mapping which sends each element of Aff(R", Eb)to b. Next, let A(n, R)

+

be the Lie group of affine-linear bijections of R" onto R", identified with the subgroup of GL(n 1, R) consisting of the matrices

+

where U E CL(n, R) and a E R" (column-vector), the action of A(n, R) on R" being

The group A(n, R) (called the afine group of R") acts on A on the right by the rule S)HU s. Show that there exists on A a unique structure of principal bundle with base B, projection a, and group A(n, R) (cf. Section 16.14, Problem 1). The set A, endowed with this structure, is called the bundle of afine frames of the vector bundle E. If we put G = A(n, R), the left action of G on R" defines a bundle of fiber-type R" associated with A, namely A x R". Show that the fibration so defined is canonically B-isomorphic to that of E.

(u,

2.

0

Let R be a principal fiber bundle with base B, group H and projection n. Suppose that H is a connecfed Lie subgroup of a Lie group G. Then H (as a subgroup of G ) acts by left translation on the homogeneous space G/H, and we may therefore form the associated bundle R x (G/H) = X with base B and fibers diffeomorphic to G/H. Let P be the canonical image in G/Hof the identity element e of G. Show that the mapping r-r * P (16.14.7) factorizes into r H n ( r ) A r . P, where u is a C" section of X over B, called the canonical section; B' = u(B) is therefore a submanifold of X diffeomorphic to B. Let V(B) be the vector bundle over B' induced on the submanifold B' by the vector bundle V(X) of vertical tangent vectors to X (16.12.1), which is a subbundle of T(X). Show that V(B') is B-isomorphic to the associated vector bundle R x (&/be), where ac and be are the Lie algebras of G and H, respectively, and t E H acts on the left on a,/b, by means of the tangent mapping at the point P of the mapping S Ht 9 (Section 19.7, Problem 7). Suppose now that dim(9) = dim(C/H). A B'-isomorphism of vector bundles of T(B') onto V(B') is called a welding of B and X (so that if such a welding exists, then the vector bundle T(B) is isomorphic to V(B'), and consequently T(B) may be considered as the vector bundle associated with a principal bundle whose group H may be larger than the group CL(n, R)(where n = dim(B))). Suppose that there exists a welding of B and X, and let (u-',a) be an isomorphism of V(B') onto T(B).Identify e./b, with R" by choosing a basis, and for each t E H let p ( t ) be the image oft in CL(n, R), which is a (linear) operation off on g e / b r , Let R =p(H),

2 PRINCIPAL CONNECTIONS O N PRINCIPAL BUNDLES

239

which is a Lie group immersed in CL(n, R), isomorphic to a quotient of H. We may therefore form the principal bundle 61 x " R,where H acts on fi on the left by means ofp. For each element r; p ( r ) of this principal bundle, the mapping x--cr(r; (p(r) x ) ) is an isomorphism u(rL .p(t)) : x-rb * x of R" onto T,(B), i.e., is an element of the fiber R(B), of the bundle of frames R(B) of B. Show that in this way we obtain an injective B-morphism (u, j ) of the principal bundle R x " fi into R(B), where j is the canonical injection of fi into GL(n, R). Conversely, such an injective B-morphism defines a well-determinded welding of B and X.

-

3. Let M be a pure differential manifold of dimension n.

(a) Show that the bundle of tangent frames R(M) of M may be canonically identified with the open submanifold of the manifold Jb(R", M) of jets of order 1 of R" into M with source 0, consisting of the inuerrible jets (Section 16.5, Problem 9). For each k 2 1, the set Rk(M) of invertiblejets of order k of R"into M, with source 0, is likewise an open submanifold of Jt(R", M) called the manifold offrames of order k of M. (b) Let Gk(n)be the group of invertible jets of order k of R" into R",with source and target 0 (Section 16.9, Problem I): it acts on the right on Rr(M) by the rule (u, S ) - U s. Show that, relative to this action, Rk(M) is a principal bundle over M with group Gk(n). (c) For h < k, the restriction to R,(M) of the canonical mapping J:(R", M) +.J$(R", M) is a submersion phk : Rk(M)-+ Rh(M). The canonical mapping phk :Gk(n)+.Gh(n)is a surjective homomorphism of Lie groups, whose kernel Nhk(n)is nilpotent and simplyconnected (and commutative if h = k - I ) ; the pair (phk,phk)is a morphism of principal bundles (16.14.3), and &(M) is a principal bundle over Rh(M), with group Nhk(n) and projection phk. (d) Let M' be a pure differential manifold. Show that the fibration (Jk(M, M'), M, n) defined in Section 16.12, Problem 6 is isomorphic to the bundle associated to Rk(M), of fiber-type J!(R", M'), on which G'(n) acts on the left by the rule (s, j ) H j s-'. Show likewise how to define the fibrations 0

0

(Jk(M, M'), M', n') and (Jk(M,M'), M x M', (n,n')) as associated fiber bundles.

2. PRINCIPAL CONNECTIONS ON PRINCIPAL BUNDLES

(20.2.1) We return to the considerations of (17.16.1) which led us to the notion of a connection in a vector bundle E of rank p over a base-space B. We shall restrict our attention to the case envisaged in (17.16.1), where B is an open subset U of R" and E is the trivial bundle U x RP.The corresponding bundle of frames is then the trivial bundle U x GL(p, R). The mapping hHF(h) considered in (17.16.1) is by hypothesis a mapping of a neighborhood V of 0 in R",into GL(p, R)(if V is taken sufficiently small), and for each h E z, F(h) defines a mapping (b, r ) H ( b + h, F(h)-' o r ) of the fiber Isom(RP, Eb) of Isom(U x RP,E) at the point b into the fiber Isom(RP, Eb+J

240

XX PRINCIPAL CONNECTIONS AND RIEMAFNIAN GEOMETRY

of this same bundle at the point b + h. Hence we obtain an indefinitely differentiable mapping (h, r ) H ( b + h, F(h)-' r) of V x GL(p, R) into 0

u x CUP, R) (where GL(p, R) is considered as an open subset of End(RP) = Rp*).Its derivative at the point (0, r ) i s therefore (k, v ) (k,~v - (DF(0) - k) r), a linear mapping of R" x End(RP)into itself. Since the tangent space 0

T(b,r)(Isom(B RP, E)) may be identified with Tb(u) x T,(End(RP)), the vectors (k, v) and (k, v - (DF(0) * k) o r )

should be considered as vectors in this tangent space, and hence alsothevector (k, -(DF(O) k) 0 r ) = f,(k, r).

On the other hand, (k, r ) should be regarded as an element of the fiber over the point b of the fiber-product T(B) x Isom(B x RP, E) over B. Replacing the bundle of frames by an arbitrary principal bundle over B, we are led to make the following general and intrinsic definition: (20.2.2) Given aprincipal bundle R with base B, projection A and group G, a connection (or principal connection) in R is defined to be a C" mapping f :T(B)

(20.2.2.1)

XB

R+T(R),

with the following properties (where b E B, k, E T,(B), r, E Rb): (20.2m2*2) T(A)(fb(kb rb)) = kb 9

Y

OR(fb(kb

rb))

= rb;

(20.2.2.3)

kbH fb(kb,rb) is a linear mapping of T,(B) into T,,(R);

(20.2.2.4)

for each S E G We have fb(kb, rb * S) = fb(kb,

rb) ' S

(where on the right-hand side, WH w * s is the tangent linear mapping of r H r * s at the point r b (16.10)). The vector fb(kb,rb) is called the horizontal lijting of the vector kbat the point rb. If Xis a vector field on B,then rb H Pb(X(b), rb) is a vector field on R, called the horizontal lifting of X . It follows from (20.2.2.2) that the linear mapping k,wf,(k,,, rb) is injective and that its image H,, is supplementary in the tangent space T,,(R) to the kernel G,, of T,,(x), which we have called the space of vertical rangent vectors (16.12.1) at the point r b E R,. The space H,,is called the space of horizontal tangent vectors at the point rb, relative to the connection f . We have Hrb, = H,, s for all s E G.


241

If R = B x G is trivial and B is an open set in R" (so that T(B) is identified with B x R"), we may write Pb((b,

k), (b, s))

= ((b, k), P(b, s)

k,

*

for (b,s) E B x G and k E R", where P(b, s) is a homomorphism of the vector space R" into T,(G) (we have identified T,,,,,(B x G ) with (6) x R" x T,(G)). For s, t E G , we derive from (20.2.2.4) the relation P(b, s t ) * k = (P(b,S) * k) . 1

(20.2.2.5)

(in the notation of (1 6.10)), whence, in particular,

P(b, S) * k = (Q(b)* k) * s,

(20.2.2.6)

where we have put Q(b) = P ( b , e), an element of Hom(R", ge). Conversely, given any C" mapping Q of B into Hom(R", ge), we may define P(b, s) by means of (20.2.2.6), then Pb((b, k), (6, s)) as above, and obtain a principal connection in B x G. (20.2.3) For each rb E R,, there is a projection p,, : Trb(R)+ T,,(R) with image H,, and kernel G,, . For each tangent vector h,, E T,,(R), we have (20.2.3.1)

= 'b(T(')

prb(hrb)

' hrb 9

'b),

which is called the horizontal component of h,, , and is the horizontal l i j k g of the projection T(n) * h,, of h,, in T,(B). It is clear that for all S E G we have (20.2.3.2)

Pr,(hr,

*

s) = (Pr,(hr,N

*

s

with the notation of (16.10). We remark that a horizontal vector h E Hr, may always be considered as the value at the point rb of a horizontal vectorjeld Y of class C"'.For there exists a C" vector field Yo on R such that YO(rb)= h (16.12.11), and we may define Y(r') to be equal to p,.( Yo('')) for all r' E R. The vector hrb

- p r b ( h r b ) = hrb

- Pb(T(')

* hrb

9

'b)

is therefore vertical. Observe now that, for each r, E R,, the mapping SH r, * s is a diffeomorphism of G onto Rb, and consequently t,, : u H r, * u

(20.2.3.3)

(in the notation of (1 6.10)) is a linear bijection of the Lie algebra ge = T,(G) onto the space of vertical tangent vectors G,, at the point r, . The mapping (20.2.3.4)

d r b )

: hrb

tE

l(hrb

- Pb(T(n) *

hrb 7

rbb))

242

XX PRINCIPAL CONNECTIONS AND RIEMANNIAN GEOMETRY

is therefore a surjective linear mapping of T,,(R) onto the Lie algebra ge: in other words, we have in this way defined a vector-valued differential l-form o on R, with values in ge (16.20.15). It is immediately verified that o is of class C": for we may assume that R = B x G is trivial, so that t;', for rb = (b, s ) E B x G, is the composition of h p s - ' h, and the second projection T(b,,)(B x G) + T,(G), from which the assertion follows immediately. The 1-form o is called the differential l-form of the connection P, or the connectionform of P. If R = B x G is trivial and B is an open set in R", the mapping o(6, e) may be identified (with the notation of (20.2.2)) with the projection (20.2.3.5)

Wb:

(k,U)HU - Q(b) * k

of R" x ge onto ge, the mapping b H wb of B into End(R" x ge) being of class C", and o(6, s) is identified with the linear mapping (20.2.3.6) ((6, k), (b, s * U))HAd(s-') . (U - Q(b) * k) = Ad(s-')

*

(wb (k, u)). 0

(20.2.4) A differential q-form a on R , with values in a finite-dimensional vector space V, is said to be vertical (resp. horizontal) if

-

a(rb) (h, A b, A * A hq) = 0 whenever one of the tangent vectors hj E T,,(R) is horizontal (resp. vertical). Notice that the definition of a horizontal q-form does not depend on the presence of a connection in R, whereas the notion of a vertical q-form is meaningless except with respect to a connection. It is clear that the 1-form o of the connection is vertical, and that the 2-form [a,0 1 (16.20.15.8) is also vertical. (20.2.5) In order that a C" vector-valued 1-form o on R, with values in ger should be the l-form of a connection in R, it is necessary and sufficient that it satisfy the following two conditions:

(1) a,considered as a mapping of T(R) into ge, is invariant (1 9.1.4) under the right action of G on T(R) induced (1 9.1.4) by the action of G on R, and the right action (u, s)t+Ad(s-') * u of G on ge (19.2.1): in other words, for h E T,,(R) and s E G we have (20.2.5.1)

w(rb * S) * (h * S)

= Ad@-')

*

(o(rb)* h).

(2) For all u E ge, if 2, is the vertical field which is the transport of Xu by the action of G (the Killingjield) (19.3.7), we have (20.2.5.2)

w(rb)

' zu(rb)

=


243

for all rb E R. (In other words, the value of w(rb) on the oertical tangent vectors in T,,(R) is determined independently of the connection P in R.)

Observe first that by definition zu(rb) = rb . u = t,,(u), hence the property (20.2.5.2) of the I-form w of the connection P follows immediately from the definition (20.2.3.1). Next, since p,,.,(h . s) = (p,,(h)) * s, we have h . s - p r b * S ( h * 3) = (h -prb(h)) * S, and (rb s) . u = (r,, (s u . s-I)) * s (16.10.1), or equivalently trb.s(u)= (t,,(Ad(s) u)) s, whence the formula (20.2.5.1) is a consequence of the definition (20.2.3.4). Conversely, suppose that the I-form o satisfies the conditions of the proposition. Then (20.2.5.2) implies that h H trb(o(rb)* h) is a projection of T,,(R) onto G,, . If we put p,,(h) = h - trb(O(rb)* h), then prb is a projection of T,,(R) with kernel G,, , and the image H,, of prb is therefore supplementary to G,, . It follows that the restriction of T,,(n) to H,, is a bijection of H,, onto T,(B). If we denote the inverse bijection by k b H P b ( k b ,rb), then P is a mapping of T(B) x R into T(R) which satisfies the conditions (20.2.2.2) and (20.2.2.3). From (20.2.5.1) and the relation trb.,(u) = (t,,(Ad(s) * u)) s it follows that p,,.,(h * s) = (prb(h)). s, and therefore H,; s = H r b . s ;and since T,,.,(II) (h * s) = Trb(n) h for h E T,,(R), it is clear that P also satisfies (20.2.2.4). Finally, P is of class C". For we may assume that R = B x G is trivial and that B is an open set in R";and then, with the notation of (20.2.2) and (20.2.3), we have Q(b) * k = -q, (k, 0), and if b H m b is of class C", the same is true of b ~ Q ( b )Hence . P is a principal connection, and o is its connection form.

-

-

PROBLEMS

If ( P , ) is a finite family of principal connections in a principal bundle R with base B, and (fi) is a finite family of real-valued C" functions defined on B, such t h a t x f i = 1, then the mapping z f , P , of T(B) x I

J

R into T(R) is a principal connection. Deduce

from this that every principal bundle carries a principal connection. (a) Let R, R be two principal bundles over the same base B, with groups G, G', respectively, and let (u, p) be a morphism of R into R corresponding to the identity mapping of B (16.14.3). Show that if P is a principal connection in R, then there exists a unique principal connection P' in R such that p;(kb u(rb)) = Trb(u) * Pb(kb 9

for all b E B, k, E Tb(B), r b E Rb.

9

rb)

244

XX

PRINCIPAL CONNECTIONS AND RIEMANNIAN GEOMETRY

If w, w' are the I-forms of the connections P, P', respectively, then p*(w(rb) ' hrb)

= w'(u(rb)) *

(Trb(u) ' hrb)

with the notation of (20.2.3). (b) Let R be a principal bundle with base B and group G, and let R = B' x R be its inverse image under a mappingf: B'+ B of class C" (16.14.6). Show that if P is a principal connection in R, there exists a unique principal connection P' in R such that, if u : R' R is the morphism corresponding to we have T,J(~,)(U)' (PL*(kb',r m , ) ) ) =Pf(bdTb*(f) ' kb,, r f w ) ) . If w is the 1-form of the connection P, then the 1-form of P' is 'u(w). The connection P' is called the inverse image of P by u. (c) Define the notion of a product of two principal connections in principal bundles R, R with bases B, B' and groups G, G', respectively, as a principal connection in the product of the principal bundles R, R (Section 16.14, Problem 7). 3. Let P be a principal connection in a principal bundle X with structure group G, base B and projection T . Show that for each unending path v : R B of class C" in B, and each point x E n-'(u(O)),there exists a unique unending path w, : R X of class C", called the horizontal lifing of u, such that w,(O) = x , T 0 w, = v, and such that the tangent vector w:(r) is horizontal for all r E R. (Using the fact that the inverse image bundle 'u(X) is trivializable (Section 16.26, Problem 71, show that there exists an unending path u : R +X of class C" such that u(0) = x and which is a lifting of u. Then write w A r ) = u ( t ) . f ( r ) , wheref(t)E G, and use Section 19.16, Problem 19.) Show that for each s E G we have w, .&) = w , ( r ) . s. The mapping X - w,(t), for a given t E R, is called the parallel displacement of the fiber XCto,onto the fiber Xu(,) along the path v. For each unending path u of class C" in X such that u(0) = x and which is a lifting of u, the path r-g(t) in G, where g(r) is the element of G such that u(f) = w,(r). g ( t ) in G, is called the development in G of the path u. With this notation, show that w(u(r))* u'(r) = - g ' ( r ) . & ) - I (in the notation of (18.1.2)), where w is the I-form of the connection P. -+

-+

4. Let R be a principal bundle with base B, group G and projection T . If dim(B) = n, show that for a C" field of n-directions r +H, on R to consist of the spaces of horG zontal tangent vectors of a principal connection in R, it is necessary and sufficient that H, . s = H, , for all r E R and s E G, and that T(T)(H,) = Tn(,)(B).

3. COVARIANT EXTERIOR DIFFERENTIATION ATTACHED TO A PRINCIPAL C O N N E C T I O N . CURVATURE FORM OF A PRINCIPAL CONNECTION

(20.3.1) Let P be a principal connection in a principal bundle R with base B, projection 3c and group G. We retain the notion of (20.2.3).Given a vectorvalued differential q-form a on R, with values in a vector space V, for each rb E R the mapping (hl,h.2*

hq)Ha(rb)

*

( p r b ( h l ) A pr,(h,?)

A

' * *

A prb(hq))

3 COVARIANT EXTERIOR DIFFERENTIATION

245

of (Tr,(R))4into V is q-linear and alternating. hence can be written in the form (hl, ..., hq)t+ap(rb).(hlA h, A * . . A hq), where a p is a differential 9-form on R with values in V, which is evidently Itorizontal (20.2.4). Moreover, if a is invariant (for the right action of G on

A T(R) and a right action of G on V), then the same is true of a p ,because 9

plb(hl * S) A * * * A P r b ( h q * S> = (prb(hi)A * A prb(hq)) * S. It is immediately checked (by reduction to the case of a trivial principal bundle) that if a is of class c' then so also is a p .

(20.3.2)

form

With the notation of (20.3.1), the horizontal differential (9 + 1)Da = (da),

(20.3.2.1)

(also denoted by D,a) is called the covariant exterior diferential of the vectorvalued 9-form a (of class C ' ) , relative to the principal connection P. It follows from (20.3.1) that if a is invariant then so also is Da, because da is invariant (19.1.9.5). If the q-form a is vertical, and if q 2 2, then Da = 0. For it follows from the formula (17.15.3.5) that the value of d a ( r , ) . (prb(ho)A A prb(hq))is a sum of terms of the form a(rb)* (k, A * . * A kq),where 9 - 1 of the vectors kj are horizontal, and of terms which are values of Lie derivatives of the form a p ,which is zero by hypothesis. The curvature form of the connection P is defined to be the C" vectorvalued 2-form with values in ge which is the covariant exterior differential of the 1-form o of the connection:

rz = D o ,

(20.3.2.2)

which is therefore horizontal and invariant (for the right action (u, s) w Ad(s-')

.u

of G on gp). The I-jiorni o of a principal connection in a principal bundle R satisfies the " structure equation" (20.3.3)

(20.3.3 .I)

dm

=

-[a,a]+ D o

=

-[w, o]

+ a.

(In other words, for all tangent vectors h, k at a point rb E R. we have (20.3.3.2) do(rb). (h

A

k) = - [ o ( r b ) . h, o ( r b ) .k]

+ Do(rb) - ( h A k).)


246

Since both sides of (20.3.3.2) are bilinear in h, k, it is enough to verify this formula in the following three cases: (i) h and k are horizontal. Since the form w is vertical, the relation (20.3.3.2) then reduces to the definition of Do (20.3.1). (ii) h and k are vertical. We have then h = tJu), k = tlb(v), where u, v E gerand dw(rb)* (h A k) is the value at the point r, of the function dw (Z, A Z,). Now we have (1 7.15.8.1)

-

But the function w . Z, with values in ge is constant on R by virtue of the condition (20.2.5.2) characterizing a connection form. The right-hand side of (20.3.3.3) therefore reduces to - w * [Z,, Z,]. Now we have [Z,, Z,] = ZLu, (19.3.7.4), and therefore the value of -o [Z, ,Z,] at the point r, is indeed equal to the right-hand side of (20.3.3.2), bearing in mind (20.2.5.2) and the facts that Do is horizontal and h, k vertical. (iii) h is vertical, k is horizontal. The vector h is again the value at the point r, of a vertical vector field of the form 2, , where u E ge. On the other hand, we have seen (20.2.3) that there exists a horizontal vector field Y on R which takes the value k at the point r, . The formula (17.15.8.1) now gives

,

Since o is vertical, we have w * Y = O . Also, as in (ii) above, we see that w . Z, is constant, and therefore the right-hand side of (20.3.3.4) reduces to - w * [Z,, Y ] . Since h is vertical and Dw horizontal, the right-hand side of (20.3.3.2) is zero; hence we are reduced to proving the following lemma; (20.3.3.5) For each u E ge and each horizontal vectorfield Y o n R, the vector field [ Z , , Y ] on R is horizontal.

It follows from the interpretation of the Lie bracket of two vector fields given in (18.2.14), and from the fact that the integral curves of the field Z, are given by t w r exp(tu), that the value of [Z, , Y ] at a point r E R is the limit, as t 0, of the tangent vector at the point r (20.3.3.6)

1

- ( Y(r * exp(tu)) exp( - tu) - Y(r)). t

By virtue of (20.2.3.2), this vector is horizontal, and therefore so also is its limit.

3 COVARIANT EXTERIOR DIFFERENTIATION

247

When the base B is reduced t o a point, so that R may be identified with the group G, there is only one connection on R, and the definition (20.2.3.4) shows that the form o is the canonical differential form defined in (19.1 6.1). Hence in this case we have (20.3.3.7)

do

+ [a,03 = 0

(Maurer-Cartan equation). (20.3.4) The curvature form identity).

satisjes the relation D n = O (Bianchi's

By virtue of the structure equation we have

D n = D ( ~ w+) D ( [ o ,a]). By definition (20.3.2. I), D(do) = 0 because d(do) = 0; and since the 2-form [o, 01 is vertical, we have D ( [ o , 01) = 0 by (20.3.2).

PROBLEMS

1. With the notation of (20.3.1), let p : G +GL(V) be the homomorphism such that (s, v)-p(s-') v is the right action of G on V under consideration. For each differential q-form a on R, with values in V, we denote by w A a the differential (q 1)-

+

form on R, with values in V,whose value is given by the formula (0 A p a ) ( r b ) * ( h lA

h2 A

... A h,+d

where u is the antisymmetrization operator (A.12.2). Show that if a is horizontal and invariant, then we have d a = - w A p a + Da.

+

(Consider, as in (20.3.3), three cases: where all the hJ (1 s / s q 1) are horizontal, where at least two of them are vertical, and where only one is vertical. In the last case, we may suppose that h, is the value of a Killing field Z , , and that the hJ ( j 2 2) are horizontal and are the values of G-invariant fields; use (19.4.4.3) and (19.8.11).) Deduce that D ( D a ) = a A, a , the right-hand side being defined by antisymmetrization as above. 2.

In the situation of Section 20.2, Problem 2(a), show that if curvature forms of P, P', then we have

a,!2' are the respective

p,,(a(rd ' ( h A k)) = a'(U(rb)) * ((Trb(U) * h) A ( T r b ( U ) ' k)). In the situation of Section 20.2, Problem 2(b), show that a'= 'u(n).

248

XX PRINCIPAL CONNECTIONS A N D RIEMANNIAN GEOMETRY

3. Let R be a principal bundle with base B, group G and projection T.Let K be a Lie group acting on R on the lefr, such that t * ( r . s ) = ( t * r) s for s E G and t E K. Then for each t E K the mapping r - f . r is an automorphisrn of the principal bundle R. Let ro be a point of R, and let S denote the stabilizer of T(ro).For each t E S, there exists a unique element p(t) E G such that f * ro = ro . p ( t ) , and p is a homomorphism of S into G. Let oO, fr , 8 , be the respective Lie algebras of G, K, S . For each vector w E f,, put Y,(r) = w . r (19.3.7). (a) Suppose that there exists a principal connection in R which is K-invariant. If w is the 1-form of the connection, then &, w = 0 (19.8.11.3). Deduce that, for each horizontal vector field X on R, and each w E fr, the vector field [ Y,, XI is horizontal. (b) For each vector w E f,, put f(w) = w(ro) . Y,(ro), so that f is a linear mapping of f. into ae. Show that f(w) = pJw) for all w E 6 = , and that 9

f(Ad(r). w) = Ad(p(t)) * f(w) for all f E S. (Use (19.3.7.6) and (20.2.5.1).) (c) If &2 is the curvature form of the connection, show that WrO) . (Y,(rO) A Yw(ro))= [f(v), fWl - f([v, wl). (Use the fact that Of,,, . w = 0.) (d) Conversely, suppose that we are given a linear mapping f : f e + g e satisfying the two conditions of (b) above, and suppose also that K acts transitively on B by the rule t v ( r ) = n(t . r). Show that there exists a unique K-invariant connection in R such that the form w of the connection satisfies the relation w(ro). Y,(ro) = f(w) for all w E f r . (The hypothesis of transitivity implies that for each point r E R and each tangent vector h, E T,(R) there exist 1 E K, s E G , w E f., and u E g. such that ro = t . r * s and f . h, . s = w . ro ro . u. Show that the formula

+

-

w ( r ) . h, = Ad($ (f(w)

+ u)

defines a connection form with the required properties. The properties of f ensure that w ( r ) is independent, first of the choices of w and u, and then also of the choices o f t and s. To prove that w is of class C", use a local section of K considered as a principal bundle over K/Sin order to choose I , s, w, and u in a neighborhood of a point of R.) (e) The connection corresponding to f is flat (20.4.1) if and only iff is a Lie algebra homomorphism o f f , into o. . 4. EXAMPLES O F PRINCIPAL C O N N E C T I O N S

(20.4.1) Given a trivial principal bundle R = B x G, we define a principal Connection P in R (called the trivial connection) by taking P,(k, (b, s)) t o be the image of k under the tangent linear mapping at b to the mapping XH ( x , s) of B into R. At a point (b, ~ ) E R the , tangent space is identified with Tb(B) x T,(G), and therefore Pb(k,(b, s)) = (k, 0). If X i s any vector field on B, its horizontal lifting 2 is therefore given by T(b,s) = (X(b),0). It follows immediately that if X , Y are vector fields on B, then [8, P]is the horizontal lifting of [X, Y]. The formula (17.15.8.1) then shows that for two horizontal

4 EXAMPLES OF PRINCIPAL CONNECTIONS

249

vectors h, k in T(b,JR) we have d o . (h A k) = 0, so that the curvature form Q = Do is zero. Conversely, suppose that R = 0 for a principal connection in a principal bundle R. If X , Y are two horizontal vector fields on R, we shall have do * ( X A Y ) = 0, and (17.15.8.1) then shows that co * [X,Y ] = 0: in other words, [ X , Y ] is also a horizontal vector field. This can also be interpreted by saying that the field of directions r H H , is completely integrable (18.14.5), and consequently (18.14.2) for each point b E B there exists an open neighborhood U of b in B and a C" section h of R over U such that the tangent space at each point to the submanifold h(U) of R is horizontal. The same will be true for the translations X H h ( x ) . s of the section h by elements s E G, by virtue of (20.2.3). It follows that cp : (x, S)H h ( x ) . s is a diffeomorphism of U x G onto n-'(U), and that the image under 'p-' of the given connection on n-'(U) is the trivial connection in U x G. A principal connection is said to beflat if its curvature form is zero. (20.4.2) Let G be a Lie group, H a Lie subgroup of G, and let ge, be be their Lie algebras. We may regard G as a principal bundle with group H (acting on G by right translations) and base G/H (16.14.2). We propose to consider when a principal connection in G (considered as a principal bundle over G/H) is G-invariant (for the action of G by left translations); in other words, if oHis the l-form of the connection, we must have (20.4.2.1)

y(s)oH= OH

for all s E G

(where G is considered as acting trivially on ge). Explicitly, we obtain (19.1.9.2) (20.4.2.2)

%(x)

*

h, = ~ H ( s x ) . (S * h,)

for all s, X E G and h,E T,(G). Now consider the form o H ( f ? ) which, by definition (20.2.5), is a projection of ge onto be; let in be its kernel, which is a supplementary subspace for be in ge. It follows from (20.2.5.1) that for each vector u E ge and each t E H we must have t ) = Ad(t-') * (u)H(e) which, in view of (20.4.2.2), can also be written as %(t)

(U

*

oH(e)* ( t - '

*

- - t ) = Ad(t-') U

*

U),

(oH(e)

or again as (20.4.2.3)

oH(e) Ad(t-') = Ad(t-') 0

0

wH(e),

and implies in particular that (20.4.2.4)

Ad@). in c m

for all t E H.

-

U)

250


Conversely, suppose that there exists a vector subspace m of ge supplementary to be, such that (20.4.2.4) is satisfied for all t E H (in which case the homogeneous space G/H is said to be weakly reductive). The projection P of ge onto be with kernel m then commutes with Ad(t), because Ad(t) is an endomorphism of the vector space gewhich stabilizes be and in. We define a 1-form oHon G with values in lje by the rule wH(s) * (s * h) = P * h

for all h in ge and s in G, and verify that oHsatisfies the conditions (20.2.5.1) and (20.2.5.2), hence is the form of a G-invariant principal connection. It follows that there is a canonical one-to-one correspondence between the set of G-invariant principal connections and the set of supplementary subspaces in of be in ge satisfying (20.4.2.4). It is clear, by transport of structure, that the curvature form QH is also invariant under G (acting by left-translations): it is given by (20.4.2.5)

a,($)( S *

'

U A S

*

V)

=

-P [U, V] *

for u, v E ge. For it is enough to calculate doH* (XuA X,) for u, v E in by the formula (17.15.8.1), observing that by transport of structure oH Xu is a constant function on G. When H is connected, the condition (20.4.2.4) is equivalent to (20.4.2.6)

[be 3 1nI c

m

by virtue of (19.11.2.4). (20.4.3) An important particular case is that in which we are given an involutory automorphism u # 1, of a Lie group G, so that u2 = 1,. It is clear that the set H of points of G fixed by u is a closed subgroup of G, hence a Lie subgroup (19.10.1). The derived homomorphism u* of the Lie algebra ge is an involutory automorphism of the Lie algebra. Since we have exp(to*(u)) = a(exp(tu)) for all t E R and u E ge (19.8.9), and since the relation

exp(ta*(u)) = exp(tu) for all t E R is equivalent to u*(u) = u (19.8.6), it follows that the set of vectors in ge fixed by u* is the Lie algebra be of H. For r E G and s E H we have, by definition, a(srs-')

= su(r)s-',

and by taking the derived homomorphisms of both sides (considered as functions of r ) we obtain (20.4.3.1)

u*

0

Ad($) = Ad(s) o * . 0

4 EXAMPLES OF PRINCIPAL CONNECTIONS

251

Regarded as an involutory automorphism of the vector space ge, a, has two eigenvalues, 1 and - I, and g, is the direct sum of Ij, and the eigenspace in corresponds to the eigenvalue - 1. The relation (20.4.3.1) then shows that Ad(s) in c in for all s E H. It follows that if H, is the identity component of H, then for each Lie subgroup H, of G such that H, c H, c H (i.e., each Lie subgroup of H having Ij, as Lie algebra) we can define in G/H, a principal connection P which is left-invariant under G, such that o H , ( e )(for this connection) is the projection of ge onto Ij, with kernel in. The principal connection P is the unique principal connection in G (considered as a principal bundle over G/H,, with group H,) which is invariant under G and the automorphism a. For by virtue of (20.4.2) the 1-form of such a connection is determined by its value at the point e, which is a projection of ge onto Ij,, with kernel in’, say. The image of the connection under a is again G-invariant, hence (since a(H,) = H,) corresponds, by transport of structure, to the projection of ge onto be with kernel o,(in’). Since the only supplement in‘ of I),in g, such that a,(in’) = in‘is the subspace in, our assertion is proved.

+

We remark that in this case, since aJu) = - u for u E in, we have a,([u, v]) = [a,(u), a,(v)] = [u, v] for all u, v E m, or in other words

(cf. Problem 2). The curvature form at the point e is therefore given by (20.4.3.3)

JZ,(e) * (u

A

v) =

- [u,v]

for u, v E in (horizontal vectors at the point e). The pair (G, H,) is said to be a symmetric pair if it arises from an involutory automorphism a of G as described above. The homogeneous space G/H, is called a symmetric homogeneous space of G (defined by a), and the connection P is called the canonical principal connection in G (considered as a principal bundle with group H, and base G/H,).

PROBLEMS 1. Let R be a principal bundle over a simply-connected base space B. If R carries a flat

connection, show that R is trivializable.

2. With the notation of (20.4.2), give an example in which the relation (20.4.2.6) is satisfied, but (20.4.3.2)is not. (Cf. Section 19.14, Problem 4.)

252

XX


3. With the notation of Section 20.3, Problem 3, suppose that there exists a supplement m of 8, in the Lie algebra f, such that Ad(t) . in c nt for all t E S. Then there exists a canonical one-to-one correspondence between the set of K-invariant principal connections in R and the set of linear mappings f,, : in -+op such that f,,(Ad(r) . w) = Ad(p(t)). f,(w) for all t E S and w E m: the mapping f considered in Section 20.3, Problem 3, has f, as its restriction to m.The connection in R corresponding to f, = 0 is called canonical (relative to the given subspace in). 4. Problem 3 applies in particular when R is the Sriefel manifold S,,,JR). Here we have K = O(n),S = O(n - p ) x O(p), G =O(p), and we may take ni to be the vector space of all n x n matrices of the form

where Y is any (n - p ) x p matrix. The point ro is chosen to be the matrix €=

(2)

(see Section 19.7, Problem 9), the elements of S.,,(R) being therefore them atrices U = S . €, where S E O(n). Equivalently, the matrices U E S,,, JR) are characterized in RnPby the relation 'U . U = I,. The tangent space t o Sn,,(R) at the point U = S E may be identified with the set of matrices S .

(t)

, where X E o(p) is a skew-symmetric

matrix and Y is any (n - p ) x p matrix. If w is the I-form of the canonical connection in Sn,,(R) (Problem 3), then w ( U ) maps the matrix S .

(t)

above to the matrix

X Eo ( p ) . Equivalently, we may write w = 'U * dU, where by abuse of notation dU is the restriction to S,,,,(R) of the differential of the identity mapping RnP+Rnp, and o ( p ) is canonically identified with the space of matrices

)(:

E R"P

with X skew-

symmetric. Discuss in the same way the complex Stiefel manifolds S,,, ,(C), where the orthogonal groups are replaced by unitary groups. Here the I-form of the canonical connection is w = ' a . dU, where 0 is the complex conjugate of U. 5. (a) Let Z be a trivializable principal bundle with base B, group U(p) and projection r,and let u be a C' section of Z over B. Let F, ( 1 5 j s N) be functions on B with

values in M,(C) such that

N

'F,* F, = I p identically

in B. Define a C" mapping H

J=l

of Z into the space CNp2= MNP.,,(C) of Np x p matrices by the formula

( ) Fi(dz))

H(z)=

i

Fddz))

*

Uz),

where V ( z )E U p ) is the unitary matrix such that z = u(r(z)). V(z). Show that If&?) * H ( z ) = I , , and hence that H is a mapping of Z into the Stiefel manifold SN,,,(C); moreover, this mapping defines a morphism of principal bundles (with

5

LINEAR CONNECTIONS

253

the same group U(p)). Show that the inverse image ' ( H u)(w) of the I-form of the canonical connection in SNP, ,(C) (Problem 4) is the vector-valued differential I-form 0

2 IFl. dF, on B, with values in the Lie algebra u ( p ) of U p ) .

I=

1

(b) Let V be an open subset of R", let A be a positive-definite Hermitian p j. p matrix, let f be a bounded C" function, defined and 1.0 on V, and let c be a coolstant such that the Hermitian matrix r l , - f 2 ( x ) A 2 is > O for all x E V. Let F A X )be the Hermitian matrix >O which is its square root (15.11.12). Let a be a real constant, and put F,(x) = f(x)eaiCLA2. Show that

'F,(x) F,(x) -t 'Fz(x) . Fz(x) = r l , and

'F, dF1 4-' F l *

'

dFl

=

u i f 2 A 2d t ' ,

a vector-valued differential I-form with values in u ( p ) . (c) Deduce from (b) that, for every bounded C" vector-valued differential I-form a on V with values in ~ ( p )there , exists an integer N depending only on the dimension tn of V, and N mappings Fl : V MJC) such that --f

and (Observe that the vector space u(p) has a basis consisting of matrices i A k , where the A r are positive-definite Hermitian matrices.) (d) Deduce from (a) and (c) that, for euery principal bundle Z with group U p ) over a pure manifold B of dimension n, and for euery principal connection P in Z, there exists an integer N depending only on n and p. and a morphism ( i t , of Z into S , , , ( C ) such that P is the inverse image under u (Section 20.2, Problem 2) of the canonical connection in SN,,(C). (Use Section 16.25, Problems 10 and II , and remark that in (16.4.1) the functions forming the partition of unity may be taken to be the squares of C" functions.)

5. LINEAR C O N N E C T I O N S ASSOCIATED WITH A PRINCIPAL CONNECTION

(20.5.1) Let R be a principal bundle with base M, group G , and projection n ; also let F be a finite-dimensional real vector space, and p a linear representation of G in F. so that G acts on the left on F by (s, Y ) H ~ ( s* )y. We have seen (20.1.3) that the fiber-bundle E = R x F of fiber-type F associated with R is canonically endowed with a structure of a real vector bundle. Suppose we are given a principal connection P in R. We shall show how to construct from it, in canonical fashion, a linear connection C (17.16.3) in E. For each point x E M and each vector u, E Ex, there exists an element r, E R, and y E F such that u, = rx y (16.14.7). For each tangent vector k, E T,(M), we put (20.5.1 .I )

C,(k,, u,) = P,(k,, r,) y

254

XX

PRINCIPAL CONNECTIONS A N D RIEMANNIAN GEOMETRY

with the notation of (16.14.7.3). First, it must be checked that this definition does not depend on the choice of the pair (r,, y) such that r, * y = u, . However, any other choice is of the form (r, * s, p ( s - ' ) . y) for some s E G , and by (20.2.2.4) and (16.14.7.4) we have

-

-

P,(k,. r, .s) ( p ( s - ' ) y) = (P,(k,, r,) = P,(k,

9

*

3)

*

( ~ ( s - ' ) Y)

r,) * Y.

Second, if nF is the canonical projection of E on M, we have nF(r* y) = n(r) for r E R, and y E F, whence T(nF) * C,(k,, u,) = T ( 4 * P,(k,, r,) = k, . Next, if we put m(r,, y) = r, y, we have m(., ay + by') = urn(. , Y) + Bm(. , Y'); taking the tangent linear mappings of these mappings of R into E, we see that, for each tangent vector h E TJR), we have h (cry

+ By') = a h y + Ph

y',

where the sum on the right-hand side is taken in the fiber (T(E))T(n).h of T(E) considered as a vector bundle over T(M). This therefore shows that u,HC,(k,, u,) is a linear mapping of Ex into (T(E)),=. Since on the other hand k,H P,(k,, r,) is a linear mapping of T,(M) into TJR), it follows that k,HC,(k,, u,) is a linear mapping of T,(M) into TUx(E).Finally, to see that C is of class C", we may assume that R = M x G is trivial and that M is an open set in R";in that situation, E may be identified with the trivial bundle M x F, and if C,((x, k), ( x , y)) = ((x, y), (k, -r,(k, y))) is the local expression of C, then with the notation of (20.2.2), and remembering that p* is a homormorphism of ge into End(F) = gI(F), we have r,(k, Y) = - P * ( Q ( X )

k)) * Y,

which proves our assertion. (20.5.2) Conversely, if E is any vector bundle over a pure manifold M and if C is any linear connection in E, then C can be obtained by the construction of (20.5.1) from a well-determined principal connection P in the bundle of frames R = Isom(M x F, E), E being identified with a bundle of fiber-type F associated with R (20.1 J). To see this, we take up again the procedure which was sketched in (20.2.1) in a nonintrinsic fashion. For this purpose, we remark that for each r,E R, = Isom(F, Ex) and each tangent vector h E TJR), the mapping U(h) : y~ h * y (in the notation of (16.14.7.3)) is a linear mapping of F into (T(E))kx,where k, = T(n) * h = T(p) . (h . y) (where n and p are the projections of the bundles R and E, respectively). For a fixed

5

LINEAR CONNECTIONS

255

k, E T,(M), the mapping h H U(h) of (T(R))kxinto Hom(F, (T(E)),) is an isomorphism. To see this, we may assume that E = M x Eo is trivial and M is an open set in R", so that R = M x Isom(F, Eo). If r, = ( x , uo) and k, = ( x , k), the vectors h E (T(R)),= are the tangent vectors of the form ( ( x , u0), (k, u,, 0 vjj, where v E End(F), and we have h * Y = ( ( x , U O ( Y N 9 (k, uo(v(Y))),

from which the assertion follows since u,, is bijective. Now the mapping y ~ C , ( k , , r, * y) is a linear mapping of F into (T(E))kx;hence, by what has been said above, there exists a unique tangent vector P,(k,, r,) in (T(R))kx satisfying (20.5.1.1) for all y E F. It is immediately verified that the mapping P thus defined satisfies (20.2.2.2) and (20.2.2.3). By reducing to the case where E is trivial, we see that P is of class C". Finally, since (r, * s) * y = r, * (s * y) for all s E GL(F), the mapping P satisfies (20.2.2.4) by virtue of its definition. (20.5.3) We shall now show how this association of a principal connection with a given linear connection enables us to reduce the operations of covariant direrentiation of a section of E (relative to C), in the direction of a tangent vector of M (17.17.2.1), and of the covariant exterior differential (relative t o C) of a differential form on M with values in E (17.19.3), t o much more elementary operations in the principal bundle R, namely, on the one hand the Lie derivative of a function on R with values in a vector space, and on the other hand covariant exterior differentiation (relative to P) (20.3.2) of a vector-valued differential form on R. Let then s be a C" section of E over a neighborhood of x , and let r, be any point in the fiber R, . Since the question is local on M, we may assume that there exists a C" section YH R(y) of R such that R(x) = r, . The section s can be written uniquely in the form (16.14.7.1)

where 0 is a C" mapping of a neighborhood of r, in R into the vector space F. Then we have the first fundamental formula, which gives Vkx s for any tangent vector h, E T,( M):

-

for all tangent vectors k E Trx(R) such that T(n) . k = h, . We recall that Ok a,the Lie derivative of the function 0 in the direction of the vector k, belongs to F (17.14,1), that o(r,j . k = u is a vector in the Lie algebra gI(F), and that the product u . y E T,(F), for y E F, was defined in (16.10.1).

256

XX


To prove (20.5.3.2), we start from the definition of z s ~ ~ ) ( v h *x s,

= Tx(s)

'

hx

s (17.17.2.1):

vhx

- Cx(hx s(x)) E Ts(.x)(E), 3

and calculate T,(s) by taking the tangent linear mappings of the two sides of (20.5.3.1) at the point x (16.14.7.5): T,(s) h,

= (Tx(R) . h,)

-

. @,(R(x))+ R ( x )

(TR(x)(@)

(TJR) * hx)).

Put k = T,( R ) h, E Trx(R), which is a vector such that T(n) * k = h,; then TR(,)(@) - k is a vector in T,,JF), equal to t$x)(& @) (17.14.1); and since rx is an isomorphism of F onto Ex,we have @)) = Tr(:)(rx

rx ' (z&:x)(ek

*

(ek

*

@))a

Hence, bearing in mind (20.5.1.l), we may write zsj:)(vhx

' S)

= (k

- P,(T(n) ' k, r,)) * @(r,) + ti:)(r,

*

( 6 , * a)).

By the definition of the vector-valued form o (20.2.3.4) we have therefore zsc.x!)(vhx

*

s, =

'

(0, * @)I + ( t r x ( a ( r x ) * k)) ' @(rx).

Now we have the following formula: (20.5.3.3)

for u ~ g [ ( F )and Y E F. This formula is obtained by taking the tangent linear mappings at the point e E GL(F) of the two functions s w r , (s y) and st+(r,. s) y, which are equal, and using the definition of trx (20.2.3.3). The formula (20.5.3.2) then follows from the fact that, for each r, E R, and each tangent vector k E TJR) such that T(n) * k = h,, there exists a section R of R over a neighborhood of x such that R(x) = r, and T,(R) h, = k (16.8.8). In particular, let us take k to be the horizontal lifting rel(h,) of h, at the point rx (20.2.2). Then we have o ( r , ) k = 0 because o is vertical, and we obtain

-

(20.5.3.4)

vhx

*

= I'x

'

(erel(hx)

. @>.

(20.5.4) Suppose now that the group GL(F) acts on three other vector spaces F,, F, , F, and that B : F, x F, --* F, is a GL(F)-inuuriunt bilinear mapping, i.e., that

for all s E GL(F), where pj ( j = I , 2 , 3 ) is the linear representation of GL( F) in Fj.

5

LINEAR CONNECTIONS

257

Let El, E l , E3 be the vector bundles associated with these three representations, and endow each of them with the linear connection obtained by the procedure of (20.5.1) from the same principal connection P in the bundle R of frames of E. Let s1 (resp. s2) be a section of El (resp. E2) over a neighborhood of x. With the notation of (20.5.3), we may write sj(y) = R ( y ) Oj(R(y))( j = 1, 2). Now define

%(Y) = R(y) * B(@l(RoJ))3W R W , which by virtue of (20.5.4.1) does not depend on the section R chosen. We write s3 = B(s,, s2). Then, for each tangent vector h, E T,(M), we have

This is an immediate consequence of the definition of B(sl, s2), the formula (20.5.3.4), and the fact that the point-distribution Orel(,,,) is a derivation (17.14.2.1). If we take E = T(M) and B to be a tensor product

T:(R") x T:(R") + T:f:(R"),

or the fundamental bilinear form on R" x (R")*, the formula (20.5.4.2) shows that the canonical extension of the covariant differentiation in the direction of a tangent vector h,, to tensor fields on M (17.18.2), coincides with the covariant differentiation defined on each of the bundles T,P(M) from the connection P on R, by the formula (20.5.3.4). (20.5.5) We shall now show how the calculation of the covariant exterior differential (relative to C ) of a differential I-form on M with values in E (17.19.3) can be reduced to the calculation of the covariant exterior diyerentiai in the sense of (20.3.2.1) of a vector-valued I-form on R, with values in F. Let & be a differential I-form on M with values in E, i.e., an M-morphism of T(M) into E. By definition, the covariant exterior differential dc is a differ2

ential 2-form on M with values in E (i.e., an M-morphism of A T(M) into E) such that, for each pair of tangent vectors h,, k, in T,(M), we have

where X , Y are any two vector fields on M which take the values h,, k,, respectively, at the point x (17.19.3). With the notation of (20.5.3), since R is an immersion of a neighborhood V of x into R, there exists a section

258

XX


R(y)wX’(R(y)) of T(R) over the submanifold R(V) such that X ’ ( R ( y ) ) = T,,(R). X(y)for all y E V, and X‘ can be extended to a vector field on R (1 6.12.1 I),which we shall also denote by X ’ . Likewise we define a vector field Y’ on R, starting with Y. We may always assume that R is such that X ’ ( R ( x ) ) = rel(h,) and V ’ ( R ( x ) )= rel(k,) (but, of course, at points y close to x, the vectors X’(R(y)) and Y‘(R(y)) will not in general be horizontal). This being so, we construct from 5 a vector-valued diyerential l-form 6 on R , with values in F, by the formula (20.5.5.2)

*

z = r;

’

*

-

( R x ) - (T(n) z))

for all tangent vectors z E TJR). We shall now prove the second fundamental formula : (20.5.5.3)

d c ( x ) * (h,

A

k,) = r, * (Dg(r,) * (rel(h,)

A

rel(k,)))

for all tangent vectors h,, k, at the point x E M. The section s = 4 * Xof E may be written in the form s ( y ) = R(y) * @ ( R ( y ) ) with @ = 5 X ’ by virtue of the above definitions. Applying the formula (20.5.3.4), we therefore obtain for the first two terms of the right-hand side of (20.5.5.1) r x . (erel(h,) . (5 Y ’ ) - erel(k,) * (5 * X ’ ) ) ; bearing in mind the definition of D (20.3.2.1) and the formula giving dg (17.15.8.1), it is enough to show that we have

However, since the vector fields X ’ and Y’ are tangent to the submanifold R(V) of R at the points of this submanifold, and since R is an isomorphism of V onto R(V), the formula (20.5.5.4) follows simply by transport of structure, bearing in mind (20.5.5.2). The formula corresponding to (20.5.5.3) when 4 is a diTerentia1 p-forrn on M with values in E may be proved similarly, by using the general formula (1 7.15.3.5).

PROBLEMS

1. Let R be a principal fiber bundle with base M, group G and projection 7.Let F be a differential manifold on which G acts differentiably on the left, and consider the associated bundle E = R x F with fiber-type F. If P is a principal connection in R, we may

5 LINEAR CONNECTIONS

259

again, for any element u, of the fiber Ex of E over a point x E M, write u, = r, * y for some rx E R, and y E F, and we can therefore define a vector C,(k,, u,) E TJE) by the formula (20.5.1.1);it does not depend on the choice of the pair ( r , , y ) such that r, . y = u,, and if xp is the projection of E onto M, then T(rJ . C,(k,, u,) =k,; finally, the mapping k,crC,(k,, u,) is linear, and its image is therefore supplementary in TJE) to the subspace VJE) = Ker(Tux(rF))of vertical tangent vectors. The vectors C,(k,, N,) are called the horizontal vectors in T.,(E) corresponding to the connection P. Let v : R + M be an unending path of class C". For each uo E Euco,,we may write uo = ro . y for some y E F and some ro E Ruco,. With the notation of Section 20.2, Problem 3, show that the unending path t-Guo(t) = w,o(r). y is the unique unending path which lifts v to E and is such that Guo(0) = uo and such that, for each f E R, the vector G:,(t) is horizontal in the tangent space to E at the point G,,(r). For each t E R, the mapping l ~ o H G u o (isr )a diffeomorphism rpI of Euto,onto Ev(r).In the particular case of (20.5.1),where F is a vector space and G acts linearly on F, the diffeomorphism rpr is a linear bijection; when E = T(M), yr is the parallel transport along u, defined in (18.6.3). Given an unending path I H ~ ( I )in E of class C" which lifts u, then for each t E R there exists a unique u ( t ) E E,,(o,such that rpI(u(t)) =f(t)(in other words, Gu&) = f ( r ) ) . The mapping t i + u(r) of R into EUco,is of class C" and is called the development off in the fiber E,,,, . Given an ro E R U c o ,we , may write u ( t ) = ro * y ( t ) ,where y ( t ) E F. Show that if G acts transitively on F, the vector y ' ( t ) is such that w,,(r). y ' ( t ) is the vertical component of f ' ( t ) in the decomposition of T,(r,(E) as the direct sum of the space of vertical vectors and the space of horizontal vectors. (Write f ( r ) locally in the form ( w , , ( t ) .g ( t ) ) . y o , where g is a C" mapping of R into G.) 2.

With the hypotheses and notation of Section 20.1,Problem 2 (so that in particular dim(B) = dim(G/H) = n), we may construct the principal bundle R x G with base B and group G (Section 16.14,Problem 17). The mapping r i + r * e is an embedding of R into R x G ; also H acts on R x G on the right (by restriction of the action of G), and X may be canonically identified with the orbit-manifold H\(R x G). A principal connection P in R x G is called a Cartan connectionfor R (relative to G ) (or, by abuse of language, a Cartan connection in R) if, for all r b E Rb, the space H,,of horizontal tangentvectors(20.2.2),which is a subspace of T,,(R x G), is such that H,, n T,,(R)= { O } . An equivalent condition is that the restriction wo to R of the 1-form w of the connection P is such that wo(rb)is an injective linear mapping (or, equivalently, boectiue since dim(B) = dim(G/H)) of T,,(R) into IL. (a) Conversely, let wo be a differential I-form on R with values in oC (not be) such that (i) wo(rb. I ) . (k . t ) = Ad(t-') . (wo(rb). k) for k E T,,(R) and r E H; (ii) wo(rb) * ZU(rb)= u for u E be; (iii) for each rb E R, the mapping w o ( r b ): T,,(R) -fae ip in injective linear mapping. Show that wo has a unique extension to the 1-form on R x G of a Cartan connection for R. (b) For each tangent vector hbETb(B), let Sb(hb) be the vertical component of Tb(u) . h, in the decomposition of Ta(b)(X)as a direct sum of the spaces of vertical and horizontal vectors (Problem I). Show that )Bb is bijective and that fl is a welding of B and X, canonically associated with the Cartan connection P.

260

XX


For each unending path r H f ( t ) in M, the development (Problem 1) in a fiber of X (isomorphic to G/H) of the path t ~ u ( f ( r ) in ) X is also called the development o f f in G/H. (c) Conversely, show that if there exists a welding of B and X, then there exists on R x G a Cartan connection for R. (Argue as in Section 20.2, Problem 1 .) (d) Show that if there exists a Cartan connection for R (or, equivalently, a welding of B and X),then the manifold R is purullelizuble. Let G be a Lie group, H a Lie subgroup of G . Consider G as a principal bundle over G/H with group H, and let 7 : G +G/H denote the canonical projection. Show that the principal bundle G x G over G/H with group G is trivializable. (Define a canonical section of this bundle by noting that for s E G, the product s . s - ' in G x G in the sense of (16.14.7) depends only on d s ) . ) The canonical differential I-form wo on G (19.16.1) extends uniquely to the I-form of a Curtun connection for G on G x " G , called the canonicul Currun connection. Show that this connection is flat (20.4.1). Let X be a principal bundle with base B, stiucture group G, and projection 7.Let H be a closed subgroup of G , and let j : H + G be the canonical injection; let Y be a principal bundle over B with group H, and let ( u , j ) be a morphism of Y into X, so that u is an embedding of Y in X (20.7.1). Suppose also that the Lie algebra ne of G contains a vector subspace m supplementary to the Lie algebra be of H, such that Ad(r). 111 c 111 for all t E H. (a) Let P be a principal connection in X, and let w be the I-form of the connection. For each yb E Y b and each tangent vector h,, E T,,(Y), let w&h) . hrband cp(yb) . h,, be the projections of w ( y b ) . h,, onto 6 , and in, respectively. Show that wo is the 1-form of a principal connection Po in Y ,and that cp is a vector-valued differential I-form on Y,with values in in, which is horizontal and such that cp(y, . t ) . (h . t ) = Ad(t-') . ((p(yb) . h) for all h E T,,(Y) and all r E H. Conversely, if we are given a principal connection Po in Y, for which wo is the connection form, and if cp is a horizontal differential I-form on Y with values in 111 which satisfies the above condition, then there exists a unique principal connection on X which gives rise to wo and cp as above. (b) If X = Y x " G and dim(B) = dim(G/H), then a connection in X with I-form w is a Cartan connection if and only if, for each y b E Y , the mapping (p(yb) of Tyb(Y) into m is surjective. (c) Suppose that G is a semidirect product N x H, so that in may be taken to be the Lie algebra n, of N (19.14). Show that if N is commutative, the curvature forms Q and SZ, of the connections P and Po are such that the restriction of SZ to Y is equal to Po Dcp (covariant exterior differential relative to Po), and that

+

dcp = - 0 0

A 'p+ D v .

(d) Suppose again that G = N x H and in = 11, (but not that N is necessarily commutative). Let u : R +. M be an unending path of class C", and let yo be a point of Yuc0, c Denote by w and wo the unending paths in X and Y,respectively, which lift u and are such that their tangent vectors are horizontal relative to P and P o , respectively, and such that w ( 0 ) = wo(0) = yo (Section 20.2, Problem 3). Show that w ( r ) = wo(f) . h(t), where h(r) E N, and that cp(wo(r)) . w b ( r ) = -h'(t) . h ( t ) - ' . (Use (20.2.5.1) and (20.2.5.2).)

6 THE METHOD OF MOVING FRAMES

261

6. T H E METHOD O F MOVING FRAMES

(20.6.1) From now on we shall consider only vector bundles E which are

tangent bundles T(M) of smooth manifolds M. In other words, we shall consider only linear connections on apure mangold M of dimension n (17.18.1). These connections are in one-to-one correspondence (20.5) with the principal connections in the bundle of frames R(M) = Isom(M x R", T(M)) of M, and it is these latter that we shall consider first.

(20.6.2) The definition (20.1. I ) of the bundle R(M) implies the existence of a canonical vector-valued differential l-jorm on R(M) with values in R" (independent of any connection in R(M)). Namely, for each x E M an element rx of the fiber R(M), is an isomorphism of R" onto T,( M); if n is the projection of the bundle R( M),then the mapping (20.6.2.1)

o(r,) : krxH r;

. (T(n) krx) *

is a surjective linear mapping of Trx(R( M)) onto R".In other words, o is a vector-valued differential 1-form on R(M), with values in R",which is called the canonical form on R(M). This fcrm is of class C". For by trivializing T(M) and R(M) over an open set U by means of a chart of M, we may assume that M is an open set in R", and hence T(M) = M x R",R(M) = M x GL(n, R). A point rx is then written as (x, U)with U E GL(n, R), and a tangent vector krxtakes the form ((x, U),(v, V ) )with v E R" and V E M,(R). The vector T(n) krxis then (x, v), and hence a(r,) is the mapping ((x,

w, (v, V ) ) H u-'

*

v;

this proves our assertion. It is clear that the canonical form o is horizontal (20.2.4), from the definition of the vertical vectors of TJR(M)). Moreover, for each s E GL(n, R) and k E TJR( M)), we have (20.6.2.2)

-

a(r, * s ) * (k s) = s-' * (a(r,) k),

because (r, . s)-' = s-' 0 r; ' and T(n) (k s) = T(n) * k. In other words, o is inuariant (19.1) under the right action of GL(n, R)on R(M) and the canonical right action (s, ~ ) H s - ' * y of GL(n, R) on R". Suppose now that we are given a principal connection P in R(M). Then the covariant exterior differential of the canonical form G is a vector-valued 2-form 0 with values in R": (20.6.2.3)

0 = DG

called the torsion form of P. It is horizontal and invariant (for the action (s, U)HS-' u of GL(n, R) on R").

262

XX


(20.6.3) The canonicalform a on R(M) satisjies the "structure equation"

for all tangent vectors h, k at a point rx E R(M), where o is the I-form of the connection P. As in the proof of (20.3.3), we distinguish three cases:

(i) h and k are horizontal. Since the connection form o is vertical, the formula (20.6.3.1) reduces to the definition of Da (20.3.2.1). (ii) h and k are vertical. Then both sides of (20.6.3.1) are zero. This is clear as far as the right-hand side is concerned; on the other hand, if u, v E gI(n, R) are such that h = Zu(rx),k = Z,(r,) , then we have

daqz,

A

2,) =ez, . ( a .z,)-e Z V . ( a z . ,)- a . z,,,,,,

and the right-hand side is 0 because a is horizontal. (iii) h is vertical; k is horizontal. Then h is the value at the point r, of a Killing vector field Z , , where u E yI(n, R). Also, there exists a vector field Y on R(M) which is invariant under the action of GL(n, R) and which takes the value k at the point r,. For if Yo is a vector field on M whose value at the point x is the projection T(n). k (16.12.11), then its horizontal lifing Y will have the required properties, because k is horizontal. We have then, by (17.15.8.1),

Now, since a is horizontal, we have a * Z , = 0 ; also, because of the choice of Y,we have [Z,,Y] = 0 (1 9.8.11); finally, since a and Y are invariant under the action of GL(n, R),the mapping a Y of R(M) into R" is also invariant under this action and under the action (s, y ) ~ s - ' y of GL(n, R) on R". Hence (1 9.4.4.3) we have Bz, * (a * Y)= - u * (a Y) (recall that u E M,(R)). By definition, the value of this expression at the point r, is --(~@x) *

-

h) * (a(rx)*

4.

On the other hand, o(r,) k = 0 since k is horizontal, and O(r,) (h A k) = 0 since h is vertical. We have therefore verified that the two sides of (20.6.3.1) are equal.

-


263

In accordance with the conventions introduced in (1 6.20.15), we shall write this equation in abridged notation in the form (20.6.3.3)

du=-oAu+0.

If the scalar-valued 1-forms o i j( I S i, j n) and oi(1 5 i 5 n) are the components of o and u with respect to the canonical bases of M,(R) and R", respectively, then the ith component (1 5 i 5 n) of o A u is (20.6.3.4)

1o i j n

(aA

=

A

aj.

j= 1

(20.6.4) The curvature and torsion 2-forms of P satisfr the relation (20.6.4.1)

D@=&~Au

(Bianchi's identity). (The vector-valued 3-form on the right-hand side of (20.6.4.1) is defined by the formula

for all h,, h,, h, in TrX(R(M)).) We obtain from (20.6.3.3), in view of the expressions (20.6.3.4) and the rules of calculation for exterior differentials (17.15.2.1), that (20.6.4.3)

dO = d m

A

u

+ o A du.

To obtain the value of DO at a trivector h, A h, A h, , we have therefore to evaluate the right-hand side of (20.6.4.3) at the trivector hi A h i A h i , where hJ is the horizontalcomponent of hi ( j= 1,2, 3) (20.3.1). Now since o is a vertical form, the value of o A do at h i A h i A h i is zero, and the formula (20.6.4.1) follows. (20.6.5) We shall now obtain expressions for the covariant derivative, the curvature and the torsion of the linear connection C on M defined (20.5.1) by the principal connection P in R(M). Since the question is local with

264

XX


respect to M, we shall suppose that there exists a C" section of R(M) over M. Such a section X H R(x) is called a moving frame on M , and the existence of such a section is equivalent to that of n C" vector fields XH ei(x) on M, such that the vectors ei(x) are linearly independent for all x E M. By definition, we have (20.6.5.1)

R(x)-'

(1 5 i 5 n),

ei(x) = e,

where ( e i ) is the canonical basis of R". We remark that the existence of a moving frame R on M implies that M is orientable, and defines an orientation on M for which R is direct at each point (16.21.2). If u, is the n-covector at x that takes the value 1 at el(x)

A

A

e,(x),

then XH u, is a C" n-form on M which is nowhere zero. To the vector-valued l-forms u and o on R(M) there correspond, under the mapping R, their inverse images (16.20.15.4) (20.6.5.2)

u(") = 'R(c),

w("' = ' R ( o ) ,

which are vector-valued 1-forms on M, with values in R" and M,(R), respectively. We may therefore write (20.6.5.3)

where the cr!") and the w!;) are scalar 1-forms on M . The forms d")and a(") (OT their scalar components .I") and are called the canonical and connection forms on M , corresponding to the moving frame R (or to the n vector fields ei). Conversely, if we are given any 1-form m on M with values in M,(R), then there exists a unique connection form o on R(M) such that a(") = m, since the value of o at one point of a fiber of R(M) determines its value at all other points (20.2.5.1). We have therefore another proof of the existence of a linear connection on M (1 7.16.8). By definition, for each tangent vector h, E T,(M), we have

~17')

u(")(x) * h, = U( R(x)) * (T,( R) * h,) E R",

and therefore, by virtue of the definition (20.6.2.1) of u, and of the fact that TR(x)(n) (T,(R) * h,) = h, (because R is a section), we obtain (20.6.5.4)

u(")(x) * h, = R(x)-'

h,.


265

In other words, the alR'(x) are the n coordinate forms on T,(M) relative to the basis (e,(x)),or equivalently,

for all h, E T,( M). We shall next express the covarianr derivative v h x * Y, for any vector h, E T,(M), of a vector field Y at the point x (17.18.1) in terms of the form a('). For this purpose it is enough to calculate the n vectors v h r . e, (1 5 i 5 n) since the e , form a basis for the module of vector fields in a neighborhood of x. We apply the fundamental formula (20.5.3.2), taking s = e, and k = T,( R ) * h,. Now @(r,) = e, is consfant on R(M), and therefore the first term on the right-hand side of (20.5.3.2) is zero. Also, we have

and therefore we obtain (20.6.5.6)

vhx *

ei =

(W:y'(x),

h,)

*

ej.

j

We shall usually omit the indication of the frame R in the notation for the components of the forms a('?) and dR). Introducing the covariant exterior differential d e , , such that de, * h, = Vhx e , (17.19.2), the formulas (20.6.5.5) and (20.6.5.6) take the forms

-

or equivalently. (a,, e j ) = d i j ; and "

(20.6.5.8)

de, =

1 w j i @ ej

j= 1

(where I,,,, is canonically identified with the Kronecker tensor field (16.18.3)); or, by abuse of notation, (20.6.5.9) n

(20.6.5.10)

de, =

C ojiej

j= 1

266

XX


From (20.6.5.9) it follows that, for each real-valued C’function f on M, we have

(df, hx)

=

i

(Ci(x), h.x)(df, ei(x)>,

which may also be written in the form

(20.6.6) In the same way, to the vector-valued 2-forms 0 and there correspond, under the mapping R, their inverse images

n on R(M)

which are vector-valued 2-forms on M, with values in R” and M,(R), respectively. Put

where the @IR) and 01,”)are scalar-valued 2-forms on M, called respectively the torsion and curvature 2-forms of the linear connection C on M, corresponding to the frame R (or to the vector fields ei). In what follows, we shall generally omit R from the notation. We have then the two structure equations (20.6.6.3)

doi =

-1wii i

A

oj

+Oi,

derived from (20.6.3.4) and (20.3.3.1) by taking inverse images with respect to R (17.15.3.2). We shall show that the torsion and curvature of the connection C , introduced in (17.19), are related to the torsion and curvature forms by the relations (20.6.6.5)

t ( X A Y ) = C (0, ,X

A

Y)ei,

i

(20.6.6.6)

-

( r . ( X A Y)) ei = C ( Q j i ,X i

where X, Yare any two C” vector fields on M.

A

Y)ej,

6 THE METHOD

OF MOVtNG FRAMES

267

is considered as a differBy definition, we have t = d( IT(,,), where ential I-form on M with values in T(M). The differential 1-form on R, with values in R",associated with 1T(M, (20.5.5.2) is precisely the canonical form cr (20.6.2.1). If we replace 6 by G in (20.5.5.3), we obtain (20.6.6.7)

t ( x ) (h,

A

k,) = r, * (O(r,) * (rel(h,)

A

reI(k,))),

and (20.6.6.5) follows immediately. be the differential 1-form Next, for each C" vector field Z on M, let on M with values in T(M) defined by h,HVh, * Z. By definition (17.19.3.1), we have

cz

(20.6.6.8)

(r

(XA Y ) ) * Z

= dcz

(XA Y ) .

Since the value of the left-hand side of (20.6.6.8) at a point x E M depends only on the value of Z at this point, we may certainly assume that 2 is a linear combination of the e, with constant coefficients. Then, by virtue of (20.6.5.6) and (20.5.5.2), the vector-valued differential 1-form 6 on R with values in R" corresponding to is given by

cz

(20.6.6.9)

Ur,) : z H W r , ) * 4 * u,

-

where u E R" is a vector such that r, u = Z(x). Hence, from the definition of $2 = Do and (20.5.5.3), we have (20.6.6.10)

d&,(x) (h,

A

-

k,) = r, * (($2(r,) (rel(h,)

A

rel(k,)))

*

u)

from which the formula (20.6.6.6) follows immediately, if we bear in mind (20.6.6.2).

From the formulas (20.6.6.5) and (20.6.6.6) we obtain directly the expressions of the 2-forms 0,and Q,, in terms of the components of the torsion and curvature tensors relative to the basis ( e i ) :if

then we have (20.6.6.1 2) (20.6.6.13)

268

XX


PROBLEMS

1. Let P be a principal connection in the bundle of frames R(M). For each vector a E R",define a horizontal vector field Ha on R(M) by the condition

for all r x E R(M). Equivalently, Ha is defined by the conditions a.H,=a,

w.H,=O.

For each s E GL(n, R), we have Ha . s = H,- I . a If (adl Sn is a basis of R" and (ul,)lSf,,Sn a basis of gL(n, R) = M.(R), then the vector fields Ha, and Z,,, form a basis of the B(R(M))-module of C" vector fields on R(M). (a) Show that, for all a, b E R" and u E yl(n, R), we have

[Zu,HaI=Hu*a, where u * a is the value at a of u E End(R"), and that

(b) Deduce that the torsion and curvature of P are zero if and only if, foreachx E M, there exists a chart c = (U, v, n) of M at x such that if R is the moving frame defined by R-' * XI =el ( I 5 i n), where the XI are the vector fields associated with the chart c (16.15.4.2), then the image of U under R is a submanifold all of whose tangent spaces are horizontal. (c) The geodesics for the connection C defined by P are exactly the projections in M of the integral curves of the vector fields H a , for all a E R". ) values in R(M) tobesuch that t ~ ~ ( r ( t ) ) (d) Deduce,that for a curve t ~ r ( t with (where T is the canonical projection of R(M) onto M) is a geodesic for C, the following condition is necessary and sufficient:

- (a(r(r)). r ' ( t ) ) + (w(r(t)). r ' ( t ) ) d

dt

(a(&))

. r ' ( t ) ) = 0.

(Argue as in Section 20.2. Problem 3, by using the structure equation (20.6.3.1).) Show that two principal connections PI, P2 in R(M) are such that the corresponding linear connections on M have the same geodesics if and only if (wl(r) * k, - w 2 ( r ) .k,) . ( 4 r ) , k,) = 0

for all r E R(M) and all tangent vectors k, at r , where w I , w2 are the I-forms of the connections PI, P 2 .


269

2. Let M be a pure differential manifold of dimension n. For 0 5 p 5 n, 0 5 q 5 n, let

and define an M-morphism of multiplication B:(M) 0BXM) B:Z(M) ( a A fi) @ ( u A v). The direct sum B(M) of the B:(M) is a bundle of algebras over M. Let B:(M) (resp. O(M)) denote the &M)-module (resp. &(M)-algebra) of C" sections of B:(M) (resp. B(M)) over M. The elements of 9;(M)may be identified with the differential p-forms on M with d u e s in T(M) (17.19.2). --f

as follows : (a0U ) 0(fi 0 V ) H

(a) Let C be a linear connection on M, corresponding to a principal connection P in R(M). There corresponds therefore to C a covariant exterior differentiation operator d which maps each B:(M) into B:+,(M) (17.19.4). Show that there exists a unique differential operator d : B(M) -+ 1 ( M ) , such that d(&JM)) C Bt+l(M), which agrees with the exterior differentiation d in each B,O(M) = &&M), and with the covariant exterior differentiation d in each J:(M), and is such that d(vw) = (dv)w

+ (- I)'v(dw)

for all v E R(M) and w E d:(M). (The product in I ( M ) is denoted by juxtaposition.) Extend the definition of d to matrices with entries in d(M): if U = (u,,) is such a matrix, with a rows and b columns, put dU = (du,,). Then we have, if V is a matrix with a rows and b columns and W is a matrix of b rows and c columns, d ( V W ) = V(dW)

+ (-I)'(dV)W

if all the elements of V belong to J;(M). (b) Let R be a moving frame. With the notation of (20.6.5), identify R with the n x 1 matrix formed by the el E&(M); u = utR)with the 1 x nmatrix formed by the u,E 9Y(M); w = wtR)with the transpose of the n x n matrix formed by the W,,E IY(M); 8 = EltR) with the 1 x n matrix formed by the 0, E @(M); and P =PtR) with the transpose of the n x n matrix formed by the a,, E a!(M). Then the formulas (20.6.5.9) and (20.6.5.10) take the form IT(^ = Q . R dR=W*R.

(E

-@:(M)),

The structure equations (20.6.6.3) and (20.6.6.9 take the form 0 =do-o.w, P =dw -w2, and by exterior differentiation we obtain Bianchi's identities (20.6.4) and (20.3.4) in the form d0=-0.W+O.P2, -P ' w . dP =w (c) Prove the identities (for r 2 I )

d(SL')= w .S' -a..0 , d(O .!X) = 8 * P' O . SL'mw, d(O .a')= o .a'+'- 8 * SL' - W.

+

270

XX


(d) Prove the identities (for r 2 1)

(e) For each vector field u E .%'A(M), prove that

for r 2 1 and s 2 1. 3. Deduce from Bianchi's identities (20.6.4) and (20.3.4) the following identities for the covariant derivation V, the torsion t and the curvature r of a linear connection C : ( r * ( X A Y ) ) . Z + ( r . ( Y A Z ) ) . X + ( r . ( Z r \X)). Y = t * ( t * ( X AY ) A Z ) + t . ( t . ( Y A Z ) AX ) + t . ( t . ( Z A X ) A Y) (v,*t) ' ( Y A Z ) ( v y ' I ) ' ( zA X) ( v z ' 1) ' (xA Y),

+

+

+

+

+

( V X. r ) * ( Y A Z ) (0, * r ) * ( ZA X) (0,. r ) . ( X A Y ) +r.(r.(XA Y)AZ)Sr.(t.(YAZ)AX)+r.(f.(ZA\)A

Y)=O

for all C" vector fields X, Y, Z on M. (Use the formula (20.5.3.4).) 4.

Letfbe a diffeomorphism of a pure differential manifold M onto a differential manifold M'. By transport of structure, f defines an isomorphism R(f) of the bundle of frames R(M) onto the bundle of frames R(M'): we have R(f)(r,) = T,(f) r, for all r, E R W , . 0

(a) Show that if u and u' are the canonical forms on R(M) and R(M), respectively, then 'R(f)(u') = u. Conversely, every isomorphism F of the principal bundle R(M) onto the principal bundle R(M') such that ,F(u') = u is of the form R(f). (b) Let P (resp. P') be a principal connection in R(M) (resp. R(M')) and let w (resp. w') be the I-form of the connection P (resp. P'). Then P' is the image of P under R(f) if and only if 'R(f)(w') = w. In this case f is said to be an isomorphism of P onto P'(or of C onto C', where C, C denote the linear connections defined by P, P'). 5. Let M be a pure differential manifold, X a C" vector field on M; let U be a relatively compact open subset of M , and a > 0 a number such that U x ] - a , a[ is contained in dom(Fx) (18.2.8). For t E ]-a, a[ and x E U, put g,(x) = Fx(x, t), so that g, is a

diffeomorphism of U onto an open subset U, of M. Then 8, defines an isomorphism GI = R(g,)ofthe bundleofframesn-'(U)onto the bundleofframesn-'(U,).Thepaths t H G , ( x ) are the integral curves of a vector field R on R(M), called the canonical lifting of X . Show that is the unique vector field on R(M) which is invariant under CL(n, R), is such that T(r) . &,) = X(x) for all x E M and all r, E R(M), , and for which fhmu= 0. The mapping X w f is a bijective homomorphism of the Lie algebra YA(M) onto a Lie subalgebra of FA(R(M)). For each point u, = r, . y of T,(M), where y E R" and r, E R(M), , the vector f ( r J * y depends only on u,. The vector field so defined on T(M) is precisely the canonical lifing of X (Section 18.6, Problem 3).


271

6. (a) Using the notation of Problem 5, show that for a C" vector field X on M and a

principal connection P in R(M), the following conditions are equivalent: (i) for each relatively compact open subset U of M, and each a > 0 such that U x ] - a , a[ c dom(Fx), and each t E ] - a , a [ , the isomorphism GI transforms the connection P [ r - ' ( U ) into the connection Plr-'(Ul); (ii) the canonical lifting f of X is such that 0% . w = 0, where w is the 1-form of the connection P; (iii) [ f ,Ha] = 0 for each a E R" (Problem 1); (iv) Ox V y -Vy & = V t x , for each vector field Yon M. (Use (18.2.14)J The field X is then said to be an infinitesimal automorphism of the connection P (or of M endowed with P). (b) For each C" vector field X on M, the mapping Y H e x Y - V x * Y of YA(M) into itself is a differential operator of order 0, hence is of the form Y HAx . Y, where Ax E 9 j ( M ) . Show that for X to be an infinitesimal automorphism of P, it is necessary and sufficient that 0

0

V y . A x = r * ( X AY )

(1)

for all C" vector fields Yon M (cf. (17.20.4.1)). If the connection is torsion-free, then

Ax. Y= -Vy* X,

The set of infinitesimal automorphisms of P is a Lie subalgebra a(P) of Yh(M). For two infinitesimal automorphisms X , Y of P we have A[x,yl = [ A X ,AYI

(c) In khe notation of (20.6.5), put

V c l . ej = so that we have Wjl

and

+r .( X A

xr

Y).

!jek

k

=qril

ok

V.,'oj=-xr{k~k. k

Deduce that the tensor A of type (1, 2) defined in (b) has components

+

A:, = -crjl t:J> in the notation of (20.6.6). (d) Show that when the connection is torsion-free, the equation (1) is equivalent to the system of scalar equations eel '

X,ah>

+cr!k -F

r:J

in the notation of (20.6.5) and (a) above, where the indices i, 1, h run independently from 1 to n. 7. (a) With the notation and hypotheses of Problem 5, let X be an infinitesimal automorphism of P. Show that if M is connected and if f vanishes at a point of R(M), Ha] = 0, show that Xis then X = 0. (Using the invariance of and the fact that [f, zero along each geodesic trajectory passing through a point x such that f ( r J = 0 for some r, E R(M),, and use (18.4.6)J

272


+ +

Deduce that dim(@)) 5 n(n 1) = dim R(M). (b) Show that if dim(@)) = n(n l), then the curvature and torsion of P are zero. (Observe that for each point r, E R(M) and each u E gL(n, R), there exists a unique ) that x(r,) =Zu(rx). Using the relations 02 + a =0 and element X E ~ ( P such [f,Ha]= 0, show that we have 0 i (8 * (HaA Hb)) = 0 in M, and in particular Oa(,x) . (e . (H, A Hb)) = 0,and therefore

-

02U(rx) '

(8 ' (Ha A

Hb)) = 0.

Next, show with the help of the structure equation that

02, ' (8 * ( H a h Hb)) = - U * ( 8' ( H a A

Hb))+

8

'([zu,

Ha1 A

Hb)

+0

'

(Ha A

[zu, H b l )

and deduce that W r , ) . (Ha@,) A Hb(r,)) = 0, by taking u to be the identity element of End(R"). Repeat the argument, replacing Q by w.) 8. A linear connection C on a connected differential manifold M (or the corresponding principal connection P ) is said to be complete if all geodesics for C are defined on the whole of R; or, equivalently (Problem I), if the integral curves of the fields Ha on R(M) are all defined on the whole of R (Section 20.2, Problem 3). For each a E R" and each r E R, let h,(r) denote the diffeomorphism r H FHa(r,1 ) of R(M) onto itself (18.2.2). Show that under these conditions, for every infinitesimal automorphism X

of M, the integral curves of the field f (and hence also those of X)are defined on the whole of R. (Use Problem 6 to show that F&(r)(r), t ' ) = ha(t)(Fx(r,1')) whenever both sides are defined. Using (18.4.6) and the connectedness of M, show that for each ro E R(M), each point r E R(M) can be written in the form r = (ha1(rd 0 haz(t2)0

* * * 0

hak(tk)k0)0

s

for suitable choices of a, E R", t~ E R, and s E GL(n, R). Then show that if Fi(ro, t ) is defined for It I < 8, then the same is true of Fi(r, r ) , and that Fdr, 1 ) = (h a~(ti)

'

"

hak(td)(Fx(ro, t ) 0 S).

9. Let M, M' be two pure differential manifolds and let P (resp. P') be a principal connection in R(M) (resp. R(M)). A mapping f :M M' is said to be a Iocal isomorphism of P into P' (or of M into M') i f f is a local diffeomorphism and if, for each x E M, there exists an open neighborhood U of x in M such that frestricted to U is an isomorphism of the restriction of P to U onto the restriction of P'tof(U). For this -+

to be so it is necessary and sufficient that f should be a local diffeomorphism and that T(R(f)) should map each horizontal tangent vector to R(M) to a horizontal tangent vector to R(M). We have then, for each x E M and all sufficiently small vectors u, E T,(M), f(exp(u,)) = exp(T(f) * 4.

(a) Suppose that M is connected, and let g be two local isomorphisms of M into M'. Show that if f ( x ) = g(x) and T,(f) = T,(g) for some point x E M, then f= g. (Consider the set of points at which fand g coincide, and use (18.4.6).) (b) Suppose that M and M' are real-analytic manifolds, and that the connections P, P' are analytic. Let f: M -+ M' be an analytic mapping; suppose that M is connected and that f is a local diffeomorphism. If there. exists a nonempty open subset U of M such that/[ U is an isomorphism of PI U onto P'lf(U). thenfis a local isomorphism. (Use Section 18.8, Problem 3.)


273

(c) Suppose that M and M’ are analytic, that the connections P and P’ are analytic and that P’ is complete. Let x E M, x’ E M’, and let U be a neighborhood of x which is the bijective image of an open subset of T,(M) under the exponential mapping, Iffis an analytic local isomorphism of a neighborhood V c U of x in M, show that fhas a unique extension to an analytic local isomorphism of U into M’, by considering TJf) and using (b) above. (d) Suppose that M and M’ are analytic, P and P’ are analytic, P’ complete, and M connected and simply-connected. Then every local isomorphism f of a nonempty connected open subset U of M into M’ has a unique extension to a local isomorphism of M into M’. (By using (c) above, extend f along a path with origin at a point xo E U, and show that its value at the endpoint of the path depends only on this point, by arguing as in (9.6.3).) (e) Suppose that M and M’ are analytic, connected and simply-connected, and that P and P’ are analytic and complete. Iff is an isomorphism of a nonempty connected open subset U of M onto an open subset of M’, thenfhas a unique extension to an isomorphism of M onto M‘. (Consider the inverse g of the isomorphism f and apply (d) tofandg.) (Cf. Section 20.18, Problem 13.) 10. Show that the group A(P) of automorphisms of a principal connection P in R(M),

where M is connected, is a Lie group acting differentiably on M . (Use Problem 4 above, and Section 19.10, Problem 6, applied to the parallelizable manifold R(M).) The Lie algebra of A(P) may be identified with a subalgebra of the Lie algebra a(P) of infinitesimal automorphisms of P (Problem 6), and is equal to a(P) if the connection P is complete (Problem 8). 11. Deduce from (19.4.4.3) that for the canonical form u on R(M) and for any u E gl(n, R), we have izu.du= - - u o u ,

u being considered as an element of End(R”). 12. Show that for each p-form a of class C1on M, we have (with the notation of (20.6.5))

if and only if the connection on M is torsion-free. (Prove that the right-hand side of ( I ) then satisfies the conditions of (17.15.2)J Deduce that when this is the case, the derivative da, considered as an antisymmetric covariant tensor of order p 1, may be identified with the antisymmetrization ( p ! ) - ’ a ( T a ) .

+

13. If two moving frames R , , Rz on a manifold M endowed with a linear connection C are such that ‘R,(o)= ‘Rz(u), show that R , = R Z . 14. Let M be a differential manifold, C a linear connection on M, and P the corresponding principal connection. Let c = (U, v, n) be a chart of M and let X i (1 5 i 5 n) be

the vector fields associated with this chart, forming a frame R = (A’,) over U. Writing ui and w I Jin place of ujR) and w$’, the local expressions of the u1 and the wIJin terms of the local expression (17.16.4.1) of the connection C are

274

XX


Likewise, writing 0,and in place of @in) and a!?, the local expressions of the 0,and the a,, in terms of the components t j k and rfJkof the torsion and curvature

tensors (17.20) are

15. Let M be. a differential manifold, C a linear connection on M and P the corresponding

principal connection. Let U be a neighborhood of a point xo E M which is of the form exp(V), where V is a starlike neighborhood of Ox, in Txo(M)in which the exponential mapping is a diffeomorphisrn onto U. If (c,)l d J d n is a basis of T,,(M), the coordinates with respect to this basis of the point exp;,'(x) for x E U are called the normal coordinates of x with respect to (cJ). (a) If

D

is the geodesic for C such that v(0) = xo and ~ ' ( 0=) h =

local expression of u in normal coordinates is

n

h,cl, then the

J= I

t H ( ho l d i s " in a neighborhood of 0 in R. Deduce that the local expression of the connection C , for rht normal coordinates in U, satisfies rjk(x0)

+ r:j(xO) = 0

for a11 i, j, k . In particular, if c is torsion-free, then rjk(x0) = 0 for all i, j, k . (b) Let ro be a frame at the point xo , and identify r o with a basis (c')~ of T,,(M). For each vector h E V , if f E R is such that th E V, then, putting x = exp(th), the frame R(x) is defined to be the value at t of the integral curve of the field Hb (Problem 1) with origin ro , where ro * b = h (in other words, the frame obtained by parallel transport of ro along the geodesic [Hexp([h) (0 5 4 5 I)). The moving frame R is said to be canonically associated with V and ro . Relative to the chart c = (U,9,n) of M, where p = ro expTO1,the canonical form and the connection form corresponding to R have local expressions of the form 0

Show that, for each point u E r;I(V), we have

(Use Problem I .) Relative to the same chart, let

6 THE METHOD

OF MOVING FRAMES

275

Show that

(I,,

w I I , 0 1 ,a,, under the mapping (Consider the inverse images of the forms . . ., u ” ) H ( ~ u ’ ., . . , run) of an open neighborhood of 0 in R’+l into rp(U), and write down the inverse images of the structure equations.) With the same notation, show that the derivatives at the point t = 0 of the funcRfkl(tu) are completely determined by the values at the tions rt-+Tjk(ru) and IH point xo of the covariant derivatives of all orders V m tand Vmr of the torsion and curvature tensors. (Use (20.5.3.4).) (1, u ‘ ,

16. Let M, M’ be two real-analytic manifolds endowed with linear connections defined by analytic principal connections P, P’. Let V , V’ denote the covariant derivatives corresponding to P, P’,respectively, and let t, t’ be the torsion tensors and r , r‘ the curvature tensors of M, M’, respectively. Let F : T,,(M) -.T,,(M‘) be a bijective linear mapping such that the image of each of the tensors Vmt(xo),Vmr(xo)under the corresponding extensions of F is V””t’(yo), V”V’(yo), respectively. Then there exists a neighborhood U of xo , a neighborhood V of yo and an analytic isomorphism fof PI U onto P’(V such that f ( x o ) = yo and T,,(f) = F. (Use Problems 15 and 13.)

17. Let M, M’ be two differential manifolds endowed with linear connections corresponding to principal connections P, P’. Let V , 0’ be the corresponding covariant derivatives, t, t’ the torsion tensors, and r, r’ the curvature tensors. Suppose that V t = 0. V’t‘ = 0, V r = 0, V’r’ = 0 . Let F : T,,(M) + T,,(M’) be a bijective linear mapping such that the image of t ( x o ) (resp. r(xo)) by the corresponding extension of F is t’(yo) (resp. r’(yo)).Show that there exists a neighborhood U of xo , a neighborhood V of yo and an isomorphism f of PI U onto P ’ / V such that f ( x o )= yo and T,,(f) = F. (Use Problems 13 and 15.) 18. Let M be a connected differential manifold. A linear connection C on M is said to be invariant under parallelism if, for any two points x , y of M and any C” path y from x to y in M (16.26.10), there exists a neighborhood U of x and an isomorphismfof U (for the connection C) onto a neighborhood of y, such that f ( x ) = y and such that T,(f) coincides with the parallel transport along (18.6.4). Show that for this to be the case it is necessary and sufficient that V t = 0 and V r = 0, where t, r are the torsion and curvature tensors of C. (Use Problem 17.) 19. Let M be a pure differential manifold of dimension n, and let R(M) be the bundle of tangent frames of M, with group GL(n, R). Let A(M) = Aff(M x R“,T(M)) be the bundle of affine frames of T(M), which is a principal bundle with group A(n, R). Recall (Section 20.1, Problem 1) that A(n, R) is canonically isomorphic to the semidirect product Rnx GL(n, R), where p is the natural action of GL(n, R) on R”.

276

XX


(a) Show that there exists a canonical isomorphism of R(M) x GL(n*R)A(n,R) onto A(M), under which the product r*(:

$3

where r E R(M), ,

+

corresponds to the affine-linear bijection y++r(U y a) of R" onto T,(M). (b) If we identify A(M) with the bundle R(M) x GL(n*R)A(n, R) obtained by extension of the structure group from CL(n, R) to A(n, R), the associated bundle

X = R(M) X GL(n.R)(A(n,R)/CL(n, R)) = R(M) X CL(n*R)Rn is canonically identified with T(M): the product r . y (where r E R(M), y E R") corresponds to the product given by the same notation in T(M). The canonical section of X (Section 20.1, Problem 2) is identified with the zero section of T(M). There exists a canonical welding h,++r, *(r;' . h,) of M into X (r;' * h, E R", for h, E T,(M) and r, E R(M),), which is identified with the identity mapping of T(M). (c) For each principal connection Po in R(M), show by using Section 20.5, Problem 4, that there exists a Cartan connection P on A(M) from which Po is obtained by the procedure described in that Problem. The connection P, for which the form cp is the canonical form a of (20.6.2), corresponds to the canonical welding defined in (b) above. This connection on A(M) is called the afine connection on M associated with the principal connection Po (or with the linear connection Co corresponding to Po). The other principal connections P on A(M) which correspond to Po are in one-to-one correspondence with the C" mappings r -u(r) of R(M) into End(R") = M.(R) such that u ( r . s) = s u ( r ) s-I for all s E GL(n, R). Hence derive another proof of the structure equation (20.6.3.1). 0

0

20. (a) The manifold RI(R") = R(R") of frames of order 1 in R" (Section 20.1, Problem 3) may be identified with the set of affine-linear mappings XH U . x a of R" into R", where a E R" and U E CL(n, R) (the jet of order I at the point 0 of a C" mapping fof a neighborhood of 0 into R" is identified with tht affine-linear mapping

+

h ++ Df(0) h

+f ( 0 ) ) .

Consequently RI(Rn) may be canonically identified with the affine group A(n, R) (Section 20.1, Problem 1). Show that this identification is an isomorphism of principal bundles with structure group GL(n, R) (where GL(n, R), as a subgroup of A h , R), acts on A(n, R) by right translation). From now on, we shall identify R,(R") with A(n, R) in this way: the identity element e of A(n, R) is therefore identified with the jet J d ( l p ) , and the tangent space to Rl(R") at the point e with the Lie algebra a h , R) of the group A(n, R), which can be canonically decomposed into R" 3 nl(n, R). (b) Let M be a pure n-dimensional differential manifold. Let p : R,(M) -+ R,(M) = R(M) and p : G2(n)-+G'(n)= GL(n, R) denote the canonical mappings (Section 20.1, Problem 3). Let f be a diffeomorphism of a neighborhood of 0 in R" onto a neighborhood of f(0) in M. By transport of structure, f defines a diffeomorphism R,(f) of a neighborhood of e in R,(R") onto a neighborhood of Jd(f) in R,(M), so that we have Rl(f)(u) = JA(f) u. Hence we obtain an isomorphism /"= T,(Rl(f)) of T,(R,(Rn)) = a(n, R) onto the tangent space Tjd,,)(R1(M)). For each tangent vector h to R2(M) at the point Ji(f), the vector T(p). h is a tangent vector to RI(M) at the point Jb(f), and thereforef-'(T(p). h) is a vector in the Lie algebra a(n, R). 0


277

Show that it depends only on the point Ji(f) and the tangent vector h at this point to Rz(M), and therefore defines a canonical vector-valued 1-form

3:hnf-'(T(p) on R,(M). The diagram

T(RAM ))

a(n, R)

T(p)

h)

T(RI( MI)

pr L

*

R"

where Q is the canonical form (20.6.2.1), is commutative. (c) Consider the case where M = R". The manifold of frames Rz(R") may be identified with the set of all mappings of R" into R" of the form x-a U . x B . (x, x), where a E R", U E GL(n. R) and B : (h, k ) n B . (h, k) is a symmetric bilinear mapping of R" x R" into R" (the jet of order 2 of a C" rnappingfof a neighborhood of 0 into R" is identified with the mapping hwf(0) + Df(0) . h iD'f(0) . (h, h) (8.14.3)). For brevity we shall denote this mapping by (a, U, 8). The projection Rz(R") + R" is then the mapping (a, U,B)H a; the group G2(n)is identified with the submanifold of Rz(R") defined by a = 0; and the right action of G2(n)on R2(Rn) is given by

+

+

+

(a,

U,6 ) . ( V ' , B') = (a, UU',U .B' i B . ( V ' , U')),

where UU' is the product of the matrices, U .B' is the quadratic mapping x w U . ( B ' . ( x , x ) ) , and B . ( U ' , U ' ) i s t h e q u a d r a t i c m a p p i n g x ~ B . ( U ' . x ,U'.x).

If e denotes the jet J:(lRn), the tangent space T,(R,(R")) may be identified with the vector space of all (v, V, W)), where v E R", V E End(R") = MJR), and W runs through the subspace of the vector space YJR", R"; R") consisting of symmetric mappings. I f f is a diffeomorphisrn of a neighborhood of 0 into R", then f defines (again by transport of structure) a diffeomorphism R,(f) of a neighborhood of e in R2(Rn) onto a neighborhood of J;(f) = (a, U,B), by the formula R2(f)(u) = Ji(f) 0 u. Show that the image under Te(Rz(f)) of the tangent vector (v, V, W) is of the form h = ( U . V, U V + B . ((v, . ) + (., v)), U .w

+ B . ( ( V , f )t-(1, v)))

and consequently that

T(p). h = ( U .V, U V S B * ((v, .)

+ (., v))).

Changing the notation, deduce that the value at the point (a, U,B) of the canonical I-form 3 is identified with the mapping (v, V, W ) - ( U - ' * v , U - ' V + U-'(B.((v,.)+(.,v)))).

(d) Suppose again that M is arbitrary. Show that for each element w of the Lie algebra 02(n) of G2(n),we have 3 . Z,,, = p,(w), and that for each s E G'(n), a ( u . s) * (h . S) = Ad(p(s-')) . ( B ( u ) . h)

for all

II E

R,(M) and all tangent vectors h E T,(Rz(M)).

278

XX


(e) Take as basis of a(n, R) the union of the canonical basis (0,) of R" and the canonical basis (&,) of d(n, R) = MAR). Then the canonical 1-form 8 on R,(M) may be written as

where the u, and us are scalar-valued differential 1-forms on R2(M). Show that

do' = (Reduce to the case M

= R",

-?

U:

A uJ.

and use ( 4 . )

7. G-STRUCTURES

(20.7.1) Let X be a principal fiber bundle with base B, structure group G , and projection p . Let H be a closed subgroup of G , a n d j : H 4 G the canonical injection. If there exists a principal bundle Y with base B, structure group H and projection q, and a morphism (u, j ) of Y into X such that u is a Bmorphism of the fibration (Y, B, q) into (X, B, p) (16.14), then u is an embedding of Y into X, and the image of u is a closed submanifold of X. Since the question is local with respect to B, we may assume that X = B x G and Y = B x H are trivial; u is then of the form (b, t ) H ( b , f ( b ,t)), where f i s a C" mapping of B x H into G such that

f ( b , tt') = f ( ht ) j ( t ' )

and consequentlyf(b, t ) = g ( b ) j ( t ) ,putting g(b) =f(b, e). Since the mapping u :(b, s ) H ( ~g(b)-'s) , is a diffeomorphism of B x G onto itself (16.12.2.1), and since u 0 u is the canonical injection (6, t ) w ( b ,j ( t ) ) , our assertion is proved. A principal bundle Y and a morphism ( u , j ) with the properties above are said to constitute a restriction of the principal bundle X to the group H. We shall generally identify Y with its image under u (cf. Problem I). (20.7.2) Consider a pure differential manifold M of dimension n, and its bundle of frames R(M), with structural group GL(n, R). For a closed subgroup G of GL(n, R),a G-structure on M is by definition a restriction S,(M) of the principal bundle R(M) to the group G (S,(M) being identified with a closed submanifold of R(M)). Let u : M + M' be a diffeomorphism. It gives rise canonically to an isomorphism of principal bundles (16.14) R(u) : R(M) + R(M'), which maps a frame r E Isom(M x R",T(M)) to the frame

T(u) r (u-' x 1) 0

0

E

Isom(M' x R",T(M')).

7 G-STRUCTURES

279

Given a G-structure Sc(M) on M and a G-structure S&(M') on M', the diffeomorphism u is said to be an isomorphism of SG(M) onto Sb(M') if the image under R(u) of S,(M) is Sb(M'). For each s E GL(n, R), &(M) * s is an (s-'Gs)-structure on M,because S,(M) is stable under the right action of G on R(M). The structure SG(M) * s is said to be conjugate to the G-structure SG(M). (20.7.3) If Sc(M) is a G-structure on M, the reasoning of (20.1.4) applies without change and shows that T(M) is canonically isomorphic to the fiber bundle S,(M) x c R" with fiber-type R", associated with SG(M). Given a principal connection P in Sc(M), we may therefore construct canonically by the procedure of (20.5.1) a linear connection C on M. Such a connection is called a G-connectioH associated with the G-structure S,(M). A C" section of Sc(M) is called a mouing G-frame. Everything in Section 20.6 remains valid for G-connections if we bear in mind that the connection form o and the curvature form R take their values in the Lie algebra ge of G. Conversely, we have seen that every linear connection C on M determines a unique principal connection P in R(M) (20.5.2). Since P is completely determined by knowledge of the space H, of horizontal tangent vectors at each point r of R(M) (20.2.2), it follows from the definitions that for C to be a G-connection, associated with the G-structure SG(M), it is necessary and SUBcient that for each frame r E SG(M), the space H, should be contained in the tangent space to Sc(M) at the point r, for the restriction of P to T(M) x SJM) will then be a principal connection on S,(M) (20.2.2).

Examples of G-structures (20.7.4) First take G = {e}: then a G-structure may be identified with the image of a C" section of the principal bundle R(M) over M. The existence of such a section is equivalent to R(M) (and therefore also T(M)) being trivializable (16.14), in which case the manifold M is said to be parallelizable, and an {e}-structure on M is called a parallelism (or total parallelism). For example, a Lie group H is parallelizable: a C" section X H R,(x) of R(H) may be obtained by taking for R,(x) the isomorphism UHX * u of the Lie algebra be onto T,(H). Another section Rd may be obtained by taking for &(x) the isomorphism u Hu x . There is here a unique {e}-connection, for which o = 0 and therefore = 0 and 0 = du. (20.7.5) Take G = SL(n, R): if u* is the n-covector e: A e; A * A e: on E = R",then G may be defined as the subgroup of all s E GL(n, R) such that n

A ('s)

u* = u*. Let S,(M) be a G-structure on

M ; we shall show that there

280

XX


corresponds to it a C" differential n-form u on M such that u(x) # 0 for all x E M ( a "volume form," cf. (16.21 .l)). For this purpose, let rx E SG(M),: since r, is an isomorphism of E = R" onto T,(M), it follows that

n

A ('r;')

is

-

an isomorphism of A (E*) onto A (T,(M)*), and we put u(x) = A ('r;') u*. This n-covector does not depend on the element rx chosen in the fiber S,(M), , because any other element of the fiber is of the form rx s with s E G, and we have n

n

n

by the definition of G. By reducing to the case where R(M) is trivial, it is immediately seen that the n-form u is of class C", and moreover it is clear that u(x) # 0 for all x E M. Conuersely, if such an n-form is given, we define SG(M), to be the set of all r, E R(M), which satisfy the relation

A ('r;') n

*

u* = u(x);

it is immediately verified that the set S,(M) so defined is a restriction of R(M) to the group G = SL(n, R), and therefore we have established a one-toone correspondence between SL(n, R)-structures and volume forms on M. In particular, the existence of an SL(n, R)-structure on M is equivalent to M being orientable (1 6.21.1). (20.7.6) Let Q, be a symmetric or alternating nondegenerate bilinear form on R", and take G to be the subgroup of GL(n, R), leaving this form invariant (16.11.2). If Q, is symmetric and of signature ( p , q ) , to be given a G-structure on M is equivalent to being given a symmetric covariant tensor$eId xt+g(x) of class C" such that, for each x E M, the symmetric bilinear form

(hl, h,)H

(B(4,

hl @ h2)

on T,(M) is nondegenerate and of signature ( p , 4). Namely, proceeding as in (20.7.5), we take an element r, E S,(M), and define g by the formula (g(x), h, C3 h2) = Wr;'

- h,, r;'

*

h2)

for h,, h, ET,(M): in other words, g(x) = T : ( r , ) . 0.As in (20.7.5) it is easily checked that this does not depend on the choice of r, in S,(M),, and that g is a C" tensor field of the type specified above. Conversely, we may define as in (20.7.5) a G-structure corresponding to such a tensor field. The procedure is the same when Q, is alternating: the assignment of a G-structure on M is in this case equivalent to that of a C" differential 2$orm on M, nondegenerate at every point.

7 G-STRUCTURES

281

Up to equivalence (20.7.2), we may always assume that

with cj = 1 for j S p and cj = - 1 for j > p , when @ is symmetric, and

c m

WX, Y)

=

j=1

( t j qj+m -tj+rn

Vj)

when n = 2m and 0 is alternating. In the first case, a G-structure on M is said to be pseudo-Riemannian of signature ( p , q ) (and Riemannian when all the cj are equal to + 1, i.e., when the signature is (n, 0)); in the second case, the structure is said to be almost-Hamiltonian. The remainder of this chapter will be devoted to the study of pseudo-Riemannian and Riemannian structures. We remark that on every pure manifold M of dimension n there exists a canonical almost-Hamiltonian structure on the cotangent bundle T(M)*, defined by the canonical (nondegenerate) 2-form -dKM (17.15.2.4). (20.7.7) Suppose n even, say n = 2m. Identify R2" with C", and let J be the endomorphism z w i z of C", considered as an endomorphism of the real vector space R2" (if (ei)lsks,,,is the canonical basis of C", and if we take ek= e ; , em+,= i e ; , then J is the endomorphism defined by

J . ek= elr+",J . ek+" = -ek

( I 5 k 5 m)).

We have J 2 = -Z, where I is the identity automorphism of R". The group G = GL(m, C ) may be considered as the subgroup of GL(n, R) consisting of all s E GL(n, R) which commute with J, and the assignment of a Gstructure is equivalent to that of a tensorfield x H j ( x ) in F i ( M ) , such that for each x E M the tensor j ( x ) E T,(M)* 0 T,(M), regarded as an endomorphism J, of T,(M), is such that J,' = -Z,, where I, is the identity automorphism of T,(M). Proceeding as before, we define J, by the condition J, h, = r, * ( J * (r; h,)) for some rx E S,( M), and all h, E T,('M). The details are left to the reader. A G-structure for this group G = GL(m, C ) is called an almost-complex structure on M. Clearly, the differential manifold underlying a pure complex-analytic manifold M of (complex) dimension m is canonically endowed with such a structure, for which J, is the endomorphism h,Hih, of the tangent space TJM) (which is canonically endowed with a structure of a complex vector space of dimension m). But there exist almost-complex structures on real differential manifolds that do not arise in this way.

282

XX


(20.7.8) Let F be a p-dimensional subspace of R", and let G be the subgroup of GL(n, R) which stabilizes F. Suppose we are given a G-structure SG(M) on M ; then for each x E M and rx E S,(M), , r, * F = L, is a p-dimensional subspace of T,(M) which does not depend on the frame r, chosen in SG(M), . It is straightforward to verify that X H L, is a C" field of p-directions on M (18.8) and that conversely such a field determines uniquely a G-structure on M. (20.7.9) Let G be a Lie group, H a Lie subgroup of G, so that G is a principal bundle over G/H with structure group H (16.14.2). Let 7c : G H G / H be the canonical projection, and put xo = x(e). The tangent space T,,(G/H) is the image under Te(n) of T,(G) = ger and since the kernel of T,(n) is Ij,, we may canonically identify ge/Ije with T,,(G/H). Since G acts transitively on G/H by left translations, for each s E G there is a canonical bijection r(s) : k H s k of ge/Qeonto Ts.xo(G/H).In order that r(s) = r(s'), first of all it is necessary that s' xo = s . x o , i.e., that s' = st for some t E H; next, if we denote by p ( t ) the automorphism k w t * k of gel$, for t E H, it is necessary that p ( t ) = 1 : in other words, t must belong to the kernel N of the homomorphism p of H into GL(g,/$,). If H = p(H) c GL(g,/Ij,), it is immediately verified that the set Sfi(G/H) of all frames r(s), as s runs through G, is a restriction t o H of the frame bundle R(G/H), that is to say, it defines an H-structure on G/H.

-

-

(20.7.10) Canonical linear connection on a symmetric homogeneous space.

With the notation of (20.7.9), suppose that G is connected and that (G, H) is a symmetric pair (20.4.3) corresponding to an involutory automorphism cr of G . Suppose moreover that G acts faithfully on G/H, or equivalently, that the intersection of the stabilizers sHs-' of the points of G/H is reduced to e, or equivalently again, that H contains no normalsubgroup of G other than {e}. Let in be the set of all vectors u E ge such that aJu) = - u, so that in is a supplement of Ije in g, and may be canonically identified with g,/$,, by projection parallel to 5,. Let us first show that, with this identification, the automorphism k w t k of ge/Ije. where t E H, is identified with the automorphism u ~ A d ( t* )u of in. Indeed, if k is the coset of u modulo $,. then f . k is the image under T(n) of f u E T,(G), and this image is the same as that o f t u - t - ' = Ad(t) * u E in. This being so, the fact that G acts faithfully on G/H implies that the homomorphism p of H into GL(g,/Ij,), defined in (20.7.9), is injective. For it may be identified with the homomorphism t H A d ( t ) of H into GL(in); if N is its kernel, then N centralizes exp(in) (19.11.6); and since exp(Ij,) c H normalizes N, it follows that exp(Ij,) exp(in)

7 G-STRUCTURES

283

normalizes N. Since exp(lj,) exp(tn) generates the connected group G (1 9.9.14), we see that N c H is normal in G, hence N = {e} by hypothesis. The mapping st+r(s) of G into the bundle of frames R(G/H), defined in (20.7.9), is therefore an isomorphism of the principal bundle ( G , G/H, n) onto (Sfi(G/H), G/H, no), where no denotes the restriction to Sfi(G/H) of the projection of R(G/H) onto G/H. By virtue of the discussion above, we may therefore canonically identify T(G/H), considered as a vector bundle associated with Sfi(G/H), with the vector bundle G x m associated with the principal bundle (G, G/H, 11)by the action ( t , u)HAd(t) * u of H on in. We may then construct, by the method of (20.5.1), from the canonicalprincipal connection P in G (20.4.3) a linear H-connection C on G/H. This connection C is called the canonical linear connection on the symmetric homogeneous space G/H. It has the following remarkable properties: (20.7.10.1) The connection C is invariant under G (acting by left translations on G/H) and under the involutory difleomorphism g o of G / H onto itself defined by ao(n(s)) = n(a(s)) for s E G (recall that the elements of H are fixed by d.

This follows from the definition of C (20.5.1.1) and the fact that P is invariant under G and under a. To avoid any confusion resulting from the identifications that have been made, if u E in and s E G, then s . u shall denote the vector in T,(G) defined in (16.9.8), so that T(II)(s u) is the vector s k in T,..,(G/H) if k is the coset of u in T,,(G/H). We have therefore, by (20.4.3), (20.7.1 0.2)

P,.,,(T(n)

*

(S

*

u), S)

=s

*

u E T,(G)

for s E G and u E in. From this we deduce (20.5.1 .I)

for s E G and u, y E in, the product on the right-hand side being that defined in (16.14.7.3) for the associated bundle G x in. (20.7.10.4) The geodesic v for C such that v(0) = xo and v’(0) = u E m (18.6.1) is given by v ( t ) = n(exp(tu)) and is defined for all t E R. The parallel transport along v of a vector y E m (1 8.6.4) is given by w ( t ) = T(II) * (exp(tu) * y).

284

XX


If u(t) = n(exp(tu)), then (19.8.1) u'(t) = T(n)* (u(t) * u). On the other hand, if cp is the mapping s ~ T ( n )(s * y) of G into T(G/H) = G x H in, then by definition (16.14.7.3) (s * u) y = T(cp) * u, and therefore w'(t) = (exp(tu) u) * y. These formulas and (20.7.10.3) show that w ' ( 0 = C"(,)(U'(t),w(t)),

and when y = u, we have w(t) = u"(t); hence u"(t) = Cv(t)(u'(t), u'(t)), and the result follows ((18.6.1) and (18.6.4)). (20.7.10.5) The torsion of C is zero, and its (G-invariant) curvature is given by

for u, v, w in in = T,,(G/H). (20.6.6.1l), The formula (20.7.10.6)followsimmediatelyfrom(20.4.3.3)and applied to the principal connection P. Furthermore, the torsion t is invariant under go, that is to say, t(X0)

*

(T(a0)

U A

T(a0) . V) = T(g0) *

.

( t ( x 0 ) ( U A V)).

Since T(ao) y = - y for all y E m, it follows that r(xo) = 0; hence t = 0. (20.7.10.7) Euery G-invariant tensor field Z on G/H satisfies VZ = 0. In particular, the curvature tensor of C satisfies Vr = 0.

If E = T:(G/H), then by (20.5.4) E may be identified with the vector bundle G x (T:(m)) associated with G, and a tensor field ZE.T~(G/H)may then be written in the form Z(s . xo) = s * Q,(s), where Q, is a mapping of G into T:(in) (20.5.3). The G-invariance of Z is then expressed by the relation

"

s' * Z(s ' xo) = Z(S'S * xo), which implies that Q,(s's) = Q,(s), so that from (20.5.3.4) that V Z = 0.

Q, so

constant on G. It now follows

Remark (20.7.11) If &(M) is a G-structure on M, and if G' =I G is a subgroup of GL(n, R),then we may obtain canonically from SG(M) a G'-structure SG,(M) by taking the frames belonging to S,,(M) to be the frames r e d , where s' E G' and r E SG(M): it is immediately verified (by reduction to the case where R(M) is trivial) that we obtain in this way a restriction of R(M) to the group G'. A linear G-connection on M is also a G'-connection. (20.7.12)

We have seen in examples above that, for a given Lie subgroup G

7 G-STRUCTURES

285

of GL(n, R), there need not exist a G-structure on M: the question depends in general on global topological properties of M. However, it is clear that if G is parallelizable (20.7.4), we can define a G-structure on M for every subgroup G of GL(n, R), by virtue of (20.7.11). Furthermore, in this case there always exist G-connections associated with a given G-structure. For if m is an arbitrary differential I-form on M, with values in the Lie algebra ge c gI(n, R), then there always exists a connection form w on S,(M) such that dR1 = m for some section R of S,(M) over M. Another important case in which no global topological condition is necessary is the case of Riemannian structures: (20.7.13) There exists a Riemannian structure on every pure differential manifold M .

Consider a denumerable family of charts c, = (U,, cp,, n ) of M such that the U, form a locally finite covering of M, and let $, : U, x R" -,oi'(U,) be the framing of T(M) over U, associated with the chart c,. We define a tensor field g, on U, which gives a Riemannian structure on U, by the formula (g,(x), $Ax, u) 0 $,kv)) = (u I v) for all x E U, and u, v E R", where (u I v) is the Euclidean scalar product on R".Let (h,) be a partition of unity subordinate to (U,),each h, being a C" mapping of M into [0, I ] (16.4.1), and put g(x) =

c h,(x)g,(x) (I

for each x E M (with the convention that h,(x)g,(x) = 0 for x 4 U,). Then g is a C" tensor field on M, because every frontier point of U, has a neighborhood on which h,(x) = 0. We assert that g defines a Riemannian structure on M. It is enough to show that if x E U,, we have (g(x), $&,

u) 0 $Ax,

4)> 0

for all u # 0 in R".Now if is an index such that h,(x) # 0, then we may write $,(x, u) = $&x, AfJ,(x) u), where AB,(x) E GL(n, R) (16.15.1 .l); consequently, 3

( g ( x ) , $,CG u) 0 $a(& =

4)

c hfJ(x)(g&). dgx1 AfJ,(X> 4 €3Il/&x, AfJ,(X) *

*

u)>

B

summed over all B such that h,(x) # 0, the norm on R" being the Euclidean hB(x)= I and IIAa,(x) . u1I2 > 0 for the indices 3/ norm. Since we have

1 B

under consideration, our assertion is proved.

286

XX


PROBLEMS 1. If Y is a restriction of a principal bundle X with group G to a closed subgroup H of G (20.7.1), then X is canonically isomorphic to the extension Y x G of Y (Section 16.14, Problem 17), and the associated bundle X x (G/H) = H\X (16.14.8) is isomorphic to Y x (G/H) (Section 16.14, Problem 16), so that X x (G/H) admits a C" section over B (Section 20.1, Problem 2). Conversely, if X x (G/H) admits a C" section over B, then there exists a restriction of X to the subgroup H, and these restrictions are in one-to-one correspondence with the C" sections of X x (G/H) over B. (If u is such a section, consider the inverse image of a(B) under the mapping

. P of X onto X x (G/H).) Consider in particular the case where G/H is diffeomorphic to R", and hence deduce another proof of (20.7.13). (Cf. Section 11.5, Problem 15.)

XHX

2.

Let K be a Lie group, H a Lie subgroup of K ; let M = K/H be the corresponding homogeneous space, T : K + M the canonical projection, and x o = ~ ( e )Suppose . that the homomorphism p of H into GL(T,,(M)) (20.7.9) is injective, so that K acts freely on the left on the bundle of frames R(M) (20.7.9): if s E K and r E R(M), so that r is a linear bijection of T,,(M) (identified with R")onto T,(M), the product s * r is the bijection k w s * r ( k ) of T,,(M) onto TS.JM), and we haves . ( r . p(r)) = ( s . r ) . p ( t ) for all t E H. If G is a Lie subgroup of GL(n, R), a G-structure S,(M) C R(M) is said to be K-invariant if for all r E Sc(M), s E K and t E G , we have s . ( r . 1 ) = (s . r ) . t . Let ro be an element of R(M),,; then for each t E H there exists a unique element h(r)E G such that t . ro = ro . h(t),and from the definition of p we have h(t)= r;' p ( t ) ro , so that h is an injective homomorphism of H into G. 0

0

(a) There exists a canonical one-to-one correspondence between the set of K-invariant principal connections in Sc(M) and the set of linear mappings f : i, + oe (the Lie algebras of K and G, respectively) such that: (i) f(w) = h,(w) for w E be (the Lie algebra of H); (ii) f(Ad(t) . w) = Ad@(()) .f(w) for w E f, and t E H. (Cf. Section 20.3, Problem 3(b).) (b) Show that the torsion and curvature of the linear connection C on M corresponding to the linear mapping f a r e given by the following formulas: t . (Z,,(XO) A ZJXO)) = ro . ( f W ' ( r o ' ~ Z v ( x 0 N -ro * (f(v) . (6'.Z,,(x0))) - Zcu, vl(~o)r

r * (Z&O) A Z,(XO)) = T o (Nu), f(v)l - f([u, vl)) r; 0

0

'

for any two vectors u, v E €=, where Z,, is the Killing field corresponding to u on M = K/H. (c) With the same notation, show that the covariant derivative (relative to C ) is given by the formula Vzv(xo). Z,, = ro . (fW . (r;'

. Z,,(x0)N + Zc,,,vl(xo).

3. The hypotheses are the same as in Problem 2. Suppose in addition that there exists a subspace in of ie supplementary to [I,, such that Ad(t) . m c m for all t E H (Section 20.4, Problem 3). Then the K-invariant principal connections in SG(M) correspond one-to-one to the linear mappings f,,, : m + oe such that fm(Ad(t)

-

W) = Ad(h(t))

*

f,(w)

7 G-STRUCTURES

287

for t E H and w E ni (loc. cit.). The formulas in Problem 2(b) then become, if we identify the vectors u in tti with the tangent vectors Zu(xo)E T,,(M), ?'

r

(U

A

V)=fm(U).

. (U A V) = [fm(U),

V--f,,(V)

fm(v)I

U-

[u, v],,

- fm([U, Vlm)

- A,([% VIb,)

for u, v E m, where [u, vl,, and [u. vIb, are the components of [u, v] in m and fj,, respectively, for the direct sum decomposition fe = m @be. The connection in SG(M) corresponding to f, = 0 is called the canonical connection (for the choice of subspace m supplementary to lie). For u E t, let 2, denote the canonical lifting to SG(M) of the vector fidd Z , (Section 20.6, Problem 5). Then the relation f,(u) = 0 for some u E m signifies that the vector &(x0) is horizontal at the point ro . Deduce that the canonical connection in SG(M)is the only K-invariant connection such that for all u E m, if we put gt(xo)= exp(tu) . xo and G , = R(gJ (loc. cit.), the orbit f b G l ( r o ) is the horizontal lifting of ft-+g,(xo)that passes through ro (Section 20.2, Problem 3). The paths t H g t ( x O ) corresponding to the vectors u E n~ are geodesics for the corresponding linear connection C on M; this connection is complete (Section 20.6, Problem 8), and we have V U = 0 for each K-invariant tensor field U on M. Show that the K-invariant principal connections in SG(M)for which the geodesics for the corresponding linear connection on M are the same as for the canonical connection, correspond to the mappings f, such that f,(u) . u = 0 for all u E m. In particular, there is just one of these linear connections which is torsion-free, and it corresponds to the mapping f, defined by f,(u) . v = #[u, v], for u, v E m. Consider the case where K = L x L, L being a connected Lie group and K acting on L by the rule ((s, r ) , x ) H s x ~ - ' ,so that L may be identified with K/H, where H is the diagonal of L x L. We may then take 111 to be any one of the subspaces (0) x I,, I, x (0}, or the image of L, under the mapping UH(U, -u), where I. is the Lie algebra of L. Calculate the torsion and curvature of the canonical connections corresponding to these three choices of m. 4.

Let M be the complement of the origin in R".The group GL(n, R) acts transitively on M, so that M may be identified with the homogeneous space K/H, where K = GL(n, R) and H is the subgroup which fixes some point #O. The restriction to M of the canonical linear connection on R" is K-invariant, but there exists no subspace i n supplementary to 11, in f, such that Ad(f)ni = m for all t E H.

5. Let M be a pure differential manifold of dimension n, let G be a Lie subgroup of GL(n, R), let (1 be its Lie algebra, and let &(MI be a G-structure on M. We shall denote again by a the restriction ?(a) to SG(M) of the canonical form u on R(M) ( j : S,(M)+R(M) being the canonical injection); a is therefore a vector-valued 1-form on SG(M)with values in R", which vanishes on vertical tangent vectors. Let r be a point of SG(M)and let H I , H2be two subspaces of T,(SG(M)), both of which are supplementary to the space of vertical tangent vectors at the point r, so that the restrictions to H I and H2 of T(T) (where T : SG(M)+ M is the canonical projection) are isomorphisms onto Tzc,,(M). For each vector x E R",let k,, k2 be the vectors in H I , H , , respectively, such that T(n). k, = T(T) . k2= r . x, or equivalently, such that a(r). kl = a ( r ) . kz= x. We may therefore write kl - k2= ZT(x)(r), where T : R"+ (1 is a linear mapping uniquely determined by HI and HI.

288

XX


For i = 1 , 2 and x, ~ E R "let , S,(x A y) = d a ( r ) . (hi A k,) E R",

where h,, k, E Hi are such that a ( r ) . h, = x, a ( r ) . ki = y. Use Section 20.6, Problem 11 to show that

(*I

SdX A y) - Si(x A y) = T ( x ) ' y - T(y) * x

(where a is identified with a Lie subalgebra of al(n, R) = End(R")). Let

a : Hom(R", o)

+ Hom(

R",R")

be the linear mapping defined as follows: if T EHom(R", o), then 6 7 is the linear 2

mapping of A R" into R" induced by the alternating bilinear mapping (x, Y)H T ( x ). Y - T(Y). x. It follows from (*) that the class of SI ~ H o m ( R", i R") modulo the subspace a(Hom(R", 0)) depends only on the point r E SG(M) and not on the choice of the subspace H I supplementary to Ker(T,(rr)) in T,(SG(M)). This class c(r) is the value at the point r of what is called thefirst-order structure function of the G-structure SG(M). The group G C GL(n, R) acts in its natural way on R",and on 0 via the adjoint

r

1

representation. Hence by transport of structure it acts on the left on Hom A R", R" and on Hom(R", a). Show that, for each s E G and each T EHom(R", (I), we have

a(s. T ) = s . aT, so that G acts linearly on H o m ( i R", Rn)/i(Hom(Rn,a)). If p is the corresponding linear representation of G, show that c(r . s) = p ( s - ' ) . c(r).

6.

With the hypotheses and notation of Problem 5 , there exists an SG(M)-morphisrn h : ( r , x) w h , . x of SG(M) x R" into the tangent bundle T(SG(M)) such that the image under h of { r } x R" is a subspace H, of T,(SG(M)) supplementary to the subspace of vertical tangent vectors. If E is a subspace of H o m ( i R", Rn) supplementary to a(Hom(Rn, a)), we may also assume that, for all rESc(M), the element E(r) of Horn(:, R", R") defined by E(r). (x A y) = d a ( r ) . (h A k) (where h, k E H, are such that a ( r ) . h = x, a ( r ) * k = y) belongs to E (by adding to P(r) if necessary an element aT,E a(Hom(R", a)), which is of class C" as a function of r ) . We then identify the coset c(r) with its representative P(r) E E. Let G") be the commutative subgroup of GL(R"$Q) consisting of automorphisms of the form (x, U)H(X, T ( x ) u), where T runs through the subspace f i ( I ) = Ker(a) of Hom(R", 0). For each frame r E Sc(M), consider the frames in the tangent space to So(M) at the point r (i.e., isomorphisms of R" @ n onto T,(SG(M))):

+

r ( ' ) : (x, U)H h, . x

+ Z&) + ZT[,&),

where T runs through the subspace 0"'. These frames constitute a G(')-structure SG(I)(SG(M))on SG(M), called the first-order prolongation of the G-structure S,(M) on M. If we replace E by another supplement E' of a(Hom(R", 0)) in H o m ( i R", R") and h by a morphism h' having the same properties relative to E' as h has relative to E, then the G(')-structure on SG(M) is replaced by a conjugate structure having the

7 G-STRUCTURES

289

same group. The Lie algebra of the commutative group G ( l ) may be identified with g"', by identifying T E Hom(R", (1) with the endomorphism of R" @ 0 which agrees with T on R" and is zero on 11. The structure function c ( I ) of the G("-structure on SG(M) is called (by abuse of language) the second-order structure fitnction of the G-structure Sti(M) on M : its values lie in

H o m ( i (R" 98). R" @3g)/?Hom(R" 3 $1, if1)). Let r : SG(M) + M. r I: S,(i)(SG(M)) + SG(M) be the canonical projections. Show that the canonical form a(') on S,(I)(SG(M)) may be written as dl)= %r1(a) w,, where w 1 is a vector-valued I-form on S G ( l ) ( S ~ ( M with ) ) values in the Lie algebra 0. The calculation of c ( ' ) ( r ( l ) )at a point of SG(I)(SG(M))is equivalent, once we have chosen a subspace H") of the tangent space to SG(l)(SG(M))at the point r"), supplementary to the subspace of vertical tangent vectors, to the evaluation of

+

d g ( l ) ( r ( l ) ) (h(1) . r\ k(I)),

where h"), k'l) are vectors in H"). Putting r = r l ( r ( l ) )E SG(M), there are three cases to consider: (i) T ( r , ) . h'l) = Z , ( r ) , T ( r , ) . k'l) = Z , ( r ) for u, v E 0, vertical vectors in T,(S,(M)) such that w I ( r ( l ) ). h'l' = u, w l ( r ( l ) ). k'l) = v. We have then d a ( l ) ( r ' I ) ) .(h") A k'") = [u, v] E 0.

(Extend h'l) and k'l) to Gc1)-invariantvector fields on a neighborhood of r C 1 )whose , projections on SG(M)are Z,, and Z, .) (ii) T ( r l ) h"' =- Z , , ( r ) , T ( r , .) k(l1= h, . x with u E (1, x E R", u ( r ) . (h, . x ) = x, w , ( r ( l ) ). h ( ' )= u, w l ( r ' " ) . k(')= 0. Then (Section 20.6, Problem 1 I ) we have d a ( I ) ( r ( I ) ) (h") . /\ k")) = - u

. x + U,(u). x ,

where U, E Hom(o, Hom(R", 0)). The group G acts by transport of structure on R(S,(M)); if it leaves invariant the structure SG(l)(SG(M))C R(S,(M)), then U, takes its values in 0"). The converse is true if G is connected. (iii) T ( r l ) . h'l) = h, . x , T ( r l ) . k(')= h, . y, where x, y E R", a ( r ) . (h, . x) = X , a ( r ) . (h, . y ) = y , wl(r'"). h(l)= w I ( r ( l ) ). k'l) = 0. In this case we have d d 1 ) ( r ( I ) )(h"' . A k"') = d a ( r ) . (h, . x /. h, . y )

+ V, .

(X

A y),

where V, E H o m ( i I, (1). If G(I1= ( e ) ,we may identify SG(I)(SC(M))with SG(M); moreover, if G leaves invariant the G("-structure on SG(M), then w 1 is the I-form of a principal connection in SG(M),and the vector-valued 2-form G?on SG(M)such that GI(r).(h,*xAh;y)=

V;(xAy)~n

is the curvature form of this connection. (For an example of this case, see 20.9.2.) By induction we can define, for each integer k > 1, the kth-order prolongation of the G-structure SG(M)to be the first-order prolongation of the G'k-l'-structure which is the (k - I)th order prolongation of SG(M). Let G Xdenote ) thecorrespondinggroup, il(k)its Lie algebra. The structure function of this G"'-structure is denoted by c('+I) and is called the (k I)th-ordcr strircturefirnction of the G-structure S d M ) .

+

(16.5.2) is a C' section of the 7. For the manifold M = R", the mapping T-' : SHT; bundle of frames R(M). If G is any closed subgroup of GL(n, R), the union of the T;'.G,

290


as x runs through M, is a G-structure SG(M)on M, called the canonicalflat G-structure on R". A G-structure on a differential manifold M' of dimension n is said to be flat if, for each n' E M', there exists a diffeomorphism of a neighborhood U of x' onto R" which (by transport of structure) transforms the induced G-structure on U into the canonical flat G-structure. For the canonical flat G-structure, if w : R(Rn) Rn is the canonical projection, the canonical form a on Sc(R") is equal to %(d(lRn)) = 'T(T),where T is considered as a vector-valued I-form on R", with values in R" (16.20.15). Deduce that the firstorder structure function of So(Rn)is zero. With the notation of Problem 6, if we take h(T; . s, y ) = TJT; . s) . (T; . y), the subspaces H, are the spaces of horizontal vectors for a principal connection in SG(R") called the canonical connection. (The corresponding linear G-connection on R" is also called the canonical linear G-connection.) Show that this connection is flat (20.4.1). If w is the differential 1-form of this connection, then (with the notation of Problem 6) w I = 'a,(w). Show that the second-order structure function d l ) is constant, and generalize to structure functions of arbitrary order. For every closed subgroup G of CL(n, R), the principal connection in R(R") which extends the canonical connection in SG(R") (20.7.1 1) is the canonical connection. The Cartan connection corresponding to this in the bundle of affine frames A(R") (Section 20.6, Problem 19(c)) is identical with the canonical Cartan connection on the group A(n, R) relative to the subgroup GL(n, R) (Section 20.5, Problem 3). --f

8. Let M, M' be two pure differential manifolds, and let SG(M)and SG(M') be G-structures on M, M', respectively. A diffeomorphismfof M onto M' is said (by abuse of langilage) to be an isomorphism of SG(M)onto SG(M') if the restriction S G ( f ) of R(f) (Section 20.6, Problem 4) to SG(M)is a bijection of Sc(M) onto SG(M').

(a) Let F : SG(M)+Sc(M') be an isomorphism of principal bundles. I n order that F should be of the form S G ( f ) , wherefis a diffeomorphism of M onto M', it is necessary and sufficient that 'F(a') =a, where a and a' are the canonical I-forms on SG;(M) and %AM'), respectively. (b) With the notation of Problem 6, show that if G is connected and if a diffeomorphism F of SG(M) onto SG(M') is an isomorphism of the G(')-structure SG(,)(SG(M))onto the G(')-structure SG(l)(SG(M')),then F = S , ( f ) , where f is an isomorphism of M onto M'. (Use (a). If Z,, and Z: are the Killing fields on SG(M)and SG(M') corresponding to a vector u E oe, begin by showing that F transforms Z,, into Z: for all u E oe, and deduce that F is an isomorphism of principal bundles.) 9. Show that the group of automorphisms of a G-connection on a connected manifold

+

M may be identified with a Lie group of dimension jdim(M) dim(G), and attains this maximum dimension only when M is the space Rn endowed with the canonical flat G-structure and the corresponding canonical G-connection (Problem 7). (Same method as in Section 20.6, Problem 7.)

10. With the notation of Problem 6, a G-structure SG(M)is said to be offirtile type if there

) (e}. Show that if a G-structure (for a connected exists an index k such that G k= group G ) is of finite type, then its automorphism group is a Lie group. (Observe that to an {+structure there is intrinsically attached a principal connection, and use Problems 8 and 9.)

7

G-STRUCTURES

291

11. Let E, F be two finite-dimensional real vector spaces, let 0 be a vector subspace of Hom(E, F), and let 0") denote the subspace of Hom(E, 0) consisting of all T such that T ( u ) .v = T ( v ) .u for all u, v E E. For each integer k > 1 define 0") inductively to be ( d k - ' ' ) ( ' )The . subspace 0 is said to be of finite rype if o(k)= {0}for some k . (Cf. Section 20.9, Problem 15.)

(a) If we identify Hom(E, F) with E* 0F (A.10.5.5), then Hom(E, 0) is identified uith a subspace of E*@E* OF, and 8'') is identified with the intersection (E* (311) n (S2(E*)0F) (A.17). Deduce that, for each integer k 2 1, 0") is identified with the intersection (SdE') @ 0)

(Sk+i(E*)@ F)

in the space Tk+'(E*)0F. (b) Deduce from (a) that if I1 is a subspace of 0, then lfk) c 0'" for all k. If F is a subspace of a vector space F , then 0'" is the same whether 0 is regarded as a subspace of Hom(E, F) or of Hom(E, F). (c) If F = E and 0 = End(E), then f l ( k ) = Sk+l(E*)@ E, and therefore 0 is of infinite type. Deduce that every subspace of End(E) which contains an endomorphism of rank I is of infinite type. (Consider the subspace generated by such an endomorphism, and observe that it may be identified with End(R).) 12.

Deduce from Problem 1 I that if G = GL(n, R) or SL(n, R), all G-structures are of infinite type. Show that the same is true if G = Sp(Q,) is the symplectic group, leaving invariant a nondegenerate alternating bilinear form @ on R2". (Notice that (D defines canonically an isomorphism of E = R2" onto E*, and that under this isomorphism the Lie algebra n = w(@)corresponds to S2(E*) C E* @ E* = Hom(E, E*); deduce that 0") may be identified with Sk+2(E*).) Show that for G = GL(n, R) or SL(n, R), all G-structures are flat (Problem 7). (b) If G is the subgroup of GL(n, R) which leaves invariant a subspace F of R", and if XH L, is the corresponding field of directions on M (20.7.8), then a G-structure is flat if and only if the field XH LAis completely integrable. Show that this condition is equivalent to the vanishing of the first-order structure function. (Observe that a(Hom(R", 0)) is the kernel of the canonical mapping

13. (a)

Horn(/; R", R") +Horn(/;

F, R"/F)

which sends an alternating bilinear mapping B : R" x R" +R" to the composition of the canonical mapping R" +Rn/F with the restriction of B to F x F.) (c) Take G to be the symplectic group Sp(n, R) (n 2m). If

so that a G-structure on M corresponds to a differential 2-form R on M such that @(r; hl, r; . h2) = 0 is the discriminant of the form Ox relative to the canonical basis of R";since this function is of class C" on cp(U), it is clear that the linear form

(1 being Lebesgue measure) is a positive Lebesgue measure on U . Consider now another chart (U, $, n ) with the same domain of definition, and let u = cp 0 $-' : $(U) + cp(U) be the transition diffeomorphism. The local ), the form expression of g relative to this second chart is X H ( X , 'I-',where Yx is given by yx(h,k) = @u(x)(Du(x) * h, D ~ x* ) k)

for all x E $( U). If J(x) is the Jacobian of u at the point x (8.1 0), the discriminant g , ( x ) of Yx is given by g,(x) = J(x)2g(u(x)).

Since the measure on U corresponding to the chart (U, $, n) is given by

f

~ ~ ~ ( u ~ ( $ - l ( x ) ) ( ~ l (dx ) w) ~, ~ 2

and since this integral can be written in the form j l c p - ' M X ) ) ) I J ( x )Ig(u(x)>'/2d

W,

298

XX


it follows from the formula for change of variables (1 6.22.1) that it is equal to the right-hand side of (20.8.5.1), as required. (20.8.6) Suppose now that the Riemannian manifold M is oriented, and let uo be a differential n-form on M belonging to the orientation of M (16.21.2). We can then associate with g a positive n-form us on M such that (20.8.6.1)

I (u,(x),

h,

A

*

- A h,) I = VJh,

A

* *

A

h,,).

For if the local expression of uo , d a t i v e to a chart (U, cp, n) of M, is u0(cp-'(x))=~(t',

...,r)dt' A d t 2

u,(cp-'(x)) = f (&)'I2

dtl

A * * *h d c ,

we put (20.8.6.2)

A

dt2 A

* *

- A dr,

the sign being that of the function w. It is immediately verified that this form does not depend on the choice of local expression. The form v, (also denoted by u ) is called the canonical volume form on the oriented Riemannian manifold M. For each v,-integrable function f on M, we have therefore (20.8.6.3)

Let R = (el, .. . , en) be a direct orthonormal moving frame (20.8.2) on an open set U in M, and let aj (1 S j 6 n) be the corresponding canonical forms. Then the form u, is given in U by U, = 6 1 A (72 A

(20.8.6.4)

For at each point

XE

*"

A

a,,.

U we can choose a chart such that the covectors

aj(x) have for local expressions the values of the dtj at the point correspond-

ing to x ; the formula (20.8.6.4) then follows from (20.8.6.2) and the fact that the discriminant of g(x) relative to the basis (ej(x))is equal to 1. (20.8.7) Let M be an oriented Riemannian manifold of dimension n, and let u be the canonical volume form on M. From exterior algebra it follows that, for each differential r-form a on M, there exists a differential (n - r ) form on M, denoted by *a, such that

8

GENERALITIES ON PSEUDO-RIEMANNIAN MANIFOLDS

299

for all differential r-forms /? (A.15.3). The form * a is called the adjoint of the form a. We have the following formulas: (20.8.7.2)

*1 = 0,

(20.8.7.3) (20.8.7.4)

PROBLEMS

1.

Let M1, M Z be two pseudo-Riemannian manifolds and gl, g, their respective metric tensors. On the product manifold M = M1 x Mz , show that g = 'prl(gl) 'prz(g2) is a pseudo-Riemannian metric tensor of signature (pl p z , q1 q2), where (p,, q,) is the signature of g, ( j = 1, 2). This metric tensor is called the product of g, and g2, and the manifold M endowed with this metric tensor is called theproducr ofthepseudoRiemannian manifolds MI, M I . At each point (xl, xz) E M1 x Mz , the tangent spaces to the submanifolds MI x {xz} and { x , } x Mz are totally orthogonal. If MI, Mz are Riemannian manifolds, the Riemannian volume uI is equal to vrl @ uIz . If N,(resp. N,) is a pseudo-Riemannian covering of M I (resp. MZ), then the product N1x N2 is a pseudo-Riemannian covering of M l x M, .

+

2.

+

+

Let M be a compact Riemannian manifold, N a Riemannian k-sheeted covering of M, where k is finite. Show that the volume of N is k times that of M.

3. Let M be a pseudo-Riemannian manifold, (h 1 k) the corresponding scalar product in T(M). Let E(h) denote (hlh).

(a) Show that for every vector h E T(M) and every verticaltangent vector v E Th(T(M)), we have

dv * E = 2(h ITh(V)). (b) With the notation of (20.8.3A let aMor a denote the differential 1-form ' G ( K M ) , KM being the canonical 1-form on T(M)* (16.20.6). Show that for every vector h E T(M) and every tangent vector kh E Th(T(M)), we have ( a W , kh)

= (hlT(od. kh).

Show that the differential 2-form da on T(M) is nondegenerate. For each vector h E T(M) and each pair of vectors kh , vh E Th(T(M)) with vh vertical, we have

(c)

0. The Lie algebra w(n, R) of this group consists of the endomorphisms V E End(R") such that

with h( V)E R.

( V .x IY)

+ (x I v. Y) = h(V)(x I Y),

(a) If we take g = oo(n, R), then the Lie algebra 0'') (Section 20.7, Problem 6) is the set of mappings u HT,, of R" into End(R") such that (Tu. x 1 y) + (x I T,, . y) = hu(x Iy), where u-h, is a linear form on E = R", so that h, = (u, p(T)\, where p(T)e E*. Show that p is a bijection of B(') onto E*. (The injectivity of p follows from (20.9.2). To prove surjectivity, observe that if C is the canonical isomorphism of E onto E*, so that (x, G . y) = (XI y), then for each a* E E*, if we put (ypr),,.x= ( ~ , a * ) x + < x , a * ) u - ( u I x ) G - ~ . a * ,

the mapping u H (ya.),, belongs to dl), and p(T) = 2a*.) (b) The Lie algebra d2)(Section 20.7, Problems 6 and 11) may be identified with the set of symmetric bilinear mappings (u, V)HT,,,,, of R" x R" into End(R") such that K , V * X I Y)

+

(XI

Tu,v . Y)

= A,,

V(XlY),

where h is a symmetric bilinear form on R" x R".Show that if (u I v) = 0, then

A,, .(v I v) = - A , v(u I u), and deduce that for n 2 3 we have A,,, = 0 for all u E R"; hence I\ = 0 and therefore = {O}, by (20.9.2). What can be said when n = 2?

10 THE RIEMANN-CHRISTOFFEL TENSOR

315

10. T H E RIEMANN-CHRISTOFFEL TENSOR

In this section, “vector field” will always mean ‘‘C“ vector field.”

On a pseudo-Riemannian manijokd M, the curvature r (17.20.4) satisjies the identit.)) (20.1 0.1)

(20.10.1.1)

(Y

.(X

A

Y ) ). Z

+ (r

(Z

A

X)) Y

+ (r

*

(Y

A

Z ) ) * X = 0.

Since the torsion of the Levi-Civita connection is zero, Bianchi’s identity (20.6.4.1) takes the form nACJ=o.

In view of the formulas (20.6.4.2) and (20.6.6.11) and the definition of CT, this gives (20.10.1.1) immediately. If we lower the contravariant index (20.8.3) of the curvature tensor of the Levi-Civita connection on M, we obtain a covariant tensor K of order 4, called the Riemaiin-Cliristo~eltensor of M : it is defined by the condition (20.10.2)

( K , X @ Y @ Z @ W ) = ( X I ( Y . ( Z AW ) ) . Y),

for any four vector fields X , Y , Z , Won M. (20.10.3) The Riernann-Christofel tensor satisfies the identities (20.10.3.1) ( K , X @ Y @ Z @ W + X @ Z @ W @ Y + X @ W @ Y @ Z ) = O . (20.10.3.2) ( K , X @ Y @ Z @ W )

=

-(K, Y @ X @ Z @ W).

(20.10.3.3) ( K , X @ Y @ Z @ W )

=

-(K, X @ Y @ W @ Z ) .

(20.10.3.4)

(K, X @ Y @ Z @ W ) = ( K , Z @ W @ X @Y).

If K‘ is another covariant tensor of order 4 on M which satisfies these four identities and is such that (20.10.3.5)

( K , X @ Y @ X @ Y ) = ( K ’ * A’@

Y @ X @Y )

for all vector fields X , Y , on M,then K’ is equal to K .

316

XX


The identity (20.10.3.1) is an immediate consequence of the definition of K and of (20.10.1 .I). Next, (20.10.3.3) is trivial. As to (20.10.3.2), it is enough to show that for each pair of indices i, j we have (20.10.3.6)

or equivalently (20.6.6.1)

(W) ei I e j ) + ( e , I W)e j ) = 0. *

But this follows from the fact that the vector-valued 2-form takes its values in the Lie algebra of the group O(@),characterized by the relation (20.9.1-1). Next, the identity (20.10.3.4) is a consequence of the three preceding ones. For if we temporarily abbreviate the expression ( K , X @ Y @ Z @ W ) to ( X Y Z W ) , then it follows from (20.10.3.1), (20.10.3.2), and (20.10.3.3) that

+

( X Y Z W )= - ( X Z W Y ) - ( X W Y Z )= ( Z X W Y ) ( W X Z Y ) = -(ZWYX)-(2YXW)-(WYZX) -(WZXY) = 2(ZWXY) ( Y Z X W ) ( Y W Z X ) = 2(ZWXY) - ( Y X W Z ) = 2(ZWXY) - ( X Y Z W ) ,

+

+

from which (20.10.3.4) follows. As to the last assertion of the proposition, by replacing K by K’ - K we may assume that ( K , X @ Y O X @ Y ) = 0 identically, and we now have to prove that K = 0. Using the same abridged notation as above, if we replace X by X Z in ( X Y X Y ) = 0 we obtain

+

+ ( Z Y Z Y ) + ( Z Y X Y ) + ( X Y Z Y ) = 0, which gives ( X Y Z Y ) + ( Z Y X Y ) = 0 and therefore, using (20.10.3.4), ( X Y Z Y ) = 0. No@replace Y by Y + W in this relation, and in the same way (XYXY)

we obtain

( X Y Z W )+ ( X W Z Y )= 0 ; hence, using (20.10.3.3), ( X Y Z W ) = ( X W Y Z ) , and so by cyclic permutation of Y , Z , W also ( X Y Z W ) = ( X Z W Y ) . But now, using (20.10.3.1), we have 3(XYZW) = 0, and the proof is complete. Given a moving frame R = ( e l , e,,

. .. , en)on M, let


317

Then the relations proved in (20.10.3) are equivalent, by linearity, to the following: (20.1 0.4.1)

K/iijk

+ KjijA i + K~ikij = 0,

(20.1 0.4.2) (20.1 0.4.3) (20.1 0.4.4)

Kkijk =

Kjkjii

9

for each quadruple of indices / I , i, j , k . If R is an ortliogonuf moving frame, such that (e,I e,) = E ~ where , E~ = 1 for i 5 p and E , = - 1 for i > p (20.8.2), the formulas (20.10.2) and (20.6.6.6) enable us to express the K I J i j imn terms of the Qij:

or equivalently. (20.1 0.4.6)

Hence, by virtue of (20.10.4.2). we have (20.1 0.4.7)

& J

QJ.J. =

-qR... JJ

(20.10.5) Given any two vector fields X , Y on a pseudo-Riemannian manifold M, we can use the curvature r of M to define a linear M-morphism of T(M) into itself: this morphism is denoted by r ’ ( X . Y ) and is defined by the condition that (20.1 0.5.1 )

r ’ ( X , Y ) .Z

= (r

(Z

A

X)). Y

for all vector fields Z on M. In particular, for each vector field X on M, we Put (20.1 0.5.2)

r”(X) = r’(X, X ) .

For each x E M, the restriction u ” ( X ) , of r ” ( X ) to T,(M) is a self-udjoint endomorphism of T,(M), relative to the bilinear form g(x) on this vector space. For if Z , W are any two vector fields on M, we have ( W l r ” ( X ) . Z ) = ( K , W @ X @ Z @ X ) = ( K , Z @ X @ W @X ) = (Zl r ” ( X ) *

by virtue of (20.10.3.4).

W)

318

XX


(20.10.6) The Ricci tensor of M is by definition the covariant tensor K’ of order 2 defined by the condition (20.10.6.1)

(K’, X @ Y ) = Tr(r‘(X, Y ) )

for any two vector fields X , Y on M. This tensor is symmetric. For if (el, e2, -.., en)

is an orthogonal moving frame such that (eil e i )= q , where ci = 1 for i 5 p and z i = - I for i > p (20.8.2), then by definition we have n

Tr(r’(X, Y ) ) =

Ei(eilr’(X,Y ) ei)

i= t =

f; E i ( ~ ,ei o Y O ei o X >

i= I n

= Cci(K,eiOX@ei@Y) i= 1

= Tr(r’(

Y, X ) )

by virtue of (20.1 0.3). An equivalent definition of K’ is that it is obtained by contraction of the contravariant index and the first covariant index in the curvature tensor of the Levi-Civita connection (17.20.5). Relative to the same moving frame, if we put (20.1 0.6.2)

KJk= (K‘,ej @ ek),

then we have (20.10.6.3)

KSk =

n

C E~ Kjiki

i= 1

with the notation of (20.10.4). (20.10.7) For each tangent vector h, E T,(M), the number (20.10.7.1)

Ric(h,) = (K’(x),h, @ h,)

= Tr(r”(X),)

(where X ( x ) = h,) is called the Ricci curvature of M in the direction of the tangent vector h,. If (e,, . .. , en) is a moving frame satisfying the same conditions as in (20.10.6), and if h, (20.10.7.2)

Ric(h,)

n

=

=

j= I

5jej(x),then we have

C Kjk(x)rjSk.

j ,k


319

(20.10.8) Since h, H Ric(h,) is a quadratic form on T,(M), it determines a self-adjoint endomorphism F, of TJM) by the rule

The number S(x) = Tr(F,)

(20.10.8.2)

is called the scalar curvature of M at the point x. If the moving frame (el,

*

..

e,,)

9

satisfies the same conditions as in (20.10.6), then with the same notation we have (20.10.8.3)

S=

C K‘..= C t i K j i i i , n

JJ

j= I

i. i

the second equality following from (20.10.4.3). (20.1 0.9) The cases n = 1, 2.

On a manifold of dimension I all 2-forms are zero; hence the curvature of a Riemannian manifold of dimension 1 is zero. Next suppose n = 2, and consider first the case of a Riemannian manifold M. Let (el,e 2 )be an orthonormal moving frame; then if we put (20.10.9.1)

KI212 =

K

relative to this frame, the only nonzero components Khijkare (20.10.9.2)

K,,,,

= K2121 = -K1221

= -K2112 = K

by virtue of (20.10.4). We have therefore for the Ricci tensor (20.10.9.3)

K;, = K i 2 = K,

K i 2 = 0,

so that the Ricci curvature is given by

(20.1 0.9.4)

Ric(h,)

=

K(x)(((’)~+ ((2)2).

320

XX


Finally, the scalar curvature is S = 2K. The number K(x) is called the Gaussian curoatirre of M at the point x. Relative to the orthonormal frame we have chosen, by virtue of (20.1 0.4.5) we have K = (Q, ,, e, A e,), so that R, = Ka, A az, and the formulas (20.6.5.10), (20.6.6.3), and (20.6.6.4) become in this case

,

de, = - a i 2 e l ,

(20.10.9.5) (20.1 0.9.6)

da,

=

-al2A a,,

do, = a12 A al,

(20.1 0.9.7)

If n = 2 and M is a pseudo-Riemannian manifold of signature (1, I), we choose an orthogonal moving frame ( e l , e,) such that (el I el) = 1, (e, I e,) = - 1. With the same notation (20.10.9.1), the relations (20.10.9.2) remain valid, but this time K;, = -K, K;, = K, Ki2 = 0, and the scalar curvature S is zero.

PROBLEMS

In the notation of (20.10.4), show that Bianchi's identity (20.3.4) takes the form Vem ' K i j k l

+ Vek K I J I m '

vet

' Kijkm

=0

for all indices i, j , k , I , m. (Cf. Section 20.6, Problem 3.) A Riemannian manifold M is said to be an Einstein manifoldif there exists a C" realvalued function h on M such that the Ricci tensor K' is equal to hg. Show that if M is a connected Einstein manifold of dimension n 2 3, then the function h is necessarily constant. (Using the formula (20.10.6.3) and Problem 1, show that we have (n - 2) v.,

.h = 0

for all indices m.) (a) Let M be an oriented differential manifold of dimension n, and let u be a C" differential n-form in the orientation of M. For each Cmvector field X on M, Ox u is a C" n-form on M; hence can be written uniquely as (div X)u, where div X is a C" scalar-valued function on M, called the dicergence of the vector field X. If X has compact support, show that (div X ) u = 0. (Use (17.15.3.4) and (17.15.5.1).)


321

(b) Suppose now that M carries a linear connection C such that Vu=O. If the tensor field A, is defined as in Section 20.6, Problem 6, show that for each C" vector field X on M, we have div X = -Tr(Ax). In particular, if the connection C is torsion-free, then div X = c:(VX).

(1)

In the notation of (20.6.5), this takes the form div X =

n 1=1

(vel. X,or).

Iffis any C" real-valued function on M, then we have div(fX) =f. div X

+ Ow .f.

(c) On a Riemannian manifold M, the definition (I) of the divergence coincides with that given in (a) by taking u to be the canonical volume form (20.8.5). If X, Y are two C" vector fields on M, show that (3)

div(Ax * Y

+ (div X)Y)= (div X)(div Y)- Tr(A, A,)

- (K',

X 0 Y)

(Yuno's formula). (Use the formula (2), with the moving frame (q) chosen so that [e, , el] = 0 for all i, j . ) (d) Deduce from Section 20.6, Problem 6(c), that if X is an infinitesimal automorphism of the Levi-Civita connection, then div X is a conslant. 4. Let M be a Riemannian manifold, X an infinitesimal isometry of M (Section 20.9,

Problem 7).

(a) Show that for each tangent vector h, E TAM) eh,

'

llXII'

= 2(h,

I AX(x)

'

X(x)).

(b) Let v be the geodesic (for the Levi-Civita connection) defined on a neighborhood of 0 in R,such that o(0) = x and v'(0) = h,. Putf(r) = Il(X*v)(t)llz. Deduce from (a) that f"(0)= (hx I vhx ' (AX . X ) ) . (Use the fact that V",. v' = 0, and Section 20.9, Problem 7.) (c) Show that at a point xo where the function llXl12 attains a relative maximum (resp. minimum), we have div(A, . X) 5 0 (resp. 2 0 ) . (Use (b), replacing h, by the values at xo of an orthonormal moving frame, together with Problem 3(b).) 5. Let M be a connected Riemannian manifold whose Ricci tensor K' is such that at each point x E M the quadratic form h, H (K'(x), h, 0h,) is negative definite. Let X

be a vector field which is an infinitesimal isometry of M. Show that if the function llXll attains a relative maximum at some point of M, then X = 0. (Use the hypothesis and Section 20.9, Problem 7, to show that at such a point x we have ++(1 -

r: -

r,2)1'2,

it follows from (1 6.8.13) that the projection of Q + on R" is a diffeomorphism; * * * Y

**.

by transporting the Riemannian structure of Q + to R" by means of this diffeomorphism, we obtain a structure of Riemannian manifold on R". The space R" with this Riemannian structure is called hyperbolic n-space and is denoted by Y,, .

11 EXAMPLES OF RIEMANNIAN MANIFOLDS

329

(20.11.8) Let G be a connected Lie group whose center is {e}. We shall show that G may be regarded as a symmetric homogeneous space. Consider the product group G x G , and let a be the involutory automorphism (s, t ) H( t , s). The set of a-invariant elements is then the diagonal subgroup D of all elements (s, s), which contains no nontrivial normal subgroup of G x G , because (s, t)(x, x)(s-', f-') = (sxs-I, t x t - I ) belongs to D for all (s, t ) E G x G only if x is in the center of G . The subspace in of ge x ge formed by the eigenvectors of CJ*for the eigenvalue - 1 is clearly the space of all vectors (u, - u), where u E ge, and we have Ad(s, s) * (u, - u) = (Ad(s) * u, -Ad(s) . u) for s E G. In order that we should be able to apply the method of (20.11.1) to the symmetric pair (G x G , D), it is therefore necessary and sufficient that there should exist on gea nondegeneratesymmetric bilinear form 0 which is invariant under the operators Ad(s) for all s E G . Let Go be the subgroup {e} x G of G x G . The mapping

-

((e,Yx- '1, ( x , 4) is a diffeomorphism of G x G onto Go x D, and the inverse diffeomorphism is cp-l : ((e, t ) , (s, s))H(s, ts). We have cp(xs, ys) = ((e, y x - I ) , (xs, xs)); hence if n : G x G --t (G x G)/D is the canonical projection, there exists a diffeomorphism II/ : (G x G)/D --f Go such that IC/(n(x,y ) ) = (e, y x - ' ) (1 6.1 0.4), and cp is therefore an isomorphism of the principal bundle cp : (x, Y )

(G x G , (G x GYD, 4 onto the trivial principal bundle (Go x D, G o ,pr,) with the same group D. The tangent linear mapping T,@, .,(cp) is (u, v) H ((0, v - u), (u, u)) (1 6.9.9), and therefore defines, by restriction to 111 and canonical identification of Lie(Go) with ge, an isomorphism y : (u, - u) H -2u of 111 onto ge. We see therefore that there exists a unique pseudo-Riemannian structure on G which is invariant under /eft translations by the elements of G and whose metric tensor g is such that g(e) = 0.The invariance of 0 under the adjoint representation shows moreover that g is also invariant under right translations, since g(x) s = x * g(e) s = (xs) * (s- * g(e) . s ) = (xs) g(e) = g(xs). Finally, the formula (20.7.1 0.6) giving the curvature of the canonical connection on (G x G)/D at the point n(e) also gives, by transport of structure via cp and p , the curvature of the connection on the pseudo-Riemannian manifold G at the point e :

-

(20.1 1.8.1)

(+9

*

(u

A

9) w = - N u , vl, wl *

for u, v, w E g p (19.8.4.2). We recall that the geodesic trajectories are the translates ojthe one-parameter subgroups of G and that, for U, v E ge, we have (20.1 1.8.2)

vxu *

xv = tx,,,v] = *[Xu

9

Xvl.

330

XX


Moreover, since the vector fields Xuand X , are by definition invariant under left translations, the function (Xu[X,) is constant on G . Hence, for each vector field Yon G we have, by (20.9.5.3), 0 =By

x,,= (V, . XulX,) + (&IVY K ) ; *

taking in particular Y = Xw , where w E ge , we obtain by virtue of (20.11.8.2) (20.11.8.3)

We remark that, when G is compact, it follows from above and from (20.11.3.1) that there exists a Riemannian structure on G , invariant under

left and right translations. Remark (20.1 1.9) With the notation of (20.11 .I), if H acts transitively via the adjoint representation on the lines in in, then the values of g(x,) in in are determined by the value at one nonzero vector; hence there exists (up to a constant factor) only one G-invariant pseudo-Riemannian metric tensor on G/H.

PROBLEMS 1. Show that Euclidean displacements (with determinant f1) are the only isometries of R" for its canonical Riemannian structure, by using Section 20.6, Problem 9(a).

2. (a) The linear connection on R" corresponding to the trivial connection (20.4.1) in

the space of frames R(R") (endowed with the canonical trivialization TH, (x, T, 0 r,)) has the property that, relative to the canonical moving frame X H T ; ~ , all the connection forms mi, are identically zero. For brevity, we shall call this connection the canonical linear connection on R".The pseudo-Riemannian structures whose LeviCivita connection is the canonical linear connectton are the canonical structures defined in (20.11.2): relative to the canonical moving frame, the components gl, of the metric tensor g are constants (cf. (20.9.6.5)). (b) Let G be a discrete group acting properly and freely on R"; then R" is a covering of M = Rn/G (Section 16.28, Problem 4). There exists a unique linear connection on M whose inverse image under the projection p : R"H M is the canonical linear connection. In order that this connection should be the Levi-Civita connection of a pseudo-Riemannian structure on M, it is necessary and sufficient that the group G should be a group of isometries for one of the canonical structures on R" defined in (20.11.2). (c) Take n = 2 in (b), and take G to be the group of affine-linear mappings (x, Y)H (x ny m, y n), where m, n E Z. Show that R2/Gis diffeomorphic to

+ +

+

11 EXAMPLES OF RIEMANNIAN MANIFOLDS

331

the torus T2,but that the (flat) linear connection induced by the canonical linear connection on RZ is not the Levi-Civita connection for any pseudo-Riemannian structure on M. Let K be a connected Lie group, H a connected Lie subgroup of K. Let f, 11, be the Lie algebras of K, H, respectively, and suppose that f, = lie @ nr, where in is a vector subspace of 1, such that Ad(/) . in C i n for all t E H. Let x : K --f K/H be the canonical projection, and put xo = a(e). (a) Show that the K-invariant pseudo-Riemannian structures on K/H are in oneto-one correspondence with the nondegenerate symmetric bilinear forms CD on in (canonically identified with T,,(K/H)) which are invariant under the adjoint action of H, or equivalently, satisfy @([w, u], v) @(u, [w, v]) = 0 for all u, v E rn and w E 11., (b) Let @ be a nondegenerate symmetric bilinear form on i n which satisfies this condition, and let G be the subgroup of CL(n, R) which leaves @ invariant (nt being identified with R"),so that the pseudo-Riemannian structure on K/H corresponding to Cr, is a G-structure. With the notation of Section 20.7, Problem 3, the K-invariant linear connections on K/H are in one-to-one correspondence with the linear mappings f,,, : in H 0, such that f,,(Ad( t ) . w) = Ad(X(r)) f,,,(w) for all t E H and w E in. Show that the Levi-Civita connection on K/H corresponds to the linear mapping f,,, given by

+

fm(u) v = t[u, vlm

for u, v E in, where B : in x in relation

--f

+ B(u,

V)

in is the symmetric bilinear mapping defined by the

2@(B(u, v), W) =WU, [w, vlm)

+ WW.ulm,

V)

for u, v, w E in. (Use the fact that Z , is an infinitesimal isometry for all u E m; equation (1) of Section 20.9, Problem 7; and Section 20.7, Problem 3.) The Levi-Civita connection corresponding to @ coincides with the torsion-free connection corresponding to the canonical connection in S,(K/H) (Section 20.7, Problem 3) if and only if @(u, [w, vim)

+

@ ([w,

ulin 3 V)

0

1

for all u, v, w E ni . The curvature of the Levi-Civita connection corresponding to @ then satisfies the relation = &@([u,Vim, [u, VIm) - @([[u,V1be 9

V1,

u)

for ail u, v E ni, where g is the pseudo-Riemannian metric tensor on K/H. Generalize the results of (20.11.4) to the spaces G:,,(R) (Section 16.21, Problem I ) by considering the symmetry s E O(n) such that s(e,) = -e, for j s p and s(e,) = e, forp -t I 5 j n, and the involutory automorphism o : t H sts-' of SO(n).The space 111 may here be identified with the space of real matrices

where X is any (n - p) x p matrix. We may take the metric tensor g to be that whose restriction to in is (axI a y )= - 4 Tr(ax ay).

332

XX


5. In complex projective space P,,(C), let Q.-l(C) denote the complex quadric defined

by the homogeneous equation

(z")2

+

(2')2

+ ... +

(z")2 = 0,

where (zJ) are the homogeneous coordinates of a point in PAC). Show that the subgroup SO(n I ) of SU(n I), which acts on PJC) by restriction of the canonical action of SU(n I), acts transitively on Qn-,(C), and that Q.-l(C) is diffeomorphic to G:+l,2 (R),and hence is endowed with a structure of Riemannian symmetric space (Problem 4).

+

+

+

6. Consider the Hermitian form on C" of signature (p, n - p )

and the unitarygroup UW),which is connected. The space Rn,,of sequences (xdl d k s , of p vectors in C" which are orfhanorrnal relative to Y' may be identified with the homogeneous space U(Y)/U(p) (argue as in (16.11.4)). If P,,,,is the subspace of Gn, ,(C) consisting of the p-dimensional subspaces of C" on which the restriction of Y is positive definite, then Pn,,may be identified with Rn,,/U(n - p ) and hence with U('k')/(U(p) x U(n- p ) ) . Define a structure of Riemannian symmetric space on P,,,, by proceeding as in Problem 4. Show that P.. ,may be canonically identified with the space of (n - p) x p matrices Z with complex entries such that the Hermitian Z is positive definite. (An element of R", ,may be identified with an matrix Z, - '2.

n xpmatrixoftheform

(9 . ,

,wherexisap x pmatrix,suchthat'X.X-'P.Y=

I,.

To this matrix corresponds Z = YX-I.) 7. (a) Let M be a connected differential manifold of dimension n, and C a linear connection on M. For each x E M,let U be a symmetric neighborhood of 0, in TJM)

contained in the domain of definition of the exponential mapping exp,, and such that exp,lU is a diffeornorphism onto an open neighborhood V of x in M. Then there exists a unique diffeomorphism s, of V onto itself such that s,(exp,(u)) = exp,(-u) for all u E U, and we have s, 0 sx = I , . The diffeomorphism s , ~is called the spnmefry with center x in V. If U' is another symmetric neighborhood of 0, having the same properties as U, and if V' is its image under exp,, then the symmetries of V and V' with center x coincide on V n V'. For every tensor field Z E .Y;(M) we have Tz(T,(s,))

Z(X)= (-

I)'+"Z(x).

(b) For each XE, M the symmetry sx, defined on a symmetric open neighborhood V, of x in M,is an automorphism of the connection induced by C on V, if andonlyif the torsion tensor t is zero and the curvature tensor r satisfies 'Cr = 0. (To show that the condition is sufficient, use Section 20.6, Problems 17 and 9(a).) The connection C is then said to be locally syrnrnefrir. 8. With the notation and hypotheses of Problem 7, the connection C is said to be symmetric if, for each x E M,the symmetry s, is the restriction of an automorphism of C (which is unique by Section 20.6, Problem 9(a)). We denote this automorphism also by s, For the rest of this Problem, assume that C is symmetric.

.

(a) Let v : I .+ M be a geodesic for C defined on an open interval I c R. Show that for each to E I, if x = o(to), the mapping u : fw s,(c(f)) of I into M is a geodesic for C such that u(to)= c(fo) and u'(fo)= -v'(fo).

12 RIEMANNIAN STRUCTURE INDUCED O N A SUBMANIFOLD

333

(b) Deduce from (a) that the connection C is complete (Section 20.6, Problem 8). (c) Deduce from (a) that if x, y E M are the endpoints of a geodesic arc, then there exists z E M such that sJx) ==.I*. Deduce that the group G of automorphisms of C is transifiw on M. (Use (20.17.51.) (d) In general, if a Lie group G acts differentiably and transitively on a connected differential manifold M, then its identity component Go also acts transitively on M. (Observe that the orbits of G o in M are open sets.) Hence deduce from (c) that if G is the identity component of the Lie group of automorphisms of C (Section 20.6, Problem lo), M may be identified with G/H,, where H, is the stabilizer of a point x E M. Furthermore, if u is the involutory automorphism f Hs, 0 t s;’ and if H is the subgroup of u-invariant elements of G, then H, is contained in H and contains the identity component of H, so that ( G , H,) is a symmetric pair (20.4.3), and C is the canonical connection on G/H,. (To show that H, contains the identity component of H, observe that the orbit of x for a one-parameter subgroup of H consists of points invariant under s, .) (e) Show that the Riemannian symmetric spaces are the Riemannian manifolds whose Levi-Civita connection is symmetric. (Use Section 20.9, Problem 8.) 0

9. Let ( G , H) be a symmetric pair and let C’ be a u-stable connected Lie group immersed in G ; then H’ = G’ n H is closed in G‘ for the proper topology of G’. Show that the canonical mapping of G’/H’ into G/H is bijective if and only if the Lie algebra of of G’contains 111. The subspace iii [iiI, iit 1 is an ideal in o,, and the Lie algebras 0: which contain 111 are those which contain this ideal. Hence construct an example of a Riemannian symmetric space G/H such that G is not the identity component of the group of isometries of G/H.

+

10. Let M be a connected differential manifold endowed with a symmetric linear connection, and let r be a discrete group acting properly and freely on M, so that M is a covering of M‘ = M / r (Section 16.28, Problem 4). Suppose moreover that r leaves invariant the connection C, so that M is canonically endowed with a linear connection C’, the canonical image of C under the projection 7r : M M‘ (17.18.6). The connection C‘ is then locally symmetric (Problem 7); for it to be symmetric, it is necessary and sufficient that, for each .Y t M, the image of any orbit of r under the symmetry s, should be an orbit of 1’. Take M to be the sphere S , , considered as the submanifold of C2 defined by the equation (2’l2 I z2 = 1 : let p, q be coprime integers, and take I’ to be the cyclic group of order p generated by the orthogonal transformation --f

+ Iz (z’,

Z*)H

(z’ exp(2vi/p), z2 exp(2niqlp)).

Show that the manifold M/1’ is not symmetric.

12. RIEMANNIAN STRUCTURE INDUCED ON A SUBMANIFOLD

(20.12.1) Let M be a pure pseudo-Riemannian manifold of dimension n, g its metric tensor. Let M‘ be a pure submanifold of M, of dimension n’ < n, and let f : M’ -+ M be the canonical injection. Consider the inverse image g’ =‘f(g) of the covariant tensor g on M, which is a symmetric covariant

334

XX


tensor of order 2 on M’. At a point X E M ‘ , the symmetric bilinear form (h,, k x b(g’(x), h, 0 k,) = (g(x), h, 0 k,) on T,(M’) = T,(M) is nondegenerate if and only if T,(M’) is not an isotropic subspace of T,(M), in which case we shall say that M’ is nonisotropic (relative to M ) at the point x. Since the nondegeneracy of g’(x) may be expressed by the nonvanishing of its discriminant, relative to a basis of T,(M’), it follows that the set of points x E M’ at which M’ is nonisotropic is open in M’. It is equal to M’ if M is a Riemannian manifold.

From now on in this section we shall consider only the case where M is a Riemannian manifold. The reader may verify for himself that the results generalize easily (at the cost of some notational complication) to pseudoRiemannian manifolds, provided that we restrict consideration to submanifolds which are nonisotropic at all points (cf. Section 20.1 3, Problem 5). Example

(20.12.1 . l ) If we equip the sphere S, with the Riemannian structure induced by the canonical structure on R”+l,we obtain the canonical structure on S, defined in (20.11.4). For since both S, and the canonical structure on R”+l are invariant under SO(n l), it is enough to verify that the two Riemannian

+

metric tensors agree at the point en+ and this is an immediate consequence of the definitions. (20.12.2) With the notation of (20.12.1), suppose that M (and therefore also M‘) is Riemannian. Let x be a point of M‘, and let R‘ = (e’,, . . . , e:.) be an orthonormal moving frame for the Riemannian manifold M ‘ (20.8.2) defined on an open neighborhood U’ of x in M’. We shall show that there exists an open neighborhood U of x in M and an orthonormal moving frame R = (el, .. , , en) for the Riemannian manifold M such that ejl M’ =e; for 1 5 j 5 n‘ on a neighborhood V of x contained in U n U’. For this purpose, extend the sections ej (1 5 j 5 n’) of T(M‘) c T(M) over U’ to C” sections aj (1 5 j 6 n’) of T(M) over a neighborhood of x in M (16.12.1 1); next, choose n - n’ vectors ck (n’ + 1 5 k 5 n) in T,(M) which together with the el($ (1 S j S n’) form an orthonormal basis of T,(M), and extend the ck to C” sections ak (n’ 1 5 k 5 n) of T(M) over a neighborhood of x in M. Then, for y E M sufficiently close to x, the ai(y) (1 5 i 5 n) will form a basis of T,(M), and we can apply to the ai the method described in (20.8.2) to obtain the frame R. At each point y E M’, let N, be the subspace of T,(M) orthogonal to T,(M’): at each point y E V n M’, the vectors ek(y) (n’ + 1 5 k 5 n) form an orthonormal basis of N,. Hence (16.17.1) the union N of the N, is a

+

12 RIEMANNIAN STRUCTURE INDUCED ON A SUBMANIFOLD

335

vector subbundle of f*(T(M)), and we have ,f*(T(M)) = T(M‘) 0 N. The bundle N is therefore canonically isomorphic to f *(T(M))/T(M’), which we have called the normal bundle of M‘ in M (1 6.19.2). We shall usually identify these two bundles. (20.12.3) The fundamental formulas of (20.6.5) and (20.6.6) for the LeviCivita connection C on M and the moving frame R are: (20.1 2.3.1) n

(20.1 2.3.2)

C ojiej,

de, =

j= 1

(20.12.3.3)

dcr, = -

C1o i j A ajr n

j=

(20.12.3.4)

Since here the basis ( e i ( y ) )of T,(M) is orthonormal, we must have (20.12.3.5)

o j i= - w i j

(1

5 i , j 5 n),

so that the matrix ( o i j ( y ) )belongs to the Lie algebra o(n, R) for all y E V. Let a; = ‘f( a,). Then, by (20.12.3.1), for each y E V n M’ and each tangent vector hi E T,( M’), we have n

hi =

( ~ K Y ) ,hi>ei(y)*

i= 1

From the definition of the frame R , this implies first that (20.12.3.6)

a;=O

for n ’ + l S a S n

and second that the a; for 1 5 j 5 n‘ are the canonical forms corresponding to the given moving frame R’. We have, therefore, n’

(20.12.3.7)

g‘ = Ca;@crf; i= 1

the tensor field g’ is also called t h e j r s t fundamental form on M‘.

336

XX


Next, let mij = ‘f(oij). Then, first of all, for 15 i,j 6 n ;

mii = -mij

(20.12.3.8)

this shows that for all y E V n M’ the matrix ( ~ f ~ ( y i,)jgn, ) ~ belongs to the Lie algebra o(n’, R), and consequently the mij define an O(n’,R)-connection C’ (20.6.5) on the Riemannian manifold V n M’. Moreover, by virtue of (20.12.3.3) and (20.12.3.6), we have

“’ (20.1 2.3.9)

for 1 5 i 5 n’, which by comparison with (20.6.6.3) shows that the torsion of C‘ is zero. In other words, C’ is the Levi-Civita connection on M‘, and the mij (1 5 i, j eu

u=n’+l

= (WLi 3 ej>,

which is a C“ function on V n M’; then we have

1a j.i = I a i j.

(20.12.4.2)

+

for 1 5 i, j 5 n’ and n’ 1 5 ct 5 n. For the formulas (20.12.3.3) and (20.12.3.6) give, for n’ 1 5 a n‘ there is therefore a symmetric covariant tensorfield I, of order 2 on M‘ such that (1, e; Q ei) = l a i j

for 1 5 i, j S n’. Equivalently, we have n’

(20.12.4.4)

1, =

1

~ i= 1

:

Qi

by virtue of (20.12.4.2). The I, (n’ + 1 5 a S n) are called the second fundamental forms on M’; their assignment is equivalent to that of the n’(n - n’) differential forms oLi. It follows immediately from (20.1 2.4.1) and (1 7.17.3.4) that if X’, Y’ are any two vector fields on M‘, then we have (20.12.4.5)

Vx. Y’- Vi,* Y‘ = *

1(I,,

a>n’

X‘

yl)ea,

which shows that the I, are independent of the choice of the moving frame R ‘ ; they do depend on the choice of the e, (the orthonormal frame of the normal bundle N), but the sum (20.1 2.4.6)

is independent of this choice, by virtue of (20.12.4.5) : it is an MI-morphism of Ti(M’) into N, and is called the (vector-valued) second fundamental form on M’. We see therefore that the Levi-Ciuita connection on M’ completely determines the vector-valued second fundamental form on M’, and conversely. We remark that the value of Vi. * Y’ at each point x E M’ belongs to T,(M‘), and the value of the right-hand side of (20.12.4.5) belongs to N,; hence (20.12.4.5) gives at each point x E M’ the canonical decomposition of the value of Vx, Y’ into its components in the two orthogonal subspaces T,(M’) and N, . These components are called respectively the tangential and normal components. (20.12.5) Let Rij denote the curvature 2Yorms of C‘, relative to the frame R’ (1 5 i,j S n’), and let bij= ‘f(Rij) for 1 5 i, j 5 n. Then we deduce from (20.12.3.4) n

(20.12.5.1)

dw!. ‘J = - 1 06A oLj k= I

+aij

(1

5 i , j 5 n)


338

and hence, by comparison with (20.6.6.4), (20.12.5.2)

nij- Q;i= C w$ A wLj a>n'

(1 6 i, j

5 n').

(20.12.6) Consider in particular the case where M = R", endowed with its canonical structure (20.1 1.2). We have then dii= 0 for all i, j , and consequently the forms wfj satisfy the structure equations n

(20.12.6.1)

h i j = -kC oh A oij = 1

(1

6 i , j 6 n).

We shall show that the assignment of the n' + n2 differential forms oi, mij determines the submanifold M' of R" (locally) up to a Euclidean displacement. More precisely, we have the following existence and uniqueness theorem : (20.12.7) Let zo be a point in R"',let U be an open neighborhood of zo , and let n be an integer >n'. Let p i ( I 5 i S n'), w i j (1 5 i, j 5 n) be a system of n' n2 direrential I-$orms on U , such that the p i are linearly independent, and satisfying the following relations:

+

(20.12.7.1)

w j i = -wij

(20.12.7.3)

dpi = -

(1

6 i,j6 n),

n'

2 w i j A pi

n'

(20.12.7.4)

0=

( I 5 i 5 n'),

j= 1

waj A p j

j= 1

(n'

+ 1 5 a S n).

Let xo be a point of R" and let (b,), siLn be an orthonormal basis of R".Then there exists a connected open neighborhood V of zo contained in U , and an embedding (16.8.4) F of V into R" having the followingproperties: (i) F(zo) = xo and Di F(z,) = bi (1 S i 5 n'). 0 ; = 'F-'(pi) (1 6 i 4 n'), wfj = 'F-'(wij) (1 5 i, j 5 n) on the submanifold M' = F(V) of R" are the differential forms induced on M' by the Brst n' canonical forms and the connection forms of an orthonormal moving frame R on R" whose first n' vectors are tangent to M' at each point and which is equal to (txo(bi))at the point xo .

(ii) The forms

12 RIEMANNIAN STRUCTURE INDUCED

ON A SUBMANIFOLD

339

Moreover, if there exists another neighborhood V, c U of zo and an embedding F , of V , into R" with these twoproperties, then F and F , coincide on the connected component of zo in V n V,. Suppose that there exists an embedding F and a frame R = (ei),6 i s-n with the desired properties. With the notation of (16.5.2), put

so that the vi are n C" mappings of V into R". Bearing in mind that R" is flat (20.4.1), the relations (20.12.3.2) at the point F(z) of M' give us n

(20.1 2.7.6)

dvi =

C wjivj j=1

(1

5 i 6 n).

Next, from the relation (20.12.3.1) at the point F(z), we have for each vector h, E T,(R"'), n'

T z ( F ) + hz =

C (ci(F(z)),TAF) i= 1

*

hz)ei(F(z)>,

from which we derive (16.5.7) n

dF= Cpivi.

(20.12.7.7)

i= 1

Conversely, we shall first show that in a neighborhood of zo there exist mappings vi (1 S i 5 n) into R" satisfying (20.1 2.7.6) and such that vi(zo) = b, for 1 5 i 5 n. I f V i h ( 1 S h 5 n ) are the components of v i , the equations (20.12.7.6) form a Pfaffian system of n2 equations (20.12.7.8)

dVih -

n

C Wjivjh = 0 j= 1

(1

5 i, h 5 n).

This system is completely integrable. For, by virtue of (20.1 2.7.2), we have

340


from which our assertion follows (1 8.14.3).The desired functions vi therefore exist in a neighborhood of zo and are uniquely determined (1 8.14.3). We seek now a mapping F of a neighborhood of zo into Rnwhich satisfies the equation (20.12.7.7)for the functions v i just determined, and which is such that F(zo) = xo. If Fh (1 5 h 5 n ) are the components of F, the equation (20.12.7.7)is equivalent to the Pfaffian system of n equations

(20.12.7.9) Again, this system is completely integrable. For by virtue of (20.12.7.3) and (20.12.7.8),we have

In this last expression, it is clear that the coefficient of V i h is zero for 1 5 i S n', and the coefficient of uah for a > n' is zero by virtue of (20.12.7.4). From this follow the existence and uniqueness of the function F in a neighborhood of zo (1 8.14.3). We shall next show that ( v i l v j )= aij for 1 S i, j 6 n. The functions w i j = (vi I vj) satisfy the relations

n

n

These n2 equations again form a completely integrable Pfaffian system: This is verified as above, using (20.12.7.2).But by virtue of(20.12.7.1) the constant functions d i j satisfy this system of equations; and since at the point zo we have wij(zo)= d i j , by hypothesis, it follows that w i j = hij throughout V, since V is connected. Since the vi(z) form a basis of Rn for all z, and since the p i are linearly independent, d y z ) is of rank n' at every point of V by (20.12.7.7); hence, replacing V if necessary by a smaller neighborhood, it follows that F is an embedding (16.8.8). We may then at each point of M' = F(V) define the n vectors ei(F(z)) by the formula (20.12.7.5),and it is clear that the embedding F and the frame R = ( e i ) (extended to a neighborhood of M' by the method of (20.8.2)) have the required properties. The uniqueness follows from the


341

uniqueness of the solutions vi and F of the Pfaffian systems (20.12.7.5) and (20.12.7.7), and the remarks made at the beginning of the proof. Q.E.D. Theorem (20.12.7) reduces many problems of the determination of Riemannian submanifolds of R" satisfying given conditions to the integration of Pfaffian systems in which the unknowns are the differential forms p i(1 5 i 5 n') and wij ( I 5 i, ,j S n), and which are obtained by adjoining to the relations in (20.12.7) additional relations expressing the conditions of the given problem. We shall now give some examples of this (see also (20.15.1)). Examples (20.12.8) Let us determine all connected submanifolds M' of R" of dimension n' c n whose second fundamental form is identically zero (cf. (20.13.7) and (20.23.6)). In the notation of (20.12.4), this means that whi = 0 for I i in' and n' 1 5 a 6 n. It follows immediately (in the notation of (20.12.7)) that each of the two Pfaffian systems

+

n'

C wjivj dv, = C mpavp dvi =

j= 1

( 1 5 i S n'), (a > n')

p'n'

is completely integrable. Now the initial value of (vilb,) is zero for i 5 n' and a > n', and the (vilb,) (1 5 i 5 n') are solutions of a completely integrable Pfaffian system which admits also the solution 0; hence (vilb,) = 0; in other words, the vector subspace E of R" spanned by the vi (1 5 i S n') is fixed. But then the projection G of F on the subspace orthogonal to E satisfies dG = 0 by virtue of (20.12.7.7), and consequently is constant. It follows that M' is a connected open subset of a linear subvariety of dimension n'. (20.12.9) A point x E M' is called an umbilic if at this point all the second fundamental forms I,(x) are scalar multiples of the first fundamental form g'(x). We shall determine all connected submanifolds M' of R", of dimension n' 2 2, such that every point of M' is an umbilic. Leaving aside the case dealt with in (20.12.8), let us assume that the I, are not all identically zero. By changing the moving frame R, we may assume that I, = 0 for n'+lSasn-l

and that In # 0. For at each point y

E

M' we may write

2 Ia(y)eab) = g'(y)an(Y), a

where o,(y) # 0 in the space N,; now choose a moving frame in N whose first vector at each point is on(y), and orthonormalize it by the method of

342


(20.8.2), beginning with a,,;by virtue of (20.12.4.5), the new frame so obtained will have the desired properties. Since the moving frame R is orthonormal, the hypothesis In(y) = A(y)g'(y)with A(y) # 0 for all y may be written in the form 0'. n i = Aof

(20.12.9.1)

5 i 5 n'),

(1

and likewise we have (20.12.9.2)

(1

wLi = 0

5 i 5 n', n'

+ 1 5 a 6 n - 1).

From the structure equations (20.12.6.1), bearing in mind (20.12.3.8) and (20.12.3.9), we now obtain (20.12.9.3)

0 = doLi = Am:,

(1

A 0;

5 i 2 n', n'

+ 1 5 a 5 n - I),

and therefore (20.1 2.9.4)

dA

A 0:

=O

(I $ i $ n ' ) .

These relations imply that dA = 0 (A,13.3.1), so that A is a constant f O on M'. Hence, from (20.12.9.3), we have Oh, A O f

=o

for 1 $ i 6 n' and n' + 1 5 LY 5 n - 1, so that ohb= 0 for these values of a. With the notation of (20.12.7), it follows that the Pfaffian system

dui =

n'

w j i oj

j= 1

dun =

+ mniu,,

(1 S i

5 n'),

n'

C

mjnvj

j= 1

is completely integrable, and as in (20.12.8) that the subspace of R" of dimension n' 1 generated by v l , . .. , v,,, and v,, is jixed. Moreover, from (20.12.9.1) we obtain

+

n'

dv, =

C

j= 1

n'

~ j , , v j=

-A

C

j= 1

UjVj

= -AdF,

so that the point v, + AF i s j x e d in R":we may assume that it is the origin, i.e., that F = - A - i v n . Thus, finally, we see that M' is a connected open subset o f a sphere A - ' S , , up to a Euclidean displacement.


343

PROBLEMS 1. Let

M be a pure submanifold of RN of dimension n.

(a) Consider a chart (U, p, n) of M, and put f = p-' in v(U) c R". Then the local expression of the first fundamental form on p(U) is, if the local coordinates are 12,

... ,u",

c

g=

(.I

where

gi j du'

du',

) ;1 ($

gfJ=

and ( x l y ) is the Euclidean scalar product. The local expressions of the second fundamental forms I, are

(Observe that (e, I H/aul)= 0.) (b) If we identify the tangent bundle T(RN)canonically with RZN,the normal bundle P of M is an N-dimensional submanifold of RZN.For each vector n, E P with origin x E M c RN, let p(nx)= x .e,(n,) E RN. The images under p of the critical points of p (16.23) are called the focal points of M, and n, is a focal vector of multiplicity p if the rank of p at the point n, is N - p. The set of focal points of M has measure zero in RN. For each unit vector n, E P normal to M at the point x, let Sn,denote the endomorphism of T,(M) defined by the relation

+

(Kx). (hx

0 kx)lnx) = (Sn,. hxl kxh

which is Hermitian relative to the scalar product in T,(M). The eigenvalues Kj(n,) ( I 2 j 5 n) of this endomorphism, each counted according to its multiplicity, are called the principal curvatures of M in the direction n, . Show that the focal points of M on the h e t H x t.e,(n,) are the points corresponding to t = Kj(nX)-' for the values of j such that K,(n,) # 0, and that the multiplicity of the focal vector K,(n,)-'n, is equal to the multiplicity of the eigenvalue K,(n,). (Take a chart of M and compute the square of the determinant of the Jacobian matrix of the mapping

+

(ul,

. ..,u", t', . . ., tN-")t+f(ul,. .., u") +ct'wa(ul,... ,u"), (I

where w,(u', . . . , u") is the local expression of .e,(e&)); use (a) and observe that we may assume that at the point x the matrix (gi,) is the unit matrix.) 2.

With the hypotheses and notation of Problem 1, for each point y be the function defined by E,(x) = IIx - ~11'.

E RN let

E, : M + R

(a) Show that x is a critical point of E, if and only if y = p(n,) for some vector n, normal to M at the point x; that x is a degenerate critical point of E, if and only if y is a focal point of M ; and that if p is the multiplicity of the focal vector n,, then the rank of the Hessian of E, at the point x is n - p. Deduce that for almost all points y E RN the critical points of E, are nondegenerate.

344

XX


(b) If x is a nondegenerate critical point of E,, the Morse index of E, at the point x (Section 16.5, Problem 3) is equal to the number of focal points (counted according to multiplicity) contained in the segment with endpoints x and y. (c) Suppose that N=2m, so that RN may be identified with C", and that M is a complex-analytic submanifold of C" of complex dimension i n . Show that at each nondegenerate critical point x of E , , the Morse index of E, (Section 16.5, Problem 3) at the is point x i s s i n . (Observe that if Q is a complex quadratic form on C4and Q' = a(Q) the real part of Q, which is a real quadratic form on R2q, then for each eigenvalue h of the Hermitian endomorphism corresponding to Q', -h is also an eigenvalue, with the same multiplicity.) 3. Let M be a differential manifold. Show that in the FrCchet space ~ R ( M of ) C" realvalued functions on M (17.1), the set of functions having no degenerate critical points , exists an embedding h = ( h l , h 2 , , h N ) of M into is dense. (If f~ ~ R ( M ) there RN such that hl = f (Section 16.25, Problem 2). Choose a point y E RN of the form (-c, e 2 , . , eN), where c is large and positive and the el are small, such that E, has no degenerate critical points (Problem 2(a)), and consider the real valued function g(x) = (E,(x) - c')/~c ou M.)

...

..

4. With the hypotheses and notation of Problem 1, consider the submanifold Q (of dimension N - 1) of P consisting of the unit vectors, and the mapping q : QH SN-I defined by q(nJ = Tx(nx).

(a) Let y €RN,y # 0 . Show that the point n, E Q is a critical point of the function n , H (ylq(nx)) if and only if q(n,) is collinear with y , and that this critical point is nondegenerate if and only if n, is a critical point of the mapping q (by using the same method as in Problem l(b)). These latter points are also the points n, E Q such that K(n,) = det(S,,) =

n II

KJ(n,) = 0

I=1

in the notation of Problem I(b). The number K(n,) is called the total curcature ofM in the direction n,. The image F under q of the set of critical points of q is a set of measure zero in SN For each pointy E S, - - F, the set q-'(y) is the discrete closed set of critical points of the function n , H (ylq(n,)) on Q (Section 16.5, Problem 4). (b) The submanifold Q of RZNis canonically endowed with a structure of a Riemannian manifold, hence also with a Lebesgue measure, namely, its Riemannian volume c Q . If u is the inverse image 'q(o"-") of the solid angle form on SN-I(16.21.10) and if t.~"is the corresponding Lebesgue measure (16.24), show that t ~=. I KI . v Q . (Take a chart of M.) (c) Suppose that the manifold M is compact and of dimension n 2 1. Then the set F is closed in SN- 1 ; the mapping q : Q + SN- 1 is surjectice (consider, for each y E SNthe function n , H (ylq(n,)) on the compact manifold Q, and use (a)); and the restriction of q to the open set q-l(SN-l - F) is a surjective local diffeornorphism of this open set onto SN - F. The integral curcature of M is the number

For every C' real-valued function f on M which has only nondegenerate critical points (and therefore cannot be locally constant anywhere) let B(M,f) denote the (finite)

13 CURVES I N RIEMANNIAN MANIFOLDS

345

number of critical points off, and let B(M) denote the infimum of P(M,f) for all such functions f. Then P(M) 2 2. Show that K(M) =

-J 1

N '

SN-I-F

B(M, (Ylq)) db4-I(Y),

where v N - l is the canonical Riemannian volume on SN-Iinduced by the solid angle form d N - l ) . (Use (16.24.8).) Deduce that

4 M )h B(M) 1 2 (Chern-Lashof theorem).

(d) With the hypotheses of (c), show that if B(M) < 3, then M is homeomorphic to S,. (Observe that there exists a point Y E such that the function (yJq)has only two critical points in M, both of which are nondegenerate, and apply Reeb's theorem (Section 20.8, Problem 6).) 5. Let f be a Cm mapping of a differential manifold N into R", and let G be a C" lifting o f f to T(R"). If Rn is endowed with its canonical connection (20.11.2), we have 7rcZ,(dG(z)) = d, G , where G(z) = T f(=)(G(.z)). We may therefore identifV G with G, and then the covariant exterior differential dG is identified with the differential dG of the vector-valued function G (16.20.15). If N is a submanifold M' of R" and f is the canonical injection, we write dx for the differential of f : M'+R". If (el, , . , en) is a moving frame having the properties of (20.12.2), then with the preceding conventions the formulas (20.12.3.1) and (20.12.3.2)

.

may be written in the form

and the structure equations (20.12.3.3) and (20.12.3.4) are obtained simply by writing down the relations d(dx) = 0 and d(dei)= 0 for the exterior differentials (17.15.3.1). The relations (20.12.3.5) are obtained by remarking that (ell e l ) = StJ and hence, taking exterior differentials, that (de,)eJ) (e,lde,) = 0. Since the Riemannian metric tensor on R" may be identified with the mapping (u, V)H (u I v) of R2"into R", the first fundamental form of M' may be written as (dxl dx), and each of the second fundamental forms as -(de,I dx). Finally, the curvature form a'of M' may be written as

+

Q' =

C (de. . de,),

a>n'

the product being that defined in Section 20.6, Problem 2.

13. CURVES IN R I E M A N N I A N MANIFOLDS

(20.13.1) We recall that a curue C in a Riemannian manifold M is a onecliniensional submanifold of M ; we regard C as endowed with the Riemannian metric tensor induced by the Riemannian metric tensor g on M. We shall study the properties of C in relation to M, in a sufhciently small neighborhood

346

XX


of a point c E C. We suppose, therefore, that there exists an embedding u of an open interval I c R into M such that #(I)= C, that 0 E I and that u(0) = c. The hypothesis that u is an embedding implies that the vector u’(t) E Tu(,)(C) (defined in (18.1.2.3)) is always fO. By transport of structure, the image under u of the canonical orientation of R is an orientation of C, called the orientation defined by u. If a, /3 E I are such that c1 c /3, and if a = u(a), b = u(p) are the corresponding points of C, we shall sometimes say that C, endowed with this orientation, is oriented in the direction from a to b. Let I be the length on C, namely, the positive Lebesgue measure defined by the Riemannian metric tensor induced by g (20.8.5). For each interval [a, /3] c I, we have therefore by definition (20.8.5.1)

) u’(5)). For each t E I, let where llu’(5)Il = (g(u(t)), ~ ’ ( 5 0 (20.13.1.2) which is therefore the length of the arc u([O, t ] ) if t 2 0, and the negative of the length of u( [ t , 01) if t S 0. This number depends only on u(t) and the choice of c and the orientation of C . For iff: I1 -,I is a diffeomorphism of an open interval I, c R containing 0 onto the open interval I, such that f preserves the orientation and f ( 0 ) = 0, then f‘(C)> 0 for all [ E I , . Now put u , = u O L sothat(16.5.4.1) u;(C) =f’(C>u’(f(O>and Ilu;(oII =f’(C)llu’(f(O)Il; if t =f ( t , ) , our assertion follows from the formula (8.7.4) for change of variable in an integral:

~;llu;(C)ll

dC =

sd’

ll~‘(f(C))llf’(C) dC

= Jju~(5)Il d5.

The number q(t)is called the curvilinear coordinate of the point x = u(t) on C , endowed with the orientation defined by u, with respect to the origin c. If we denote it by $(x), then $ is a chart of C on the interval q(1) = J (an interval which is therefore entirely determined by the curve C, the point c, and the chosen orientation), and we have q ( t ) = +(u(t)). The mapping s H u(s) = u(q -I(s))

1 3 CURVES I N RIEMANNIAN MANIFOLDS

347

of J into M is called the parametrization of C by arc length (relative to the given choices of c and the orientation). We have by definition

( d $ ( 4 0 ) , WD2= (4w2 = IIU’(t)112

= (g(u(t)), 2 4 ) @u’(t)>;

since C is 1-dimensional, this may also be written in the form d$ @ d $ = ‘h(g), where h : C + M is the canonical injection. In other words, d$ @ d$ is the metric tensor induced on C. If (U, [, n ) is a chart of a neighborhood of c in M, then writing the metric tensor g in the form (20.8.2.5), we have on C,

d$ @ d$ =

C g i j d(ci

0

i, i

h) @ d(c’ 0 h ) ;

or, equivalently, on I,

dq @ d q =

1( g i j

0

U) du’ @I du’,

i.j

where we have put u i= ti0 u (1 5 i n). By abuse of notation, we shall sometimes write ds in place of dip, and abbreviate the above formula to

Because of this formula, the metric tensor g on M is sometimes referred to as ds2 on M.”

“

(20.13.2) In this section we shall assume that the point c and the orientation of C have been chosen once for all, and we retain the previous notation. By definition, we have IIu‘(s)II = 1 for s E J; the vector u’(s) is called the unit tangent vector to C at the point z@). It depends only on the choice of orientation (and changes sign if the opposite orientation is chosen). We recall that if a mapping w of J into T(M) is a lifting of u, the covariant exterior differential dw is the differential 1-form on J, with values in T(M), such that dw(s) * E(s) = VE(s) * w

for all s E J (17.19.4.2), where E is the field of unit vectors on R (18.1.1). Recall also that, by abuse of notation, we write V, * w in place of VE w (18.7.2). With this notation established, a Frenet frame of C is by definition a C“ mapping sH(fi(s),. . ., fn(s)) of J into the frame bundle R(M) of M, such that the fj(s) belong to Tv(s)(M)and form an orthonormal basis (relative to g )

348

XX


of this space, and such that the following relations (Frenet’s formulas) hold in J :

(20.1 3.2.1)

where the k j are C” real-valued functions on J, everywhere strictly positive. (20.13.3) We shall show that in general there exists a unique Frenet frame in a sufficiently small open subset of C: the meaning of the words “in general” will be made clear by the proof. From the relation (fl I fl) = 1, by taking the covariant derivatives of both sides and using (20.9.5.4), we obtain 2(V, . fi 1 fl) = 0. We recall that V, fl = 0 at all points of C if and only if v is a geodesic of M (1 8.6.1.2). Assume that this is not so (the “ general case”); then there exists an open set U1in C in which V, . fl does not vanish, and we may therefore write V, * fi = k i f 2 , where f 2 is orthogonal to f l , of length 1, and kl > 0 at all points of U1. Suppose that we have determined a decreasing sequence (Uj)l s j s i - i of open sets in C such that in U i - l the first i of the formulas (20.13.2.1) hold, the f j for j i being mutually orthogonal and of length 1, and the functions k j ( j 5 i - 1) strictly positive at each point of U i - 1. By taking the covariant derivatives of the relations ( f j 1 f i ) = 0 for j < i and (fiI f i ) = 1, we obtain immediately

s

(V;filfj)=0

(1 s j S i - 2 ) ,

(Vs*fiIfi-l)+ki-l =O, (Vs

*

fil fi)

=O,

which shows that V, * f i + k i - l f i - l is orthogonal to f , , . . . , f i - l , f i at each point of U i - l . If i c n, it may happen that V, f i + k i U lf i - l vanishes identically on U i v l .Assume that this is not so (the “general case”); then there exists an open subset Ui c U i - , of C on which this vector field does not vanish, and then we may write V, . f i k i - l f i - l = k i f i + , , where f i + , is orthogonal to fl, ... , f i , of length I , and ki> 0 a t all points of U i . If i = n, we necessarily have V, * f. k, - f, - = 0. 3

+

+


349

If there exists a Frenet frame of C, the number kj(s) > 0 is called thejth curvature of C at the point x = u(s), and the number I/kj(s) thejth radius of curvature; the vector fj+,(s) is called thejth unit normal vector to C at x, the line Rfj+,(s) in T,(M) thejth normal, and the plane Rf,(s) Rfj+,(s) thejth osculating plane. When n = 2, k , is simply called the curvature, and the first normal is called the normal. When n = 3, k , and k2 are called, respectively, the curvature and the torsion; the first and second normals are called, respectively, the principal normal and the binormal, and the first and second osculating planes are called, respectively, the osculating plane and the rectifying plane. When M = R", this terminology is commonly taken to refer, not to the lines and planes in T,(R") just defined, but to their images under the composition of the translation h H h + x and the canonical bijection T,, so that they are taken to be affine lines and planes contained in R" (cf. 16.8.6).

+

(20.13.4) When M = R" and the curve C possesses a Frenet frame, the n - 1 functions k j ( 1 5.j 5 n - 1) determine the curve C up to a Euclidean displacement. Precisely, we have: (20.13.5) Let J c R be an interval containing 0, and let k j (1 5 j 5 n - 1) be C" functions on J which are everywhere >O. Let c be a point of R" and let bi ( 1 5 i n) be n uectors in TJR") forming an orthonormal basis. Then there exists an interval I c J containing 0, a mapping u : I -,R", and n mappings fi : I -,T(R") ( 1 5 i 5 n), such that:

(i) 1' is an embedding of I in R", s H v(s) is the arc length parametrization of C = v(1) (for the orientation nvhich is the image of the canonical orientation of R), and (f,), is a Frener frame of C satisfying the formulas (20.13.2.1) for the giuen junctions k j ; (ii) v(0) = c and fi(0)= bi (1 5 i 5 n).

Furthermore, if I , is an interval containing 0 and contained in J , and i f v('),f i ( l ) ( 1 =< i 2 n) are functions satisfying the above conditions, then v = d l ) andfi =fit' ( I 5 i = < n) in I n I,. This is a particular case of (20.12.7), with n' = 1. In this case there are no compatibility conditions, nor is it necessary to invoke Frobenius' theorem since we are dealing with ordinary differential equations. Henceforth, whenever we consider a curve (other than a geodesic trajectory) in a Riemannian manifold, we shall always assume that it possesses a Frenet frame, unless the contrary is expressly stated.

350

XX


Remark (20.13.5) When the manifold M is oriented, it is often convenient to modify the definition of a Frenet frame, replacing f,, by Ef,, , where E = f I is chosen so that ( f l , . , . , f n - l , Ef,,) is a direct frame (16.21.2). The Frenet formulas

remain valid provided that we replace k,,.-l by Ek,of either sign).

(which therefore can be

(20.13.6) Let M be a Riemannian manifold of dimension n , and M' a sub-

manifold of dimension n', where 1 < n' < n. Let C be a curve in M'. Applying the formula (20.12.4.4) with X' = Y' = fl o v - I (in the notation of (20.13.2)), we obtain (20.1 3.6.1)

Vs*fl=V:.fl

+C(Ia,flOfi)eu. a

Since the two terms on the right-hand side are the components of the lefthand side in T,(M') and the normal space N,, we deduce immediately: Every curve C in M ' which is u geodesic trajectory of M is also a geodesic trajectory of M'. (ii) Let C be a curve in M' which is riot u geodesic trajectory of M ; in order that C should be a geodesic lrajectory of M', it is necessary and suficient that its first normal (relative to M ) should be normal to M'. (20.13.6.2) (i)

(20.1 3.6.3) The normal component of V, . fi (called the normal curvature vector of C ) at a point x is the same for all curves in M' having the same tangent at the point x (Meusnier's theorem). Thejirst curvature at x is the same for all curves in M having the same tcngent and the same first normal at x .

The latter assertion is a consequence of the following formula, which comes from (20.13.6.1) and (20.13.2.1): (20.13.6.4)

kl(f2 I e,) = (Ia,

fi

0 fi)

(n'

+ 1 5 a 6 n).

A curve C c M' is called an asymptotic line of M' if its normal curvature vector is zero, or equivalently, if (I,, fl 8 fl ) = 0 for n' 1 5 a 5 n. If C is not a geodesic trajectory of M, another equivalent definition is that the first osculating plane of C should be tangent to M'.

+

(20.13.7) A submanifold M' of M is said to be geodesic at the point x if all the geodesic trajectories of M' passing through x are also geodesic trajectories of M. The formula (20.13.6.1) shows that this condition signifies that all the second fundamentalforms of M' are zero at the point x. The submanifold M' is said to be totally geodesic if it is geodesic at every point, or equivalently if all the second fundamental forms of M' are identically zero.


351

PROBLEMS 1. Let C be a curve in R”.With the notation of (20.13.3). suppose that we have the first i formulas (20.13.2.1), but that C , . f, $- k , - l f , - , =: 0 identically in U i - l . Show that in this case U,-,is contained in an affine-linear subspace of R” of dimension i. (Observe that, for a fixed vector a E R”,the i functions (a If,) for j 5 i satisfy a homogeneous

linear system of differential equations.) Generalize to the case where C is a curve in an arbitrary Riemannian manifold: With the same hypotheses, show that the i-vector which is the product of the first i unit normal vectors at any point of C is obtained from its value at a point xo E C by parallel transport along C.

2.

Show that if there exists a Frenet frame for a curve C defined by an embedding E I being any parameter), then the jth curvature of C is the following function o f t :

u : IH u ( t ) ( r

where V, . u = u‘ and Vik). I I areas” defined in (20.8.4).

=

V, . (V;”-”

. u), the norms being the

“

k-dimensional

3. (a) The second fundamental forms (20.12.4) of a curve C, relative to the Frenet frame of C, are given by 1 2 1 1 = k l,

1,11=0

for a L 3 .

(b) Show that for a compact curve C in RN, the integral curvature (Section 20.12, Problem 4) is given by

where L is the length of the curve. Deduce that )s(l)!

dS 2 (N - I)QN-I/QN-z

(=277 if N = 3) (Fenchel’s inequality). (c) A compact connected curve C in R3 is said to be a knot if there exists no homeomorphism f o f the disk llxil =< I in RZ onto a subspace of R3 such that the image underfof the circle IIx 11 = 1 is C. Show that if C is a knot, we have

(Supposing that

1:

kl(s) ds < 47r. deduce as in Section 20.12, Problem 4(d), that there

exists a unit vector y E S2 such that the function s ~ ( yv(s)) l has a derivative which vanishes at only two points of the interval [0, L[, which we may assume to be 0 and

352


L' < L. Then define a homeomorphism fwith the properties described above.) (FuryMilnor theorem). 4. A curve C in the plane R2 is said to be strictly conuex (by abuse of language, cf. Section 8.5, Problem 8) if C meets any line in at most two distinct points.

(a) Let C be a connected, strictly convex curve. Show that for any point x E C the curve C lies entirely on one side of the tangent to C at the point x . (Prove first that this is the case for a sufficiently small neighborhood of x in C, arguing by contradition.) Deduce that if C is compact, then C is the frontier of the convex set D which is the intersection of the closed half-planes containing C, or equivalently, is the convex hull of C (Section 12.14, Problem 13). (b) Let C be a compact, strictly convex curve in RZ (identified with C ) ; let L be its length and S H U ( S ) its arc length parametrization (0 5 s < L) for some choice of orientation. Show that u'(s) = e r e ( S where ), 0 is a strictly monotone mapping of [0, L[ onto [0, 277[ or ] - 2 n , 01. Conversely, if 0 is a C" mapping satisfying this condition, and such that Let'(')ds = 0, then the image of [0, L[ under the mapping s H elo(')dr

I:

0

is a compact strictly convex curve. (c) Show that on a compact, strictly convex curve C there exist at least four points at which the curvature (considered as a function s H k ( s ) of the arc length) is stationary (Mukhopadhyaya's theorem). (Assume that the result is false and show that, up to a displacement, the curve is the image of a mapping S H ( [ ' ( S ) , ['(s) of [0, L[ into R2 such that, for some so €10, L[, the function k is strictly decreasing on [0, so], strictly increasing on [SO, L[, and such that ['(s) > 0 for s €10, so[ and ['(.) < 0 for s E ]so, L[. Deduce from Frenet's formulas that IoLk(s)

split this integral into

ds = 0 ;

s,'" +I: andintegrateeachof these by parts; then use the mean-

value theorem to obtain a contradiction.) Consider in particular the case where C is an ellipse. (d) Give an example of an immersion u of an interval I C R into R2 which is not injective but is such that each point to E I has a neighborhood J c I such that the restriction of u to J is an embedding and u(J) is strictly convex.

R, periodic with period 277, and taking only values > O Show that there exists a compact strictly convex curve C in RZ (Problem 4)such that the origin is an interior point of the convex hull D of C and such that the function of support of D (Section 16.5, Problem 7) is given by H(rere)= rh(t9). (Assuming that the problem has been solved and that C is defined as the image fh(x(f?), y(@) of [0, 2n[ under a C" mapping, express that the tangent at the point with parameter 0 is orthogonal to the vector ere.) Let U be the open subset of S1 x S1 consisting of pairs (11, such that (2 f For each point ( e t B 1ere') , E U , let P(&, 8,) be the intersection of the lines of support (Section 5.8, Problem 3) of D with equations (zle'") = h(&), (zle1'') = h(eZ).Show that this mapping P makes U a two-sheeted covering of R2 - D, and that its Jacobian at the point (el", ele2)isequal in absolute value to tl(P)t2(P)/lsin w(P)I, where tl(P) and

5. Let h be a Cm function on

c2)


353

rz(P) are the distances from P to the points of contact of the two tangents to C which pass through P, and w(P) is an angle between these two tangents. Deduce that

(Crofton'sformula). 6. (a) Let G be the group of displacements of R2 with determinant +1, so that G is the semidirect product of the group R2 and SO(2, R). Any element u E G is of the form (X,Y)H(U

+ x cos I9 --y

sin 19,u

+ x sin I9 + y cos 0).

The forms w 1 = cos 0 du

+ sin 0 du,

w2=

-sin 0 du

+ cos 0 du,

w 3 = dB

constitute a basis of the space of left-invariant differential forms on G, so that uG = w 1 A w , A w j = du A dil A dB is a left- and right-invariant 3-form on G, and the corresponding measure p is a Haar measure on G . Consider two curves C , , C , in the plane R2,of finite lengths I t , 12, parametrized by their curvilinear coordinates s,, s,. For each pair ( s , , 3,) of parameters and each angle T E [ 0 , 2 a [ ,let u = g(s,, s2 , ~JI) be the element of G which maps the point Mz of C, with parameter sz to the point MI of C , with parameter s,, and the unit tangent vector at M, to C , to a vector making the angle CJI with the unit tangent vector at MI to C,. Show that 'g(uG)= fsin v ds, A ds, A d v . Using Sard's theorem (16.23.1), deduce that for almost all u E G (relative to the measure p) the set C1 n (u . C,) is finite, and that if n(u) is the number of elements in this set, then

j, n(u)d p ( 4

= 4I1l2

(Poincark'sformula). (b) Let C be a compact strictly convex curve in R2,of length L; let D be its convex hull, and V,(D) the set of points of R2 whose distance from D is s r , for some r > 0. For each integer k > 0, let t n k be the Lebesgue measure of the set of points (x,y ) E R2 such that the circle with center (x, JJ) and radius r meets C in k points. Show that mk = 0 if k is odd. By applying Poincart's formula with C1 = C and C2 a circle of radius r, show that (1)

m2

+ 2m4 + 3m6 + ... = 2rL.

Let A be the Lebesgue measure of D. Using the formula of Steiner-Minkowski to evaluate the measure of V,(D) (Section 16.24, Problem 7(b)), prove that (2)

m; + r L

-A

- ar2 = m4 +2m6

+ 3ms f...,

where mb is the measure of the set of points of V,(D) such that the circle with center at the point and radius r does not intersect C . Let ri be the maximum of the radii of closed disks contained in D, and re the minimum of the radii of the closed disks containing D. Deduce from (2) that L2 - 4aA

2 +(re

- r,)'

(Banneserz's inequality). (Observe that mh = 0 if ri 5 r 2 re .)

354

XX


7. (a) With the notation of Problem 6, let H be the subgroup of G consisting of the displacements which leave the line y = 0 invariant as a whole, so that H consists of the (I E G for which u = 0 and 0 = 0 or ‘ I. The homogeneous space G / H may be identified with the space of affine lines in R2 (a submanifold of the Grassmannian C1, If ‘ I :IG +G/H is thecanonical submersion, show that there existsa G-invariant differential 2-form uGIHon G / H such that ‘‘II(UG,H)= w 2 A w3 (Section 19.16, Problem 16). Let Y be the corresponding Lebesgue measure on G / H (16.24.1). Consider a curve C in the plane RZ,of finite length I, parametrized by its arc length s. For each parameter s and angle Q, E [0, ‘II[, let A(s, y ) be the line through the point M of C with parameters, inclined at the angle Q, to the tangent at M to C. Show that ‘A(uC/H) = &sin Q, ds A d p Deduce that for almost all lines y in RZthe set y n C is finite, and that if n(y) is the number of elements in this set, then

(Crofton’s jormula).

(b) Deduce from (a) that if C is contained in the convex hull D1 of a compact strictly convex curve C1 of length I , , then there exists at least one line y such that the number of points of y n C is 3 1 / 1 1 (Apply Crofton’s formula to C and Cl). (c) Generalize Crofton’s formula to curves in R” and their intersections with hyperplanes in R”. 8. (a) The existence “ in general” of a Frenet frame for a curve C may be expressed in the following terms. Let So(.,(M) be the bundle of orthonormal frames of M,‘Iits I projection on M I and uI, w,, the canonical and connection forms on So(.,(M). Then there

exist two sections of So(.)(M) over C (corresponding to the two orientations of C ) such that the (n I) x n matrix of the forms 0: , miJ which are the inverse images of al,mi,, respectively, under one of these sections is of the form

+

,

where w ; + ~ , = k,u;, the kJ being functions >O. These frames may be determined by the following systematic inductive procedure: the principal bundle Po over ‘II-I(C) induced by So(.)(M), with group O(n), has dimension $n(n - I) I , hence for each r E ‘II-I(C) the kernel KO@)of the canonical surjective mapping

+

: Tr(So(n,(M))* Tr(Po)*, ‘T~C~O)

corresponding to the canonical injection j o : Po -+SO(~,(M),has dimension n - 1. The frames r E Po for which Ko(r) is the vector subspace spanned by the covectors u2(r), . . . ,un(r)form a principol bundle PI over C with group O ( n - 1) x {&I}, hence of dimension $(n - I)(n - 2) 1. For each r E Po, the kernel Kl(r) of the canonical surjective mapping

+

‘T,(jJ :Tr(Po)* -+T,(pd*,

13 CURVES IN RIEMANNIAN MANIFOLDS

355

corresponding to the canonical injection j, : PI + Po, has dimension n - 1. The frames ) klal(r), r E PI for which Kl(r) is spanned by the covectors wjI(r) forj 2 3 and w Z 1 ( rfor some number k , depending only on n(r) E C, form aprincipal bundle P2 over C with group O(n - 2) x { * I)z. If j2 is the canonical injection P2 +PI, we consider at the next stage the kernel of 'T,(j2), of dimension n - 2; and so on. (b) Define in the same way the Frenet frame for a curve C in a pseudo-Riemannian manifold M,assuming that C is not isotropic at any point. (c) Let M be a pseudo-Riemannian manifold of signature (n - 1, I), so that the ) a symmetric bilinear form @ on R" corresponding group G (20.7.6) is the group O ( @ of such that @(el,e l ) = @ ( e 2 , e2) = 0, @(el, e2) = @(el, el) = 1, @ ( e l ,e,) = for i,j 2 2. A curve C c M is said to be a curue ofrero length if it isisotropicateverypoint. Prove the existence " in general" of a " Frenet frame" for C in the following sense: If &(M) is the bundle of frames of the G-structure of M, then the connection forms satisfy the relations

a,,

Ull UJI

+ +

u 2 2

= 0,

U z j

=O

w2,

=

u , 2

for j > 3,

= 0,

w,,

for j 2 3 ,

UIJ+U,~=O

+wji=o

for i, j z 3 .

+

There exist a firrite number of sections of S,(M) over C, such that the (n 1) x n matrix formed by the inverse images of 0,, w,, under one of these sections is of the form

-w;2

0 0

-0;

0

wi2

0

0

0

Wk3

0

0

0

0

0

0:

0

0 0

0 0

0:

0 -wk,

0 0 0 0 -w;4

0

...

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

0

0 0 0 0 0

0 0 0 0

(reduce to the case n = 3). Generalize to the case of a pseudo-Riemannian metric tensor of arbitrary signature. Calculate the number of sections having the above property. Suppose that M = R" and that there exists an embedding u of an open interval I c R (containing 0) into R" such that u(1) = C, thereby defining on C an orientation and an origin c = u(0). The pseudo-arc-length of a point x = u( t ) on C is then defined to be the value of the integral over C of the forrnf,~; (16.24.2), wheref, is the charac-

teristic function of the set u([O, 11) if t 20,and the negative of the characteristic function of the set a ( [ / ,01) if r 5 0. If s is the real-valued function on C so defined, then 0;= ds,and we may write w ; , = K,-&) ds for 3 5 j 5 n. Show that the assignment of n - 2 functions K,-&) of class C" and everywhere >O on an open interval of R defines a curve of zero length in R" satisfying the relations above, which is uniquely determined up to a displacement. 9. Endow R" with the canonical flat G-structure, where G = SL(n, corresponding canonical connection (Section 20.7, Problem 7). Let

R), and with the CJ,

canonical and connection forms on S,(M) (which satisfy the relation

and w,, be the

356


Show that for a curve C C R" there exist in general a finite number of sections of S,(M) over C such that the (n 1) x n matrix formed by the inverse images of the u,and w , , under one of these sections is of the form

+

0;

w:l

0

...

0

0

... ..'

0

0

wi2

w;3

0;

(The notational conventions of Section 20.12, Problem 5, will be found useful.) Define as in Problem 8 a pseudo-arc-length s on C, and show that if we put w;, = K,(s) ds, then the assignment 01 ::.en - I functions K, in an open interval of R defines a curve C satisfying the relations above; moreover, every other curve satisfying these relations is obtained from C by a translation followed by an automorphism belonging to SL(n, R).

14. HYPERSURFACES IN RIEMANNIAN MANIFOLDS

(20.14.1) We recall that a hypersurjace in a Riemannian manifold M of dimension 17 is a submanifold V of dimension 17 - 1. Suppose that M is oriented (which involves no loss of generality if we are concerned only with local properties). Since locally M is diffeomorphic to the product V x R, an orthonormal moving frame R' = ( e l . . . . , en-l) on the Riemannian manifold V determines uniquely a direct orthonormal frame (el,. . ., en , en) for M. The vector field n = en is then called the oriented unit nor~??al vector field of V, relative to the orientations of M and V (the latter relative to R'). There is only one second fundamental form I = I,, on V, which is uniquely determined up to sign (the sign depending on the orientations of M and V) and is given by the formula

-,

z

n-1

(20.1 4.2)

I=

w:i@a;.

i=1

As soon as the orientation of M is fixed, that of V is fixed by the choice of - 1 eigenvalues of I(x) (counted according to multiplicity) with respect to the positive definite form g'(x) are real (of either sign), and are called the principal curraturcs of V at the point x ; they change sign with n. Denoting them by

n, and the form I changes sign with n. The n

P , ( 4 iP A X )

5 . . . 5 P"-l(x).

the numbers

+ ... +

1 H(x) = -( P ~ ( x )

n-l

Pn-

1(xN

14 HYPERSURFACES I N RIEMANNIAN MANIFOLDS

357

and N x ) = PI(x)PZ(x)

... P,-l(X)

are called, respectively, the niean curratwe and the total curvature of V at the point x. The inverses ( p j ( x ) ) - ' for those indices , j such that pj(x) # 0 are called the principal radii of currwture of V at the point x . When M = R", the points x + ~ ~ ( x ) - ' ~ . ~ ( n(for ( x pj(x) )) # 0) are called the principal centers qfcurvature of V at the point x; they do not depend on the sign of n. Suppose that the principal curvatures p j ( s ) are all distinct. Since they are the roots of the equation det(/ij(.u) - t ) = 0, and these roots are all simple, it follows that the derivative of this polynomial with respect to t is nonzero at each root; hence the pi ( I 2.j 2 n - i ) are well-defined functions of class C" in a sufficiently small neighborhood of x (1 0.2.3) and their values at each point of this neighborhood are all distinct. I t follows therefore from Cramer's formulas that there exist n - 1 fields of unit vectors cj (1 5.j 5 n - 1) o f class C" on V such that at each point y the lines Rcj(y) are the principal axe3 of I(y) with respect to g ' ( y ) .These lines are called the principaldirections of V at the point y . A curve in V which at each of its points is tangent to a principal direction is called a line qfciiri~atureof V. We may always assume that the cj (which are determined only up to sign) ~ orthonormal frame. have been chosen so that (cl, .... c ~ n)- is ~a direct Relative to this frame, the second fundamental form has the expression n- 1

I=

(20.14.2.1)

CPjOJ@Oj j = 1

For any curve C in V, the number k , ( fz In) (in the notation of (20.1 3.2)) is called the normal curvature of C (relative to the chosen normal vector n to V). It follows from (20.14.2.1) and (20.13.6.4) that we have n- 1

(20.14.3)

k , ( fi I n ) =

1 pj( f, I c,)'

(Euler'sformula).

j= 1

The relation (20.14.2.1) also signifies that relative to the frame

we have, with the notation of (20.12.3),

358

XX


and hence the formula (20.12.5.2) may be written as (20.14.5)

In particular, when n = 3 and M

= R3,we

(20.14.6)

K = PIP2

have (assuming that p, # p z ) 7

i.e., the product of the principal curvatures is equal to the Gaussian curvature (and therefore depends only on the induced Riemannian metric tensor g') (Gauss's theorema egregium). It should be remarked that this result remains true when p, = pz . This is clear by continuity at an umbilic which is a limit of nonumbilical points; the only other cases are the sphere and the plane, for which the theorem is immediately verified (20.12.9). (20.14.7) In general, for n 2 4, hypersurfaces in R" are rigid: the second fundamental form is in general uniquely determined by the Riemannian metric tensor g'. To be precise: (20.14.8) Let V, V, be two connected hypersurfaces in R",where n 2 4, and suppose that the principal curvatures of V are distinct and nonzero at every point of V. Then every isometry f of V onto V, is a Euclidean displacement (Beez's theorem).

Let us retain the notation introduced above for the second fundamental form of V, and let (fij(x)) be the matrix at the point x, relative to the frame (C~(X))~ of theinverse image under f of the second fundamental form of V,. Since the curvature of the connection on R" is zero, the hypothesis that f is an isometry implies, by virtue of (20.14.5) and (20.12.5.2), that (20.14.8.1)

( -nij, c,

A

c,) = p i p j ( a j , 6 ,

- 6 , d j s ) = Ijrlis - lirIjs

for all indices i,j,r, s. This signifies that if u, u are the endomorphisms of R" whose matrices are ( f i j ( x ) ) and diag(p,(x), . . . , pn-,(x)), respectively, then 2

2

A u = A v ; we have to show that u = + u .

Since by hypothesis v is invertible, it is enough to show that u-'(u(z)) is a scalar multiple of z, for all z E R"-'. Suppose not; then there exists z E R"-' such that u(z) and v(z) are linearly independent. Since n - 1 2 3, there exists a vector y such that u(z), v(y), and

14 HYPERSURFACES I N RIEMANNIAN MANIFOLDS

359

u(z) are linearly independent, and therefore u(z) A u(y) A u(z) # 0; but by hypothesis we have v(y) A u(z) = u(y) A u(z), which leads to a contradiction.

Hence u = au for some scalar a ; since A u = A u, this implies that a' and completes the proof, in view of (20.12.7). 2

2

=

1

By contrast, in R3 there exist surfaces whose principal curvatures are distinct and #0, and for which there exist isometric surfaces not obtainable by any Euclidean displacement (Problem 8).

PROBLEMS 1. With the hypotheses and notation of (20.14.1), for each x E V let S(x) denote the self-adjoint endomorphism of TJV) associated (Section 11.5, Problem 3) with the symmetric bilinear form I(x), i.e., such that ( S ( x ) . h,l k,) = =j (a complex cube root of unity), a3 = i = j 2 . Let D be the open set in R2 which is the union of the three angles" AJk defined by (XI aJ)< 1 and (XI ar)< 1. In the angle A Z 3 , define a function z I to be equal to 0 if (XI at) 2 1, and equal to exp(-((XI al) - 1)-2) if (XI al) > 1 ; define z2 in A31 and 23 in AI2 analogously. Then these three functions coincide in the triangle D' which is the intersection of any two of A Z3,A31rA,, , and therefore define a C" function z in D. Let C be the hypersurface in R3 which is the 'I

362

XX


graph of the function z. With the notation of (c) above, show that Vo = a, V1 = V -fit, V 1 = B', and that through a point of D' there passes no line wholly contained in V. (e) For each point x E V, let X ( x ) denote the affine tangent hyperplane to V. If R" is identified with an open subset of P.(R). then X ( x ) is an element of the Grassmannian G , , 1, . The mapping x + X ( x ) of V into G,+ 1 . is of class C" (it is called the tangenrial image of V). With the notation of (c) show that if V is compact then X(V0) is dense in X(V). 5. Let V be a compact hypersurface in R". Then the integral curvature of V (Section 20.12,

Problem 4) is equal to

where K(x) is the total curvature at the point x of v.

EV

and u is the Riemannian volume

(a) Suppose that K(V)= 2 (cf. lor. cit.). Show that if a point x E V belongs to Vo (notation of Problem 4(c)), then V lies entirely to one side of the tangent hyperplane to V at the point x . (Observe that n is a local diffeomorphism at the point x : if V were not to one side of H, there would exist a neighborhood of n(x) in S,-, such that, for all points z in this neighborhood, there would be at least three distinct points of V at which n took the value z. Then use Section 20.12, Problem 4.) (b) Deduce that if K ( V )= 2, then V is the frontier of a convex body in R". (Use Problem 4(c).) 6. Let M be a submanifold of R", let r be its curvature morphism and1 the second (vector-

valued) fundamental form on M. (a) Show that for each x we have (1)

E

M and each system of tangent vectors u, v, w, t in T,(M),

(tl(r(x) * (u A v)) . w) = ( I M x ) , v 0t>) 0 w> I M x ) , 0t>).

-( n - 1 ; hence none of the subspaces Ej (1 s j n - 1) is empty, and in view of (20.15.1.8) this establishes our assertion. Furthermore, for n > 3, the inequality f n ( n - 1) > (n - 1) (n - 2) shows that v2 E T,,(P) may be chosen so that the vectors

=-

+

W,(V~),

e a . 9

Wn-I(VI)j

w2(~2>9* * * > wn-I(v2)

are linearly independent. We now assume as inductive hypothesis that, for some p 5 n, we have determined vectors vlr . . . , vp-l in T,,(P) such that

(20.1 5.1.14)

(i) if uj = (ej(xo),vj) for 1 s j S p - 1, the uj generate an integral element of dimension p - 1, and the uj with index 5 p - 2 a regular integral element; (ii) the (n - 1) + (n - 2) + * + (n - p + 1) vectors 9

wk(vj)

(lgksn-1,

are linearly independent in R"'"- I)''.

1SjSp-1,

jrk)

372

XX


Let uJ= (cJ(x), vJ) be a system of p - 1 vectors such that the cj(x)

(1

s j 6 p - 1) form an orthonormal system and such that (z, u;, . . . , u6-

1)

lies in (M x P)p-l (notation of (18.10.4)). Since we have only to consider a neighborhood of ( z o , ul, . . . , uPdl) in (M x P)p-l, we may assume that the vectors wk(vJ) (1 5 k S n - 1, 1 5 j S p - 1, j 6 k) remain linearly independent. In order that a vector ub = (c;(x), v;) should be such that the cJ(x) (1 S j 5 p ) form an orthonormal system and that ub determines together with u',, . . ., ub- an integral element, it is necessary and sufficient that the 2-forms (20.15.1.3) and (20.15.1.4) should vanish at all the bivectors (c;(x), v;) A (c;(x), vb) (1 S k 5 p - 1). Let R' be an orthonormal moving frame on M, defined in a neighborhood of x,, , and such that its first p vectors at the point x are the cJ(x) ( 1 S j S p). Denoting again by K;lijk(x)the corresponding components of the Riemann-Christoffel tensor K at the point x , the conditions which V; has to satisfy may be expressed as

(20.1 5.1.16)

(Wj(v6) IWi(V;)) - (wi(v6) Iw~(v;)) - Kfjk,(~)= 0

(1

s i < j 6 n,

1S k

5p -

1).

For brevity we shall put

for 1 2 i 5 n, 1 S j S n, 1 6 k S p , 1 5 h S p . The wk(v;) are completely determined by (20.15.1.15) for k p - 1. We shall first show that for i 5 p - 1 and j S p - 1, the equations (20.15.1.16) are identically satisfied by these values of wi(vb) and wj(v;). Namely, we have wi(v;) = wk(vi) and wj(v;) = wk(vJ) for these values of i. j and for k S p - 1, by virtue of the inductive hypothesis; bearing in mind the identity Kjjkp= KLpij(20.10.4.4), it is therefore evident that ([jkp)= (kpij) = 0 by virtue of the inductive hypothesis. We need therefore to consider only those of the equations (20.15.1.16) in which one of rhe indices i, j is 2.1. Suppose first that i ,
( n - 1)

+ (n - 2) +

+ (n - p ) ,

and moreover we can choose v p such that the wk(vj)are again linearly independent for 1 5 k S n - 1, 1 2.j S p and j S k ; hence the induction can continue, and the proof of the theorem (20.1 5.1) is complete. (20.15.2) By using much deeper results from the theory of partial differential equations, it can be shown that every Riemannian manifold of dimension n is isometric to a submanifold of RN,for some larger value of N.

16. T H E METRIC SPACE STRUCTURE O F A R I E M A N N I A N M A N I F O L D : L O C A L PROPERTIES

If M is a Riemannian manifold, a path (9.6, 16.7) y : [a, b]-+M is said to be piecewise of class c' or piecewise-Cr (r an integer 2 1, or + co) if there exists a strictly increasing sequence a = a, < a , < * . . < a,,, = b of points in [a, b ] such that the restriction of y to each interval [ a j - , , a j ] (1 5 j 2 rn) is equal to the restriction to this interval of a C' mapping of an open interval (containing [ a j - , , a j ] )into M. If y is a piecewise-C' path, the vector y ' ( t ) E T(M) (18.1.2.3) is therefore defined on each of the open intervals ] a j - l . a j [ , on which y is of class C', and has a right (resp. left) limit at the point a j - (resp. aj). Hence, assigning arbitrary values to the function y' at the subdivision points a j , the function t Hy ' ( t ) is a regulated function on [a, b ] . Generalizing (20.13.1 .I), the length qf the path y in M is defined to be the number (20.1 6.1)

374

XX


It is immediately obvious that L(yo) = L(y), where yo is the opposite of the path y , and that for a,juxtaposition (9.6) of two piecewise-C' paths yl, y 2 , we have L(Y, 72) = UYl) L(Y2). The length L(y) of the path y is also called the length of the image C of the path in M, and is denoted also by L(C).

"

+

(20.16.2) Let M be a connected Riemannian manifold of dimension 2 1, and x, y two points of M. By (1 6.26.10) there exists a C" path in M with end-

points x and y . The number (20.16.2.1)

where y runs through the set of piecewise-C' paths with endpoints x, y, is therefore finite and 20. We shall prove the following proposition: The .function d defined by (20.16.2.1) is a distance defining the topology of M.

(20.16.3)

Clearly d(x, x) = 0, and by virtue of (20.16.1) we have d ( y , x) = d(x, y ) and d(x,z) 5 d ( x , y ) d ( y , z ) for x, y , Z E M. It remains to show that d(x, y ) > 0 if x # y , and that the open balls in M with center x (relative to d ) form a fundamental system of neighborhoods of x in M. We shall obtain these results by showing that, for y E M sufficiently close to x, the lower bound d(x, y) is attained by a geodesic path (i.e., the restriction of a geodesic to a compact interval contained in its interval of definition) with endpoints x and y . In what follows, we shall denote by exp the exponential mapping defined by the geodesic field of the Levi-Civita connection, by R c T(M) its domain of definition (18.4.3 and 18.6.1), and by exp, its restriction to R n T,(M), so that for x E M and h, # 0 in T,(M), the unending path t ~ e x p ( t h , )= exp,(th,) is a geodesic through x, tangent to the vector h, at x, and defined o n the open interval consisting of all t E R such that th, E R n T,(M). We shall first prove the following lemmas:

+

(20.16.3.1) Let x be a point of M,let h,, k,Kbe two tangent vectors in T,(M) such that h,ER, and let h i , kk be their images in Thx(T,(M)) under the canonical bijection T;~' (16.5.2).

(i)

We have

16 METRIC SPACE STRUCTURE: LOCAL PROPERTIES

375

(ii) If h, , k, are orthogonal in T,(M), then the vectors Thx(exp,) * h: and Thx(exp,) . k: are orthogonal in Tsxp(hx)(M) (Gauss’s lemma).

+

Consider the family of geodesics t~ f ( t , t) = exp(t(h, tk,)) (18.7.9), and the corresponding Jacobi field tt+ w(t) =&(t, 0). If u(r) =f ( t , 0), we have, therefore, for all t E R such that th, E R, v ‘ ( r ) = Trhx(expx) ’

w ( r )= t(Tthx(expx)’ (tth:(kx))).

To prove (20.16.3.2) it is enough to show that the function ( u ’ l v ’ ) is constant: and this is immediate since by (20.9.5.4), the derivative D(u‘I v’) = 2(V, * v‘ 10’ ) is zero because u is a geodesic (notation of (1 8.7.1)). Again, to prove the second assertion of the proposition, it is enough to show that the function (0’I w ) is identically zero. Now, by (20.9.5.4), we have

D(u’I W ) = (V, *

I

U‘ W )

+ (0’I V,

*

W)

= (0’I V , . W )

because u is a geodesic; hence

D2(v’I W ) = (V, * U’ I V , * W ) + (v’ I V, * (V, * w)) = (u’ I v, * (V, . w ) ) = (V’I(Y

. (0’ A

W ) ) * V‘),

because w is a Jacobi field (18.7.5.1). This can be written in the form

D2(v’I W ) = ( K , V ‘

@ V’ @

w @ 0’) = 0

by virtue of (20.1 0.2) and (20.10.3.2). The function D(u’ I w ) is therefore constant. Since its value at t = 0 is (h,l k,) = 0 by hypothesis (by virtue of (18.7.9)), it follows that (0’1 w ) is constant; but it too is zero for t = 0, and the proof is complete. (20.16.3.3) Let x E M and let y : [a, b ] + R n T,(M) be a piecewise-c’ path. Then

Suppose in addition that exp, is a local diyeomorphism at allpoints of y ([a,b]). Then the two sides of (20.16.3.4) are equal if and only i f y ( t ) = p(t)h,, where h, is a $xed unit vector in T,(M), and t H p ( t ) is an increasing, piecewise-C’ function on [a, b]. Replacing y by its opposite yo if necessary, we may assume that IlY(b)Il 2 Ily(a)II

376


and Ily(b)II > 0. Let F be the closed set of points t E [a,b ] such that y ( t ) = 0,. It is clear that L(exp, 0 y) is greater than or equal to the sum of lengths of the restrictions of exp, 0 y to the intervals which are the connected components of I F (3.1 9.6). On the other hand, if F is not empty and if c < b is the largest element of F, we have y(t) # 0, for all t E ]c, b ] , and if F is empty we have y(t) # 0, for all t E ]a, b ] . To prove (20.16.3.4), we may therefore assume that y(t) # 0, for all t E ]a, b ] . By definition we have (exp,

O

7)”

= T,(&XP,)

*

(T&DY(~))).

Since y(t) # 0, in ]a, b], we may put y(t) = p(t)u(t) in this interval, where > 0 is piecewise of class C’, u(t) a unit vector in T,(M), and t H u(t) piecewise of class C’. We have then

p(t) = Ily(t)II

Dr(0 = (DP(t))U(t) + p(t)Du(t) at all points where Dy is defined; on the other hand, by differentiating the relation (u(t) I u(t)) = 1 we have 2(u(t) I Du(t)) = 0 (8.1.4) at all points where Dy is defined. Consequently (expx y)’(t) = (Dp(t) *

(dt))-‘)Th,(expx> ‘h: + Thx(expx)’ k:

in the notation of (20.16.3.2), with h, = y ( t ) and k, = p(t)Du(t). Since the vectors h, and k, are orthogonal, we can apply (20.1 6.3.1) and obtain

and consequently II(exp, 0 y)’(t)II 2 I Dp(r) 1. The inequality (20.1 6.3.4) now follows from the mean value theorem (8.7.7). Since any connected component of I F contains a nonempty open interval in which IIDy(t)II > 0, the same argument shows that equality cannot occur in (20.16.3.4) unless F = 0 or {a}. Moreover, since both sides of (20.16.3.5) are regulated functions oft, it follows from (8.5.3) that equality can only occur in (20.16.3.4) if Thx(exp,) * k, = 0 except at points where Dy is undefined; but since by hypothesis Thx(eXp,) is a bijection, this relation is equivalent to Du(t) = 0 except at the point a and the points where Dy is undefined. By virtue of (8.5.3), this completes the proof of the lemma (20.16.3.3). (20.16.3.6) We are now in a position to complete the proof of (20.16.3). For each point x E M and each neighborhood V of x in M, there exists (18.4.6) in T,(M) an open ball B : IIh,ll < r with center 0, such that B c f2 and the restriction of exp, to B is a diyeomorphism of B onto an open neighborhood exp,(B) c V of x in M. We shall show that for each h, E B we have

377


For this purpose let cp : [a, 61+ M be a piecewise-C' path in M, with origin x and endpoint exp,(h,). For each r, such that 0 < r, < IIh,II, let B1 c B be the open ball with center 0, and radius rI in the normed space T,(M), and let to be the least number in [a, b ] such that cp(ro) E M - exp,(B,). Then there exists a unique piecewise-C' mapping y : [a, to]

-.

B; c R n T,(M)

such that cp(t) = exp,(y(t)) for all t E [a, t o ] .Clearly we have L(cp) 2 L(exp, 0 y ) and hence, by (20.16.3.3), Ucp) this shows that d(x, exp,(h,))

2 IIY(t0)ll - Ilr(a)ll = r1; I II h,ll. On the other hand, the path t H exp(th,)

is defined on [0, 1 1 , of class C", with origin a and endpoint exp,(h,); by virtue of (20.16.3.2), its length is Ilh.J, which proves (20.16.3.7). With the same notation (rl being any number such that 0 < r, < r ) we - B1 is the frontier of B, in B, therefore exp,(& B,) remark that, since is the frontier of exp,(B,) in exp,(B); but since exp,(&) is compact and therefore closed in M (3.17.2), exp,(& - B,) is also the frontier of exp,(B,) in M. For each frontier point z of exp,(B,) in M, we have therefore d(x, z ) = rl by (20.1 6.3.7). On the other hand, if y E M - exp,(B,), then for each piecewiseC' path cp : [a, b ] + M with origin x and endpoint y , the closed set of points t E [a, b ] such that cp(t) E exp,(F - B,) is not empty (3.19.9); if to is the smallest element of this set, and cpo the restriction of cp to [a, t o ] , we have L(cp) 2 L(cpo) 2 r, from what has already been proved, and therefore d(x, y ) 2 r l . We have thus shown that exp,(B,) is exactly the set of points y E M such that d(x, y) < r , ; and this, together with (20.16.3.7), completes the proof of (20.16.3).

-

F

The function d defined by (20.16.2.1) is called the Riemannian distance on the Riemannian manifold M. Whenever we refer to M as a metric space, it is always to be understood that the distance is the Riemannian distance, unless the contrary is expressly stated. (20.1 6.4) For each x E M and each r > 0 such that the open ball

BP,; r ) : Ilhxll < r

in T,(M) is contained in R and such that exp, is a difeomorphism of this ball onto an open subset of M (1 8.4.6), this open set exp,(B(O,; r ) ) is the open baN B(x; r ) relative to the Riemannian distance on M. Moreover, in order that a piecewise-C' path cp : [a,b ] -P M withoriginxandendpointy = exp,(h,) E B(x; r)

378

XX


should satisfv d(x, y ) = L(cp), it is necessary and suflcient that there should exist apiecewise-C', surjective, increasing mapping p : [a, b ] + [0, 13, such that d t ) = exPXmh3. The first assertion follows from the proof of (20.16.3). If L(cp) = d(x, y ) , then cp( [a, 61) is wholly contained in B(x; r ) : because otherwise it would contain frontier points of B(x; r), hence points y' E B(x; r), such that

4x9 Y ' ) > 4x9 Y ) , and a fortiori we should have L(cp) > d(x, y). We can therefore put cp = exp, y, where y : [a, b ] + B(0,; r ) is a piecewise-C' path with origin 0, and endpoint h,, and the result now follows from (20.1 6.3.3). 0

A path cp : [a, b ] 3 M of the form t Hexp,(p(t)h,), where p is a piecewise-

C' increasing function, is called a rectilinear path in M. We have

by virtue of (20.1 6.3.4).

'

(20.1 6.5) Under the hypotheses of (20.1 6.4), the diffeomorphism exp; of B(x; r) onto B(0,; r), followed by a linear bijection of T,(M) onto R", defines a chart of B(x; r ) for which the images of the geodesic arcs with origin x are straight-line segments with origin 0 in R".The local coordinates corresponding to such a chart are called normal coordinates at x .

Remark (20.16.6) Given a piecewise-C' path cp : [a, b ] + M, for each E > 0 there exists a C'-path cpl : [a, b ] + M that coincides with cp except in the intervals [aj - E , a j + E ] (where the a j E [a, b ] are the points at which the rth derivative ofcpis undefined), is such that d(cp(t),cpl(t)) 5 &forall t E [a, b ] , and for which I L(cp,) - L(cp)l 5 CE.where C is a constant independent of E. It is immediately seen that it is sufficient to consider the case in which M is an open subset of R", the interval of definition of cp is an open interval containing [ - a , a ] , and the origin is the only point at which Dc' p is undefined: we then have to find cp, which is equal to cp except in the interval [ - E , E ] , such that IIcpl - cpII S E in this interval and such that IIDcp,II is bounded in the interval [ - E , E ] by a number independent of E. For this purpose, let h be a C" function on R, with support contained in the interval [ - 1 , 1 ] , with values in


379

[0, 11, and such that h(t) = 1 for ( t l 5 3 (16.4.1); if

I+m h(t) dt put

p(t) = h(t)/c, so

JI, defined by

that

1'" -m

p(t) dt = 1. Then the function =m

JI,

--oo

= C,

cp(s)p(m(t - s)) ds

is of class C" in an open interval containing [-a, a ] (17.12.2) and converges uniformly to cp as m + 03 (14.11.2); hence by taking m large enough we shall have I[+,, - cpII 6 E in [ - a , a ] . Moreover (17.11.11), in the intervals [-a, O[ and 30, a] we have DJI,(t) = m

Jy,

Ddt

- s)p(ms) ds,

and therefore there exists a constant A independent of m,such that

IIDJImll 6 A for all m and IIDcpII 5 A. Now take cpl(t) = $ A t )

so that cpl(t)

= cp(t)

- (1 - h(t/&))($m(t)

-dt))

for I t I > E, and

IIcpl(t)

- ~ ( t ) l I= h(t/E)IIJIm(t)- dt)II 5 E

for t E [ - E , E ] ,and cpl(t) = Jl,(t) for t E [-&, $51, so that cpl is of class Cr. Moreover, if B is an upper bound of lh'(r)l, one sees immediately that IIDp,(t)ll 5 3A + B. This completes the proof.

PROBLEMS 1. With the hypotheses and notation of (20.16.4), for each rl such that 0 < rl < r, let S(x; r , ) denote the sphere with center x and radius rl relative to the distance function

d; it is a hypersurface in M, diffeomorphic to Show that for each point y = exp(h,) of S(x; rl), the normal to S(x; r l ) at the point y coincides with the tangent to the geodesic t H exp(th,) at the point y. (UseGauss's Lemma.)

2.

Under the hypotheses of (20.16.4), let ( c , ) ~ be ~ , an ~ ~orthonormal basis of T,(M). The normal coordinates of a point y E B(x; r ) relative to (cJ) may be written as y1 = p i t , where (p'),L16n is a point of and 0 6 t < r. Choose as moving frame on B(x; r ) the frame R canonically associated with this neighborhood and the basis (cJ) (Section 20.6, Problem 15). Show that for this frame the canonical and connection forms are such that (i) 01 = P dt tl, where t1does not contain the dp'; (ii) W ~ contains J only the dpl, not dt (loc. cit.)

+


380

Furthermore, the forms

and w I Jvanish at r = 0. Show that

41= --(dp'+ 1 WIJP' J ) dull =

A dt

+

61,

-c r h P ' t I A dt + P ~ J , k. I

where the rjklare the components of the curvature tensor relative to the moving frame R, and at,PlJare 2-forms containing only the dp'. Finally, the metric tensor, relative to the frame R, is given by

0dt + C 51 05 1 .

I = dt (Use Problem 1.)

3. We propose to calculate the Riemannian volume of the sphere S ( x ; rl) of dimension n - 1, relative to the Riemannian distance on M (Problem 1). With the notation of Problem 2, consider the diffeomorphism f of S.-l onto S ( x ; rl) which sends each p cJ . The inverse image ( ' on S(x; rl) (oriented by transport of the canon-

point p = (p') of S.-l to the point y =f(p) = exp, rl ~ ( I Jof) the canonical volume form

'J:1

IJ

ical orientation of S,-l by/) may be written as Fo. where u is the solid angle form (16.24.7) and F is a C" function. on

(a) Let (pl, . .., p,) be an orthonormal basis of Rn such that pI = p, and let (hl,. ,h,) be its image in T,(M) under the linear mapping ( p J ) w zpJcJ.The vectors

..

J

T,,h(eXP,)

*

(2 5 i 5 n),

(T = 1,

show that = 0, the open ball B(a;r ) in M is connected, and its closure in M is the closed ball B'(a; r ) with same center and radius.

Since B'(a; r ) is closed in M, we have B(a; r ) c B'(a; r ) ; hence the second assertion will be proved if we can show that the sphere S(a; r ) with center a and radius r is contained in B(a; r). Now, if x E S(a; r), then for each E such that 0 < E < r, there exists a path cp : [LY, p] -,M with origin a and endpoint x such that L(cp) S r + E. Since t Hd(a, cp(t))is continuous on [ r , p ] and varies from 0 to r, there exists f o e]a, /3[ such that d(a, cp(to)) = r - E (3.19.8). If cpI, (p2 are the restrictions of cp to the intervals [a, t o ]and [ t o , PI, respectively, we have L(cp) = L(cp,) L(cp2), and d(a, cp(to)) S L(cp,); consequently

+

d(cp(td, x ) 5 J-(cp2)

=

Ucp) - L(cp,) 5 r + - (r - 4

= 2-5,

from which the result follows. The fact that for each x E B(a; r ) there exists a path cp with origin a and endpoint x, of length L(cp) < r, implies that this path is contained in B(a; r), and hence that B(a; r ) is connected (3.19.3). (20.18.2) Let M be a connected Riemannian manifold, d the Riemannian distance on M , and a, b two distinct points of M.If cp : [a, p ] + M is apiecewiseC' path with origin a and endpoint b such that L(cp) = d(a, b) = I, then cp is a rectilinear path tHexp,(p(t)h,) (20.16.4) with 11 hall = 1, p(a) = 0, p ( p ) = 1, and Ih, E R. Furthermore, there exists an open interval 1 3 [0, I ] in R such that v : s ~ e x p , ( s h )is an embedding of I in M, where s is the curvilinear coordinate

386

XX


of u(s) on the curue C = u(1) with respect to the origin a and the orientation defined by u. If A < p are any two points of [a, p], let p = q ( A ) , q = q ( p ) be their images; if p # q, then we must have L(q,, p ) = d(p, q), where q,, denotes the restriction of cp to [A, p]; for otherwise we should have d(p,q ) < L(cp,, p ) , and therefore by definition there would exist a path $ : [A, p ] -,M with origin p and endpoint q such that d(p, q) S L($) < L(cp,,,), and then the path which is equal to cp on [a, A] and b, p ] and to $ on [A, p ] would have length a such that p = q ( A ) is distinct from a and contained in B(a; r). From the previous paragraph and from (20.16.4), there exists a unit vector h, E T,(M) such that cp(t) = exp,(p(t)h,) in [a, A], where p is increasing and piecewise of class C ' ; furthermore, h, does not depend on the choice of p in B(a; r). Let to denote the least upper bound of the numbers p E [a, p ] such that q ( t ) = exp,(p(t)h,) for all t E [a, p ] ; since p(t) = d(a, q(t)) by the preceding remarks and (20.16.4.1), the function p is well-defined. We have to show that to = p. Suppose not, and put xo = q ( t o ) ; also let r f > 0 be such that every open ball of radius r f with center lying in B(xo; r') is strictly geodesically convex (20.17.5). If we put fo = d(a, xo), then we have sh, E R for 0 S s < fo , and q(t) = uo(p(t)) for a 5 t < t o , where uo(s) = exp,(sh,) for 0 5 s < fo . Let t, < to < t2 be two values of t such that x1 = cp(tl) and x2 = cp(t2) are distinct from xo and contained in B(xo;j r ' ) ; put I, = p(tl) = d(a, x,) < lo, and f, = d(a, x,) > fo , Also let h,, = u&), which is a unit vector in TJM) (20.16.3.2). By virtue of the choice of tl, t , and the remark at the beginning, the path q t l , , ,is rectilinear, of the form twexp,,(p,(t)h,,) = vl(s), where ul(s) = exp,,(sh,,) is defined for

,

0 5 s 5 f, - f, (which implies that sh,,

+

ER

for these values of s) (20.16.4). But if

0 5 s < f, - f,,

we have vl(s) = uo(Il s) by virtue of (18.2.3.2); hence (18.2.2) we have sh, E R for 0 5 s < f,, and the function st+ exp,(sh,) is defined in this interval. Putting p(t) = fl pl(t) for t , 5 t 5 t 2 , we have therefore q(t)= exp,(p(t)h,) for a 5 t < t , , contradicting the definition of t o . To complete the proof, it is enough to show that SH u(s) = exp,(sh,) is injective in [0, I]. Now, if u(sl) = u(s2) for 0 5 s1 < s2 5 I, the juxtaposition o f t h e p a t h s s ~ u ( s ) f o r O 5 s ~ s , a n d s ~ u ( s + s , , s , ) f o rSs ~S ~ S + I-S, , would be a piecewise-C' path with endpoints a, b and length 0. We shall prove that (a) * (b) (c) =.(d) *(a). Since every Cauchy sequence is bounded, the implication (a) * (b) is immediate (3.16.1), and so also is (c)*(d). The remaining implications and assertions of the theorem will result from the following lemmas.

388

XX


(20.1 8.5.2) Let a, b be two distinct points of a connected Riemannian manifold M. Then there exists a vector ha # 0 in R n T,(M) such that

IIhall = d(a, exPa(ha)), llhall + d(exPa(h3, b) = d(a, b). Let r > 0 be sufficiently small so that exp, is a diffeomorphism of B(0,; r) onto B(a; r) (20.16.4), and choose p E 10, r[ sufficiently small so that b # B(a; p). Let S denote the sphere with center 0, and radius p in T,(M), so that exp,(S) is the sphere with center a and radius p in M. Since exp,(S) is compact, there exists c E exp,(S) such that d(b, c) = d(b, exp,(S)) (3.17.10). Let ha E T,(M) n R

be such that c = exp,(h,); then we have d(a, b) 5 d(a, C) + d(c, 6) = IIh,ll

+ d(b, C)

by (20.16.4). On the other hand, S being the frontier of B(0,; p), exp,(S) is the frontier of B(a; p ) (20.16.3.6); a path y with endpoints a and b must meet exp,(S) (3.19.10); hence there exists c’ E exp,(S) such that L(y) 2 d(a, c’)

+ d(b, c‘)

2 d(a, exp,(S)) + 4 b , exp,(S)) = d(a, exPa(S)) =p =

+ d(b, C)

IIhall

+ d(b, C)

+ d(b, 4.

Hence, by definition, d(a, b) L (1 hall

+ d(b, c), which proves the lemma.

(20.1 8.5.3) Let a, b be two distinct points in a connected Riemannian manifold M. If ha E R n T,(M) is a nonzero vector such that

IIhall

+ d(expa(ha), b) = d(a, b),

then for each t E R such that th, Ilthall

ER

and 0 5 Ilrh,ll S &a, b), we have

+ d(eXPa(tha), b) = d(a, b)*

First observe that if t 2 0 and th, E R, then exp,(th,) is defined and d(a, exp(th,)) S )Ithall by (20.16.4.1), so that

II thall + d(exPa(tha), b) 2 d(a, exp,(th,)) 2 d(a, b).

+ d(exp,(th,),

b)

389

18 METRIC SPACE STRUCTURE: GLOBAL PROPERTIES

Hence, if Ilth,ll have IIt’hAl

+ d(exp,(th,),

b) = d(a, b), it follows that for 0

t’

IIWI + d(exp,(t’h,), exp,(th,)) + d(exp,(th,), 5 llt’hall + Il(t - t‘)h,Il + d(a, b) - IIthaII

+ d(exp,(t’h,),

b) 2

t

we b)

= d(a, b)

by (20.16.4.1), and therefore from above llt’h,ll

+ d(exp,(t‘h,),

In other terms, the set I of numbers such that

t

6) = d(a, b).

>= 0 such that th, E R n T,(M)

and

is a bounded interval in R, with endpoints 0 and ro 2 1. The lemma will be established if we can show that it is not possible that to h, E R and

II t o ha I1 < d(a, b) simultaneously. Assume the contrary; then by continuity it follows from

(20.18.5.4) that to E 1. Since d(a, b) > lltoh,ll I d(a. c), where c = exp,(to ha), we have b # c. Lemma (20.18.5.2) then shows that there exists a vector h, # 0 in R n T,(M) such that

IIhCll

= dk-7 exp,(h,)),

Ilh,ll

+ d(exp,(h,),

b) = d(c, 4,

and therefore Ilto holl

+ IIhCll + d(eXp,(h,), b) = d(a, 6).

By (20.16.3.3) and the triangle inequality, this implies that Ilto MI

+ llh,ll

= 4 a , exp,(h,)).

It follows now from (20.16.3.3) and (20.18.2), applied to the juxtaposition of the geodesic paths twexp,(th,) (0 2 t 5 t o ) and tHexp,((t - to)h,) (0 6 t - to 5 l ) , that exp,(h,) = exp,(t‘h,), where t’ = to + At, and Consequently, we have

A = IIhclllllto h,ll.

ll~’h,ll + d(exp,(t‘h,), b) = d(a, b)

and t’h, E R, contrary to the definition of t o . (20.18.5.5) Let M be a connected Riemannian manifold, a a point of M. If the closed ball B’(0,; r> in T,(M) is contained in R, then the closed ball B’(a; r ) (resp. the open ball B(a: r ) ) in M is equal to exp,(B’(O,; r)) (resp. exp,(B(Oa; r))).

390

XX


Clearly we have B‘(a; r) 3 exp,(B‘(O,; r)), by (20.16.4.1). Since on the other hand B(a; r) is the union of the B’(a; r’) for r‘ < r, it is enough to show that B’(a; r) is contained in exp,(B’(O,; r)). Let b be a point of M such that b 4 exp,(B’(O,; r)); we shall show that d(a, b) > r . By virtue of (20.18.5.2), there exists a vector ha # 0 in R n T,(M) such that

+ d(exPa(ha), b).

d(a, b) = II hall

Let t = inf(r/llh,,ll, d(a, b)/[lhall); then the hypotheses imply that th, 0 < Ilth,ll 5 d(a, b), and by virtue of (20.18.5.3) we have d(a, b) = IIth,lI

+ d(exp,(th,),

E

R and

b);

but since b 4 exp,(B’(O,; r)) and this set is compact, we have d(exp,(th,), b) > 0 (3.17.10), hence Ilth,ll < d(a, b). This implies that Ilth,ll = r; hence d(a, b) =

r

+ d(exp,(th,),

b) > r, which proves the lemma.

(20.18.5.6) Let M be a connected Riemannian manifold, and let a be a point of M sucli thar T,(M) c R. Then for each x E M there exists a uector h, E T,(M) such that 11 hall = d(a, x) andx = exp,(h,), andfor each r > 0 we have B(a; r) = exp,(B(O,; r)) and B’(a; r) = exp,(B’(O,; r)).

We may assume that x # a; then, by virtue of (20.18.5.2), there exists a vector ha # 0 in T,(M) c Iz such that

II hall

+ d(exp,(h,),

x) = d(a9 XI.

But then, by virtue of the hypothesis and (20.18.5.3), if t > 0 is such that Ilth,ll = d(a, x), we have Ilthall which is possible only if x

+ d(exp,(th,),

= exp,(th,).

X) = d(a, XI,

The remaining assertions follow from

(20.1 8.5.5). (20.18.5.7) These lemmas already show that (d)*(a) in (20.18.5) and establish the last two assertions of the theorem. It remains to prove that (b) implies (c). Let a E M and ha E T,(M), and let to be the least upper bound of the r E R such that th, E R ; we have to show that to = 00. Suppose not, and put u(t) = exp,(th,). The sequence (o(to - (l/n)))n,l is a Cauchy sequence in M, because

+

by virtue of (20.16.4.1). Let b be the limit of this sequence in M. There exists a


391

neighborhood W of b in M and a real number r > 0 such that R contains the closed ball B’(0,; r ) for all x E W. Choose n sufficiently large so that c = v(to - (l/n)) E Wandr/llh,ll > l / n . Then the vector h, = v’(to - (I/n)) belongs to T,(M), and its norm is equal to IIhJ (20.16.3.2), and therefore we have

-.r

II ha II

h,

E R.

By virtue of (18.2.3.1) applied to the geodesic field, it follows that

which contradicts the definition of t o . This completes the proof of (20.1 8.5). A connected Riemannian manifold M is said to be complete if it satisfies the equivalent conditions of (20.18.5).

Examples (20.18.6) Every compact connected Riemannian manifold is obviously complete. Real n-space R”,endowed with its canonical Riemannian structure, is a complete Riemannian manifold. If (G, H) is a symmetric pair with H compact, then the Riemannian symmetric space G/H (20.11.3) is a complete Riemannian mangold: for it follows from the properties of the canonical linear connection on G/H (20.7.10.4) that a geodesic with origin x, = n(e) is defined on the whole of R,hence condition (d) of (20.18.5) is satisfied. In a Riemannian manifold M, an open subset U, which is a complete Riemannian manifold for the structure induced by that of M, is closed in M, as follows from condition (a) of (20.1 8.5), which shows that the frontier of U is empty. (20.18.7) Let MI be a complete connected Riemannian manifold, M a connected covering of MI, and a : M + M, the canonical projection. If we

endow M with the Riemannian structure canonically induced from that of M, (20.8.1), then M is also a complete Riemannian manifold. For since n is a local isometry, each geodesic t H v(t) of M determines a geodesic ti+ n(v(t)) of M, (by transport of structure), and conversely the lifting to M (16.28.1) of a geodesic in M, is a geodesic in M. Since the geodesics in M, are defined on the whole of R, the same is true of the geodesics in M,, by reason of the uniqueness of liftings (16.28.1). Conversely, we have the following proposition : (20.18.8) Let M be a complete connected Riemnnnian manifold, and let f be a local isometry of M into a connected Riemannian manifold M,. Then M is a Riemannian covering of M,,with f as projection, and M, is complete.

392

XX


The setf(M) is open in M,. It is enough (20.1 8.6) to prove (20.1 8.8) with M1 replaced byf(M). Let a, be a point off(M), and V1 a geodesically convex (20.1 7) neighborhood of al. By (1 8.5.3), there exists a C" mapping (x,,y,) H s(xl,y , ) of V1 x V, into T(M,) such that oM,(s(xl,y,)) = x1 and exp,,(s(x,, y,)) = y,. Let u denote the mapping y , I+ s(a,, y,) of V1into Ta,(Ml), which is a diffeomorphism of V, onto an open neighborhood of 0,, in T,,(M,), the inverse of which is the restriction of exp,, to this open set (18.5.2). For ) ,linear mapping T,(f) is by hypothesis an isomoreach point a ~ f - l ( a ~the phism of T,(M) onto T,,(M,). Let u, denote the inverse of this isomorphism. Since M is complete, the mapping g, = exp, 0 u , o u is defined and of class C" on V,. On the other hand, for each ha E T,(M) and each t E R, exp,,(r(T,(f) * ha)) is defined, and we have f(exp,(th,)) = expa,( t ( T a ( f ) * ha)).

(20.1 8.8.1)

For, sincefis a local isometry, it is clear that t wf(exp,(th,)) is (by transport of structure) a geodesic in M, defined on the whole of R,whence our assertion follows. We have therefore, in V,, fo

g, = exp,,

0

T,(f)

0

u,

0

u = exp,,

0

u = lVI,

from which it follows (16.8.8) that g, is a diffeomorphism of V, onto an open set g,(V,), the inverse mapping being the restriction off to g,(V,). Let a' be a point of f-'(a1), other than a ; then we have g,(Vl) n g,.(V,) = 0. For if ga(xl)= gat(xl) for some x,, then also g,(y,) = g,.(y,) for all y, sufficiently close to x,, because the restriction offto a neighborhood of g,(xl) is injective, The set of points x1E V1 such that g,(xi) = g,.(xl) is therefore both open and closed (3.15.1), hence is either the whole of V, or the empty set, because V, is connected. Since g,(al) # g,,(a,), it is the second alternative which holds. Again, if x1 # y,, then we have ga(xl)# ga.(y,), because f(g,(x,)) = x1 and f(g,.(yl)) = y,. Consequently g,(V,) n g,,(Vl) = 0,as asserted. To complete the proof, it is enough to show that f -'(V,) is the union of the sets ga(Vl) as a runs throughf-'(a,) (16.12.4.1). Let x~f-'(V,), x1 =f(x), and consider the tangent vector h,, = s(xl,a,) at the point x,; there exists a unique tangent vector h, E T,(M) such that T,(f) * h, = hxl. By definition, we have exp,,(h,,) = a,, and the formula (20.18.8.1) shows that f(exp,(h,)) = a,, so that exp,(h,) = a Ef -'(a,). The definition ofg, now shows that x = g,(xl). The fact that f(M) is complete has been established during the course of the proof, because we have shown thatf(M) satisfies condition (d) of (20.18.5). PROBLEMS 1. In a Riemannian manifold M, let U be a strictly geodesically convex open set. If x, y are any two distinct points of U, show that the unique geodesic arc with endpoints x , y contained in U is the only geodesic arc in M with endpoints x , y , of length d(x, y).


393

(Assume the result false, and use the strict triangle inequality (20.18.3) to obtain a contradiction.) Deduce that every finite intersection of strictly geodesically convex open sets is strictly geodesically convex. 2.

Give an example of a noncomplete Riemannian manifold with the property that through any two distinct points there passes a unique geodesic trajectory.

3. Give an example of an unbounded connected open set U in R2 such that no two points x, y E U such that d(x, y ) > I can be joined by a geodesic arc contained in U. 4.

Show that the product of two complete Riemannian manifolds (Section 20.8, Problem 1) is a complete Riemannian manifold.

5.

Let M be a complete, connected, non-simply-connected Riemannian manifold. Show that the function (x, y ) (d(x, ~ y))' on M x M cannot be of class C". (Consider the simply-connected Riemannian covering R of M, and two distinct points a, b E which project to the same point x of M. If d(a, b) = r in R,consider the sphere with center x and radius I r in M.)

6. Let M be a noncomplete connected Riemannian manifold, g its metric tensor. For each x E M, let p(x) be the least upper bound of the real numbers r such that the closed ball B'(x; r) is compact. The hypothesis on M implies that 0 < p(x) < co for each x E M.

+

+

(a) Show that p(y) 5 p(x) d(x, y ) for all x, y in M, and hence that p is a continuous function on M. (b) Letfbe a C" function on M such thatf(x) > I / p ( x ) for all x E M, and consider the metric tensor gI = f 2 g . Let y : I + M be a piecewise-C" path in M. If L, L1 are its lengths relative to g, g,, respectively, then

Deduce that if d, d , are the Riemannian distances corresponding to g, g,, respectively, and if the endpoints a, b of y satisfy d(a,b) 2 $p(a),then we have LI 2 4. (Use (a).) (c) Deduce from (b) that the closed ball with center a and radius 4 (relative to the distance d,) is contained in the closed ball with center a and radius tp(a) (relative to the distance d). Consequently, the manifold M is complete relative to the metric tensor gl. 7.

Let M be a complete, noncompact, connected Riemannian manifold, and let g be its metric tensor. Given a point a E M, let h be a C" function on M such that h(x) > d(a,x ) for all x E M, where d is the Riemannian distance. Show that for the Riemannian metric tensor gz = e-2hg on M, the diameter of M is 51, and hence that M is not complete relative to g2.

8.

Let M be a complete Riemannian manifold, d the Riemannian distance on M, and let M' be a submanifold on M, endowed with the Riemannian structure induced by that of M. If d' is the Riemannian distance on M', then d'(x, y) 2 d(x, y) for all x, y E M', and consequently every Cauchy sequence in M' (relative to d')converges to a point

394


of M (relative to d). Suppose that every point of M has a neighborhood U such that every connected component of U n M' is closed in U. Show that M' is then complete. (Argue by contradiction, by supposing that there exists a geodesic in M' which is not defined on the whole of R.)Give an example in which the condition above is satisfied, but the distances d and d' are not uniformly equivalent (3.1 4) on M'.

M be a connected Riemannian manifold. For each x E M, let a(x) denote the supremum of the radii of open balls with center 0, contained in TAM) n a, and by r(x) the supremum of the real numbers r such that (relative to the Riemannian distance d o n M) every open ball contained in the ball with center x and radius r is strictly geodesically convex. Then r ( x ) 5 a(x).

9. Let

+

(a) Show that if r(xo) = co for some xo E M, then r(x) = +a, for all x E M. If r(x) is finite for all x E M,then Ir(x) - r(y)l 5 d(x, y ) , and therefore r(x) is a continuous function of x on M. (b) We have a(xo) = 03 if and only if M is complete; hence u(xo) = co implies that a(x) = co for all x E M. If a(x) is finite for all x E M, show that

+

+

+

and hence a(x) is a continuous function of x on M. (Argue by contradiction, using the fact that for r < a(x) the closed ball B(x; r ) is compact, and proceeding as in the proof of (b) => (c) in (20.18.5).)

R2,consider the connected Riemannian submanifold R2- {0}, and its simplyconnected universal Riemannian covering M, which is not complete. Show that there exists no connected Riemannian manifold N containing M, in which M is a proper open subset of N. (Argue by contradiction. Let a be a frontier point of M in N, and U a strictly geodesicallyconvex neighborhood of a in M. Observe that through each point x E M there passes only one geodesic not defined on the whole of R, and deduce that the complement of U n M in U consists of the single point a. Then consider in U the set S(a; p) of points whose distance from a is constant and equal to p, where p is sufficiently small, and obtain a contradiction by observing that this set is homeomorphic to a circle, and must be contained in the set of points of M which project onto the circle with center 0 and radius p in R2- {O}.)

10. In

11. Consider on R the Riemannian metric tensor p = ex dx 0 dx, which is not complete. The mapping x I+ x 1 is a homometry (Section 20.9, Problem 5) and hence an auto-

+

morphism for the Levi-Civita connection. Hence construct an example of a linear connection on T = R/Z,which is not complete in the sense of Section 20.6, Problem 8, and by extension (17.18.5) a noncomplete linear connection on Sz. 12. (a) Let M be a connected Riemannian manifold, d the Riemannian distance on M. Let y :I --+ M be a path of class Co such that for all tl < t z < t3 in I, we have

+

d(y(rJ, y(t3)) = d(y(t,), Y ( t 2 ) ) d(y(h),y(t3)).

Show that y(I) is a geodesic arc. (By considering a strictly geodesically convex neigh, that there exists E > 0 such that y([t - E. I €1) is a geodesic borhood of ~ ( t )show arc.) (b) Let M, M' be two connected Riemannian manifolds and d, d' the Riemannian distances on M, M', respectively. Let f be a mapping of M onto M' such that d'(f(x), f(y)) = d(x, y ) for all x, y E M. Show that f is an isometry in the sense of (20.8.1). (Using (a), prove first that for each geodesic y :I + M in M, the path

+

19 PERIODIC GEODESICS

395

y : I -+ M' is a geodesic in M'. Deduce that for each x E M there exists a bijection F, of T,(M) onto Tf(,)(M') such that F,(ch,) = cF,(h,) for all c E R and h, E T,(M), and such that f o exp, = expf,,, F, in a neighborhood of 0, in T,(M). Finally show that (F,(h,)I F,(k,)) = (h,l k,) for all h,, k, E T,(M), by using Section 20.16, Problem 10.)

fo

0

R3 defined by t 2= 0.6' E I, where I is an open interval in R, and t3=f(l'), wherefis a C" function on I which is everywhere >O. If a is an endpoint of 1 andf(t) +0, f'(t ) + & co as t + a,show that if S is the surface of revolution with axis Re, generated by C (Section 20.14, Problem 7), then S u {gel} is a differential manifold. Show that there exists a functionf(of class Cm)defined on the interval I = 1- m, 1[ such that f(") = 1 for 6' 5 0,f(6') > 0 throughout I, and f"(& < 0 in 10, 1[, and such that f(6') +O and f'(6') -+ - 03 as 6' 1. Let V be the corresponding surface, which is closed in R3. Let V' be the union of the set of points of V such that 6' 2 0 and the mirror-image of this set with respect to the plane 6' = 0. Show that V' is also a differential manifold and that there exists an isometry of a neighborhood of el in V onto a neighborhood of el in V', which cannot be extended in an isometry of V onto V', although V and V' are complete and simply-connected. (Compare with Section 20.6, Problem 9(e) and Section 20.9, Problem 8.)

13. Consider a curve C in

-+

14. Let M, N be two connected Riemannian manifolds of the same dimension. A C" mapping f :M + N is said to be complete if there exists a continuous positive-valued ) all tangent vectors h, E T,(M), function h on N such that, for ally E N, all x ~ f - ' ( y and

we have IIT(f) . h,(lL h(y)llh,ll. This condition implies thatfis a local diffeomorphism.

(a) Show that iffis complete and if the manifold M is complete, thenf(M) = N and fmakes M a covering of N. (For the first assertion, argue by contradiction, by supposing that there exists a frontier point yo off(M) in N. Deduce that there exists a frontier ) endpoint y, pointy off(M) and a geodesic path u : [0, I ] -+ N with origin b E ~ ( Mand such that u ( t ) E/(M) for 0 5 t < 1. Show that there exists a C" mapping u : [0,1[ + M which lifts u, such that u ( t ) tends to a limit as t + 1, by using the completeness offand M. To show that M is a covering of N, use Section 16.29, Problem 5.) (b) The mappingfis said to be uniformly complete if h(x) is bounded in every bounded subset of N (relative to the Riemannian distance). Show that iffis uniformly complete and M is complete, then N is complete. (Consider a Cauchy sequence in a strictly geodesically convex ball in N, and show that it can be lifted to a Cauchy sequence in M.) Consider the case wherefis a local isometry. (c) Under the hypotheses of (a), if we suppose in addition that the fundamental group of N is finite, thenfis proper and N is complete (cf. Section 16.12, Problem 1).

19. PERIODIC GEODESICS

(20.19.1) Let M be a complete connected Riemannian manifold, a and b two points of M,and y : [u, 81+ M a piecewise-C' path with origin a and endpoint b. Then there exists a piecewise-C' path yo : [u, 81 + M, with origin a and endpoint b, of length d(a, b) (so that yo is rectilinear (20.18.2)) which is homotopic to y under a homotopy leaving a and bfixed.

396

XX


Let M’ be the universal covering of M, endowed with the Riemannian structure canonically induced by that of M (20.8.2), so that M‘ is a Riemannian covering of M. Let y’ be a lifting of y to M‘ (1 6.28.1) with origin a’ and endpoint b’. Since M’ is complete (20.18.7), there exists a rectilinear path y b in M’ with origin a’ and endpoint b‘, whose length is equal to the distance from a‘ to b’. The projection yo of yb has the desired properties since the length of a path in M is equal to that of any lifting of the path to M’, and since the paths with origin a’ and endpoint b‘ are precisely the liftings of paths with origin a and endpoint b which are homotopic to y under a homotopy leaving a and b fixed (16.27). (20.1 9.2) In a Riemannian manifold M, a periodic geodesic is by definition a geodesic t H q ( t ) in M, defined on the whole of R, not reduced to a single point, such that cp is a periodic mapping with period #O. If T is the smallest period of q, and if cp restricted to [0, T[ is injective, then the image of cp is diffeomorphic to the circle S , . (20.19.3) Let M be a compact connected Riemannian manifold, and let y be a piecewise-C’ loop in M which is not homotopic (as a loop) to a point. Then (after a change of parameters) there exists a loop yo in M which is loophomotopic to y and which is the restriction of a periodic geodesic (to an interval of length equal to a period); moreover, L(yo) is the minimum of the lengths of piecewise-C’ loops which are loop-homotopic to ‘y.

Let xo be the origin of y. For each x E M there exists a piecewise-C’ loop with origin x which is loophomotopic to y (16.27.3.1). Let Hy,xdenote the set of all piecewise-C’ loops with origin x which are loop-homotopic to y, and let A(x) denote the greatest lower bound of the lengths of the loops belonging to HY,+. Let M‘ be the Riemannian universal covering of M, d, d’ the Riemannian distances on M, M’, respectively, and p : M’ -,M the projection. The loops in Hy,xare precisely the projections of the piecewise-C’ paths in M’ with endpoints x‘ and x”, where x‘ is any point of p - ’ ( x ) , and x” is the image of x’ under the element s of n , ( M ) which is the class of the loop y (16.29.2). Since the length of a path in M’ is equal to that of its projection by p , it follows that A(x) = d‘(x’, s * x’). Hence A(x) is a continuous function of x on M: for if x1 is any point of M, there exists a neighborhood V‘ of xi ~ p - ’ ( x ~ ) such that the restriction of p to V’ is a diffeomorphism of V onto a neighborhood V of x1 in M; if q is the inverse diffeomorphism, we have A(x) = d(q(x), s .p(x)) for x E V,, and our assertion is now obvious because s acts

20 FIRST AND SECOND VARIATION OF ARC LENGTH

397

continuously on M’. Since M is compact, there exists a point a E M at which f O because y is not homotopic to a point. Moreover, since M’ is complete, I is the length of a loop yo with origin a which is a geodesic path and the projection of a geodesic path with endpoints a’ and s . a‘ in M’ (where a’ ~ p - ’ ( a )of) length &(a‘, s * a‘). We shall show that yo has the required properties. Let t Hy o ( t ) be the arc length parametrization of y o ; we have to show that yb(0) = ?;(I). Suppose that this is not the case, and let B be a strictly geodesically convex open ball with center a (20.17.5). Choose E > 0 sufficiently small so that yo(&) and yo(l - E ) both belong to B. Then there exists a rectilinear path w : [I - E , I + E ] + M contained in B, with origin yo(l - E ) and endpoint yo(&), and length equal to d(yo(I - E ) , yo(&)). Now the hypothesis implies that

L attains its minimum I, which is

d(Y0V - 4 7

Yo(4) < 4 Y o ( l - 4, a ) + d(a9 YO(&))

by virtue of (20.1 8.3). The loop y1 : [I - E , 21 - E ] --* M which is equal to w(t) in [I - E , I + E ] and to yo(r - I) in [I E , 21 - E ] then has length I(z, z )

unless w = z. Let ( u ~ be )a basis ~ of ~ T"(,)(M), ~ ~ and ~ for each j let zj be the Jacobi field along v such that zj(a) = 0, and (V, zj)(a) = uj (18.7.6). The hypotheses imply that for each t E ]a, b ] the n vectors zj(t) (1 S j S n) form a

400

XX


basis of TV(,)(M); for if Jacobi field

1I., zj(t) = 0 with scalars

not all zero, then the

c iLjzj is not identically zero but vanishes at two distinct points i

i

a and t, which is impossible (18.7.15). We may therefore write, for each

t E la, bl,

where the hj are piecewise of class C’ on ]a, b ] and are constant on ]a, c] (1 8.7.7). It follows (1 7.17.3.4) that

v, * w =

c

+ c hj(V,

hJZi

i

*

Zj).

j

Hence, putting

f=ChJzi,

(20.20.3.3)

u=~hj(vr.zi),

i

i

we have

IIV,

(20.20.3.4)

*

W1l2 =

IIfllt + 2(f I u ) + IIuI12.

On the other hand, we have

-

((r (v’ A w)) * u‘ I w ) =

C hj((r -

(0’ A

z j ) )*

11’

i

IW)

and since the zi are Jacobi fields and therefore satisfy (18.7.5.1), we have ((r

- (0’A w ) ) . dI w ) =

hi(V, (V, . zj)l w ) j

= (V,

and therefore, using (20.9.5.4). (20.20.3.5)

((r . (u’

A

w)) * c ‘ ( w)

*

UI

w) -

ch p , i

*

Zil

w)

401


But since the zj are Jacobi fields, we have by (20.9.5.4) and (18.7.5.1), d dt

- ((vt

*

' j

I 'k) - ( z j I vt

' 'k))

= (vt

= ((r

*

*

(vt

(0'

I z k ) - ( z j I vr (v, z k ) ) h z j ) ) 0' I z k ) - ((r (0' A zk)) ' zj)

*

'

*

*

*

u'

I 2,)

= 0,

by virtue of the symmetry properties (20.10.3.2), (20.10.3.3), and (20.10.3.4) of the Riemann-Christoffel tensor. The function (V, zj I z k ) - ( z j I V, * zk) is therefore constant on [a, b], and since it vanishes at the point a, it follows that (V, *

(20.20.3.6)

Z j I Z k ) - (ZjlV,

*

Z k ) = 0.

Consequent 1y,

and finally the relations (20.20.3.4) and (20.20.3.5) give

IIv, . Wl12 4- ( ( r

*

(0' A W ) ) 0'1 W )

d dt

= -( U l

W)

-k

IIfl12.

Now f a n d u both tend to limits at the point a, by virtue of the hypothesis on w , and therefore we may integrate both sides of this relation from a to b, thus obtaining the following expression for the index form:

We may repeat the same calculation with w replaced by z. This time the hj are constants in ]a, b] (18.7.7), and f is replaced by 0. Since w(b) = z(b), we obtain (20.20.3.8)

I(w,

W)

- I(z,

Z)

=

Since the right-hand side vanishes only when the hj are constants (8.5.3), the proof is complete. Remark (20.20.3.9) The assertion of (20.20.3) remains valid if we assume only that w(a) = Ov("). Let sj be the parallel transport of uj along the path u (1 8.6.4)

402

XX


(1 S j 5 n); then the vectors sj(t) form a basis of TV(,)(M)for all t and we may write

E

[a, b ] ,

-

since V , sj = 0 (18.6.3.1), we have n

n

By hypothesis, the functions zjk and wJ are continuous at the point a, and the matrix (Zjk(t))tends to the unit matrix Inas t + a. Since Z,k(a) = 0 for all j , k, the matrix ( ( t - a)-'zj,(t)) also tends to I, as t + a, and the same argument shows that the functions ( t - a)-'wj(t) tend to finite limits as t + a. This being so, we have

and Cramer's formulas show that the hk(t) tend tofinite limits as t + a . It follows that the limit of (u(t)I w(t)), as t -+ a, exists and is finite; the formula (20.20.3.7) is therefore still valid, and the integral IIf1I2 dt is finite (13.8.1), hence the conclusion is unaltered.

1;

(20.20.4) With the hypotheses and notation of (20.20.3), suppose that the parameter t is the curvilinear coordinate of the geodesic t tt u(t) =f'(t, 0 ) , with origin v(a), and that f ( a . 5) = v(a) and f ( b , 5) = o(b) for all 5 E J . Then, i f there exists no point on v conjugate to v(a) (18.7.12) and iff;(?, 0) is not identically zero in [a, 61, we have

L(5) > L(0) = b - a

for all suficiently small 5 # 0. Since lift'(?, 0)II = 1 for all t E [a, 61, we may assume thatfj(t, 5) # 0 for all ( t , 5) E I x J, and the change of parameter (t, 0; and since E'(0) = 0 by (20.20.2.1), we have E(5) > E(0) for all sufficiently small 5 # 0, by virtue of Taylor's formula (8.14.2). Since

E(5) = (L(o>2/(b -4 by virtue of our choice of the parameter t , the result is proved. (20.20.5) Let c be a geodesic in M defined on an open interval I c R containing [a, 61, and suppose that v(6) is thejrst point on v conjugate to v(a). Then, f o r

all su8ciently small E > 0, there exists a C" path defined on [a, b + E ] whose distance from ir (relative to the topology of uniform convergence and the Riemannian distance d on M) is E , with endpoints v(a) and v( b + E ) , and of length strictly less than that of the restriction of u to [a, b + E ] .

s

In view of (20.1 6.6), it will suffice to establish the existence of a piecewiseC" path with the properties asserted. We shall first show that there exists a piecewise-C" lifting w of u (restricted to [a, b + E ] ) such that

I( w, w)< 0 , w(a) = 0, and w(b + E ) = O b + E . For each interval [a, p ] c I, let I a , p ( ~ ,w ) denote the value of the index form relative to the restrictions of v and w to [a, 81. Let p be the radius of a geodesically convex ball with center v(b) (20.17.5), and let E > 0 be suficiently small so that v ( t ) lies in this ball f o r b - E 9 t b + E . By hypothesis, there exists a Jacobi field z 1 along u which is not identically zero on [a, b ] and is such that z,(a) = 0, and z , (b ) = 0, (18.7.11). On the other hand, the choice of E implies that there exists no point conjugate t o v(6 + E ) on the restriction of v to [b - E , 6 + E ] (18.5.2); consequently, there exists a Jacobi field z2 along this restriction of v such that z2(b + E ) = O b + & and z2(b- E ) = z l ( b - E ) (18.7.11). Put w(t) = z , ( t ) for a S t 5 b - E , and w(t) = z2(r)for b - E 5 t b + E ; also let u denote the lifting of v such that u(t) = w(t) for a 5 t 2 b and u(t) = 0 for b 5 t 5 b + E . The formula (20.20.3.7) applied to z I on an interval [a, b'] with b' < b gives I,, b ( z 1 , z l ) = 0 on letting b' tend to b, and this may also be written as

s

I,, b + d U ,

=

IQ,b ( z l ,

zl>

= O*

But also we have la,b+&(U,

u, =

+ Ib-c, w, + l b - e . b + e ( Z ,

la,b-a(Zlr ' 1 )

= l,,b-e(W,

z)*

404

XX

PRINCIPAL CONNECTlONS AND RIEMANNIAN GEOMETRY

Finally, we may apply (20.20.3) in the interval [b - E , b + E ] to the Jacobi field z2 and the lifting u (since the latter is zero in a neighborhood of b + E ) , and hence we obtain Ib-&,b+&(U,

u, > 1 b - & , b + & ( Z 2>

'2)

= lb-&.h+E(W1

w)9

and thus 0 = I,,b+&(U,U) > I , , b + & ( W , w). The conclusion of (20.20.5) will therefore result from the following proposition (in which the notation has been slightly altered): (20.20.5.1) Let u be a geodesic dejned on an open interval 113 [a, b ] , and let w be a piecewise-C3 lifting of u to T(M), dejned on [a, b ] and such that w(a) = o,, w(b) = o b , and I,, b( w, w ) < 0. Then f i r each E > 0 there exists a piecewise-C3 path cp : [a, b ] + M such that

for all t E [a, b ] ,and L(q) < L(v). There exists a real number 6 > 0 such that 6w(t)E R for all t E [a, b ] , so that the functionf(t, t) = exp,(,,(tw(t)) is defined and piecewise of class C3 in [a, b ] x 3-S, 6 [ ; moreover f; is continuous on this set, and for each t E [a, b ] the function < H f ; ( r , t) is of class C" and satisfies the equation (V, 5 ) = 0 and the boundary condition f;(t, 0) = w(t). Applying the formulas (20.20.2.1) and (20.20.2.2) in each of the intervals [ a j , on which w is of class C3, we obtain a&)(?,

E'(0) = 0,

E"(0) = ~ J ( w , W ) < 0.

For sufficiently small t > 0 we have therefore d ( f ( t , t), u(r)) S E for t E [a. b ] , and E(5) < E(O), by Taylor's formula. By virtue of (20.20.1.3), this implies L(5) < L(O), whence the result. It should be carefully noted that the length of a geodesic arc with endpoints p , q in M may well be >d(p, q ) , even if there exists no point conjugate t o p or q on the arc, as the example of the cylinder (20.17.3) shows. The property of minimizing the length of an arc with the same endpoints, when the geodesic arc under consideration contains no point conjugate to the endpoints, holds only for " neighboring" arcs. (20.20.6)

(20.20.7) Let M be a complete connected Riemannian manvold. I f a point a E M is such that no geodesic with origin a contains a point conjugate to a, then (T,(M), M , exp,) is the universal covering of M .


405

The hypothesis that M is complete implies that exp, is a surjection of T,(M) onto M (20.18.5), and the hypothesis that no geodesic contains any point conjugate to a implies that exp, is a local diffeomorphism (18.7.12). If g is the Riemannian metric tensor on M, consider the Riemannian metric tensor ‘exp,(g) = g, on T,(M), relative to which exp, is a local isometry. Since for each ha E T,(M) the curve tt-+exp,(th,) is a geodesic in M defined on the whole of R, it follows that it-+ th, is a geodesic in T,(M) relative to the metric tensor g,, defined on the whole of R ; by virtue of (20.18.5), T,(M) is coniplete relative to g, and the result follows from (20.18.8). (20.20.8) Let M be a connected Riemannian manuold of’ dimension n. let a be a point of M. and let B(0,; r ) be an open bail in T,(M) n R on which exp, is injective. Then exp, is a difeomorphism of B(0,; r ) onto the open ball B(a; r ) in M, such that d(a, exp,(h,)) = IIh,Il .for lIh,II r.

-=

Suppose that there exists a point ha E B(0,; r ) at which the rank of exp, is < n . Then, by virtue of (20.20.5), there exists a point th, E B(0,; r ) such that d(a, exp,(th,)) < ll~h,II.If r’ is such that d(a, exp,(th,)) < r’ Ilth,ll, then it follows from the fact that B(a; r’) is the image under exp, of B(0,; r ) (20.18.5.5) that there exists a vector hi E T,(M) such that exp,(h;) = exp,(th,) and I/ h:II < r’ < IIthJ. contrary to the hypothesis that exp, is injective on B(0,; r ) . Since exp,, restricted to B(0,; r ) , is therefore a bijective local diffeomorphism of this ball onto B(a; r ) , it follows (16.8.8) that it is a diffeomorphism.

-=

(20.20.9) Let M be a complete connected Riemannian manifold and a, a point of M. For exp, to be injective on T,(M), it is necessary and suficient that M should be simply-connected and that no geodesic with origin a should contain any point conjugate to a. The mapping exp, is then a digeomorphism of T,(M) onto M.

The necessity of the condition follows from (20.20.8) and (18.7.12), and the sufficiency from (20.20.7), since M is simply-connected.

PROBLEMS 1.

Let Pn(K) be projective n-space over K = R, C, or H, endowed with the Riemannian structure defined in (20.11.5) and (20.11.6). All geodesics in P,(K) are periodic with period T (relative to the curvilinear coordinate). If two geodesics u , , uz have the same origin xo E PJK) and if (relative to the curvilinear coordinate with origin xo) we put h , = c;(O) and hz = c;(O), then these two geodesics have no common point # x o unless the vectors h,, h2 are linearly dependent for the K-rector-space structure o n

406

XX


T.ro(Pn(K));in which case they have a second common point, with curvilinear coordinate An.The maximum radius d ( x o )of a ball B(O,,; r), such that exp,, is injective when restricted to this ball, is in-, and the ball B(xo; An) is strictly geodesically convex. 2.

Let M = G / H be a Riemannian symmetric space (20.11.3), with H compact, and let u : R + M be a geodesic parametrized by the curvilinear coordinate with origin xo = u(0). Put u’(0) = h (so that u ( t ) = exp,,(fh)), and let R(h) denote the endomorphism k H ( r ( x o ) . (k A h)) . h of T,,(M), which is self-adjoint (20.10.5). If, for every Jacobi field w along u, we denote by G(r), the vector in T,,(M) obtained by parallel transport of w ( t )along o, show that the mappings & of R into T,,(M) are the solutions of the equation 3“ = R(h) . G. (Use the fact that the connection on M is G-invariant.) There exists an orthonormal basis (hi)ls1gnof T,,(M) consisting of eigenvectors of R(h), with h, = h: if hl is the eigenvalue corresponding to h,, then h , 10. Show that the Jacobi fields along u which are zero at the point xo are linear combinations of the n fields w , given by the conditions w,(O) = 0, (C, . w i ) ( 0 )= hi ( I 5 i 5 n). We have

if hr < 0, if hr 0, if hi > 0.

sin((-Af)l/zf)h,

w,(t)=

w , ( t ) = thi

w,(f) = hr1l2sinh(h:”f)h,

3. (a) In Problem 2, take M to be one of the projective spaces PJK) of Problem I . Use Problem I to show that the eigenvalues h, are equal to - I or -4. (b) With the notation of Problem 2, consider the family of geodesics ( t , a ) ++exp,,(f (cos a . h

+ sin a . h,)) =f(t,

a),

so that f ( t , 0) = u(f ). Show that fi(br, a) = w,(&a), where w. is the Jacobi field along u such that w.(O) = Ox, and (C, . w,)(O) = -sin CL . h cos a . h i . (c) Deduce from (b) and Problem I that hr = - 4 if and only if h and hi are linearly dependent over the field K. (To show that the condition is necessary, remark that -sin CL . h cos a . hl is an eigenvector of R(cos a . h + sin CL . h,) (cf. (20.21.2)) and deduce that if hi = - 4 we must have fL(in,a) = 0 for all a , ) Hence find the values of the /\,in each of the three cases K = R, K = C, and K = H.

+

+

4.

Let M be a Riemannian manifold all of whose geodesics are periodic with the some minimum period I (when parametrized by arc length). (a) Show that M is a complete manifold and that for all x E M we have M = exp,(B’(O,; A[)), and hence that M is compact. (b) Show that two geodesic paths with origin x and length I (and therefore with endpoint x ) are loop-homotopic in M. (If u l , u z are the paths, defined on [0, I ] , consider a path in the unit sphere with center 0,. with endpoints u;(O) and ~ ~ ~ ( 0 ) . ) Show that the fundamental group n-,(M) has one or two elements. (Use (20.19.1) for the Riemannian universal covering of M.)

5. Let f b e a C“ function defined on 10, I[. Let S c R3 be the surface of revolution with

axis Re,, given by the equation

6, =f(((P)’+ (6z)z)1~z)


407

(Section 20.18, Problem 13). The mapping (r, v) ~ (cosrp,, r sin v,f(r)) is a diffeomorphisni of the open subset or R 2 defined by 0 < r < 1, 0 < v < 27r onto a dense open subset U of S. Show that, relative to the corresponding chart of S with domain U, we have g=(I

+ ( f ’ ( r ) ) ’ ) dr 0dr + r 2 dp, 0d v .

Deduce that, for each geodesic in U, parametrized by arc-length with a suitable orientation, there exists a constant a 0 such that

and

implying that r 2 a at all points of this geodesic. 6. Letfl,f2 be two C“ functions defined on [0, I [, such rhatfi(1) = O , f i ( O ) = O,f;(r) 5 0 for 0 5 / < 1 and limfi(t) = - rn ( i = 1 , 2). Consider the two surfaces of revolution 1-

1

S1, Sz obtained by the procedure of Problem 5 , by taking f = f l and f = tively.

-f2,

respec-

(a) Let h(r) be a polynomial such that h(0)= 0 and (1 - f 2 ) - 1 / 2 + A ( t ) 2 1

for 0 6 t < 1. Choose fl and f 2 so that

(I + f : y = ( I ( 1 +f;2)1’2

-r2)-’/2+A(t),

( I - / 2 ) - ” 2 -A(/).

Show that the closure S of S, u S , in R3 is a compact analytic surface diffeomorphic to S2 (Zoll’s surface). (If u = ( I - r2)’/’ on S1, u = -(I - r2)I/’ on S 2 , show that du/df3= F(u) in a neighborhood of t3= 0 in S , and in Sz. for the same analytic function F.) (b) Show that all the geodesics of S are periodic and have the same minimum period. ( I f a is the minimum value of r on a geodesic, calculate the variation of p, as 5’ varies from -f2(a) toftb).) (c) When A(/): it4, show that the total curvature of S takes opposite signs (cf. Section 20.14, Problem 7).

7. Let M be a complete connected Riemannian manifold, and let x be a point of M. Consider a geodesic / H u ( / ) = exp,(th), parametrized by arc length (where h E T,(M) and llhll = 1). (a) Show that the set I of numbers s > 0 such that d(x, u(s)) = s is a n interval, either 10, -1 m[ or 10, r] with r finite. In the latter case, the point u(r) is called the cut-point on the positive geodesic ray with origin x defined by h.

408

XX


(b) If there exists a cut-point u(r) on u, show that one of the following two alternatives holds : (a) u(r) is the first point on u conjugate to x (18.7.12); (6) There exists at least one geodesic u1 :t Hexp,(thl) through x with hl # h, ul(r) = u(r), and d(x, ul(r)) = r. (Let (u.) be a decreasing sequence of numbers > r and tending to r, and for each n let t ~+exp,(th,) be a geodesic parametrized by arc length and such that d(x, mPx(bn h.)) = &, e x p h . h)) < a. .

We may assume that the sequence (b. h.) tends to a limit k in TJM). Examine the two possibilities k # r h and k = rh; in the second case, argue by contradiction, observing that exp, cannot be a local diffeomorphism at the point k.) (c) Show that if y = u(r) is the cut-point on the positive geodesic ray f Hu ( t ) with origin x, then x is the cut-point on the positive geodesic ray t Hu(r - t ) with origin y . (Use (b).) (d) Let S, be the unit sphere llhxll = 1 in T,(M). For each h, E S,, let p(h,) be the number r if exp,(rh,) is the cut-point on the positive geodesic ray with origin x defined by h, , and let p(h,) = co if there is no cut-point on this geodesic ray. Show that the mapping p of S, into [0, + a ]c R is continuous. (Argue by contradiction, by considering a sequence (h,) of points of S, with limit h such that the sequence (p(h,)) tends to a limit c # p(h); distinguish two cases according as c < p(h) or c > p(h). In the first case, observe that exp,(ch) is not conjugate to x , and hence that exp, is a diffeomorphism of a neighborhood of ch onto a neighborhood of exp,(ch); now use (b) to obtain a contradiction. In the second case, we may assume that p(h,) 2 p(h) + b for some b > 0; observe that there exists a vector h’ # h such that exp,((p(h) + b)h) = exp,((p(h) b’)h‘) for some b’ < 6. Consider the path obtained by juxtaposition of thepatht-exp,(th‘)forO 5 t 5 p(h) b’andageodesicpathfromexp,((p(h) b’)h‘) to exp,((p(h) b)h,,), of length equal to the distance between these two points, and hence arrive at a contradiction.) (e) The set P(x) of points exp,(p(h,)h,), where h, runs through the set of points of S, such that p(hJ < to, is called the cut locus of x . If E, is the set of all th, E T,(M) such that t < p(h,), show that Ex is homeomorphic to T,(M), that exp, is a diffeomorphism of Ex onto an open subset exp,(E,) of M, and that M is the disjoint union of exp,(E,) and P(x). (Use (a) and the Hopf-Rinow theorem.)

+

+

+

+

+

+

8. Determine the cut locus of a point when M is a projective space P.(K) (Problem 1)

or the flat torus TZ.

9. (a) Let M be a complete connected Riemannian manifold and xo a point of M. Show that M is compact if and only if there is a cut-point on each geodesic ray with origin xo . (Use Problem 7(d) to show that the condition is sufficient.) (b) Let M be a complete connected Riemannian manifold whose universal covering is not compact. Show that for each point x E M there exists a geodesic ray t H exp,(r h) (f 2 0) with origin x which contains no point conjugate to x. (Reduce to the case where M is simply-connected, and then use (a).) If M is simply-connected, we may assume that d(x, exp,(th)) = t along a geodesic ray.

10. Let M be a complete connected Riemannian manifold. For each x E M let d(x) denote the radius of the largest open ball B(0,; r) on which exp, is injective.

21 SECTIONAL CURVATURE

409

(a) Show that d ( x ) is also the radius of the largest open ball with center 0, contained in the set Ex (Problem 7(e)). (Observe that the proof of Problem 7(d) shows that the function p is continuous, not only on each S,, but on the submanifold U(M) of T(M) which is the union of the S, .) (b) Deduce from (a) that the set of x E M such that no geodesic through x contains a point conjugate to x is closed in M. (Reduce to the case where M is simply-connected, and then use (a) and (20.20.7).) (c) Take M to be the surface given by the equation = i((tl)* (5')')in R3.Show that the origin is the only point x of M such that no geodesic through x contains a point conjugate to x . (Use Problem 5 . )

t3

+

11. Let M be a complete pure Riemannian manifold, and let X be a n infinitesimal isometry of M (Section 20.9, Problem 7). Let S be the set of points x E M such that X ( x ) = 0.

(a) The flow Fx of X has M x R as domain (Section 20.6, Problem 8); if we put p,(x) = F x ( x , t ) , then p 1 is an isometry of M onto itself for all t E R,and leaves fixed the points of S. For each x E S, the mapping t HTJv,) is a homomorphism of R into the orthogonal group relative to the scalar product (u Iv) defined on T,(M) by the metric tensor of M, and hence TJM) is the Hilbert sum of suspaces E, (1 < j 5 r ) of dimension 2 and a subspace N of dimension dim(M) - 2r, stable under T,(pI) for all / E R (cf. (21.8.1)). Show that the geodesics through x all of whose points are fixed by the isometries v, are those whose tangent vector at x lies in N, and that the union of these geodesics is a totally geodesic submanifold of M. (b) Deduce from (a) that the connected components of S are totally geodesic submanifolds of M. If V,, V, are two distinct components of s, show that there exists a n infinite number of distinct geodesic trajectories of length d ( x l , x 2 ) , with endpoints x1 and x , , for each point x , E V, and each point x , E V, . In particular, if for each point x E M and each plane P, C T,(M) we have A(P,) 0 (20.22.1), then S must be connected.

21. SECTIONAL C U R V A T U R E

(20.21.1) Let M be a Riemannian manifold, K its Riemann-Christoffel tensor (20.10.2). It follows immediately from the symmetry properties of K (20.10.3) that, for any two vectors h, , k, in T,(M), the number

depends only on the biuector h, A k,, and is multiplied by A' when this bivector is multiplied by a scalar 1,. If h, A k, # O , i.e., if the two vectors h,, k, are linearly independent, the number

therefore depends only on the plane P, spanned by h, and k, . This number is called the sectional curirature (or Riemannian curvature) o f M for the plane

410


P,, and is denoted by A(P,). We have seen in (20.10.3) that knowledge of g(x) and A(P,) for all planes P, in T,(M) completely determines the RiemannChristoffel tensor K(x). For an orthonormal moving frame (el, . ., en), with the notation of (20.1 0.4) we have

.

(20.21.1.21

A(ei(x), ej(x)) = Kijij(x) = (Qij(x), ei(x) A ej(x)>

for i # j , since by definition /lei A ejll = 1 (20.8.4.2). When n = 2, T,(M) is itself a plane, and A(T,(M)) is the Gaussian curvature of M at the point x. The formulas (20.10.6.3) and (20.10.7.2) give the values of the Ricci curvature in the directions of the vectors ei(x) in terms of the sectional curvature:

and likewise for the scalar curvature we have (20.21.1.4)

Examples (20.21.2) The sectional curvature of a flat Riemannian manifold is evidently zero, and conversely. The formula (20.7.10.6) giving the curvature of the canonical linear connection on a Riemannian symmetric space G/H also enables us to compute the sectional curvatures for such a space. In view of the invariance of the metric tensor under G, it is enough to perform the calculation at the point x,, = n(e);identifying T,,(G/H) with m, we have then

for all u , v E in. For example, consider the complex projective space P,(C), identified with SU(n + l)/U(n); take for u and v endomorphisms whose matrices are of the type (20.11.6.1); since H = U(n) acts transitively on the real lines in in, we may assume that u = a,,,. If

21 SECTIONAL CURVATURE

411

then an elementary calculation with matrices (bearing in mind that in is to be considered as a real vector space, so that lie, A (iel)ll = 1) gives the result 4v: A(ac,, a,) =

(20.21.2.2)

2

+ j = 2 (5

v: +

i(5:f +vS)

:- 1 J-*

For the sphere S, , the imaginary parts are replaced by zero, and we obtain the constant 1. (The fact that A(u, v) is here independent of u and v could have been foreseen without calculation from the fact that in this case H = SO(n) acts transitively on the planes in in.) Likewise, for hyperbolic space Y, (20.11.7), we obtain for A(u, v) the constant value -1. Finally, for a compact Liegroup G with center {e},the formulas (20.11.8.1) and (20.11.8.3) give

for all u, v in ge. (20.21.3) Let M be a pure Riemannian manifold of dimension n, and let

M‘ be a pure submanifold of M of dimension n’. We wish to express the sectional curvature A’(P,) of M’ for a plane P, c T,(M‘) in terms of the sectional curvature A(P,) of M for P, , and the second fundamental forms. We shall use the notation of Section 20.12. By a suitable choice of the moving frame R‘, we may suppose that P, is spanned by the vectors ei(x), ej(x) ( i # j ) . It then follows from the formulas (20.21.1.2), (20.10.4.5), and (20.12.5.2) that A(Px) - A’(P.r) =

-C

(ei(x>)(wAj(x),ej(x))

U

- <whi(x),ej(x>> , ei(x>>>

or (20.21.3.1)

A(P,) - A’(P,) = - C (Zaii(x)lajj(x)- (l,i,(~))~) a

(Gauss’s formula). From this formula we can derive a simple geometrical

interpretation of the sectional curvature. Consider an open ball B(0,; r ) contained in R n T,(M), and take M’ to be the surface exp,(P, n B(0,; r ) ) which is the union of the geodesic trajectories passing through x E M whose tangent vectors at x lie in P,. This signifies that the values (lu(x), h, @ h,)

412

XX


are zero for all h, E P, (20.13.6.2), which is possible only if the restriction (h,, k,)H (lu(x),h, 0 k,) of the symmetric bilinear form Iu(x) to P, is identically zero for all a. The right-hand side of (20.21.3.1) therefore vanishes; bearing in mind (20.21.1.2), it follows that A(P,) is the Gaussian curoature of the surface M‘. The formula (20.21.3.1) also has as a consequence the following proposition : (20.21.4) Let M’ be a submanifold of M and suppose that M‘ contains a geodesic trajectory C of M.Then for each x E C and each plane P, c T,(M) containing the tangent vector to C at x we have

(Synge’s lemma). We may assume that the frame R’ has been chosen such that e l ( x ) is tangent to C and that P, is spanned by e l ( x ) and e2(x). Then we have lall(x)= 0 for all a (20.1 3.6.2), and consequently A(P,) - A’(Px) =

1

(1u12(x))2

U

giving the result.

22. MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE O R NEGATIVE SECT10 NAL CURVATURE

(20.22.1) Let M be a Riemannian manifold and v a geodesic in M . Suppose that at each point x of this geodesic and each plane P, c T,(M) we have A(P,) 5 0. Then there exists no pair of conjugate points on v.

Let w be a Jacobi field along u, not identically zero. Then for each t in the interval of definition I of u, we have (1 8.7.5)

V, (V, *

w ) = ( r (u’

A

w ) ) . u‘.

We shall see that w cannot vanish at two distinct points of I. First let us show that the function t H ( w ( t ) I V, w ) is increasing. By (20.9.5.4), we have (20.22.1 .I)

d

( w I V, * w ) = IIV, * wII dt = IIV,

*

W1I2

+ ( w I( r - A(d,

*

(u’

A

w ) ) u’)

W ) [ ~ UA‘ W1I2

20

22 MANIFOLDS WITH POSITIVE OR NEGATIVE SECTIONAL CURVATURE

413

by virtue of the hypothesis, which proves our assertion. Moreover, if (WlV, * w )

were to vanish on an interval J c I of length >O, then we should have V, . w = 0 on this interval by the formula above, and it follows immediately from (18.7.6) that if w vanished at a point in J, then w would vanish identically on I. This proves the proposition. (20.22.2) (Hadamard-Cartan theorem) Let M be a complete connected Riemannian manifold of dimension n, such that A(P,) 6 0 for each x E M and each plane P, c T,(M). Then the universal covering of M is difeomorphic to R",and ifM is simply-connected then M is strictly geodesically conuex.

This follows immediately from (20.22.1) and (20.20.7). (20.22.3) (Myers's theorem) Let M be a complete connected Riemannian manifold of dimension n 2 2. If there exists a number c > 0 such that the Ricci curvature of M (20.10.7) satisfies the inequality Ric(h,) 2 cI(h,1I2 for all h, E T(M), then M is compact, the diameter of M satisfies the inequality (20.22.3.1)

and the fundamental group n,(M) is finite. Let a, b be two distinct points of M, and put I = d(a, b). Since M is complete, there exists a geodesic path v : tHexp,(th,) defined on [0, I], with origin a and endpoint b, such that IIh,ll = I (20.18.5). Let ( e i ) l s i 6 nbe an orthonormal basis of T,(M) such that h, = len, and let uj denote the parallel transport of ej along u (18.6.4), so that lu, = v' and (ujl U k ) = 6 j k for all pairs of indices j , k (20.9.5.4). Put w j ( t ) = u j ( t ) sin nt for 1 5 j S n and 0 5 t 6 1, and let us calculate the index form I ( w j , w j ) (20.20.2.3). We have V, * wj = (V,

u j ) sin nt

+ nuj(t) cos nt = nuj(t)cos nt

by the definition of a parallel transport; since IIuj(t)))= 1, we obtain (20.22.3.2)

r(wj, wj) =

s,'

+

(nZ ( ( r (v'

A

u j ) ) * u'I u j ) ) sin2 nt dt.

By definition (20.10.7.2), we have Ric(u') = -

n

((r' (v'

j= 1

n- 1

A

u j ) ) * u ' ) u j )=

-

j= I

( ( r * (u'

A

uj)) * v'luj)

414

XX


since lu, = v’; hence we obtain from (20.22.3.2) the formula n- 1

C I( w j , w j ) =

j=1

sd

((n - 1)x2 - Ric(o’(t))) sin2 xt d t

and consequently the inequality n- 1

(20.22.3.3)

1 I ( w j , w j ) 5 jol((n- 1)x’

- cl’)

sin’ xt dt.

j= 1

This being so, if 1’ > (n - l)x’/c, at least one of the numbers I ( w j ,w j )

(1 S j S n - 1)

would be negative, and it would follow from (20.20.5.1) that d(a, 6) < 1, which is absurd. This establishes the inequality (20.22.3.1), and since M is complete it follows that M, being equal to a closed ball, is compact (20.18.5). Now let be the universal covering of M, endowed with the Riemannian metric canonically induced by that of M (20.8.1). Clearly $I satisfies the same hypotheses as M (20.18.7), hence $I is compact. Every fiber in $I over a point of M is therefore compact and discrete, hence finite (3.16.3); but by (16.28.3) the fibers of the covering fi are each in one-to-one correspondence with x,(M), and therefore xl(M) is finite. (20.22.4) Let M be a complete connected Riemannian manifold of dimension n 2 2. If there exists c > 0 such that A(P,) 2 c for all x E M and all planes P, c T,(M), then M is compact, its diameter satisfies the inequality (20.22.4.1 )

and the fundamental group R , ( M ) is finite.

For it follows from (20.21.1.3) that Ric(h,) 2 (n - l)cllh,11’ h, E T(M), and the result therefore follows from (20.22.3).

for all

(H. Weyl’s theorem) I f G is a compact connected Lie group with discrete (or equivalently (3.16.3) finite) center, then the universal covering of G is compact.

(20.22.5)

We shall begin by reduction to the case where the center of G is {e}. For this purpose we need the following topological lemma : (20.22.5.1) Let G be a connected Hausdorff topological group, D a discrete subgroup of the center Z of G. Then the center of G / D is Z/D.


415

Let 7c : G -+ G/D be the canonical mapping. If s E G is such that n(s) is in the center of G/D, we have ~ ~ ( s x s - ~ = x -n(e) ~ ) for all x E G, so that sxs-'x-' E D. Now the mapping X H S X S - ~ X - ' is continuous, and since G is connected and D is discrete, it follows that S X S - ~ X - ~takes the same value for all x E G (3.19.7). Taking x = e, we see that sx = xs for all x E G, and therefore s E Z. This lemma being established, let be the universal covering of G, and let Z, be the center o f c , wihch is discrete becausee is locally isomorphic with G, so that its Lie algebra has a trivial center (19.11.7). The group G is isomorphic toG/D, where D is a subgroup of Z, (1 6.30.2); by virtue of (20.22.54, the center Z of G is Z,/D, and G/Z is isomorphic to e / Z , (16.10.8). Since G is compact and connected, the same is true of G/Z, and the center of G/Z is trivial by virtue of (20.22.5.1); but is the universal covering of e / Z , , and we may therefore assume henceforth that the center of G is trivial. The group G being endowed with a Riemannian structure as in (20.11.8), if ( e i ) , is an orthonormal basis ofg,, it cannot be the case that [ei, e j ] = 0 for all , j # i, since this would imply that ei belonged to the center of ge, which by hypothesis is trivial (1 9.11.7). The formulas (20.21.1.3) and (20.21.2.3) therefore show that Ric(e,) > 0 ; but since e, can be any unit vector in ge, and since the unit sphere in ge is compact and the function u ~ R i c ( u )is continuous on this sphere, we see (3.17.10) that there exists c > 0 such that Ric(h,) 2 cII h,llz for all h, E T(G), having regard to the fact that the Riemannian metric is translation-invariant. The result now follows from (20.22.3).

e

(20.22.6) (Synge's theorem) Let M be an orientable, compact, connected Riemannian manifold of even dimension. If A(P,) > 0 for all x E M and all

planes P, c TJM), then M is simply-connected.

Suppose that n,(M) is not the trivial group. Then there exists (20.19.3) a periodic geodesic v : t Hexp(th,,) of period 1 , so that v(0) = v(1) = a and v'(0) = v'(1) = h,,; moreover, 1 = (Ih,(I is the smallest of the lengths of loops homotopic to v. As in the proof of (20.22.3), we define a sequence ( u ~ of parallel transports along v, such that v' = f u n and (ujl u k )= 6,. Since v(1) = v(0) = a, the sequence ( u j ( l > )b,j s n is an orthonormal basis of T,,(M), and therefore there exists an orthogonaltransformation S of T,,(M) such that u j ( l ) = S . uj(0) for 1 5 j S n. We assert that S is a rotation. To prove this, it is enough to show that the n-vectors u,(O) A A u,(O) and

)

~

~

416


are equal. Now if u is a volume-form on M, the function i

+

(u(dr)), ~

ul(t) A

*

*

A un(t))

is continuous on [0, 11, and never zero because u(x) # 0 for all x E M and the uj(t) form a basis of T”(,)(M) for all t E [0, 11; consequently the numbers (u(a), u,(O) A A u,(O)) and (u(a), u,(l) A * . * A u,(l)) have the same sign and are therefore equal. This proves our assertion. We remark next that un(l) = u,(O), and hence the rotation S stabilizes the hyperplane H in T,(M) spanned by u,(O), . .. , U , - ~ ( O ) . The restriction S’ of S to H is therefore again a rotation, and since H is odd-dimensional, there exists a vector k, # 0 in H which is invariant under S. The parallel transport w along v such that w(0) = k, therefore satisfies w(1) = k,. Now putf(r, t) = exp,(,,(tw(t)), as in (20.20.5.1), so thatf;(r, 0) = w(r) and (V, .f;)(r, 5 ) = 0. With the notation of (20.20.2) we have again E’(0) = 0 by virtue of the periodicity of w, and +E”(O)= I(w, w); but because w is a parallel transport, V, w = 0, and hence 9 . .

I(w, w) = - llu’

A

w (I2

sd

A(v’, w) dt,

because (Id A wII is constant along u (20.9.5.4). By construction, we have u‘ A w # 0; hence A(v‘, w) is a continuous function which by hypothesis is > O at all points of [0, 11, and therefore I(w, w) < 0 (8.5.3). Taylor’s formula now shows that for sufficiently small to> 0, the length of the path t H f ( t , l o ) is < I ; but it is clear that ( t , l ) ~ f ( tt), (0 5 f 5 1, 0 5 5 5 to)is a loophomotopy of u to f( . , to).We have therefore obtained a contradiction and the proof is complete. The example of an odd-dimensional real projective space Pz,+l(R), which is orientable (16.21.11), for which A(P,) = 1 (20.21.2), but which is not simplyconnected, shows that the hypothesis of even-dimensionality in (20.22.6) cannot be dispensed with.

PROBLEMS

1. Let M be a Riemannian manifold, x a point of M, B(0,; ro) an open ball contained in Cl n T,(M), and P, a plane in T,(M). For 0 < r < ro , let C(P,; r) be the image under exp, of the circle llhxll= r in P,. Show that L(C(P,; r)) = 2nr - fwr3A(P,)

+ oa(r),


417

where 03(r)/r3-+O as r -+ 0. (If hl, h2 form an orthonormal basis of P,, consider the family of geodesics f ( r , a) = exp,(r(cos a * hl

+ sin

o!

. hJ).

Argue as in Section 20.21, Problem 3(b), and use Section 20.16, Section 7.) 2.

Let M be a Riemannian manifold of dimension n 2 2, such that Ric(h,) 2 cllh, 112 for all h, E T(M), where c is a constant >O. Show that on every geodesic, the length of an arc not containing conjugate points is 6 w((n - I)/C)"~. (Argue as in (20.22.3).)

3. Let M I and M2 be two Riemannian manifolds of dimension n, and let u1 (resp. u 2 ) be a geodesic in M I (resp. M2) whose interval of definition contains [a, b]. Let w I (rap. w z )be a Jacobi field along ul (resp. u 2 ) orthogonal to v ; (resp. v;). Assume that: (1) w l ( a )= w2(a )= 0; (2) II(V, . w,)(a)II = ll(VI . wd(a)II; (3) u k r ) is not conjugate to ur(a)for r E [a, b ] and i = I , 2; (4) for all f E [a, 61 we have A(P,,cI,) 2 A(Q,a(I,for all planes PvlcI)in T,,tI,(MI) confaining u ; ( f ) , and all planes Q,(,) in Tuz(,)(M2)containing v i ( f ) . Then Ilw,(r)II 5 Ijw2(f)II for a =< t 5 b (Rauch's comparison theorem). (Put u,(r) = llw,(t)l12, u2(r)= llw2(r)l12, and A(r) = Ig,I(w,, wl)/ul(r) (notation of 20.20.5) for i = 1, 2. Using the formula (20.20.3.7), show that "I

I

sf2(r)

and hence that it is enough to prove thatfl(r) for f E [a, b]. Fix to E [a, b] and put zl(r) = wl~r)/llw,~ro)llfor i = I, 2. Show that there exists a lifting s of u1 such that Ils(r)ll= Ilz2(t)II and II(V, .s)(r)ll = ll(VI . z,)(t)ll for r E [a, b] (use parallel transports from to to t along v1 and u2). Then use (20.20.3) to show that IP.I0(Z1' 21)

and deduce that f,(r,)

2 I~,I&, s I..to(z2. 4,

6fz(to).)

4. With the notation of Problem 3, suppose that v , ( r ) is not conjugate to ul(a) for a < f 5 b, and that A(P,,(J 2 A(Qu8(tJfor a 5 r 5 6, where Pvlc1)is any plane containing u;(r), and Qua(,)any plane containing u;(f). Then u 2 ( t ) is not conjugate to 02(a) for a < t 5 6. (Argue by contradiction, using Problem 3.) 5. Let M be a complete Riemannian manifold such that A(P,) 5 0 for all x E M and all planes P, c TJM). With the notation of (20.16.3.1), show that

lIThx(exP,) kill 2 llk,Il. a

(Use Section 20.16, Problem 7.) Deduce that if M is simply-connected, we have d(exp,(h), exp,(k))

L ll k - h II

for any two vectors h, k in Tx(M). 6. (a) Let M be a complete, simply-connected Riemannian manifold, such that A(P,) I_ 0 for all x E M and all planes P, C T,(M). Let Z be a compact metric space

418

XX


and let p be a positive measure f O on Z; finally, let f:Z + M be a continuous mapping. For each x E M, put

h(x) =

J-,

d(x, f(z))' dp(z),

where d is the Riemannian distance. Show that h attains its minimum at exactly one point of M. (To show the existence of such a point, observe that f ( Z ) is contained in a closed ball in M and that such a ball is compact. To show uniqueness, consider a point xo at which h attains its minimum. Since exp,, is a diffeomorphism of T,,(M) onto M (20.223, we may writef= exp,, f,, where f, is a continuous mapping of Z into T,,(M). If we put 0

ho(u) =

s,

II u

- fo(Z)llZ

dp(z)

for u E T,,(M), show that h(xo) = ho(0) < h,(u) 5 h(exp,,(u)) for all u # 0 in T,,(M), by using Problem 5 . ) (b) With the same assumptions on M, let G be a compact subgroup of the group of isometries I(M). Show that G has a fixed point in M. (Apply (a) to the function s Hs . xo on G , with p a Haar measure on G.) (c) With the same assumptions on M, let G be. a closed subgroup of I(M) which acts transitively on M, so that M may be identified with G/K, where K is the stabilizer of a point of M. The subgroup K is compact (Section 20.16, Problem 11). Show that every compact subgroup of G is conjugate in G to a subgroup of K (use (b)). 7. Let M be a complete connected Riemannian manifold, such that A(P,) 5 0 for all x E M and all planes P, c T,(M). Show that every element (other than the identity element) of the fundamental group rr,(M) has infinite order. (Observe that if fi is the Riemannian universal covering of M, the elements of rr,(M) may be identified with isometries of and use Problem 6(b).)

a,

8. Let M be a compact Riemannian manifold such that A(P,) 2 0 for all x planes P, c TJM).

E

M and all

(a) Show that if M is even-dimensional and nonorientable, then rr,(M) is of order 2. (b) Show that if M is odd-dimensional, then M is orientable. (Argue by contradiction, as in (20.22.6)) 9.

Let M be a compact submanifold of R".Suppose that for each x E M there exists in T,(M) a vector subspace E, of dimension rn such that A(P,) 5 0 for all planes P, c Ex. Show that n 2 rn dim(M). In particular, if A(P,) 5 0 for all x E M and all planes P, c T,(M), then n 2 2 dim(M). (Observe that there exists at least one point x E M such that I(x) . (h, 0 h,) # 0 for all h, st 0 in T,(M) (Section 20.14, Problem 3(c)). On the other hand, deduce from Problem 5(a) of Section 20.14 that

+

( 4 x ) . (hx 0 h,) I I(x) . (k, 0 kXNS II I(x) . (h, 0 k,)112 for all h,, k, E E,, and use the algebraic lemma of Section 20.14, Problem 5(c) to obtain a contradiction.)

22

MANIFOLDS WITH POSITIVE OR NEGATIVE SECTIONAL CURVATURE

419

10. Let M be a Riemannian manifold such that A(P,) 5 0 for all x E M and all planes P, c T,(M). Let U be a strictly geodesically convex open set in M. For each pair of points x, y in U and each r E [O, I], let u(x, y, t) be the point on the unique geodesic arc with endpoints x, y contained in U, such that d(x, u(x, y, I))= t * d(x, y). Show that for each u E U we have

c l W , x, t ) , 40,y . t ) ) 5 f * d(x, Y ) . (Let I = d(x, y). For 0 2 6 5 I, consider the functionf(f, 5) = u(u, u(x, y, [/f), t), and observe that t ~-+f(/, 5) is a geodesic. Use Section 20.16, Problem 7, to show that

M r , 01~ t iIf31,011.) 6

11. With the hypotheses and notation of Section 20.16, Problem 3, consider a basis of TJM) consisting of h and any 12 - 1 vectors k z , . . . , k, orthogonal to h, and let z, denote the Jacobi field along u : t HexpJth) such that ~ ~ (=0 0) and (V, . z,)(O) = k, (2 5 j 5 n). If we put f ( r ) = F(u(t)), then we have f ( t )=

ilZ,(t)

A ... A

ZAf)ll/Ct”-l

f o r O s t < r , w h e r e c = Ilk2 A . . - A k,ll (a) Suppose that the k, have been chosen so that when f = rl, the z,(t) and v’(t) form an orthonormal basis of Tu(,,(M). Show that we have

where the index form is calculated for liftings of u in [0, rl]. (Use (20.22.1.1).) (b) Suppose that the Ricci curvature of M satisfies the inequality Ric(h,) 2 ( n - l)aZllh,l12

for all h,

E T(M).

Show that if g is a piecewise-C’ function defined on [O,rl], such that

g(0) = 0 and g(rl) = 1, then

(Consider the parallel transport of zJ(rl) along u for 2 5 j 5 n and use (20.20.3.9)) (c) Deduce from (b) that the functionf(t)(at/sin at)”-’ is decreasing in [O,r[. (Choose the function g suitably in (b).) Deduce that the function S(x, t)(a/sin at)”-’ is decreasing in [O,r[. (d) Suppose that the sectional curvature of M satisfies A(P,) 5 b2 for all x E M and all planes P, c T,(M). Show that the functionf(r)(bt/sin bt)”-’ is increasing in [0, r [ . (Follow the proof of Rauch’s comparison theorem (Problem 3), by taking as comparison manifold a sphere of radius l/b.) Deduce that the function S(x, t)(b/sin bt)”-’ is increasing in [O,r[.

420


23. R I E M A N N I A N M A N I F O L D S O F C O N S T A N T CURVATURE

(20.23.1) Let M be a Riemannian manifold of dimension n 2 3. At a point x E M, the following conditions are equivalent:

(a) The sectional curvature A(P,) is the same for all planes P, c T,(M). (b) Relative to an orthonormal moving frame (e,, . .., en) defined in a neighborhood of x, the curvature forms satisfy the relations (20.23.1.1)

Rij(x) = A(x)ai(x)

A

oj(x)

(1 6 i, j

5 n),

where A(x) is a constant independent of i, j . When this condition is satisfied, the relations (20.23.1.1) are true for all orthonormal moving frames, with the same constant A(x).

First of all, it is immediately verified that the relations (20.23.1.l)signify 2

that the vector-valued 2-form i2'R)(x), which is a mapping of A T,(M) into End(T,(M)), identified with T,(M)* Q T,(M), is the linear mapping defined by Q(')(x) (h,

A

k,) = A(x)((G,

*

h,) 0 k,

- (G,

*

k,) 0 hJ,

where the linear mapping G, : T,(M) T,(M)* is that which is canonically defined by g(x) (20.8.3). Since this mapping does not depend on a choice of basis in T,(M), the last assertion of the proposition is proved. This also shows that (b) implies (a), since we may always choose a frame R in which the vectors ei(x) and ej(x) span a given plane P,. Let us show conversely that (a) implies (b). If h, = tiei(x), k, = qiei(x), we have --f

c

c

i

i

and this is equal to A(x)llh,

kJ2 = A(x)

A

C ((ti)z(qj)2+ (tj)2(qi)2- 2titjqiq9;

i
0, we have

(20.23.3.3) From this calculation we derive an explicit expression for the scalar product in T,.(T,(M)) defined by the metric tensor g , . It follows already from Gauss’s lemma (20.16.3.1)that ((ha, h,)l(h,, ha)),, = Ilh,lI:, and that ((h”, h,)\(h“, k,)),, = O if h, is orthogonal to k, (relative to g(a)); finally, from the preceding work, if k: and k: are orthogonal to h, (relative to g(a)), then we have

The following proposition is an immediate consequence of this :

(20.23.4) Let M, M‘ be two Riemannian mangolds of equal constant curvature, and g , g’ their respective metric tensors. Let a (resp. a‘) be a point of M (resp. MI), and let F be a linear isometry of T,(M) (endowed with the scalar product g(a)) onto T,,(M’) (endowed with g’(a’)). Then, if r is sufffcientlysmall, f = exp,, 0 F 0 (expo)-’ is the unique isometry of the open ball uith center a and radius r in M onto the open ball with center a‘ and radius r in M’ such that T,(f) = F.

23

RIEMANNIAN MANIFOLDS OF CONSTANT CURVATURE

423

If the metric tensor of a Riemannian manifold is multiplied by a constant

c

> 0, the Levi-Civita connection remains unchanged (this follows from

(20.7.6) and (20.9.4), since the group G is unaltered); the Riemann-Christoffel tensor is multiplied by c, and the sectional curvature A(PJ is multiplied by c-’. Hence in the study of manifolds of constant curvature we may restrict our considerations to the three cases A = I , A = - 1, and A = 0.

A simply-connected, complete Riemannian manifold M of constant curvature A = 0 (resp. A = I . A = -1) is isometric to R” (resp. S , , resp. hyperbolic space Y,, (20.11.7)). (20.23.5)

If A = 0 or A = - 1, and if we take M‘ = R” or M‘ = Y, , respectively, then for a E M and a’ E M‘ the mappings exp, and exp,, are respectively diffeomorphisms of T,(M) onto M and of T,,(M’) onto M’, by virtue of (2022.1) and (20.20.9); if F is an isometry of T,(M) onto T,,(M’), then it follows from (20.23.4) that exp,. 0 F (expo)-’ is an isometry of M onto M’. Consider now the case A = 1, Then the formulas (20.23.3.3) and (20.23.3.4) show that exp, is injective on B(0,; n), hence is a diffeomorphism onto B(a; n) because M is complete (20.20.Q and similarly for M’ = S, . Now let u’,b’ be two points of M’ such that d(a’, b’) < n, and let F be a linear isometry of TJM’) onto T,(M); then, by (20.23.4), the mapping 0

f

= exp,

0

F 0 (expo.)-’

is an isometry of B(a’; n) onto B(a; n). By hypothesis we have b’ E B(a’; n); put b =,f’(b’)and G = Tht(,f),so that G is a linear isometry of T,.(M’) onto T,(M); then the mapping g = exp, G 0 (exp,.)-’ is likewise an isometry of B(b’; n) onto B(b; n). Now the mappings f and g coincide in the intersection U = B(a’;n) n B(b’; n). For if C is the great circle arc of length .

Likewise, putting F =fi f2, we have F(g,, . . * 9") =fi(s1-. . gJfz(s1, * * * 9"). 5

. ?

9

This may be proved directly as in (A.21.4), or deduced from (A.21.4) by remarking that it is a particular case of this result iffi, f , are without constant terms; and that in general we may write f,= a + F , , f, = b F , , where a, b E K and F,, F , are without constant terms, and

+

fif,=ab+aF,+bF,

+F1F2.

REFERENCES

VOLUME I

[l] Ahlfors, L., “Complex Analysis,” McGraw-Hill, New York, 1953. [2] Bachmann, H., “Transfinite Zahlen” (Ergebnisse der Math., Neue Folge. Heft I). Springer, Berlin, 1955. [3] Bourbaki, N., “ ElCments de Mathtmatique,” Livre I, “Thtorie des ensembles” (Actual. Scient. Ind., Chaps. I, 11, No. 1212; Chap. 111, No. 1243). Herrnann, Paris, 1954-1956. [4] Bourbaki, N., “ ElCments de Mathtmatique,” Livre 11, “Algebre” (Actual Scient. Ind., Chap. 11, Nos. 1032, 1236, 3rd ed.). Herrnann, Paris, 1962. [5] Bourbaki, N., “ Eltnients de Mathtmatique,” Livre 111, “ Topologie gtntrale” (Actual. Scient. h d . , Chaps. I, 11, Nos. 858, 1142, 4th ed.; Chap. IX, No. 1045, 2nd ed.; Chap. X,No. 1084,2nd ed.). Hermann, Paris, 1958-1961. [6] Bourbaki, N., “ ElCments de Mathkrnatique,” Livre, V, “ Espaces vectoriels topologiques” (Actual. Scient. Ind., Chap. I, 11, No. 1189, 2nd ed.; Chaps. 111-V, No. 1229). Hermann, Paris, 1953-1955. [7] Cartan, H., Skminaire de 1’Ecole Normale Suptrieure, 1951-1952: “ Fonctions analytiques et faisceaux analytiques.” [8] Cartan, H., “Thtorie l%mentaire des Fonctions Analytiques.” Hermann, Paris, 1961. [9] Coddington, E., and Levinson, N., “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. [lo] Courant, R., and Hilbert, D., “ Methoden der mathematischen Physik,” Vol. I, 2nd ed. Springer, Berlin, 1931. [ l l ] Halmos, P., “Finite Dimensional Vector Spaces,” 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1958. [I21 Ince, E., “Ordinary Differential Equations,” Dover, New York, 1949. [13] Jacobson, N., “ Lectures in Abstract Algebra,” Vol. 11, “ Linear algebra.” Van Nostrand-Reinhold, Princeton, New Jersey, 1953. [ 141 Kamke, E., “ Differentialgleichungen reeller Funktionen.” Akad. Verlag, Leipzig, 1930. [I51 Kelley, J., “General Topology.” Van Nostrand-Reinhold, Princeton, New Jersey, 1955. [I61 Landau, E., “Foundations of Analysis.” Chelsea, New York, 1951. 432

REFERENCES

433

[I71 Springer, G., “Introduction to Riemann Surfaces.” Addison-Wesley, Reading, Massachusetts, 1957. [I81 Weil, A., “Introduction k I’Etude des Varietes Klhleriennes” (Actual. Scient. Ind., No. 1267). Hermann, Paris, 1958. [I91 Weyl, H., “Die ldee der Riemannschen Flache,” 3rd ed. Teubner, Stuttgart, 1955.

VOLUME II

[20] Akhiezer, N., “The Classical Moment Problem.” Oliver and Boyd, EdinburghLondon, 1965. [21] Arnold, V. and Avez, A., “Thtorie Ergodique des Systkmes Dynamiques.” GauthierVillars, Paris, 1967. [22] Bourbaki, N., ‘‘ Elements de Mathtmatique,” Livre VI, “Intkgration ” (Actual. Scient. Ind., Chap. I-IV, No. 1175, 2nd ed., Chap. V, No. 1244, 2nd ed., Chap. VLI-VIII, No. 1306). Hermann, Paris, 1963-67. [23] Bourbaki, N., *‘ Elements de Mathematique: Theories Spectrales” (Actual. Scient. Ind., Chap. I, 11, No. 1332). Hermann, Paris, 1967. [24] Dixmier, J., “ L e s Algkbres d’optrateurs dans I’Espace Hilbertien.” Gauthier-Villars, Paris, 1957. [25] Dixmier, J., “ Les C*-Alg6bres et leurs Representations.” Gauthier-Villars, Paris, 1964. [26] Dunford, N. and Schwartz, J.. “Linear Operators. Part 11: Spectral Theory.” Wiley (Interscience), New York, 1963. [27] Hadwiger, H., ‘‘ Vorlesungen iiber Inhalt, Oberflache und Isoperimetrie.” Springer, Berlin, 1957. (281 Halmos, P., “Lectures on Ergodic Theory.” Math. Soc. of Japan, 1956. [29] Hoffman, K., “Banach Spaces of Analytic Functions.” New York, 1962. [30] Jacobs, K., ‘‘ Neuere Methoden und Ergebnisse der Ergodentheorie” (Ergebnisse der Math., Neue Folge, Heft 29). Springer, Berlin, 1960. [31 J Kaczmarz, S. and Steinhaus, H., “Theorie der Orthogonalreihen.” New York, 1951. [32] Kato, T., ’‘Perturbation Theory for Linear Operators.” Springer, Berlin, 1966. [33] Montgomery, D. and Zippin, L., “Topological Transformation Groups.” Wiley (Interscience), New York, 1955. [34] Naimark, M., Normal Rings.” P. Nordhoff, Groningen, 1959. [35] Rickart, C., “General Theory of Banach Algebras.” Van Nostrand-Reinhold, New York, 1960. [36] Weil, A., “Adeles and Algebraic Groups.’’ The Institute for Advanced Study, Princeton, New Jersey, 1961. ‘I

VOLUME 111

[37] Abraham, R . , ‘‘ Foundations of Mechanics.” Benjamin, New York, 1967. [38] Cartan, H., Seminaire de I’Jkole Normale Supkrieure, 1949-50: ” Homotopie: espaces fibres.” [39] Chern, S. S., “Complex Manifolds” (Textos de matematica, No. 5). Univ. do Recife, Brazil, 1959.

434

REFERENCES

[40]Gelfand, I. M.and Shilov, G. E., “Les Distributions,” Vols. 1 and 2. Dunod, Paris, 1962. [41]Gunning, R., “ Lectures on Riemann Surfaces.” Princeton Univ. Press, Princeton, New Jei:exr, 1966. [42]Gunning, R., “Lectures on Vector Bundles over Riemann Surfaces.” Princeton Univ. Press, Princeton, New Jersey, 1967. [43] Hu, S. T., “Homotopy Theory.” Academic Press, New York, 1969. [44]Husemoller, D., “Fiber Bundles.” McGraw-Hill, New York, 1966. [45] Kobayashi, S., and Nomizu, K., “Foundations of Differential Geometry,” Vols. 1 and 2. Wiley (Interscience), New York, 1963 and 1969. [46] Lang, S.,“Introduction to Differentiable Manifolds.” Wiley (Interscience), New York, 1962. [47] Porteous, I. R., “ Topological Geometry.” Van Nostrand-Reinhold, Princeton, * New Jersey, 1969. [48] Schwartz, L., ‘‘ ThCorie des Distributions,” New ed. Hermann, Paris, 1966. [49] Steenrod, N., “The Topology of Fiber Bundles.” Princeton Univ. Press, Princeton, New Jersey, 1951. [50] Sternberg, S.,“Lectures on Differential Geometry.” Prentice-Hall, Englewood Cliffs. New Jersey, 1964.

VOLUME IV

[51] Abraham, R. and Robbin, J., “Transversal Mappings and Flows.” Benjamin, New

York. 1967. [52]Berger, M., “Lectures on Geodesics in Riemannian Geometry.” Tata Institute of Fundamental Research, Bombay, 1965. [53]Carathtodory, C., “Calculus of Variations and Partial Differential Equations of the First Order,” Vols. 1 and 2. Holden-Day, San Francisco, 1965. [54]Cartan, E., “Oeuvres Complktes,” Vols. lI to 311. Gauthier-Villars, Paris, 1952-1955. [55]Cartan, E., ‘‘LeGonssur la ThCorie des Espaces a Connexion Projective.” GauthierVillars, Paris, 1937. [56] Cartan, E., “La ThCorie des Groupes Finis et Continus et la GkomCtrie DiffCrentielle traittes par la Mtthode du Repkre Mobile.” Gauthier-Villars. Paris, 1937. [57]Cartan, E., “ LesSystkmesDifftrentiels Exttrieurs et leurs Applications GComCtriques.” Hermann, Paris, 1945. [58]Gelfand, I. and Fomin, S., “Calculus of Variations.” Prentice Hall, Englewood Cliffs, New Jersey, 1963. [59]Godbillon, C., “ GComCtrie Diffkrentielle et Mkanique Analytique.” Hermann, Paris, 1969. [60] Gromoll, D., Klingenberg, W. and Meyer, W.,“Riemannsche Geometrie im Grossen,” Lecture Notes in Mathematics No. 55. Springer, Berlin, 1968. [61] Guggenheimer, H., “Differential Geometry.” McGraw-Hill, New York, 1963. [62] Helgason, S., “ Differential Geometry and Symmetric Spaces.” Academic Press, New York, 1962. [63] Hermann, R., “ Differential Geometry and the Calculus of Variations.” Academic Press, New York, 1968. [64]Hochschild, G., “The Structure of Lie Groups.” Holden-Day, San Francisco, 1965.

REFERENCES

435

[65] Klotzler, R., “ Mehrdimensionale Variationsrechnung.” Birkhauser, Basle, 1970. [66] Loos, O., “Symmetric Spaces,” Vols. 1 and 2. Benjamin, New York, 1969. [67] Milnor, J., “ Morse Theory,” Princeton University Press, Princeton, New Jersey, 1963. (681 Morrey, C., “ Multiple Integrals in the Calculus of Variations.” Springer, Berlin, 1966. [69] Reeb, G., “Sur les VariCtCs FeuilletCes.” Hermann, Paris, 1952. [70] Rund, H., “The Differential Geometry of Finsler Spaces.” Springer, Berlin, 1959. [71] Schirokow, P. and Schirokow, A., “ Affine Differentialgeometrie.” Teubner, Leipzig, 1962. [72] Serre, J. P., “ Lie Algebras and Lie Groups.” Benjamin, New York, 1965. [73] Wolf, J., “ Spaces of Constant Curvature.” McGraw-Hill, New York, 1967.


INDEX

In the following index the first reference number refers to the chapter in which the subject may be found and the second to the section within that chapter.

A

Absolute integral invariant of a vector field: 18.2, prob. 13 Adjoint of a differential r-form: 20.8 Adjoint representation: 19.2, 19.11 M n e connection: 20.6, prob. 19 M n e group of Rn: 20.1, prob. 1 Almost-complex structure : 20.7 Almost-Hamiltonian structure: 20.7 Analytic Riemannian manifold: 20.8 Angle between two tangent vectors at a point on a Riemannian manifold: 20.8 Arc of trajectory with origin a and endpoint b : 18.4 Area : 20.8 Ascending central series of a Lie algebra: 19.12, prob. 3 Asymptotic line: 20.13

B

Beez’s theorem: 20.14 Bianchi‘s identities: 20.3, 20.6, 20.9, prob. 13

Bieberbach’s theorem on crystallographic groups: 19.14, prob. 16 Binormal to a curve: 20.13

Bochner’s theorem: 19.1, prob. 6 Bochner-Yano theorem: 20.10, prob. 8 Bonnesen’s inequality: 20.13, prob. 6 Bonnet’s theorem: 20.23, prob. 1 Bundle of affine frames of a vector bundle: 20.1, prob. 1 Bundle of frames of a vector bundle: 20.1 Bundle of frames of order k of a differential manifold: 20.1, prob. 3 Bundle of tangent frames of a differential manifold: 20.1

C

Campbell-Hausdorff formula: 19.6, prob. 5 Canonical and connection 1-forms relative to a moving frame: 20.6 Canonical Cartan connection in G X ” G : 20.5, prob. 3 Canonical chart of a Lie group: 19.8 Canonical coordinates of the first kind: 19.8 Canonical coordinates of the second kind: 19.9

Canonical differential 1-form on a Lie group: 19.16 Canonical flat G-structure on R”: 20.7, prob. 7 437

438

INDEX

Canonical 1-form on the bundle of frames of a manifold: 20.6 Canonical lifting of a vector field on M to a vector field on R(M): 20.6, prob. 5 Canonical lifting of a vector field on M to a vector field on T(M): 18.6, prob. 3 Canonical linear connection on a symmetric homogeneous space: 20.7 Canonical linear connection on R" : 20.1 1, prob. 2 Canonical principal connection on a symmetric homogeneous space: 20.4 Canonical projective structure on P,,(R): 20.9, prob. 14 Canonical Riemannian structure on S., R": 20.11

Canonical vector fields on R2: 18.7 Canonical volume form on an oriented Riemannian manifold: 20.8 Canonical welding of M into R(M) xCLlnsR)R": 20.6, prob. 19 CarathCodory's inaccessibility theorem: 18.16, prob. 18 Cartan connection: 20.5, prob. 2 Cartan-Kahler theorem: 18.13 Cartan-Kuranishi theorem: 18.13, prob. 5 Cartan quadruple: 18.13, prob. 3 Cartan's theorem on closed subgroups of Lie groups: 19.10 Cartan's theorem on manifolds of constant curvature: 20.23 Cauchy characteristic, Cauchy characteristic vector: 18.16 Cauchy-Kowalewska theorem: 18.12 Cauchy's method of majorants: 18.12 Cauchy's problem: 18.12 Characteristic submanifold: 18.15 Characteristic vector field of a first-order partial differential equation: 18.17 Chern-Lashof theorem: 20.12, prob. 4 Circular cylinder : 20.1 1 Clairaut's equation : 18.15 Co-compact subgroup of the group of isometries of R": 19.14, prob. 16 Cohn-Vossen's theorem: 20.14, prob. 9 Commutator group: 19.12 Complete connection: 20.6, prob. 8 Complete C" mapping from one Riemannian manifold to another: 20.8, prob. 14

Complete integral of a first-order partial differential equation: 18.17, prob. 2 Complete Riemannian manifold : 20.18 Complete system of first-order partial differential equations: 18.17, prob. 1 Complete vector field: 19.9, prob. 9 Completely integrable Pfaffian system: 18.14 Complex-analytic Riemannian manifold: 20.8

Complex quadric: 20.11, prob. 5 Complexification ofa real Lie algebra: 19.17 Conformal mapping: 20.8 Conjugate G-structures: 20.7 Conjugate point (of a point on a geodesic): 18.7

Connection 1-form of a principal connection: 20.2 Constant term of a formal power series: A.21.2

Convex open set (relative to a spray): 18.5 Covariant exterior differential in a principal bundle: 20.3 Critical point of a vector field: 18.2, prob. 5 Crofton's formulas: 20.13, probs. 5 and 7 Crystallographic subgroup of the group of isometries of R": 19.14, prob. 16 Curvature 2-form of a principal connection: 20.3

Curvature 2-form relative to a moving frame: 20.6 Curvatures of a curve: 20.13 Curve of zero length: 20.13, prob. 8 Curve oriented from a to b : 20.13 Curvilinear coordinate: 20.13 Cut locus, cut-point: 20.20, prob. 7 Cm-connectedsubset of a manifold: 19.11, prob. 8 D

Darboux's theorem: 18.16, prob. 3 Degree of a term in a formal power series: A.21.2

Demiquadric: 20.11 Derivatives of a Pfaffian system: 18.16. prob. 8 Derived group of a group: 19.12 Derived homomorphism (of a homomorphism of Lie groups): 19.3

INDEX

Derived ideal of a Lie algebra: 19.12 Derived length of a differential ideal: 18.16, prob. 8 Descending central series of a connected Lie group: 19.16, prob. 6 Descending central series of a Lie algebra: 19.12, prob. 3 Development of a path in a homogeneous space: 20.5, prob. 2 Development of a path on a fiber: 20.5, prob. 1 Differential ideal: 18.9 Differential of a mapping into a Lie group: 19.15 Differential system: 18.9 Differential system in involution relative to p coordinates: 18.10, prob. 2 Distinguished integral field: 18.13, prob. 2 Divergence of a vector field: 20.10, prob. 3 Domain of the flow of a vector field: 18.2 Dynamical system : 18.1 E

Ehresmann’s theorem: 20.8, prob. 4 Einstein manifold: 20.10, prob. 2 Enveloping algebra of the Lie algebra of a Lie group: 19.6 Equilinear actions of a Lie group on a vector bundle and its base: 19.1 Equivariant action of a Lie group G on a bundle over G : 19.2 Euler’s formula: 20.14 Euler’s identity: 18.6, prob. 2 Exponential mapping defined by a spray: 18.4 Exponential mapping for a Lie group: 19.8 F

Fary-Milnor theorem: 20.13, prob. 3 Fenchel’s inequality: 20.13, prob. 3 Field of p-directions: 18.8 Finite type (G-structure): 20.7, prob. 10 Finite type (subspace of Hom(E, F)): 20.7, prob. 11 First fundamental form on a submanifold of a Riemannian manifold : 20.12

439

First integral of a vector field (or of a firstorder differential equation): 18.2, prob. 12 First-order differential equation on a manifold: 18.1 First-order prolongation of a G-structure: 20.7, prob. 6 First-order structure function: 20.7, prob. 5 Flat connection: 20.4 Flat G-structure: 20.7, prob. 7 Flat projective structure: 20.9, prob. 14 Flat torus: 20.1 1 Flow of a vector field: 18.2 Focal point, focal vector at a point of a submanifold of R”: 20.12, prob. 1 Formal power series in n variables: A.21.2 Frenet frame: 20.13 Frenet’s formulas: 20.13

G Gaussian curvature of a surface: 20.10 Gauss’s formula: 20.21 Gauss’s lemma: 20.16 Gauss’s theorema egregium: 20.14 G-connection associated with a G-structure: 20.7 Geodesic, geodesic arc, geodesic spray, geodesic trajectory of a linear connection: 18.6 Geodesic at a point (submanifold): 20.13 Geodesic in a pseudo-Riemannian manifold: 20.9 Geodesic path: 20.16 Geodesically convex Riemannian manifold : 20.17 Gleason-Yamabe theorem: 19.8, prob. 9 Gradient of a function on a pseudoRiemannian manifold: 20.8 G-structure: 20.7

H

Hadamard-Cartan theorem: 20.22 Hamiltonian structure: 20.7, prob. 13 Heinz’s inequality: 20.14, prob. 2 Hessian of a vector field: 18.2, prob. 5

440

INDEX

Homogeneous function on a vector bundle: 18.6, prob. 2 Homogeneous part of degree m of a formal power series: A.21.2 Homogeneous vector field on a vector bundle: 18.6, prob. 2 Homometry: 20.9, prob. 5 Hopf-Rinow theorem: 20.18 Horizontal component of a tangent vector, relative to a principal connection: 20.2 Horizontal lifting of a path relative to a principal connection: 20.2, prob. 3 Horizontal lifting of a tangent vector, of a vector field, relative to a principal connection: 20.2 Horizontal q-form on a principal bundle 20.2

Horizontal vector, relative to a principal connection: 20.2, 20.5, prob. I Hyperbolic n-space 20.11 ypersurface of revolution : 20.14, prob. 7

I

Image of a left-invariant field of pointdistributions under a homomorphism of Lie groups: 19.3 Immersed connected Lie group: 19.7 Imprimitive action of a Lie group on a manifold: 19.3, prob. 5 Index form on a space of liftings: 20.20 Infinitesimal action of a Lie algebra on a differential manifold: 19.3, prob. 2 Infinitesimal algebra of a Lie group: 19.3, 19.17

Infinitesimal automorphism of a principal connection: 20.6, prob. 6 Infinitesimal automorphism of a projective connection: 20.9, prob. 14 lnfinitesimal isometry of a pseudo-Riemannian manifold: 20.9, prob. 7 Integral curvature of a submanifold of RN: 20.12, prob. 4 Integral curve of a vector field : 18.1 Integral element of dimension p of a differential system: 18.10 Integral manifold of a differential system: 18.9

Integral manifold of a field of p-directions: 18.8

Integral manifold of a Pfaffian system: 18.8 Invariant differential form (with respect to a vector field): 18.2, prob. 12 Invariant linear connection on a Lie group: 19.8

Invariant under parallelism (connection): 20.6, prob. 18 Inverse image of a principal connection: 20.2, prob. 2 Isochronous differential equation: 18.4 Isochronous vector field: 18.4 Isometry of one pseudo-Riemannian manifold onto another: 20.8 Isomorphism of G-structures: 20.7 Isomorphism of principal connections: 20.6, prob. 4 1

Jacobi field along a geodesic: 18.7 Jordan’s theorem on finite subgroups of U(n): 19.9, prob. 3 K

Killing field on a manifold acted on by a Lie group: 19.3 Knot: 20.13, prob. 3 k-ply transitive action of a Lie group on a manifold: 19.3, prob. 1 I 1

Left differential of a function with values in a Lie group: 19. I5 Left-equivariant action of a Lie group G on a bundle over G: 19.2 Left-invariant field of point-distributions on a Lie group: 19.2 Length of a path: 20.16 Length of a tangent vector to a Riemannian manifold: 20.8 Levi-Civita connection: 20.9 Lewy’s example: 18.11, prob. Lie algebra of a Lie group: 19.3, 19.17

INDEX Lie bracket of two tangent vectors in TJG): 19.3 Lie derivative of a lifting of an integral curve: 18.7, prob. 1 Lie series: 18.12, prob. 4 Line of curvature on a hypersurface: 20.14 Liouville field: 18.6, prob. 2 Liouville’s theorem on conformal mappings : 20.19, prob. 1 I Local expression of a differential equation on a manifold: 18.1 Local expression of the multiplication law in a Lie group: 19.5 Local isometry of one pseudo-Riemannian manifold into another: 20.8 Local isomorphism of one principal connection into another: 20.6, prob. 9 Locally isometric pseudo-Riemannian manifolds: 20.8 Locally symmetric connection: 20.11, prob. -

7

Logarithmic differential of a function with values in an algebra: 19.15 Lowering of indices: 20.8

441

N

Nilpotent connected Lie group: 19,14, prob. 6 Nilpotent Lie algebra: 19.12, prob. 3 Nondegenerate critical point of a vector field: 18.2, prob. 5 Nonisotropic submanifold of a pseudoRiemannian manifold : 20.12 Normal bundle of a submanifold in a Riemannian manifold: 20.12 Norma, component of a tangent vector: 20.12 Normal coordinates: 20.6, probs. 15 and 16 Normal coordinates at a point of a Riemannian manifold: 20. 16 Normal curvature of a curve on a hypersurface : 20, 14 Normal curvature vector of a curve on a submanifold: 20.13 Normal projective connection: 20.9, prob. 14

0

M Malcev’s theorem 19.14, prob. 15 Maurer-Cartan equation: 19.16, prob. 12, and 20.3 Maximal integral curve with origin X O : 18.2 Maximal integral manifold of a completely integrable Pfafian system: 18.14 Maximal solution of a second-order differential equation: 18.3 Maximal trajectory of an isochronous equation (or of a spray): 18.4 Mean curvature of a hypersurface: 20.14 Metric tensor on a pseudo-Riemannian manifold: 20.8 Meusnier’s theorem: 20.13 Minkowski’s formulas: 20.14, prob. 9 Moving frame: 20.6 Moving G-frame: 20.7 Mukhopadhyaya’s theorem: 20.13, prob. 4 Multidegree of a term in a formal power series: A.21.2 Myers’s theorem: 20.22

One-parameter family of curves, geodesics: 18.7 One-parameter group of diffeomorphisms defined by a vector field: 18.2 One-parameter subgroup of a Lie group: 19.8 Orientation of a curve defined by a parametrization: 20.13 Oriented unit normal vector field of a hypersurface: 20.14 Osculating planes to a curve: 20.13 Outward normal to a closed set in R” at a frontier point: 18.2, prob. 1

P

Palais’s theorem: 19.10, prob. 5 Parallel displacement of a fiber: 20.2, prob. 3

442

INDEX

Parallel transport along a path, relative to a linear connection: 18.6 Parallelism, parallelizable manifold: 20.7 Parametrization of a curve by arc-length: 20.13 Partial right action of a Lie group on a manifold: 19.3 prob. 1 p-direction: 18.8 Periodic geodesic: 20.19 Pfaffian class of a differential I-form: 18.16, prob. 3 Pfaffian system: 18.8 Piecewise-C' path: 20.16 PoincarC's formula : 20.13, prob. 6 Primitive action of a Lie group on a manifold: 19.3, prob. 5 Primitive at a point: 19.3, prob. 6 Principal affine curvatures at a point of a hypersurface in R": 20.14, prob. 12 Principal centers of curvature at a point of a hypersurface : 20.14 Principal connection: 20.2 Principal curvatures at a point of a hypersurface: 20.14 Principal curvatures in a direction normal to a submanifold of RN:20.12, prob. 1 Principal directions at a point of a hypersurface: 20.14 Principal normal to a curve: 20.13 Principal radii of curvature at a point of a hypersurface: 20.14 Product of principal connections: 20.2, prob. 2 Product of pseudo-Riemannian manifolds: 20.8, prob. 1 Projective connection: 20.9, prob. 14 Projective structure: 20.7, prob. 16 Prolongation of a differential system: 18.13, prob. 3 Prolongation of a G-structure: 20.7, prob. 6 Proper topology of an immersed Lie group: 19.7 Pseudo-arc-length: 20.13, prob. 8 Pseudo-normal at a point to a hypersurface: 20.14, prob. 1 1 Pseudo-Riemannian covering: 20.8 Pseudo-Riemannian manifold: 20.8 Pseudo-Riemannian metric tensor: 20.8 Pseudo-Riemannian structure: 20.7

R Radii of curvature of a curve: 20.13 Raising of indices: 20.8 Rank of a Pfaffian system: 18.16, prob. 9 Rauch's comparison theorem: 20.22, prob. 3 r-dimensional area: 20.8 Real form of a complex Lie algebra: 19.17 Rectifying plane to a curve: 20.13 Rectilinear path: 20.16 Reeb foliation: 18.14, prob. 1 1 Reeb's theorem: 20.8, prob. 6 Regular integral element: 18.10 Relative integral invariant of a vector field: 18.2, prob. 14 Restriction of a principal bundle to a subgroup of the structure group: 20.7 and 20.7, prob. 1 Ricci curvature, Ricci tensor: 20.10 Riemann-Christoffel tensor: 20.10 Riemannian covering of a Riemannian manifold: 20.8 Riemannian curvature: 20.21 Riemannian distance: 20.16 Riemannian manifold: 20.8 Riemannian manifold of constant curvature: 20.23 Riemannian metric tensor: 20.8 Riemannian structure: 20.7 Riemannian symmetric space: 20.1 1, and 20.11. prob. 8 Riemannian volume: 20.8 Right differential of a function with values in a Lie group: 19.15 Right-equivariant action of a Lie group G on a bundle over G : 19.2 5

Scalar curvature of a Riemannian manifold: 20.10 Scalar product of two vectors in T,(M): 20.8 Schur's theorem: 20.23 Second fundamental form on a hypersurface: 20.14 Second fundamental formb) on a submanifold : 20.I2

INDEX

Second-order differential equation on a manifold: 18.3 Second-order G-structure: 20.7, prob. 15 Second-order structure function: 20.7, prob. 6 Sectional curvature of a Riemannian manifold: 20.21 Semidirect product of two groups: 19.14 Semidirect product of two Lie subalgebras: 19.14 Signature of a pseudo-Riemannian structure: 20.7 Singular integral element: 18.10 Singular integral manifold of a differential system: 18.15 Singular integral of a first-order differential equation: 18.15 Singular point (of the exponential mapping) : 18.7 Singular solution of a first-order differential equation: 18.15 Solvable group, solvable Lie algebra: 19.12 Spray: 18.4 Strict triangle inequality: 20.18 Strictly convex curve in RZ:20.13, prob. 4 Strictly convex hypersurface: 20.14, prob. 3 Strictly geodesically convex open set in a Riemannian manifold: 20.17 Structure equation of a canonical form: 20.6 Structure equation of a connection form: 20.3 Structure equations, relative to a moving frame: 20.6 Structure functions of a G-structure: 20.7, prob. 5 Symmetric connection: 20.1 1, prob. 8 Symmetric homogeneous space: 20.4 Symmetric pair: 20.4 Symmetry with center x, relative to a linear connection: 20.1 I , prob. 7 Synge’s lemma : 20.2 1 Synge’s theorem: 20.22 System of first-order partial differential equations: 18.8 T

Tangent p-direction to a manifold: 18.8 Tangent vector field to a submanifold: 18.2

443

Tangent vector to a closed set in R”:18.2, prob. 1 Tangential component of a tangent vector: 20.12 Tangential image of a hypersurface: 20.14, prob. 4 Taylor expansion: 19.5 Taylor polynomial of degree $rn of a function: 19.5 Taylor’s formula in a Lie group: 19.5 Tensor parallel transport: 18.6 Tensor product of algebras over a field: A.20.4 Tensor product of vector spaces over a field: A.20.1 Torsion of a curve: 20.13 Torsion 2-form of a principal connection: 20.6 Torsion 2-form relative to a moving frame: 20.6 Torsion 2-form of a Cartan connection: 20.9, prob. 13 Total curvature in a direction normal to a submanifold of RN:20.12, prob. 4 Total curvature of a hypersurface: 20.14 Total parallelism on a manifold: 20.7 Totally geodesic submanifold: 20.13 Trajectories of a second-order differential equation: 18.4 Transitive at a point: 19.3, prob. 6 Transport of a differential operator P: 19.1 Transversally isolated point of an integral curve: 18.2, prob. 7 Trivial connection: 20.4

U

Umbilic: 20.12 Uniformly complete C“ mapping between Riemannian manifolds: 20.18, prob. 14 Unit normals to a curve: 20.13 Unit tangent vector to an oriented curve: 20.13 Unit vector field on R: 18.1 Unit vector field on a Riemannian manifold: 20.8

444 INDEX V

Vector field definining a second-order differential equation: 18.3 Velocity vector: 18.1 Vertical q-form on a principal bundle: 20.2 Volume of an n-vector to a Riemannian manifold: 20.8 W

Weakly reductive homogeneous space: 20.4

Welding: 20.1, prob. 2 Weyl projective curvature 2-form on a bundle of frames: 20.6 Weyl projective curvature tensor of a projective structure: 20.9, prob. 14 Weyl's theorem: 20.22

x, Y, z X-invariant lifting of an integral curve of X: 18.7, prob. 1 Yano's formula: 20.10, prob. 3 Zoll's surface: 20.20, prob. 6

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors

Samuel Ellenberg and Hyman Bars

Columbia University, New York

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