Lie Equations, Vol. I: General Theory. (AM-73) 9781400881734

Table of contents :
TABLE OF CONTENTS
FOREWORD
GLOSSARY OF SYMBOLS
INTRODUCTION
A. INTEGRABILITY OF LIE STRUCTURES
EXAMPLE (1): The Frobenius theorem
EXAMPLE (2): Fields of p-frames, parallelism
EXAMPLE (3): Riemannian structures
EXAMPLE (4): Linear connections
EXAMPLE (5): Canonical differential forms
EXAMPLE (6): Almost-product structure
EXAMPLE (7): Almost-complex structure
EXAMPLE (8): The complex Frobenius theorem
Example (9): G-structures
EXAMPLE (10): Structures defined by fields of endomorphisms
B. DEFORMATION THEORY OF LIE STRUCTURES
THEOREM 1 (General mechanism for deformation)
EXAMPLE (1): Volume structure
EXAMPLE (2): Symplectic structure
EXAMPLE (3): ΓQ-structure
EXAMPLE (4): Multifoliate structure
EXAMPLE (5): Cohen-Macaulay structure
EXAMPLE (6): Complex analytic structure
CHAPTER I – JET SHEAVES AND DIFFERENTIAL EQUATIONS
1. Notation
2. Jet bundles
3. The prolongation space X(k)
4. Prolongation of sheaves
5. Coordinates
6. The first linear complex
7. The second linear complex
8. Homogeneous linear partial differential equations
9. Linear differential operators
CHAPTER II – LINEAR LIE EQUATIONS
10. Brackets in Jk^T
11. Coordinates
12. Lie equations
CHAPTER III – DERIVATIONS AND BRACKETS
13. Derivations of scalar differential forms
14. The bracket [,]
15. The adjoint representation
16. The fundamental identities
CHAPTER IV – NON-LINEAR COMPLEXES
17. Non-linear jet sheaves
18. Coordinates
19. The Lie algebra of the groupoid ΓkX
20. Actions of ΓkX
21. The first non-linear complex
22. The Od representation and the operator D
23. Invariance of [,] and the second non-linear complex
24. Partial exactness of the second non-linear complex
25. The third non-linear complex
CHAPTER V – DERIVATIONS OF JET FORMS
26. The sheaves
27. The derivations
28. The D-complex
29. The restricted D-complex
30. The D-complex and the twisting of d
31. The projective limits
APPENDIX – LIE GROUPOIDS
32. Definitions
33. The Exponential map
34. Prolongation of Lie groupoids
35. Examples
REFERENCES
INDEX

Citation preview

Annals of Mathematics Studies Number 73

LIE EQUATIONS VOLUME I: GENERAL THEORY BY

ANTONIO KUMPERA AND DONALD SPENCER

P R IN C E T O N

U N IV E R S IT Y

PRESS

AND U N IV E R S IT Y

OF

TOKYO

PRESS

P R IN C E T O N , NEW JERSEY 1972

Copyright © 1972, by Princeton University Press A L L RIGHTS RESERVED. NO PART OF THIS BOOK M A Y BE REPRODUCED IN A N Y FO R M OR BY A N Y ELECTRONIC OR M E C H A N IC A L

M EANS

IN C LU D IN G

IN F O R M A T IO N

AND RETRIEVAL SYSTEM S W IT H O U T PER M ISSIO N

STORAGE

IN W R IT IN G

F R O M THE PUBLISH ER, EX CEPT BY A REVIEW ER W H O M A Y Q UO TE BRIEF PASSAGES IN A R EVIEW .

LC C ard: 77-39055 ISBN: 0-691-08111-5 AMS 1970: 58H05

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

FOREWORD

In h is p a p e rs [1 3 (a ), ( b ) ], Spencer d ev e lo p e d a gen eral m echanism for the lo c a l deform ation o f structures on m an ifo ld s d efin ed by tran sitive con­ tinuous p se u d og ro u p s.

A new versio n o f this theory, b a s e d on the d iffe r­

en tial c a lc u lu s in the a n a ly tic s p a c e s o f G rothendieck, h a s been given by M alg ran g e [9 (c )] in h is proof o f the in tegrab ility (e x is t e n c e o f lo c a l co o rd i­ n a te s) o f alm o st-stru ctu res d efin ed by e llip t ic tra n sitiv e continuous p seu d ogro u p s (o r e llip t ic L i e eq u a tio n s), under a certain in tegrability condition. T h e authors here re d eve lo p the theory by two d ifferen t ap pro ach es. T h e startin g point o f one approach is b a s e d on the id e a o f B . M algran ge in w hich the je t s h e a v e s and the operators on them are d efin ed by facto r­ ing s h e a v e s and operators d efin ed on the product of the m anifold with it­ s e lf modulo p o w e rs o f the id e a l d efin in g the d ia g o n a l.

T h is id e a is e x ­

p lo ited in a com p letely sy stem atic w ay , w hich b rin gs com putational sim p licity.

T h e seco n d approach is d ev e lo p e d in the context of d e riv a ­

tions w here the theory fin d s its most natural e x p re s s io n .

T h e e q u iv a ­

le n c e o f the two a p p ro ach e s is shown. In C hapter I w e d efin e the vario u s je t s h e a v e s and the lin e a r com­ p le x e s and g iv e a b r ie f outlin e o f formal in te g ra b ility for p artial d iffe re n ­ tia l eq u atio n s w hich is e s s e n t ia lly borrow ed from M alg ran g e [9 (a ), (b )]. T h e theory is d e s c rib e d on the s h e a f le v e l and the only e s s e n t ia l in nova­ tion (b etter re tro g re s s io n ) is a d irect p ro o f o f the e x a c tn e s s o f the first lin ea r com plex without u s in g

S.

We sh o w that

D

is e q u iv alen t to the

s h e a f map (on the le v e l o f germ s) a s s o c ia t e d to the bun dle map Jk ( A p T * ) -> J k _ i ( A p + 1 T * )

d efin ed by

j k&>(x) h> j k _

1 do)(x).

T h e e x a c t-

n e s s o f the e n su in g vecto r bun dle com plex is then a fib r e w is e problem

v

FOREWORD

vi

w hich tran scrib ed in co o rd in a tes is sim ply the P o in c a r e lemma for exterior d iffe ren tial forms with p olynom ial c o e ffic ie n ts . In C hapter II w e d e fin e lin ea r L i e equations. [

,

1

in

J^T

or

J^T

T h e com pensated bracket

is d efin ed fo llo w in g M alg ran g e [9 (c )] and w e

prove that L i e eq u a tio n s are invariant by prolongation. In C hapter III w e introduce the com pensated bracket

[

,

1

for vector

v alu ed d iffe re n tia l forms (je t form s) and w e p rove the main id en tities which relate this bracket with the lin e a r operator

D.

T h e defin ition o f

[

,

1

fo llo w s a pattern sim ila r to the o rig in a l defin ition given by Spencer in [1 3 (a )]. In C hapter I V w e d e s c rib e the n o n -lin ear co m p lexes which are fin ite forms o f (the in itia l portions o f ) the lin e a r o n es and w e prove the main n o n -lin ear id en tities.

T h e n o n -lin ear operator

S p en cer’s id ea [ 1 3 ( c ) ] o f twisting

d

B

is d efin ed fo llo w in g

w hich, in the p resen t context, ap ­

p e a rs a s the tw istin g o f the vector 1-form

X

by the representation

&d.

In C hapter V the theory is transform ed into the context o f d erivation s o f s h e a v e s o f je t forms, a technique e x te n s iv e ly u s e d in the o rigin a l papers [1 3 (a ), (b ) ].

Many ad hoc d efin itio n s and constructions o f e a rlie r

chapters ap pear n atu rally in this setting.

F o r exam ple, the tw istin g o f

X

ap p e ars in th is context a s the tw istin g o f the exterior d iffe re n tia l operator d.

T h is chapter is , in fact, the b e s t e x p re s s io n o f our work and p ro vid es

ad dition al insight. T h e ap pend ix g iv e s an introduction to L i e groupoids.

We hope this

w ill b en e fit the re ad e rs who are acq u ain ted with the work o f Ehresm ann [2 ] and the p o in tw is e je t theory.

Some con struction s o f C hapter IV are

presented in this context. L e t us fin a lly make a few comments on the a n a lo g ie s and d iffe re n c e s betw een M a lg ra n g e ’s approach [ 9 ( c ) ] and the p resen t one.

T o start w ith,

the lin e a r co m p lexes a s w e ll a s the com pensated bracket in vio u sly the sam e. sated bracket in

J^T

are o b ­

T h e d iffe re n c e b e g in s in the d efin itio n o f the com pen­ AT*

( A ( J ° T ) * (J ^ T + J ^ T )

two ap p ro ach e s are en tirely d ifferent.

in M alg ran g e).

It is p o s s ib le to re late the two

The

FOREWORD

v ii

p re v io u sly m entioned s p a c e s by a “ tra n s p o s itio n ” procedure w hich, rough­ ly s p e a k in g , c o n s is t s in tran sp o sin g h orizon tal forms into v e rtic a l ones on

X

. H o w ev er, one b ra ck et d o e s not tran sp o se into the other.

our representation

ad

operation, a certain L i e d e riv a tiv e , in M alg ran g e, s in c e G ^ -lin e a r on

In fact,

d o e s not a gree with the co rresp o n d in g tran sp osed

A T ^ ^ ^ T

but only on

A T ^ q ^ T .

point the p re c is e d iffe re n c e in the two d efin itio n s.

ad

is not left

It is not e a s y to p in ­ Our approach som ehow

d is c a rd s the high er order terms, in the co varian t part, which are present in M a lg ra n g e ’s treatment due to a d iffe re n tia l

d'.

H o w ev er, the two bra ck ­

ets a g ree, by tran sp o sitio n , on a s u ffic ie n tly la rg e domain, nam ely (A T *< 8> g 5 k T )

0

^ k *

T h is accou nts for the fact that the two ap pro ach es

w ill e v en tu a lly meet at the Buttin form ula (2 2 .1 2 ), h en ce the no n -lin ear operators

3)

and

is the fo llo w in g :

w ill b e the sam e.

A nother s u b s ta n tia l d iffe ren c e

M alg ran g e claim s that h is bracket is in variant by

F

_1

w hich m eans, after tran sp ositio n , that our bracket should be in variant by (JdF.

T h is h o w ev er is not the c a s e .

In our approach, the in varian ce only

h olds for elem ents in a certain s u b s h e a f w hich, h o w ever, is la rg e enough to meet a ll the requirem ents o f the theoryv F in a lly , w e do not claim to g iv e com plete re fe re n c e s for a ll the known re su lts in the text.

T h e reader can find many re fe re n c e s and an e x te n s iv e

b ib lio grap h y in S p e n c e r’s survey a rtic le [1 3 (d )]. T h e authors are greatly indebted to B. M alg ran g e w h o s e id e a s p rovide the foundation for part o f this monograph.

T h e second-nam ed author is

e s p e c ia lly in debted to him for many s u g g e s tio n s com m unicated in a co rre­ spond en ce ex ten d in g o v er two y e a rs; in fact, the startin g point of this work w a s a m anuscript w hich w a s o rig in a lly intended a s a c o llab o ratio n o f B . M alg ran g e and the second -nam ed author.

T h e seco n d author is a ls o

grateful to C. Buttin for many h e lp fu l d is c u s s io n s and c la rific a tio n s .

( * ) During the course of proof-reading this manuscript, a defin itive version of M algran ge’ s work [ 9 ( c ) ] appeared, namely “ E qu ations de L i e , ” U n iv e rs ity de Grenoble (to appear in the Journal of Differential Geometry), which is liste d as [9 (d )] in the references.

v iii

FOREWORD

T h e a u t h o r s * '* a l s o w is h to e x p re s s their ack n ow ledgem en ts to C. M. de B arro s and T . K lem o la for many h elp fu l d is c u s s io n s on the s u b ­ je c t o f this monograph.

( * * ) The first-nam ed author w as partially supported by the N a tio n a l R esearch C ou ncil of C anada, grant A-5604, and the second-nam ed author by N a tio n a l S cience Foundation Grant G P-31917X.

GLOSSARY OF SYMBOLS E, E, 6

, 6(a), G, Gx

49

F

49

a’ I ak+1 e A, i 3

50

A

52

J kE , J k E

51

JkX ,J k X

136

I

k+1,4

4 E- §kE

83 84

53

56, 91

5kE

57

4 E = 4k» e 6

59

4 E

60

g ®©4

g ® e 4 ® © 3: < T « < ¥ % e p

n kx , n k x

4 T’ 4 T< 4 t L

t

% 4+e

136, 141

r kX

137, 141, 144, 156 57

F k’ F k C °°x 2

140

e °°x

136 2

138

dut X

136

S u ty X 2

139

61 75 x

86

202