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

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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