10.

V. Guillemin and S. Sternber g

The Lie bracke t defines a bilinea r ma p of J

k

(T(M)) X J

k

(T(M))

—— J, (T(M)) . It therefor e induces a ma p (which we continue to denote

by [ , ]) of

T

p

(D

k

(0;M) ) x T

p

(D

k

(0;M) ) T

q

(D

k

_

1

(0;M))

wher e q = TT .(p) . This ma p can be explicitly describe d as follows: Let

4 , T\ be element s of T (D, (0;M)) . Let X and Y be vector fields on M

P

K

(defined nea r x = TT (p)) with X, (p) = £ a n d Y,(p) = fj . Then

[*.!)] = [X

k - 1

, Yk-1](q) .

A consequence of (2.4) is that

(2-5) ( % ) * ( V = (P*X)k .

In p a r t i c u l a r , evaluating (2.5) at the point p shows that

(2-6) ^ (*(Jk(X)(x))) = «[J

k + 1

((p)(x)^J

k

(X)(x)) .

wher e a = a. on the left of (2. 6) and a -OL ^ \ on the right .

We have see n that a local diffeomorphism, p , of M induces a

local diffeomorphism, p, , on D, (0;M) . We now want to discus s the

convers e question: Given a local diffeomorphism, ij) , of D, (0;M) , what

conditions mus t 0 satisfy in o r d e r that 0 = pt for som e local diffeomorph-

i s m , (p , of M ?