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 ?
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