V. Guillemin and S. Sternber g
j
k
(ex p tX)(x) = j
k
(id)(x) for all t if and only if Jk(X)(x) = Jk(0)(x)
Now let O be som e fixed point of M . Let us denote by
D, (0;M) the spac e of k-jet s of local diffeom orphism s with s o u r c e O and
targe t anywhere in M . It is easy to se e that D, (0;M) is a differentiable
manifold and that ther e is a natura l differentiable m a p , T T . = TT . of
D
k
(0;M) - D^(0;M) for k t given by ^ (Jk(p)) = j/cp)
In case i = 0, we can identify D
0
(O;M) with M itself (since a
zer o jet is uniquely determine d by its sourc e and target) . We thus have
^0 = U
0
,D
k
(0;M )
M = D
0
(O;M)
"i
= U
i
(Notice that D, (0;M) is actually a principal bundle over M
whose s t r u c t u r e group is the group of all k-jet s of diffeomorphisms with
s o u r c e and targe t O . F o r any such j , (ty) we have the action j , (p)
" * Jk(p) ° Jk(0) )
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