Deformation Theor y of Pseudogrou p Structure s 7.
We will denote by J ^ M ^ M ^ the se t of al l k jets from Ml to
M
?
. It is easil y see n that (for k oo) the set J, (M,, M
?
) has a natura l
differential s t r u c t u r e .
If i k then j* (f)(x) clearl y depends only on j , (f)(x) . We thus
obtain a mapping IT - of Ji(M , ,IVU) onto J - ( M , , M
2
) . It is eas y to se e
that this is a differentiable m a p and that IT. behave s right under c o m p o s i -
,.
c
. .. . k m _ m
tion. Similarly , TT- O TT, = TT . etc.
Instead of having two manifolds M, and M-, , we could generaliz e
all of the above to the c a s e of differentiable fiber bundles. We thus conside r
jets of c r o s s - s e c t i o n s of a fiber bundle, etc . F o r instance , if E is a
vecto r bundle over M then we can conside r the spac e J, (E) of k-jet s of
c r o s s - s e c t i o n s of E (which is also a vecto r bundle). In particular , if
E = T(M) a section of E is a vector field and we can consider k-jet s of
vecto r fields.
If X is a vecto r field on M and p is a diffeomorphism of M
with itself then j , , (p;,, X)(p(x)) depends only on j,(p)(x) and j , _,(X)(x) .
We can thus writ e
(2-2) J
k
_
1
KX)(p(x)) = [Jk(V)(x)]!f!Jk_1(X)(x) .
F u r t h e r m o r e , it is easy to se e that j (p)(x) is determine d by its
action on J, ,(T(M)) in the following s e n s e : if p and 0 a r e two differ-
entiable map s with sourc e x and targe t y then j , (p)(x) = j , ($)(x) if and
only if [ ^ M l ^ ^ . ^ X j f x ) = [J
k
(0)]
#
J
k
_
1
(X)(x) for all X . (Just check in
t e r m s of local coordinates taking X = r- . ) Notice also that
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