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