Deformation Theory of Pseudogroup Structure s 9.

Let p be any local diffeomorphism of M . Then p induces a

local diffeomorphism of D,(0;M), which we denote by A , defined as

k -1

follows: If domain p = U then d o m a i n A = (irn ) (U) . Fo r any

P £ (IT Q) (U) w i t h ¥ o ' P ' = X '

(

2

-3) ^(p) = jk(cp)(x) o p .

Notice that for two local diffeomorphisms p and 0 we have

( Z - 4 )

(* °

P)k

= 0

k

° %

whereve r either side is defined.

Let X be a vecto r field defined about x € M. Then{exptX J gives

a local one p a r a m e t e r group { (exp tX), / . In p a r t i c u l a r , X induces a

local vecto r field XR on D (0;M) .

Let p be any point of D, (0;M) with Trn(p) = x . Then it is eas y

to check that the value of the vecto r field X, at p depends only on the

k-je t of X at x . Also if XR(p) = Yk(p) then Jk(X)(x)=Jk(Y)(x) . We thus

have a correspondenc e of the spac e of k-jet s of vecto r fields at x with

T (D,(0;M)) . We will denote this ma p going from k-jet s to tangent vector s

by a = OL . Thus

K, p

a : J

k

(T(M))

x

T

p

(D

k

(0;M) )

is an injection. Since J, (T(M)) and D, (0;M) have the s a m e dimensions ,

K X K

a is an i s o m o r p h i s m of J,(T(M)) with T (D

k

(0;M)) .