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)) .
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