Deformation Theory of Pseudogrou p Structure s 11.
In o r d e r to answe r this question, we introduce the fundamental
form o\ , on D, (0;M) . It is a vector-value d linea r differential form with
values in the vecto r spac e J, , ( T ( M ) ) Q (the spac e of (k-l)-jets of vecto r
fields at ©). It is defined as follows:
Let p be a point of D, (0;M) with "n"n(p) = x and let
4 T (D
k
(0;M)) . Then a~l(£) = Jk(X)(x) for som e local vecto r field X ,
and p"l is the jet of a diffeomorphism with sourc e x and targe t O . Then
p"l j , _, (X) is a well-defined element of J , , ( T ( M ) ) ~ depending only on |
(and p) . It is ( | , j) .
In short ,
(2- 7) «,a
k
= P ^ ^ o f ^ ) if i = T (D
k
(0;M)) .
PROPOSITION 2. 1. Let 0 be a local diffeomorphism of
D
k
(0;M ) . Then (locally) 0 = p for s o m e local diffeomorphism, cp , of M,
if and only if
( 2 -8) 0*crk = a
k
, (L_e_. , 0*C,ark = «,(Xk for all | ) .
Proof: If 0 = p then for any £ T (D (0;M)) with
I = Q!(Jk(X)(x)) , we have , by (2. 6)
* * t = p
k
* * = C([j
k + l
(cp)(x)].j
k
(x)(x)) .
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