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)) .