12.

V. Guillemin and S. Sternber g

Let p = j , (ff)(0) , for som e diffeomorphism rj • Then

***.Tk = t(Jk(V)(x) °P)" 1 ]

; :

^k_

1

([J

k + 1

M(x)]

: ; !

J

k

(X)(x) )

= Jk_i((p °n)~l*v*(x))(o)

J k - i f o _ 1 * x ) ( 0 ) = P _ I * J

k

- i ( x ) ( x ) = ^' C T k

Thus condition (2. 8) is n e c e s s a r y .

To show that (2. 8) is sufficient we will procee d by induction on k .

LEMMA 2. 1. ^ _Let ij) be a local diffeomorphism of D^OjM)

with ^ (a.) - a, . Then we can locally writ e lj) - (p, for som e local diffeo-

m o r p h i s m p of M .

In this cas e let us denote the projection IT : D.(0;M) ~» M by T T .

Observ e that for any 4 € : T (D^OjM)) we have

*

0

(«"l(«))

- n.r -

Consider § with TT^ £ = 0 . Then (2. 8) implie s that IT O. 0... | = 0 . Thus 0

sends a tangent spac e to the fiber of D,(0;M) onto a tangent spac e to the

fiber. This mean s that locally, ^ is a fiber ma p and induces a ma p tp on

M . In other words ther e is a local diffeomorphism, p , of M satisfying

(2. 9) p o

TT

=

TT

o 0 .

(0 The bundle D|((D;M) can be identified with what was called the bundle

of frame s in [23] or [25], In this cas e 0\ coincides with the form a

introduced t h e r e .