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 .
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