2.

V. Guillemin and S. Sternber g

wor d but r a t h e r a "pseudogroup". A pseudogroup is a collection of t r a n s -

formations which is closed under invers e and composition whenever thes e

a r e defined (there being som e troubl e about domains of definition). Her e

a r e the p r e c i s e definitions:

DEFINITION i. 1. Let M and N be C°°-differentiable

manifolds. A diffeomorphism, cp , of an open se t U c M onto an open set

V c N is called a local diffeomorphism of M _to N ; ii_ M - N we shall

simpl y call cp a local diffeomorphism of M .

DEFINITION 1. 2. A pseudogroup , T , is a collection of local

diffeomorphisms of M satisfying the following axioms :

1) If p G r and 0 6 T, and the domain of p equals the rang e of

lj) then cp o if) is in T .

2) _If p is in r , (p is in T •

3) _If cp is in T and U is an open set contained in the domain

o£ cp , (P|U (the restrictio n of p _to_ U) is in T -

4) _If_ cp is a local diffeomorphism of M and ever y point in the

domain of cp admits a neighborhood, U , such that cp|U is in T then cp

is in P •

5) The identity diffeomorphism is in T •

DEFINITION I 3. A pseudogroup is called transitiv e if for

ever y pair of points x and y in M ther e is a cp £ T with cp(x) = y .