Deformation Theor y of Pseudogrou p Structure s

3.

Let M be a differentiabie manifold and U an open subset of M

If r is a pseudogroup on M we set

Tj = {p £ r | domain ( p c U and rang e p c U | .

It is eas y to see that Tijj is a pseudogroup on U .

Let a be a diffeomorphism of a differentiable manifold M onto

a differentiable manifold M . If cp is a local diffeomorphism of M then

Oi o (p o a. is a local diffeomorphism of M . If § = {cp} is a collection of

local diffeomorphisms of M then we denote by a&(X the set of all diffeo-

m o r p h i s m s of M of the form capa. , (p G § .

DEFINITION I. 4. Let T be a pseudogroup on M and J? _a

pseudogrou p on M . A diffeomorphism, a , of M onto M is said to be

an equivalence of F with T* if dTOL - T4 • If a : U -* U is a local

diffeomorphism of M with M then a is called a local equivalence of T

with r' ii_ aOTijjjcT = rf ly' .

Notice that if p is a pseudogroup and p £ T then cp is a local

equivalence of T with itself. Thus if P is a transitiv e pseudogroup, then

ever y x and y in M have neighborhoods U and V such that TITT and

T\y a r e equivalent In other words , a transitiv e pseudogroup looks the

s a m e about ever y point of M The first proble m we discus s is that of

classifying transitiv e pseudogroups up to local equivalence. The (formal)

infinitesimal versio n of this proble m was treate d in [13] .