4.
V. Guillemin and S. Sternber g
We will then take up the question of when two pseudogroups a r e
globally equivalent. This require s firs t of all that the two manifolds on
which they a r e defined be diff eomorphic , so we might as well think of all of
our pseudogroups as defined on the s a m e manifold. We also might as well
a s s u m e that we have handled the local problem . In other w o r d s , we want
to look at the following problem : Given a pseudogroup T on M , "how
many " pseudogroups a r e ther e on M which a r e locally equivalent to T but
fail to be globally equivalent to r o r to each o t h e r ? Can this se t be
effectively p a r a m e t r i z e d ? Only in ver y specia l case s has anyone bee n able
to mak e any headway with this problem , the cas e mos t exhaustively studied
being the complex pseudogroup in one dimension. A m o r e elementar y
proble m along the s a m e lines is the deformation proble m which we will now
briefly describe .
Let V be a neighborhood of the origin in R and let \T , T £ V}
be a family of pseudogroups on M each of which is locally equivalent to I\
We say that the family depends differentiably on r if ever y point of M
admit s a neighborhood U and a p a r a m e t r i z e d diffeomorphism a : U M ,
depending differentiably on r , such that OL is a local equivalence of T
with F .
DEFINITION 1. 5. A deformation of T is a p a r a m e t r i z e d family
J F , T £ VJ of pseudogroups , depending differentiably on r , with I \ = T .
We shall discus s the proble m of describin g all deformations of a
pseudogroup and also that of describin g all "infinitesimal deformations" ,
a notion which will be introduced later . We will not solve any of thes e
problem s (local equivalence, equivalence, deformation, infinitesimal
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