Victor Guillemin Shlomo Sternber g
Columbia University Harvar d University
§1. Introduction. In this pape r we study the geometri c and
forma l problem s involved in the theor y of deformations of transitiv e
pseudogroup s t r u c t u r e s . It essentiall y consists of an exposition (with s o m e
simplifications and extensions) of som e of the result s of Spencer [24] and
Kodaira-Spence r [14;15]. In addition, we will state som e result s and
conjectures concerning elliptic pseudogroups . Our p r e s e n t pape r should be
regarde d as a continuation of our previous pape r [13]. Although m o s t of the
result s h e r e will be logically independent of [13], we shall refe r to [13] for
examples and motivation. We als o refe r to [23] for an alternat e descriptio n
of the foundations. We expres s our thanks to R. Bott, D. Quillen and
D. C. Spencer for helpful conversation s in the cours e of this paper. We
also mention our specia l appreciatio n to Charle s Friefel d and Hubert
Golds chmidt for help with s o m e of the foundational questions and useful
critica l comment s on the whole manuscript .
We now begin by giving a genera l descriptio n of what the subject
is about. In studying a "geometrica l s t r u c t u r e " it is fruitful to study its
"group of a u t o m o r p h i s m s . " Usually, thes e automorphism s a r e not globally
defined. They therefor e do not for m a group in the presen t sens e of the
* The r e s e a r c h of both author s was supported in par t by a gran t from
the National Science Foundation. The second author was also partiall y
supported by a grant from the Sloan Foundation.
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