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Geometric structures on filtered manifolds. (English) Zbl 0801.53019
A filtered manifold is a differential manifold $$M$$ equipped with a tangential filtration $$F$$, which is by definition a sequence $$\{F^ p\}_{p\in \mathbb{Z}}$$ of subbundles of the tangent bundle $$TM$$ of $$M$$ such that: (i) $$F^ p \supset F^{p+1}$$, (ii) $$F^ 0 = 0$$, $$\bigcup_{p\in z} F^ p = TM$$, and (iii) $$[\underline{F}^ p,\underline{F}^ q] \subset \underline{F}^{p+q}$$ for all $$p,q \in \mathbb{Z}$$, where $$\underline{F}^ p$$ denotes the sheaf of the germs of sections of $$F^ p$$. In this paper the author develops a general scheme to treat the geometric structures on filtered manifolds and gives a unified method into general equivalence problems. First it is introduced the notion of a tower $$(P,M,G,\theta)$$ on a filtered manifold $$(M,F)$$, which is a principal fiber bundle $$P$$ over $$M$$ with structure group $$G$$ possibly infinite dimensional and with an absolute parallelism $$\theta$$ on $$P$$ satisfying certain conditions. Then, based on this notion, it is developed a method to find the invariants of a given geometric structure in a geometric and group-theoretical way. In particular, the invariants of a transitive geometric structure are described in terms of generalized Spencer cohomology groups. It is also given a general criterion and method for constructing a Cartan connection associated with a given geometric structure.
Reviewer: T.Morimoto (Kyoto)

MSC:
 53C10 $$G$$-structures 57S25 Groups acting on specific manifolds
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