Geometric structures on filtered manifolds. (English) Zbl 0801.53019

Morimoto, Tohru

Hokkaido Math. J. 22, No. 3, 263-347 (1993).

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

Keywords:

tower; structure group; absolute parallelism; Spencer cohomology groups; Cartan connection