Contractibility of geometric structures (Q1082618)

Let \(B\subset {\mathcal F}(M)\) be a given G-structure on a manifold M (G\(\subset GL(V)\), V a vector space). The author considers the problem of the existence (and construction) of a family of local diffeomorphisms preserving this structure and contracting a piece of the manifold to a given point (as an example, one can contract the Euclidean space by homotheties). For a given subgroup N of the normalizer of G the notion N- contractibility of the G-structure B is defined. Some necessary condition for that contractibility is obtained. In particular: (a) Transitive contact structures are G-contractible, (b) A Riemannian structure is N(G)-contractible if and only if it is locally Euclidean. In the case when G is algebraic and the image of the exponential map for it (or, for a maximal solvable subgroup) is dense, a necessary and sufficient condition for a locally N-transitive structure to be N-contractible is obtained.