An analog of the Vaisman-Molino cohomology for manifolds modelled on some types of modules over Weil algebras and its application
(Lobachevskii Journal of Mathematics, Volume IX)
An epimorphism μ: A → B of local Weil algebras induces the functor Tμ from the category of fibered manifolds to itself which assigns to a fibered manifold p:M → N the fibered product pμ:T AN×T BNT BM→ T AN. In this paper we show that the manifold T AN×T BNT BM can be naturally endowed with a structure of an A-smooth manifold modelled on the A-module L=An ⊕ Bm, where n=dim N, n+m=dim M. We extend the functor Tμ to the category of foliated manifolds (M, F). Then we study A-smooth manifolds ML whose foliated structure is locally equivalent to that of T AN×T BNT BM. For such manifolds ML we construct bigraduated cohomology groups which are similar to the bigraduated cohomology groups of foliated manifolds and generalize the bigraduated cohomology groups of A-smooth manifolds modelled on A-modules of the type An. As an application, we express the obstructions for existence of an A-smooth linear connection on ML in terms of the introduced cohomology groups.