# 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 *T*^{μ} from the category of fibered manifolds to itself which assigns to a fibered manifold *p:M → N* the fibered product *p*^{μ}:T^{ A}N×_{T BN}T^{ B}M→ T^{ A}N. In this paper we show that the manifold *T*^{ A}N×_{T BN}T^{ B}M can be naturally endowed with a structure of an *A*-smooth manifold modelled on the *A*-module *L=A*^{n} ⊕ B^{m}, where *n=dim N*, *n+m=dim M*. We extend the functor *T*^{μ} to the category of foliated manifolds *(M, F)*. Then we study *A*-smooth manifolds *M*^{L} whose foliated structure is locally equivalent to that of *T*^{ A}N×_{T BN}T^{ B}M. For such manifolds *M*^{L} 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 *A*^{n}. As an application, we express the obstructions for existence of an *A*-smooth linear connection on *M*^{L} in terms of the introduced cohomology groups.