An analog of the Vaisman–Molino cohomology for manifolds Modelled on some types of modules over weil algebras and its application

An epimorphism $\mu:\mathbf A\to\mathbf B$ of local Weil algebras induces the functor $T^\mu$ from the category of fibered manifolds to itself which assigns to a fibered manifold $p\colon M\to N$ the fibered product $p^\mu\colon T^{\mathbf A}N\times{}_{{T^B}N}T^{\mathbf B}M\to T^{\mathbf A}N$. In this paper we show that the manifold $T^{\mathbf A}N\times{}_{{T^B}N}T^{\mathbf B}M$ can be naturally endowed with a structure of an $\mathbf A$-smooth manifold modelled on the $\mathbf A$-module $\mathbf L={\mathbf A}^n\oplus{\mathbf B}^m$, where $n=\dim N$, $n+m=\dim M$. We extend the functor $T^\mu$ to the category of foliated manifolds $(M,\mathcal F)$. Then we study $\mathbf A$-smooth manifolds $M^\mathbf L$ whose foliated structure is locally equivalent to that of $T^{\mathbf A}N\times{}_{{T^B}N}T^{\mathbf B}M$. For such manifolds $M^\mathbf L$ we construct bigraduated cohomology groups which are similar to the bigraduated cohomology groups of foliated manifolds and generalize the bigraduated cohomology groups of $\mathbf A$-smooth manifolds modelled on $\mathbf A$-modules of the type ${\mathbf A}^n$. As an application, we express the obstructions for existence of an $\mathbf A$-smooth linear connection on $M^\mathbf L$ in terms of the introduced cohomology groups.