## Vadim V. Shurygin, Larisa B. Smolyakova

## 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, Vol.9, pp.55-75)

An epimorphism
$\mu:\A\to\B$ of local Weil algebras induces the functor
$T^\mu$ from the category of fibered manifolds to itself
which assigns to a fibered
manifold $p:M\to N$ the fibered product
$p^\mu:T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M\to T^{\bf A}N$.
In this paper we show that the manifold
$T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M$ can be naturally
endowed with a structure of an $\A$-smooth manifold modelled on the
$\A$-mod\-ule $\L=\A^n\oplus\B^m$, where $n=\dim N$, $n+m=\dim M$.
We extend the functor $T^\mu$ to the category of
foliated manifolds $(M,{\cal F})$.
Then we study $\A$-smooth manifolds $M^\L$ whose foliated
structure is locally equivalent to that of
$T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M$.
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 $\A$-mod\-ules 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.

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