## An analog of the Vaisman-Molino cohomology for manifolds modelled on some types of modules over Weil algebras and its application.(English)Zbl 0995.58001

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 fibred manifold $$P: M\to N$$ the fibred product $$p^\mu : T^{\mathbf A} N \times_{T^{\mathbf B} N} T^{\mathbf B} M \to T^{\mathbf A} N$$.
The authors show that the manifold $$T^{\mathbf A} N\times_{T^{\mathbf B} N} T^{\mathbf B} M$$ can be endowed with a canonical structure of an $$\mathbf A$$-smooth manifold modeled on the $$\mathbf A$$-module $$\mathbf L= \mathbf A^n \oplus \mathbf B^m$$, $$n=\dim N$$, $$n+m=\dim M$$. The functor $$T^\mu$$ is extended to the category of foliated manifolds. $$\mathbf A$$-smooth manifolds $$M^{\mathbf L}$$ whose foliated structure is locally equivalent to that of $$T^{\mathbf A} N\times_{T^{\mathbf B} N} T^{\mathbf B} M$$ are studied. For manifolds $$M^{\mathbf L}$$ bigraduated cohomology groups are constructed. The obstructions for existence of an $$\mathbf A$$-smooth linear connection on $$M^{\mathbf L}$$ are described in terms of the introduced cohomology groups.

### MSC:

 58A32 Natural bundles 53C12 Foliations (differential geometric aspects)

### Keywords:

Weil algebra; cohomology group
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