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

Shurygin, Vadim V.; Smolyakova, Larisa B.

Lobachevskii J. Math. 9, 55-75 (2001).

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.

Reviewer: Josef Janyška (Brno)

MSC:

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

Keywords:

Weil algebra; cohomology group