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 Lobachevskii J. Math., 2001, Volume 9, Pages 55–75 (Mi ljm129)

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

V. V. Shurygin, L. Â. Smolyakova

Kazan State University

Abstract: 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.

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Citation: V. V. Shurygin, L. Â. Smolyakova, “An analog of the Vaisman–Molino cohomology for manifolds Modelled on some types of modules over weil algebras and its application”, Lobachevskii J. Math., 9 (2001), 55–75

Citation in format AMSBIB
\Bibitem{ShuSmo01} \by V.~V.~Shurygin, L.~Â.~Smolyakova \paper An analog of the Vaisman--Molino cohomology for manifolds Modelled on some types of modules over weil algebras and its application \jour Lobachevskii J. Math. \yr 2001 \vol 9 \pages 55--75 \mathnet{http://mi.mathnet.ru/ljm129} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1884110} \zmath{https://zbmath.org/?q=an:0995.58001} 

• http://mi.mathnet.ru/eng/ljm129
• http://mi.mathnet.ru/eng/ljm/v9/p55

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. Kureš, W. M. Mikulski, “Liftings of linear vector fields to product preserving gauge bundle functors on vector bundles”, Lobachevskii J. Math., 12 (2003), 51–61
2. L. Â. Smolyakova, “The structure of complete radiant manifolds modeled by modules over Weil algebras”, Russian Math. (Iz. VUZ), 48:5 (2004), 71–78
3. A. S. Podkovyrin, A. A. Salimov, V. V. Shurygin, “Ocherk nauchnoi i pedagogicheskoi deyatelnosti V. V. Vishnevskogo (k 75-letiyu so dnya rozhdeniya)”, Trudy geometricheskogo seminara, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 147, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2005, 26–36
4. L. Â. Smolyakova, V. V. Shurygin, “Lifts of geometric objects to the Weil bundle $T^\mu M$ of a foliated manifold defined by an epimorphism $\mu$ of Weil algebras”, Russian Math. (Iz. VUZ), 51:10 (2007), 76–88