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

This article is cited in 4 scientific papers (total in 4 papers)

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. . 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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Received: 01.12.2001
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Citation: V. V. Shurygin, L. . 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.~.~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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  mathscinet  zmath
    2. L. . Smolyakova, “The structure of complete radiant manifolds modeled by modules over Weil algebras”, Russian Math. (Iz. VUZ), 48:5 (2004), 71–78  mathnet  mathscinet  zmath
    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  mathnet
    4. L. . 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  mathnet  crossref  mathscinet
  • Lobachevskii Journal of Mathematics
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