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Lobachevskii J. Math., 1999, Volume 5, Pages 29–55 (Mi ljm146)  

This article is cited in 10 scientific papers (total in 10 papers)

The structure of smooth mappings over weil algebras and the category of manifolds over algebras

V. V. Shurygin

Kazan State University, Faculty of Mechanics and Mathematics

Abstract: As is known, the bundle $T^{\mathbf A}M_n$ of infinitely near points of $\mathbf A$-type defined for any local Weil algebra $\mathbf A$ and smooth real manifold $M_n$ is one of basic examples of smooth manifolds over $\mathbf A$. In the present paper we give a description of the local structure of smooth mappings in the category of smooth manifolds over local algebras and consider various examples of such manifolds. Next we study the homotopy and holonomy groupoids of a smooth manifold $M^{\mathbf A}_n$ over a local algebra $\mathbf A$ associated with canonical foliations corresponding to ideals of $\mathbf A$. In particular, it is proved that a complete manifold $M^{\mathbf A}_n$ has neither homotopy nor holonomy vanishing cycles.

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Citation: V. V. Shurygin, “The structure of smooth mappings over weil algebras and the category of manifolds over algebras”, Lobachevskii J. Math., 5 (1999), 29–55

Citation in format AMSBIB
\by V.~V.~Shurygin
\paper The structure of smooth mappings over weil algebras and the category of manifolds over algebras
\jour Lobachevskii J. Math.
\yr 1999
\vol 5
\pages 29--55

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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. V. V. Shurygin (Jr.), “On the structure of complete varieties over Weil algebras”, Russian Math. (Iz. VUZ), 47:11 (2003), 84–93  mathnet  mathscinet  zmath  elib
    3. G. N. Bushueva, “Weil functors and product-preserving functors on the category of parameter-dependent manifolds”, Russian Math. (Iz. VUZ), 49:5 (2005), 11–18  mathnet  mathscinet  zmath
    4. V. V. Shurygin (Jr.), “Radiance obstructions for smooth manifolds over Weil algebras”, Russian Math. (Iz. VUZ), 49:5 (2005), 67–79  mathnet  mathscinet  zmath
    5. G. N. Bushueva, V. V. Shurygin, “On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds”, Lobachevskii J. Math., 18 (2005), 53–105  mathnet  mathscinet  zmath  elib
    6. V. V. Shurygin (Jr.), “Poisson structures on Weil bundles”, Lobachevskii J. Math., 17 (2005), 231–258  mathnet  mathscinet  zmath
    7. G. N. Bushueva, “Funktory tipa Veilya na kategorii mnogoobrazii, zavisyaschikh ot parametrov”, Trudy geometricheskogo seminara, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 147, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2005, 37–49  mathnet
    8. V. V. Shurygin (ml.), “Lifty struktur Puassona–Neienkheisa v rassloeniya Veilya”, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 151, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2009, 203–214  mathnet
    9. A. A. Malyugina, V. V. Shurygin, “Psevdogruppy golonomii kak prepyatstviya k ekvivalentnosti mnogoobrazii nad algebroi dualnykh chisel”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 161, no. 3, Izd-vo Kazanskogo un-ta, Kazan, 2019, 438–455  mathnet  crossref
    10. Andrew James Bruce, Eduardo Ibarguengoytia, Norbert Poncin, “The Schwarz–Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds”, SIGMA, 16 (2020), 002, 47 pp.  mathnet  crossref
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