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Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, Volume 161, Book 3, Pages 438–455 (Mi uzku1529)  

Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers

A. A. Malyugina, V. V. Shurygin

Kazan Federal University, Kazan, 420008 Russia

Abstract: A smooth manifold over the algebra of dual numbers $\mathbb{D}$ (a $\mathbb{D}$-smooth manifold) carries the canonical foliation whose leaves are affine manifolds. Extension of charts on a $\mathbb{D}$-smooth manifold along leaf paths allows ones to associate with an immersed transversal of the canonical foliation a pseudogroup of local $\mathbb{D}$-diffeomorphisms called the holonomy pseudogroup. In the present paper, holonomy pseudogroups are applied to the study of $\mathbb{D}$-diffeomorphisms between quotient manifolds of the algebra $\mathbb{D}$ by lattices. In particular, it is shown that a $\mathbb{D}$-diffeomorphism between two such manifolds exists if and only if one of the lattices is obtained from the other by the multiplication by a dual number. In addition, it is shown that some $\mathbb{D}$-smooth manifolds naturally associated with an affine manifold are $\mathbb{D}$-diffeomorphic if and only if this manifold is radiant.

Keywords: affine manifold, manifold over algebra of dual numbers, foliation, foliated bundle, tangent bundle, tangent manifold, torus over the algebra of dual numbers, Weil bundle.

DOI: https://doi.org/10.26907/2541-7746.2019.3.438-455

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Bibliographic databases:

UDC: 514.76
Received: 08.05.2019

Citation: A. A. Malyugina, V. V. Shurygin, “Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 161, no. 3, Kazan University, Kazan, 2019, 438–455

Citation in format AMSBIB
\Bibitem{MalShu19}
\by A.~A.~Malyugina, V.~V.~Shurygin
\paper Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2019
\vol 161
\issue 3
\pages 438--455
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1529}
\crossref{https://doi.org/10.26907/2541-7746.2019.3.438-455}


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