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
E-mail: *alexandra.malyugina@gmail.com, **Vadim.Shurygin@kpfu.ru
Received May 8, 2019

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DOI: 10.26907/2541-7746.2019.3.438-455

Abstract

A smooth manifold over the algebra of dual numbers D (a D-smooth manifold) carries the canonical foliation whose leaves are affine manifolds. Extension of charts on a D-smooth manifold along leaf paths allows ones to associate with an immersed transversal of the canonical foliation a pseudogroup of local D-diffeomorphisms called the holonomy pseudogroup. In the present paper, holonomy pseudogroups are applied to the study of D-diffeomorphisms between quotient manifolds of the algebra  by lattices. In particular, it is shown that a 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 D-smooth manifolds naturally associated with an affine manifold are 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

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