Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers
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...
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| Language: | English |
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Kazan Federal University
2019-09-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/uz-eng-phm-2019-3-9.html |
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| author | A.A. Malyugina V.V. Shurygin |
| author_facet | A.A. Malyugina V.V. Shurygin |
| author_sort | A.A. Malyugina |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-655e98debd5e48b6aa4e551b2c25ab28 |
| institution | OA Journals |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2019-09-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-655e98debd5e48b6aa4e551b2c25ab282025-08-20T02:18:19ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982019-09-01161343845510.26907/2541-7746.2019.3.438-455Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbersA.A. Malyugina0V.V. Shurygin1Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaA 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.https://kpfu.ru/uz-eng-phm-2019-3-9.htmlaffine manifoldmanifold over algebra of dual numbersfoliationfoliated bundletangent bundletangent manifoldtorus over the algebra of dual numbersweil bundle |
| spellingShingle | A.A. Malyugina V.V. Shurygin Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers Учёные записки Казанского университета: Серия Физико-математические науки affine manifold manifold over algebra of dual numbers foliation foliated bundle tangent bundle tangent manifold torus over the algebra of dual numbers weil bundle |
| title | Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| title_full | Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| title_fullStr | Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| title_full_unstemmed | Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| title_short | Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| title_sort | holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers |
| topic | affine manifold manifold over algebra of dual numbers foliation foliated bundle tangent bundle tangent manifold torus over the algebra of dual numbers weil bundle |
| url | https://kpfu.ru/uz-eng-phm-2019-3-9.html |
| work_keys_str_mv | AT aamalyugina holonomypseudogroupsasobstructionstoequivalenceofmanifoldsoverthealgebraofdualnumbers AT vvshurygin holonomypseudogroupsasobstructionstoequivalenceofmanifoldsoverthealgebraofdualnumbers |