Advertisement

Russian Mathematics

, Volume 63, Issue 2, pp 74–79 | Cite as

Holonomy Pseudogroup of a Manifold Over the Algebra of Dual Numbers and Some its Applications

  • A. A. MalyuginaEmail author
  • V. V. ShuryginEmail author
Brief Communications
  • 2 Downloads

Abstract

We study properties of the holonomy pseudogroup on a total immersed transversal of the canonical foliation on a smooth manifold over the algebra of dual numbers ⅅ. We apply holonomy pseudogroups to the investigation of ⅅ-diffeomorphisms between quotient manifolds of the algebra ⅅ by lattices and between ⅅ-smooth manifolds naturally associated with an affine manifold.

Keywords

affine manifold manifold over the algebra of dual numbers foliation foliated bundle tangent bundle tangent manifold torus over the algebra of dual numbers Weil bundle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Evtushik, L. E., Lumiste, Yu. G., Ostianu, N. M., and Shirokov, A. P. “Differential-Geometric Structures on Manifolds”, Probl. geometrii 9 (Itogi nauki i tekhniki VINITI, Moscow, 1979), pp. 5–247.Google Scholar
  2. 2.
    Molino, P. Riemannian Foliations (Birkhäuser, 1988).CrossRefzbMATHGoogle Scholar
  3. 3.
    Thompson, G. and Schwardmann, U. “Almost Tangent and Cotangent Structures in the Large”, Trans. AMS 327, No. 1, 313–328 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Vaisman, I. “Lagrange Geometry on Tangent Manifolds”, Int. J. of Math. and Math. Sci. 51, 3241–3266 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Thurston, W. P. Three-Dimensional Geometry and Topology (Princeton University Press, Princeton, 1997; MTsCME, Moscow, 2001), Vol. 1.CrossRefzbMATHGoogle Scholar
  6. 6.
    Goldman, W. and Hirsch, M. W. “The Radiance Obstruction and Parallel Forms on Affine Manifolds”, Trans. Amer. Math. Soc. 26 (2), 629–649 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Phillips, J. “The Holonomic Imperative and the Homotopy Groupoid of a Foliated Manifold”, Rocky Mountain J. of Math. 17, No. 1, 151–165 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shurygin, V. V. “On Structure of Complete Manifolds over Weil Algebras”, Russian Mathematics, No. 11, 88–97 (2003).Google Scholar
  9. 9.
    Shurygin, V. V. “Smooth Manifolds over Local Algebras and Weil Bundles”, J. Math. Sci. 108, No. 2, 249–294 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diamond, F. and Shurman, J. A First Course in Modular Forms (Springer, 2005).zbMATHGoogle Scholar
  11. 11.
    Malakhaltsev, M. A. “Structures of a Manifold over the Algebra of Dual Numbers on the Torus”, Trudy geom. semin. 22, 47–62 (1994).MathSciNetGoogle Scholar
  12. 12.
    Malakhaltsev, M. A. “On a Class of Manifolds over the Algebra of Dual Numbers”, Trudy geom. semin. 21, 70–79 (1991).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Malyugina, A. A. and Shurygin, V. V. “Holonomy Representations of a Class of Manifolds over the Algebra of Dual Numbers”, Izv. Penza gos. ped. univ, No. 26, 128–136 (2011).Google Scholar
  14. 14.
    Shurygin, V. V. “Radiance Obstructions for Smooth Manifolds over Weil Algebras”, Russian Mathematics, No. 5, 71–83 (2005).Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia

Personalised recommendations