Advertisement

Russian Mathematics

, Volume 62, Issue 11, pp 45–52 | Cite as

The Contact Metric Connection on the Heisenberg Group

  • V. I. Pan’zhenskiiEmail author
  • T. R. Klimova
Article
  • 6 Downloads

Abstract

We prove that there is only one contact metric connection with skew-torsion on the Heisenberg group endowed with a left-invariant Sasakian structure. We obtain the expression of this connection via the contact form and the metric tensor, and show that the torsion tensor and the curvature tensor are constant and the sectional curvature varies between −1 and 0.

Keywords

Heisenberg group Sasakian structure connection with skew-torsion sectional curvature 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Scott, P. The Geometries of 3–Manifolds (Mir,Moscow, 1986). (in Russian)zbMATHGoogle Scholar
  2. 2.
    Gonzalez, J. C., Chinea, D. “Quasi–Sasakian Homogeneous Structures on the Generalized Heisenberg Group H(p, 1)”, Proc. Amer.Math. Soc. 105, No. 1, 173–184 (1989).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Binz, E., Podc, S. Geometry of Heisenberg Groups (Amer.Math. Soc., 2008).CrossRefGoogle Scholar
  4. 4.
    Boyer, C. P. The Sasakian Geometry of the Heisenberg Group (Albuquerque, 2009).zbMATHGoogle Scholar
  5. 5.
    Tanno S. “The Automorphism Groups of Almost Contact Riemannian Manifolds”, Tôhoku Math. J. (2. 21, No. 1, 21–38 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Vershik, A. M., Gershkovich, V. Ya. “Nonholonomic Dynamical Systems. Geometry of Distributions and Variational Problems”, Dynamical systems–7, Itogi Nauki i Tekhniki. Ser. Sovrem. probl. matem. Fundam. napr. 16, 5–85 (VINITI,Moscow, 1987) [in Russian].Google Scholar
  7. 7.
    Agrachev, A., Barilari, D., Boscain, U. Introduction to Riemannian and Sub–Riemannian Geometry (SISSA, Italy, 2012).zbMATHGoogle Scholar
  8. 8.
    Sachkov, Yu. L. “Control Theory on Lie Groups”, Sovrem. matem. Fundam. napr. 26, 5–59 (2008).[in Russian].Google Scholar
  9. 9.
    Agrachev, A. A. “Topics in Sub–Riemannian Geometry”, Russian Math. Survey. 71, No. 6, 989–1019 (2016).CrossRefzbMATHGoogle Scholar
  10. 10.
    Blair, D. E. Contact Manifolds in Riemannian Geometry, Lecture Notes Math. 509 (Springer–Verlag, Berlin, New York, 1976).CrossRefGoogle Scholar
  11. 11.
    Kirichenko, V. F. Differential–Geometric Structures on Manifolds (“Pechatnyi Dom”, Odessa, 2013) [in Russian].Google Scholar
  12. 12.
    Sasaki, S. “On Differentiable Manifolds With Certain Structures Which are Closely Related to Almost Contact Structure”, I, TôhokuMath. J. (2. 12, No. 3, 459–476 (1960).MathSciNetzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Pedagogical Institute named after V. G. BelinskiiPenzaRussia

Personalised recommendations