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Journal of Dynamical and Control Systems

, Volume 18, Issue 4, pp 479–498 | Cite as

Geodesics in the Heisenberg group H n with a Lorentzian metric

  • Tiren Huang
  • Xiaoping Yang
Article

Abstract

Let \( {\mathbb{H}^n} \) be the Heisenberg group in \( {\mathbb{R}^{2n + 1 }} \) and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study the reachable sets by the time-like future directed curves. Second, we give a complete description of the Hamiltonian geodesics. Third, we compute the time-like conjugate locus of the origin.

Key words and phrases

Geodesic Heisenberg Group Lorentzian metric Conjugate Locus 

2000 Mathematics Subject Classification

58E10 53C50 

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References

  1. 1.
    A.A. Agrachev, El-H. Chakir, El-Alaoui, and J. P. Gauthier, Sub-Riemannian metrics on R 3. Proc. Conf. Canad. Math. Soc. 25 (1998).Google Scholar
  2. 2.
    A. Agrachev, D. Barilari, and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry. Calculus of Variations and Partial Differential Equations  10.1007/s00526-011-0414-y (2011), 1–34.
  3. 3.
    A. Agrachev, U. Boscain, J.-P. Gauthier, and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256 (2009), No. 8, 2621–2655.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. A. Agrachev, G. Charlot, J. P. A. Gauthier, and V. M. Zakalyukin, On sub-Riemannian caustics and wave fronts for contact distributions in the three-space. J. Dynam. Control Systems 6 (2000), No. 3, 365–395.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    El-H. Alaoui, J-P. Gauthier, and I. Kupka, Small sub-Riemannian balls in \( {\mathbb{R}^3} \). J. Dynam. Control Sys. 2 (1996), No.3, 359–421.MATHCrossRefGoogle Scholar
  6. 6.
    R. Beals, B. Gaveau, and P.C. Greiner, Hamilton-Jacobi theory and the Heat Kernal on Heisenberg groups. J. Math. Pures Appl. 79, 7 (2000), 633–689.MathSciNetMATHGoogle Scholar
  7. 7.
    J.K. Beem, P.E. Ehrlich, and K.l. Easley, Global Lorentzian geometry. Marcel Dekker (1996).Google Scholar
  8. 8.
    D. C. Chang, I. Markina, and A. Vasiliev, Sub-Lorentzian geometry on anti-de sitter space. J. Math. Pures Appl. 90 (2008), No. 1, 82–110.MathSciNetMATHGoogle Scholar
  9. 9.
    M. Golubitsky and V. Guillemin, Stable mappings and their singularities. Spinger-Verlag, New York (1973).MATHCrossRefGoogle Scholar
  10. 10.
    M. Grochowski, Differential properties of the sub-Riemannian distance function. Bull. Polish. Acad. Sci. 50 (2002), No. 1.Google Scholar
  11. 11.
    M. Grochowski, Geodesics in the sub-Lorentzian geometry. Bull. Polish. Acad. Sci. 50 (2002), No. 2.Google Scholar
  12. 12.
    M. Grochowski, Normal forms of germs of Contact sub-Lorentzian structures on \( {\mathbb{R}^3} \), Differentiability of the sub-Lorentzian distance function. J. Dynam. Control Sys. 9 (2003), No. 4, 531–547.MathSciNetMATHGoogle Scholar
  13. 13.
    M. Grochowski, Reachable sets for the Heisenberg sub-Lorentzian structure on \( {\mathbb{R}^3} \), An estimate for the distance function. J. Dynam. Control Sys. 12 (2006), No. 2, 145–160.MathSciNetMATHGoogle Scholar
  14. 14.
    M. Grochowski, On the Heisenberg sub- Lorentzian Metric on R 3. Geometric Singularity Theory, Banach Center Publications 65 (2004).Google Scholar
  15. 15.
    M. Gromov. Carnot-Caratheodory spaces seen from within. Progr. Math. 144, Birkhauser, Boston (1996), 79–323.Google Scholar
  16. 16.
    F. Monroy-Pérez and A. Anzaldo-Meneses, Optimal control on the Heisenberg group. J. Dynam. Control Sys. 5 (1996), No. 4, 473–499.CrossRefGoogle Scholar
  17. 17.
    A. Korolko and I. Markina, Non-Holonomic Lorentzian geometry on some \( \mathbb{H} \)-type groups, (preprint).Google Scholar
  18. 18.
    R. Montgomery, Singular extremals on Lie groups. Math. Control, Signals and Systems 7(3) (1994), 217–234.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications. Math. Surveys and Monographs 91, American Math. Soc., Providence (2002).Google Scholar
  20. 20.
    P. Piccione and D.V. Tausk, Variational aspects of the geodesic problem in sub-Riemannian geometry. J. Geometry and Physics. 39 (2001), 183–206.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Yu. L. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calc. Var. 16 (2010), 1018–1039.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    H. J. Sussmann, An extension of a theorem of Nagano on transitive Lie algebras. Proc. Am. Math. Soc. 45 (1974), 349–356.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R. Strichartz, Sub-Riemannian geometry. J. Diff. Geom. 24 (1986), 221–263.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsNanjing University of Science & TechnologyNanjingP.R. China
  2. 2.School of Mathematical SciencesUSTCNanjingP.R. China

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