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Geodesics in the Engel Group with a Sub-Lorentzian Metric — the Space-Like Case

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Abstract

Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.

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References

  1. Agrachev, A. A., Chakir, El-A. and Gauthier, J. P., Sub-Riemannian metrics on ℝ3, Conference on Geometric Control and Non-holonomic Mechanics, 25, 1998, 29–78.

    MathSciNet  MATH  Google Scholar 

  2. Ardentov, A. and Sachkov, Yu., Extremal trajectories in a nilponent sub-Riemannian problem on the Engel group, Matematicheskii Sbornik, 202(11), 2011, 31–54.

    Article  MathSciNet  Google Scholar 

  3. Beals, R., Gaveau, B. and Greiner, P. C., Hamilton-Jacobi theory and the Heat Kernal on Heisenberg groups, J. Math. Pures Appl., 79(7), 2000, 633–689.

    Article  MathSciNet  Google Scholar 

  4. Beem, J. K., Ehrlich, P. E. and Easley, K. L., Global Lorentzian Geometry, Marcel Dekker, New York, 1996.

    MATH  Google Scholar 

  5. Cai, Q., Huang, T., Yang, X. and Sachkov, Yu., Geodesics in the Engel group with a sub-Lorentzian metric, J. Dynam. Control System, 22(3), 2016, 465–483.

    Article  MathSciNet  Google Scholar 

  6. Cartan, E., Sur quelques quadratures dont l’élément différentiel contient des fonctions arbitraires, Bull. Soc. Math. France, 29, 1901, 118–130.

    Article  MathSciNet  Google Scholar 

  7. Chang, D. C., Markina, I. and Vasiliev, A., Sub-Lorentzian geometry on anti-de Sitter space, J. Math. Pures Appl., 90(1), 2008, 82–110.

    Article  MathSciNet  Google Scholar 

  8. Grochowski, M., Geodesics in the sub-Lorentzian geometry, Bull. Polish. Acad. Sci., 50(2), 2002, 161–178.

    MathSciNet  MATH  Google Scholar 

  9. Grochowski, M., Reachable sets for the Heisenberg sub-Lorentzian structure on ℝ3, an estimate for the distance function, J. Dynam. Control Sys., 12(2), 2006, 145–160.

    Article  MathSciNet  Google Scholar 

  10. Grochowski, M., Normal forms and reachable sets for analytic Martinet sub-Lorentzian structures of Hamiltonian type, J. Dynam. Control Sys., 17(1), 2011, 49–75.

    Article  MathSciNet  Google Scholar 

  11. Gromov, M., Carnot-Carathseodory spaces seen from within, Bellaïche, A., Risler, J.J. (eds.) Sub-Riemannian Geometry, Progress in Mathematics, 144, Birkhauser, Boston, 1996, 79–323.

    Chapter  Google Scholar 

  12. Huang, T. and Yang, X., Geodesics in the Heisenberg Group H n with a Lorentzian metric, J. Dynam. Control Sys., 18(1), 2012, 21–40.

    Article  Google Scholar 

  13. Korolko, A. and Markina, I., Non-holonomic Lorentzian geometry on some ℍ-type groups, J. Geom. Anal., 19, 2009, 864–889.

    Article  MathSciNet  Google Scholar 

  14. Molina, M. G., Korolko, A. and Markina, I., Sub-semi-Riemannian geometry of general H-type groups, Original Research Article Bulletin des Sciences Mathématiques, 137(6), 2013, 805–833.

    Article  MathSciNet  Google Scholar 

  15. Montgomery, R., A tour of sub-Riemannian geometries, their geodesics and applications, Math. Surveys and Monographs, 91, Amer. Math. Soc., Providence, RI, 2002.

    MATH  Google Scholar 

  16. O’Neill, B., Semi-Riemannian Geometry: with Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, New York, 1983.

    MATH  Google Scholar 

  17. Piccione, P. and Tausk, D. V., Variational aspects of the geodesic problem in sub-Riemannian geometry, J. Geom, Phys., 39, 2001, 183–206.

    Article  MathSciNet  Google Scholar 

  18. Strichartz, R., Sub-Riemannian geometry, J. Diff. Geom., 24, 1986, 221–263.

    Article  MathSciNet  Google Scholar 

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Correspondence to Qihui Cai.

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This work was supported by the Science and Technology Development Fund of Nanjing Medical University (No. 2017NJMU005).

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Cai, Q. Geodesics in the Engel Group with a Sub-Lorentzian Metric — the Space-Like Case. Chin. Ann. Math. Ser. B 41, 147–162 (2020). https://doi.org/10.1007/s11401-019-0191-z

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  • DOI: https://doi.org/10.1007/s11401-019-0191-z

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