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On integrable non-canonical geodesic flow on two-dimensional torus

  • School of Mathematical and Computational Sciences, Anqing Normal University, Anqing 246133, China

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https://doi.org/10.3969/j.issn.0253-2778.2016.12.002
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  • Author Bio:

    ZHANG Hai, male, born in 1982, PhD/lecturer. Research field: Integrable geodesic flows.

  • Received Date: 19 March 2015
  • Accepted Date: 05 July 2015
  • Rev Recd Date: 05 July 2015
  • Publish Date: 30 December 2016
    Abstract

  • Abstract

    A non-canonical metric on two-dimensional torus was introduced. It was proved that its geodesic flow is Liouville integrable and has vanishing topological entropy when restricted onto invariant hypersurface.

    Abstract

    A non-canonical metric on two-dimensional torus was introduced. It was proved that its geodesic flow is Liouville integrable and has vanishing topological entropy when restricted onto invariant hypersurface.
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    • References(9)
  • [1]
    TAIMANOV I A. Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds[J]. Math USSR Izv, 1988, 30: 403-409.
    [2]
    BOLSINOV A V, JOVANOVIC B. Integrable geodesic flows on Riemannian manifolds: Construction and obstructions[C]// Contemporary Geometry and Related Topics. River Edge, NJ: World Scientic, 2004: 57-103.
    [3]
    MATVEEV V S. Quadratically integrable geodesic flows on the torus and on the Klein bottle[J]. Regul Chaotic Dyn, 1997, 2: 96-102.
    [4]
    WALTERS P. An Introduction to Ergodic Theory[M]. New York: Springer-Verlag, 1982.
    [5]
    CHEN C, LIU F, ZHANG X. Orthogonal separable Hamiltonian systems on T2[J]. Science in China Series A: Mathematics, 2007, 50: 1 735-1 747.
    [6]
    THOMPSON G. Killing tensors in spaces of constant curvature[J]. J Math Phys, 1986, 27: 2 693-2 699.
    [7]
    CRAMPIN M. Hidden symmetries and Killing tensors[J]. Reports on Mathematical Physics, 1984, 20: 31-40.
    [8]
    BRUCE A T, MCLENAGHAN R G, SMIRNOV R G. A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables[J]. Journal of Geometry and Physics, 2001, 39: 301-322.
    [9]
    PATERNAIN G. Entropy and completely integrable Hamiltonian systems[J]. Proc Amer Math Soc, 1991, 113: 871-873.
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    [1]
    TAIMANOV I A. Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds[J]. Math USSR Izv, 1988, 30: 403-409.
    [2]
    BOLSINOV A V, JOVANOVIC B. Integrable geodesic flows on Riemannian manifolds: Construction and obstructions[C]// Contemporary Geometry and Related Topics. River Edge, NJ: World Scientic, 2004: 57-103.
    [3]
    MATVEEV V S. Quadratically integrable geodesic flows on the torus and on the Klein bottle[J]. Regul Chaotic Dyn, 1997, 2: 96-102.
    [4]
    WALTERS P. An Introduction to Ergodic Theory[M]. New York: Springer-Verlag, 1982.
    [5]
    CHEN C, LIU F, ZHANG X. Orthogonal separable Hamiltonian systems on T2[J]. Science in China Series A: Mathematics, 2007, 50: 1 735-1 747.
    [6]
    THOMPSON G. Killing tensors in spaces of constant curvature[J]. J Math Phys, 1986, 27: 2 693-2 699.
    [7]
    CRAMPIN M. Hidden symmetries and Killing tensors[J]. Reports on Mathematical Physics, 1984, 20: 31-40.
    [8]
    BRUCE A T, MCLENAGHAN R G, SMIRNOV R G. A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables[J]. Journal of Geometry and Physics, 2001, 39: 301-322.
    [9]
    PATERNAIN G. Entropy and completely integrable Hamiltonian systems[J]. Proc Amer Math Soc, 1991, 113: 871-873.
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