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On Geodesic Flow and Energy Functional on Riemannian Manifolds

Received: 20 August 2021     Accepted: 2 September 2021     Published: 12 October 2021
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Abstract

In this paper, we study the geodesic flow and the energy functional on a Riemannian manifold and show that the geodesics have minimal energy, in other words, are the minimizers of the energy functional, from the new perspective of involution, and that the geodesic flow is a Hamiltonian flow which has a close connection with the canonical symplectic structure on the tangent bundle of a Riemannian manifold.

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 5)
DOI 10.11648/j.pamj.20211005.11
Page(s) 104-106
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Geodesic Flow, Energy Functional, Hamiltonian Flow, Vector Bundle

References
[1] Naeem Alkoumi and Felix Schlenk. Shortest closed billiard orbits on convex tables. Manuscripta Mathematica, 147 (3): 365–380, 2015.
[2] Georgios Arvanitidis, Soren Hauberg, Philipp Hennig, and Michael Schober. Fast and robust shortest paths on manifolds learned from data. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 1506– 1515. PMLR, 2019.
[3] Luther Pfahler Eisenhart. Riemannian geometry. Princeton university press, 1997.
[4] Wei Guo Foo and Joel Merker. Differential e-structures for equivalences of 2-nondegenerate levi rank 1 hypersurfaces m5 in c3. arXiv preprint arXiv: 1901.02028, 2019.
[5] Guilherme França, Alessandro Barp, Mark Girolami, and Michael I Jordan. Optimization on manifolds: A symplectic approach. arXiv preprint arXiv: 2107.11231, 2021.
[6] Hajime Fujita, Yu Kitabeppu, and Ayato Mitsuishi. Distance functions on convex bodies and symplectic toric manifolds. arXiv preprint arXiv: 2003.02293, 2020.
[7] Jürgen Jost and Jèurgen Jost. Riemannian geometry and geometric analysis, volume 42005. Springer, 2008.
[8] Roman Karasev. Mahlers conjecture for some hyperplane sections. Israel Journal of Mathematics, 241 (2): 795–815, 2021.
[9] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry, volume 1. New York, 1963.
[10] Yang Liu. On the range of cosine transform of distributions for torusinvariant complex minkowski spaces. Far East Journal of Mathematical Sciences (FJMS), 39 (2): 137–157, 2010.
[11] Yang Liu. On the lagrangian subspaces of complex minkowski space. J. Math. Sci. Adv. Appl, 7 (2): 87–93, 2011.
[12] Yang Liu. On the kähler form of complex lp space and its lagrangian subspaces. Journal of Pseudo-Differential Operators and Applications, 6 (2): 265–277, 2015.
[13] Peter Petersen, S Axler, and KA Ribet. Riemannian geometry, volume 171. Springer, 2006.
[14] Vandana Rani and Jasleen Kaur. On the structure of lightlike hypersurfaces of an indefinite kaehler statistical manifold. Differential Geometry-Dynamical Systems, 23: 229–242, 2021.
[15] Felix Schlenk. Symplectic embedding problems, old and new. Bulletin of the American Mathematical Society, 55 (2): 139–182, 2018.
[16] Fengzhen Tang, Mengling Fan, and Peter Tiňo. Generalized learning riemannian space quantization: A case study on riemannian manifold of spd matrices. IEEE transactions on neural networks and learning systems, 32 (1): 281–292, 2020.
[17] Edoardo Vesentini. Complex geodesics. Compositio mathematica, 44 (1-3): 375– 394, 1981.
Cite This Article
  • APA Style

    Yang Liu. (2021). On Geodesic Flow and Energy Functional on Riemannian Manifolds. Pure and Applied Mathematics Journal, 10(5), 104-106. https://doi.org/10.11648/j.pamj.20211005.11

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    ACS Style

    Yang Liu. On Geodesic Flow and Energy Functional on Riemannian Manifolds. Pure Appl. Math. J. 2021, 10(5), 104-106. doi: 10.11648/j.pamj.20211005.11

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    AMA Style

    Yang Liu. On Geodesic Flow and Energy Functional on Riemannian Manifolds. Pure Appl Math J. 2021;10(5):104-106. doi: 10.11648/j.pamj.20211005.11

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  • @article{10.11648/j.pamj.20211005.11,
      author = {Yang Liu},
      title = {On Geodesic Flow and Energy Functional on Riemannian Manifolds},
      journal = {Pure and Applied Mathematics Journal},
      volume = {10},
      number = {5},
      pages = {104-106},
      doi = {10.11648/j.pamj.20211005.11},
      url = {https://doi.org/10.11648/j.pamj.20211005.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211005.11},
      abstract = {In this paper, we study the geodesic flow and the energy functional on a Riemannian manifold and show that the geodesics have minimal energy, in other words, are the minimizers of the energy functional, from the new perspective of involution, and that the geodesic flow is a Hamiltonian flow which has a close connection with the canonical symplectic structure on the tangent bundle of a Riemannian manifold.},
     year = {2021}
    }
    

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    AB  - In this paper, we study the geodesic flow and the energy functional on a Riemannian manifold and show that the geodesics have minimal energy, in other words, are the minimizers of the energy functional, from the new perspective of involution, and that the geodesic flow is a Hamiltonian flow which has a close connection with the canonical symplectic structure on the tangent bundle of a Riemannian manifold.
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Author Information
  • Shenzhen Technology University, Shenzhen, China

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