Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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HOMOGENEOUS GEODESICS IN HOMOGENEOUS SUB-FINSLER MANIFOLDS

  • Zaili Yan (School of Mathematics and Statistics Ningbo University) ;
  • Tao Zhou (School of Mathematics and Statistics Ningbo University)
  • Received : 2022.10.31
  • Accepted : 2023.03.30
  • Published : 2023.11.30
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Abstract

In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of ℓp-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.

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Acknowledgement

We are deeply grateful to the reviewers of this paper for very careful reading and useful suggestions.

References

  1. A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov. Izdat. Akad. Nauk SSSR, Moscow.38 (1951), 5-23.
  2. D. Bao, S. S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer, New York, 2000. https://doi.org/10.1007/978-1-4612-1268-3
  3. V. Berestovskii, Homogeneous manifolds with intrinsic metric. II, Siberian Math. J. 30 (1989), no. 2, 180-191; translated from Sibirsk. Mat. Zh. 30 (1989), no. 2, 14-28, 225. https://doi.org/10.1007/BF00971372
  4. V. Berestovskii and Y. Nikonorov, Riemannian manifolds and homogeneous geodesics, Springer Monographs in Mathematics, Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-56658-6
  5. S. Bochner and D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639-653. https://doi.org/10.2307/1969226
  6. D. Y. Burago, Y. D. Burago, and S. V. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc., Providence, RI, 2001. https://doi.org/10.1090/gsm/033
  7. L. Capogna and E. Le Donne, Smoothness of sub Riemannian isometries, Amer. J. Math. 138 (2016), no. 5, 1439-1454. https://doi.org/10.1353/ajm.2016.0043
  8. S. S. Chern and Z. Shen, Riemann-Finsler geometry, Nankai Tracts in Mathematics, 6, World Sci. Publ., Hackensack, NJ, 2005. https://doi.org/10.1142/5263
  9. D. Dantzig and B. L. Waerden, Uber metrisch homogene raume, Abh. Math. Sem. Univ. Hamburg 6 (1928), no. 1, 367-376. https://doi.org/10.1007/BF02940622
  10. S. Q. Deng, Homogeneous Finsler spaces, Springer Monographs in Mathematics, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-4244-8
  11. S. Q. Deng and Z. X. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (2002), no. 1, 149-155. https://doi.org/10.2140/pjm.2002.207.149
  12. Z. Dusek, Survey on homogeneous geodesics, Note Mat. 28 (2009), suppl. 1, 147-168.
  13. Z. Dusek, The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, J. Geom. Phys. 60 (2010), no. 5, 687-689. https://doi.org/10.1016/j.geomphys.2009.12.015
  14. M. Gromov, Hyperbolic groups, in Essays in group theory, 75-263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. https://doi.org/10.1007/978-1-4613-9586-7_3
  15. O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), no. 1-3, 209-214. https://doi.org/10.1023/A:1005287907806
  16. O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7) 5 (1991), no. 1, 189-246.
  17. S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), no. 2, 400-416. https://doi.org/10.2307/1968928
  18. A. Podobryaev, Homogeneous geodesics in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var. 29 (2023), Paper No. 11, 17 pp. https://doi.org/10.1051/cocv/2022086
  19. Z. Shen, Lectures on Finsler Geometry, World Sci. Publishing, Singapore, 2001. https://doi.org/10.1142/9789812811622
  20. Z. Yan and L. B. Huang, On the existence of homogeneous geodesic in homogeneous Finsler spaces, J. Geom. Phys. 124 (2018), 264-267. https://doi.org/10.1016/j.geomphys.2017.10.005