The minimal number of homogeneous geodesics depending on the signature of the Killing form
DOI:
https://doi.org/10.33044/revuma.3132Abstract
The existence of at least two homogeneous geodesics in any homogeneous Finsler manifold was proved in a previous paper by the author. The examples of solvable Lie groups with invariant Finsler metric which admit just two homogeneous geodesics were presented in another paper. In the present work, it is shown that a homogeneous Finsler manifold with indefinite Killing form admits at least four homogeneous geodesics. Examples of invariant Randers metrics on Lie groups with definite Killing form admitting just two homogeneous geodesics and examples with indefinite Killing form admitting just four homogeneous geodesics are presented.
Downloads
References
D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer-Verlag, New York, 2000. DOI MR Zbl
S. Deng, Homogeneous Finsler Spaces, Springer Monographs in Mathematics, Springer, New York, 2012. DOI MR Zbl
Z. Dušek, The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, J. Geom. Phys. 60 no. 5 (2010), 687–689. DOI MR Zbl
Z. Dušek, The affine approach to homogeneous geodesics in homogeneous Finsler spaces, Arch. Math. (Brno) 54 no. 5 (2018), 257–263. DOI MR Zbl
Z. Dušek, The existence of homogeneous geodesics in special homogeneous Finsler spaces, Mat. Vesnik 71 no. 1-2 (2019), 16–22. MR Zbl Available at http://www.vesnik.math.rs/landing.php?p=mv1912.cap&name=mv191202.
Z. Dušek, The existence of two homogeneous geodesics in Finsler geometry, Symmetry 11(7) (2019), Paper No. 850, 5 pp. DOI
Z. Dušek, Homogeneous Randers spaces admitting just two homogeneous geodesics, Arch. Math. (Brno) 55 no. 5 (2019), 281–288. DOI MR Zbl
Z. Dušek, Geodesic graphs in Randers g.o. spaces, Comment. Math. Univ. Carolin. 61 no. 2 (2020), 195–211. DOI MR Zbl
O. Kowalski, S. Nikčević, and Z. Vlášek, Homogeneous geodesics in homogeneous Riemannian manifolds—examples, in Geometry and Topology of Submanifolds, X (Beijing/Berlin, 1999), World Scientific, River Edge, NJ, 2000, pp. 104–112. DOI MR Zbl
O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 no. 1-3 (2000), 209–214, Erratum: Geom. Dedicata 84 no. 1-3 (2001), 331–332. DOI MR Zbl
O. Kowalski and Z. Vlášek, Homogeneous Riemannian manifolds with only one homogeneous geodesic, Publ. Math. Debrecen 62 no. 3-4 (2003), 437–446. DOI MR Zbl
D. Latifi, Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys. 57 no. 5 (2007), 1421–1433. DOI MR Zbl
B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics 103, Academic Press, New York, 1983. MR Zbl
Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. DOI MR Zbl
Z. Yan and S. Deng, Existence of homogeneous geodesics on homogeneous Randers spaces, Houston J. Math. 44 no. 2 (2018), 481–493. MR Zbl
Z. Yan and L. Huang, On the existence of homogeneous geodesic in homogeneous Finsler spaces, J. Geom. Phys. 124 (2018), 264–267. DOI MR Zbl
Downloads
Published
Issue
Section
License
Copyright (c) 2023 Zdeněk Dušek
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.