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Keywords:
homogeneous space; Finsler space; Killing vector field; homogeneous geodesic
homogeneous space; Finsler space; Killing vector field; homogeneous geodesic
Summary:
In the recent paper [Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension, Monatsh. Math. 182,1, 165–171 (2017)], it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. However, the proof contains a serious gap. The situation is a bit delicate, because the statement is correct. In the present paper, the incorrect part in this proof is indicated. Further, it is shown that homogeneous geodesics in homogeneous Finsler spaces can be studied by another method developed in earlier works by the author for homogeneous affine manifolds. This method is adapted for Finsler geometry and the statement is proved correctly.
References:
[1] Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Springer Science+Business Media, New York, 2000. MR 1747675 | Zbl 0954.53001
[2] Deng, S.: Homogeneous Finsler Spaces. Springer Science+Business Media, New York, 2012. MR 2962626
[3] Dušek, Z.: On the reparametrization of affine homogeneous geodesics. Differential Geometry, Proceedings of the VIII International Colloquium World Scientific (Singapore) (López, J.A. Álvarez, García-Río, E., eds.), 2009, pp. 217–226. MR 2523507
[4] Dušek, Z.: Existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds. J. Geom. Phys. 60 (2010), 687–689. DOI 10.1016/j.geomphys.2009.12.015 | MR 2608520
[5] Dušek, Z.: The existence of homogeneous geodesics in special homogeneous Finsler spaces. Matematički Vesnik (2018). MR 3895904
[6] Dušek, Z., Kowalski, O.: Light-like homogeneous geodesics and the Geodesic Lemma for any signature. Publ. Math. Debrecen 71 (2007), 245–252. MR 2340046
[7] Dušek, Z., Kowalski, O., Vlášek, Z.: Homogeneous geodesics in homogeneous affine manifolds. Result. Math. 54 (2009), 273–288. DOI 10.1007/s00025-009-0373-1 | MR 2534447
[8] Figueroa-O’Farrill, J., Meessen, P., Philip, S.: Homogeneity and plane-wave limits. J. High Energy Physics 05, 050 (2005). DOI 10.1088/1126-6708/2005/05/050 | MR 2155055
[9] Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata 81 (2000), 209–214, Erratum: Geom. Dedicata 84, 331–332 (2001). DOI 10.1023/A:1005287907806 | MR 1825363 | Zbl 0980.53061
[10] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital. B (7) 5 (1991), 189–246. MR 1110676 | Zbl 0731.53046
[11] Latifi, D.: Homogeneous geodesics in homogeneous Finsler spaces. J. Geom. Phys. 57 (2007), 1421–1433. DOI 10.1016/j.geomphys.2006.11.004 | MR 2289656
[12] Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension. Monatsh. Math. 182 (1) (2017), 165–171. DOI 10.1007/s00605-016-0933-x | MR 3592127
[13] Yan, Z., Huang, L.: On the existence of homogeneous geodesic in homogeneous Finsler space. J. Geom. Phys. 124 (2018), 264–267. DOI 10.1016/j.geomphys.2017.10.005 | MR 3754513
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