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Homogeneous Randers spaces admitting just two homogeneous geodesics. (English) Zbl 07144743
In the present paper the author considers the problem about the number of homogeneous geodesics on a homogeneous Finsler manifold. A geodesic in a homogeneous manifold is called homogeneous geodesic if it is an orbit of a one-parameter group of isometries. The existence of at least one homogeneous geodesics in arbitrary homogeneous Riemannian manifold was proved by O. Kowalski and J. Szenthe in [Geom. Dedicata 84, No. 1–3, 331–332 (2001; Zbl 0980.53061)]. The existence of at least one homogeneous geodesics in the Finsler setting has been shown more recently by Z. Yan and S. Deng for Randers metrics [Houston J. Math. 44, No. 2, 481–493 (2018; Zbl 1423.53095)], by the author for odd-dimensional Finsler metrics [Arch. Math., Brno 54, No. 5, 257–263 (2018; Zbl 1424.53076)] and for Berwald or reversible Finsler metrics [“The existence of homogeneous geodesics in special homogeneous Finsler spaces”, Mat. Vesn. 71, No. 1–2, 16–22 (2019)], and by Z. Yan and L. Huang in general [J. Geom. Phys. 124, 264–267 (2018; Zbl 1388.53036)]. It is conjectured that an arbitrary homogeneous Finsler manifold admits at least two homogeneous geodesics through arbitrary point. In the present paper, the author gives examples of invariant Randers metrics which admit just two homogeneous geodesics.
MSC:
53C22 Geodesics in global differential geometry
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C30 Differential geometry of homogeneous manifolds
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