Structure of geodesics in weakly symmetric Finsler metrics on H-type groups. (English) Zbl 07285964

A Finsler g.o. space is a homogeneous Finsler space so that each geodesic of (with respect to the Chern connection) is an orbit of a one-parameter subgroup of the group of isometries. A tool to study g.o. spaces in Riemannian manifolds is the notion of geodesic graph, introduced by O. Kowalski and the author.
In the present paper special families of weakly symmetric Finsler metrics are considered. It is known that these are g.o. spaces (e.g. [S. Deng, Homogeneous Finsler spaces. New York: Springer (2012; Zbl 1253.53002)]).
The author considers weakly symmetric metrics on modified H-type groups which were studied in [Z. Yan and S. Deng, Differ. Geom. Appl. 36, 1–23 (2014; Zbl 1308.53114)].
He then constructs geodesic graphs on 5-dimensional and 6-dimensional H-type groups with these Finsler metrics and compare them with Riemannian geodesic graphs constructed in [O. Kowalski and S. Ž. Nikčević, Arch. Math. 73, No. 3, 223–234 (1999; Zbl 0940.53027); Arch. Math. 79, No. 2, 158–160 (2002; Zbl 1029.53063)], and with Randers geodesic graphs constructed in [Z. Dušek, Commentat. Math. Univ. Carol. 61, No. 2, 195–211 (2020; Zbl 1474.53206)].


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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