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

Dušek, Zdeněk

Arch. Math., Brno 56, No. 5, 265-275 (2020).

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)].

Reviewer: Andreas Arvanitoyeorgos (Pátra)

MSC:

53C22	Geodesics in global differential geometry
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C30	Differential geometry of homogeneous manifolds

Keywords:

Finsler space; weakly symmetric space; g.o. space; homogeneous geodesic; geodesic graph

Citations:

Zbl 1253.53002; Zbl 1308.53114; Zbl 0940.53027; Zbl 1029.53063; Zbl 1474.53206

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