arXiv subjects

Zdenek Dusek

Publications and source records attributed to Zdenek Dusek.

4 recordsLinked to original sources

arXiv2025

The natural reductivity in Finsler geometry in terms of geodesic graphs

A new geometrical definition of naturally reductive Finsler manifold using geodeic graph is proposed, with a possible generalization. Based on a construction from a recent paper by the authors, Finsler metrics based on naturally reductive Riemannian metrics $g_i$ are studied. Explicit examples of purely Finsler naturally reductive $\alpha_i$-type metrics are constructed. Geodesic graphs on broad classes of Finsler $\alpha_i$-type metrics $F$ which are derived from naturally reductive Riemannian metrics and which are not naturally reductive are described. The influence of one-forms $\beta_j$ to the structure of geodesics of the metric $F$ is also demonstrated and explicit construction of families of Finsler naturally reductive metrics of the $(\alpha_i,\beta_j)$-type is described.

math.DG↗

arXiv2024

Structure of geodesics for Finsler metrics arising from Riemannian g.o. metrics

Homogeneous geodesics of homogeneous Finsler metrics derived from two or more Riemannian geodesic orbit metrics are investigated. For a broad newly defined family of positively related Riemannian geodesic orbit metrics, geodesic lemma is proved and it is shown that the derived Finsler metrics have also geodesic orbit property. These Finsler metrics belong to the newly defined class of the $\alpha_i$-type metrics which includes in particular the $(\alpha_1,\alpha_2)$ metrics. Geodesic graph for the sphere ${\mathrm{S}}^7={\mathrm{Sp(2)}}{\mathrm{U}}(1)/{\mathrm{Sp(1)}{\mathrm{diag}}{\mathrm{U}}(1)}$ with geodesic orbit Finsler metrics of the new type $(\alpha_1,\alpha_2,\alpha_3)$, arising from two or more Riemannian geodesic orbit metrics, is analyzed in detail. This type of metrics on $S^7$ is one of the missing cases in a previously published classification of geodesic orbit metrics on spheres.

math.DG↗

arXiv2023

Geodesic graphs for geodesic orbit Finsler $(α,β)$ metrics on spheres

Invariant geodesic orbit Finsler $(α,β)$ metrics $F$ which arise from Riemannian geodesic orbit metrics $α$ on spheres are determined. The relation of Riemannian geodesic graphs with Finslerian geodesic graphs proved in a previous work is now illustrated with explicit constructions. Interesting examples are found such that $(G/H,α)$ is Riemannian geodesic orbit space, but for the geodesic orbit property of $(G/H,F)$ the isometry group has to be extended. It is also shown that projective spaces other than ${\mathbb{R}}P^n$ do not admit invariant purely Finsler $(α,β)$ metrics.

math.DG↗

arXiv2021

The minimal number of homogeneous geodesics depending on the signature of the Killing form

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.

math.DG↗