The natural reductivity in Finsler geometry in terms of geodesic graphs

TL;DR

This paper introduces a new geometric definition of naturally reductive Finsler manifolds using geodesic graphs, providing explicit examples and analyzing the influence of one-forms on geodesic structures.

Contribution

It proposes a novel geometric definition of naturally reductive Finsler manifolds via geodesic graphs and constructs explicit examples of such metrics, including their generalizations.

Findings

Explicit examples of purely Finsler naturally reductive metrics are constructed.

Geodesic graphs for broad classes of Finsler metrics are described.

The influence of one-forms on geodesic structures is demonstrated.

Abstract

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.