The natural reductivity in Finsler geometry in terms of geodesic graphs

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.