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 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.