The natural reductivity in Finsler geometry in terms of geodesic graphs
Teresa Arias-Marco, Zdenek Dusek
TL;DR
The paper addresses extending natural reductivity from Riemannian to Finsler homogeneous spaces by using geodesic graphs. It constructs purely Finsler $\alpha_i$-type metrics from positively related naturally reductive Riemannian metrics via $F=\sqrt{L(\sqrt{g_1},\dots,\sqrt{g_k})}$ and studies geodesic graphs to determine g.o. properties, including explicit results on the Heisenberg group $H_3$. It further derives a general framework for geodesic graphs of $F=\sqrt{L(\sqrt{g_1},\dots,\sqrt{g_k})}$ and $F=\sqrt{L(\sqrt{g},\beta)}$, showing when such spaces are geodesic-orbit and when they fail to be naturally reductive, depending on the interaction of the $g_j$ and the one-forms $\beta$. The findings yield new classes of g.o. Finsler spaces, including purely Finsler $(\alpha_1,\alpha_2)$ metrics on irreducible manifolds, and clarify the role of positively related metrics and parallelism of $\beta$ in preserving or destroying natural reductivity.
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 $α_i$-type metrics are constructed. Geodesic graphs on broad classes of Finsler $α_i$-type metrics $F$ which are derived from naturally reductive Riemannian metrics and which are not naturally reductive are described. The influence of one-forms $β_j$ to the structure of geodesics of the metric $F$ is also demonstrated and explicit construction of families of Finsler naturally reductive metrics of the $(α_i,β_j)$-type is described.