Metric operator and geodesic orbit property for a standard homogeneous Finsler metric

TL;DR

This paper introduces the metric operator for compact homogeneous Finsler spaces and classifies when certain standard homogeneous metrics exhibit the geodesic orbit property, revealing new insights into their geometric structure.

Contribution

It defines a generalized notion of standard homogeneous $(oldsymbol{ ext{ extalpha}}_1,oldsymbol{ ext{ extalpha}}_s)$-metrics and classifies g.o. properties for these metrics on specific homogeneous manifolds.

Findings

Classified homogeneous manifolds with g.o. but not naturally reductive $( ext{ extalpha}_1, ext{ extalpha}_2)$-metrics.

Proved that on certain Wallach spaces, g.o. metrics are only normal homogeneous Riemannian metrics.

Extended the understanding of geodesic orbit properties in generalized Finsler geometry.

Abstract

In this paper, we introduce the metric operator for a compact homogeneous Finsler space, and use it to investigate the geodesic orbit property. We define the notion of standard homogeneous $(\alpha_1,\cdots,\alpha_s)$-metric which generalizes the notion of standard homogeneous $(\alpha_1,\alpha_2)$-metric. We classify all connected simply connected homogeneous manifold $G/H$ with a compact connected simple Lie group $G$ and two irreducible summands in its isotropy representation, such that there exists a standard homogeneous $(\alpha_1,\alpha_2)$-metric which is g.o. but not naturally reductive on $G/H$. We also prove that on a generalized Wallach space which is not a product of three symmetric spaces, any standard homogeneous $(\alpha_1,\alpha_2,\alpha_3)$-metric $F$ with respect to the canonical decomposition is g.o. on $G/H$ if and only if $F$ is a normal homogeneous Riemannian metric.