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

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.