Standard homogeneous $(\alpha_1,\alpha_2)$-metrics and geodesic orbit property

TL;DR

This paper introduces standard homogeneous $(,)$-metrics as a non-Riemannian deformation of normal homogeneous metrics, establishing conditions for geodesic orbit properties and identifying new non-Riemannian g.o. Finsler spaces.

Contribution

It defines standard homogeneous $(,)$-metrics, proves that one g.o. metric implies all are g.o., and characterizes g.o. metrics on certain Lie group triples and Wallach spaces.

Findings

Existence of a generic standard g.o. $(,)$-metric implies all such metrics are g.o.

Derived algebraic criteria for g.o. property in Lie group triples.

Identified new non-Riemannian g.o. Finsler spaces not weakly symmetric.

Abstract

In this paper, we introduce the notion of standard homogeneous $(\alpha_1,\alpha_2)$-metrics, as a natural non-Riemannian deformation for the normal homogeneous Riemannian metrics. We prove that with respect to the given bi-invariant inner product and orthogonal decompositions for $\mathfrak{g}$, if there exists one generic standard g.o. $(\alpha_1,\alpha_2)$-metric, then all other standard homogeneous $(\alpha_1,\alpha_2)$-metrics are also g.o.. For standard homogeneous $(\alpha_1,\alpha_2)$-metrics associated with a triple of compact connected Lie groups, we can refine our theorem and get some simple algebraic equations as the criterion for the g.o. property. As the application of this criterion, we discuss standard g.o. $(\alpha_1,\alpha_2)$-metric from H. Tamaru's classication work, and find some new examples of non-Riemannian g.o. Finsler spaces which are not weakly symmetric. On the other hand, we also prove that all standard g.o. $(\alpha_1,\alpha_2)$-metrics on the three Wallach spaces, $W^6=SU(3)/T^2$, $W^{12}=Sp(3)/Sp(1)^3$ and $W^{24}=F_4/Spin(8)$, must be the normal homogeneous Riemannian metrics.