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

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.