Structure of geodesics for Finsler metrics arising from Riemannian g.o. metrics

TL;DR

This paper investigates the structure of geodesics in a new class of Finsler metrics derived from Riemannian geodesic orbit metrics, proving their properties and analyzing specific examples on spheres.

Contribution

It introduces a new class of ta-type Finsler metrics derived from Riemannian geodesic orbit metrics and proves their geodesic orbit property, filling gaps in sphere classifications.

Findings

Proved geodesic lemma for a broad family of Riemannian geodesic orbit metrics.

Showed derived Finsler metrics also possess geodesic orbit property.

Analyzed geodesic graph for 7 on spheres with new Finsler metrics.

Abstract

Homogeneous geodesics of homogeneous Finsler metrics derived from two or more Riemannian geodesic orbit metrics are investigated. For a broad newly defined family of positively related Riemannian geodesic orbit metrics, geodesic lemma is proved and it is shown that the derived Finsler metrics have also geodesic orbit property. These Finsler metrics belong to the newly defined class of the $\alpha_i$-type metrics which includes in particular the $(\alpha_1,\alpha_2)$ metrics. Geodesic graph for the sphere ${\mathrm{S}}^7={\mathrm{Sp(2)}}{\mathrm{U}}(1)/{\mathrm{Sp(1)}{\mathrm{diag}}{\mathrm{U}}(1)}$ with geodesic orbit Finsler metrics of the new type $(\alpha_1,\alpha_2,\alpha_3)$, arising from two or more Riemannian geodesic orbit metrics, is analyzed in detail. This type of metrics on $S^7$ is one of the missing cases in a previously published classification of geodesic orbit metrics on spheres.