Geodesic Graphs for Geodesic Orbit Finsler \((\alpha ,\beta )\) Metrics on Spheres

The relation between Riemannian geodesic graphs and Finslerian geodesic graphs proved in a previous work is illustrated with explicit constructions. Interesting examples are found such that \((G/H,\alpha )\) $(G/H,\alpha )$ is Riemannian geodesic orbit space, but for the geodesic orbit property of (G/H, F) the isometry group has to be extended. Consequently, an alternative approach to the classification of invariant geodesic orbit Finsler \((\alpha ,\beta )\) $(\alpha ,\beta )$ metrics F which arise from Riemannian geodesic orbit metrics \(\alpha \) $\alpha$ on spheres is obtained. It is also shown that projective spaces other than \(\mathbb {R}P^n\) $\mathbb{R}{P}^n$ do not admit invariant purely Finsler \((\alpha ,\beta )\) $(\alpha ,\beta )$ metrics.