Structure of Geodesics for Finsler Metrics Arising from Riemannian g.o. Metrics

Abstract

We investigate homogeneous geodesics of homogeneous Finsler metrics derived from two or more Riemannian geodesic orbit metrics. A geodesic lemma is proved for a broad, newly defined, family of positively related Riemannian geodesic orbit metrics, and it is shown that the derived Finsler metrics have also the geodesic orbit property. These Finsler metrics belong to the newly defined class of αi-type metrics which includes, in particular, (α1, α2) metrics. We give a detailed analysis of the geodesic graph for the sphere S7 = Sp(2)U(1)/Sp(1)diagU(1) with geodesic orbit Finsler metrics of the new type (α1, α2, α3), arising from two or more Riemannian geodesic orbit metrics. This type of metric on S7 is one of the missing cases in a previously published classification of geodesic orbit metrics on spheres.