Homogeneous Finsler sphere with constant flag curvature

Abstract

We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv 1$ and a prime closed geodesic of length $2\pi$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Finsler sphere with $K\equiv 1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv 1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv 1$ can be generalized to the non-Randers case.

Changes in this verseion: a reference (see [5]) and a paragraph for remark is added after Corollary 1.3

References in corpus (1)

• The number of geometrically distinct reversible closed geodesics on a Finsler sphere with $K\equiv 1$