# Homogeneous Finsler sphere with constant flag curvature

## TL;DR

This paper proves that homogeneous Finsler spheres with constant flag curvature 1 and certain geodesic properties are necessarily Riemannian, and explores geodesic behavior on such spheres, extending results beyond Randers metrics.

## Contribution

It establishes that such spheres with specific geodesic conditions are Riemannian and extends geodesic properties to non-Randers homogeneous Finsler spheres.

## Key findings

- Homogeneous Finsler spheres with constant flag curvature 1 and a prime closed geodesic of length 2π are Riemannian.
- Provides evidence for the non-existence of homogeneous Bryant spheres.
- Generalizes geodesic properties from Randers to non-Randers homogeneous Finsler spheres.

## Abstract

We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ 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 $K\equiv1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv1$ can be generalized to the non-Randers case.