On homogeneous Landsberg surfaces

Abstract

In this paper, we study the well-known unicorn problem for Finsler metrics. First, we prove that every homogeneous Landsberg surface has isotropic flag curvature. Then by using this particular form of the flag curvature, we prove a rigidity result on the homogeneous Landsberg surfaces. Indeed, we prove that every homogeneous Landsberg surface is Riemannian or locally Minkowskian. Thus, we give an affirmative answer to the Xu-Deng's well-known conjecture in two-dimensional homogeneous Finsler manifolds.