On Homogeneous Landsberg Surfaces

TL;DR

This paper proves that all homogeneous Landsberg surfaces have isotropic flag curvature and are either Riemannian or locally Minkowskian, confirming a longstanding conjecture in 2D homogeneous Finsler geometry.

Contribution

It establishes that homogeneous Landsberg surfaces are either Riemannian or locally Minkowskian, resolving the Xu-Deng conjecture in two dimensions.

Findings

Homogeneous Landsberg surfaces have isotropic flag curvature.

Such surfaces are either Riemannian or locally Minkowskian.

The result confirms the Xu-Deng conjecture in 2D.

Abstract

In this paper, we prove that every homogeneous Landsberg surface has isotropic flag curvature. Using this special form of the flag curvature, we prove a rigidity result on homogeneous Landsberg surface. Indeed, we prove that every homogeneous Landsberg surface is Riemannian or locally Minkowskian. This gives a positive answer to the Xu-Deng's well-known conjecture in 2-dimensional homogeneous Finsler manifolds.