Special left invariant conic Finsler metrics and homogeneous conic Landsberg Problem in two dimension

TL;DR

This paper classifies special left invariant conic Finsler metrics on 2D non-Abelian Lie groups, showing Landsberg metrics are Berwald, and proves the 2D case of a broader conjecture linking Landsberg and Berwald metrics.

Contribution

It establishes that all 2D homogeneous conic Landsberg metrics are Berwald and supports the conjecture that this holds in higher dimensions.

Findings

Landsberg metrics on the group are necessarily Berwald.

Classification of constant curvature, Landsberg, and Berwald conditions.

Proof of the 2D case of the homogeneous conic Landsberg conjecture.

Abstract

In this paper, we study left invariant conic Finsler metrics on the 2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vector fields, and classify those satisfying the constant curvature condition, the Landsberg condition or the Berwald condition respectively. We prove that any left invariant conic Landsberg metric on $G$ must be Berwald. This discovery enable us to propose a homogeneous conic Landsberg Conjecture, which guesses that every homogeneous conic Landsberg metric is Berwald, and prove the 2-dimensional case for it.