On Landsberg general $(\alpha,\beta)$-metrics with a conformal 1-form

TL;DR

This paper classifies Landsberg general $(eta)$-metrics in Finsler geometry, showing that regular cases are Berwaldian in higher dimensions and discovering new non-Berwaldian examples in the almost regular case.

Contribution

It provides a classification of Landsberg general $(eta)$-metrics under conformal conditions, including new non-Berwaldian examples.

Findings

Regular Landsberg metrics are Berwaldian in dimensions > 2

Classification of almost regular Landsberg metrics

Discovery of new non-Berwaldian Landsberg metrics

Abstract

In this paper, we study almost regular Landsberg general $(\alpha,\beta)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general $(\alpha,\beta)$-metrics under the conditon that $\beta$ is closed and conformal to $\alpha$. Under this condition, we prove that regular Landsberg general $(\alpha,\beta)$-metrics must be Berwaldian when the dimension is greater than two. For the almost regular case, the classification also is given and some new non-Berwaldian Landsberg metrics are found.