# On general $(\alpha, \beta)$-metrics of weak Landsberg type

## TL;DR

This paper proves that all weak Landsberg general $(eta,eta)$-metrics are Berwald metrics with closed, conformal one-forms, and establishes the equivalence between Landsberg and weak Landsberg metrics in this class.

## Contribution

It demonstrates that weak Landsberg general $(eta,eta)$-metrics are Berwald and characterizes when such metrics are Landsberg, revealing no existence of generalized unicorn metrics.

## Key findings

- Weak Landsberg general $(eta,eta)$-metrics are Berwald.
- Landsberg and weak Landsberg conditions are equivalent for these metrics.
- No generalized unicorn metrics exist in this class.

## Abstract

In this paper, we study general $(\alpha,\beta)$-metrics which $\alpha$ is a Riemannian metric and $\beta$ is an one-form. We have proven that every weak Landsberg general $(\alpha,\beta)$-metric is a Berwald metric, where $\beta$ is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(\alpha,\beta)$-metric. Further, We show that $F$ is a Landsberg general $(\alpha,\beta)$-metric if and only if it is weak Landsberg general $(\alpha,\beta)$-metric, where $\beta$ is a closed and conformal one-form.