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

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.