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.