Some Ricci-Flat $(\alpha ,\beta )$-Metrics

Abstract

In this paper, we study a special class of Finsler metrics, $(\alpha ,\beta )$-metrics, defined by $F=\alpha \varphi (\frac{\beta }{\alpha })$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form. We find an equation that characterizes Ricci-flat $(\alpha ,\beta )$-metrics under the condition that the length of $\beta$ with respect to $\alpha$ is constant.