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

In this paper, we study a special class of Finsler metrics, $(\alpha,\beta)$-metrics, defined by $F = \alpha \phi(\frac{\beta}{\alpha})$, 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.