On the class of Einstein exponential-type Finsler metrics

In this paper, a special class of Finsler metrics, the so-called $(\alpha,\beta)$-metrics, which are defined by $F=\alpha \phi(s)$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form, is studied. First we show that the class of almost regular metrics obtained by Shen is Einstein if and only if it reduces to the class of Berwald metrics. In this case, the Riemannian metrics are Ricci-flat. Then we prove that an exponential metric is Einstein if and only if it is Ricci-flat.