On Einstein Matsumoto metrics

Abstract

In this paper, the necessary and sufficient conditions for Matsumoto metrics $F=\frac{{\alpha }^2}{\alpha -\beta }$ to be Einstein are given. It is shown that if the length of $\beta$ with respect to $\alpha$ is constant, then the Matsumoto metric $F$ is an Einstein metric if and only if $\alpha$ is Ricci-flat and $\beta$ is parallel with respect to $\alpha$. A nontrivial example of Ricci flat Matsumoto metrics is given.