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.
Download