On Einstein Matsumoto metrics

TL;DR

This paper characterizes when Matsumoto metrics are Einstein, showing that under certain conditions, they are Ricci-flat and have parallel 1-forms, with explicit examples provided.

Contribution

It provides necessary and sufficient conditions for Matsumoto metrics to be Einstein, including a characterization involving Ricci-flatness and parallelism, and constructs explicit examples.

Findings

Matsumoto metrics are Einstein iff α is Ricci-flat and β is parallel when β's length is constant.

A nontrivial Ricci-flat Matsumoto metric example is constructed.

Conditions for Einstein Matsumoto metrics are fully characterized.

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.