On Einstein Matsumoto metrics

Abstract

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the ( α , β ) -metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n -dimensional ( n ≥ 3 ) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.