On Einstein Matsumoto metrics

We study a special class of Finsler metrics, namely, Matsumoto metrics \(F = \tfrac{{\alpha ^2 }} {{\alpha - \beta }}\) , where α is a Riemannian metric and β is a 1-form on a manifold M. We prove that F is a (weak) Einstein metric if and only if α is Ricci flat and β is a parallel 1-form with respect to α. In this case, F is Ricci flat and Berwaldian. As an application, we determine the local structure and prove the 3-dimensional rigidity theorem for a (weak) Einstein Matsumoto metric.