On projective Ricci curvature of ‎Matsumoto‎ ‎metrics

Abstract

In this paper, we study Finsler metrics with weak, isotropic or flat projective Ricci curvature (briefly, $\textbf{PRic}$-curvature). First, we prove a rigidity result that shows for a complete Finsler manifold inequality \ ${\bf PRic}\geq{\bf Ric}$ holds if and only if ${\bf S}=0$. Then, we show that the Matsumoto metric is of weak $\textbf{PRic}$-curvature if and only if it is a $\textbf{PRic}$-flat metric. We characterize projective Ricci flat Matsumoto metrics with constant length one-forms. In this case, we show that the Matsumoto metric reduces to a Ricci flat metric. Finally, we prove that a Matsumoto metric is $\textbf{PRic}$-reversible if and only if it is $\textbf{PRic}$-quadratic.