On projective Ricci curvature of cubic metrics

To study the projective Ricci curvature (PRic-curvature) in Finsler geometry is interesting because it reflects the geometric properties that are invariant under the projective transformation. In this paper, we firstly derived an expression of S-curvature for the cubic Finsler metric and proved that this S-curvature vanishes if and only if $\beta$ is a constant Killing form. Next, we obtain an explicit expression of projective Ricci curvature for the cubic metric. We also proved that for the projective Ricci-flat Finsler space, the $1$-form $\beta$ is closed, and then the Riemannian metric of $\alpha$ is also Ricci-flat. Finally, we show that the cubic Finsler metric is of weak projective Ricci curvature if and only if it is projectively Ricci-flat.