Characteristics of Randers Metrics of Isotropic Projective Ricci Curvature

In this paper, we delve into the exploration of projective Ricci curvature, with a specific focus on characterizing Finsler metrics possessing isotropic projective Ricci curvature and isotropic S-curvature. Notably, our investigation reveals a compelling result: every Randers metric featuring isotropic S-curvature and constant projective Ricci curvature emerges as a weak Einstein metric. Furthermore, we pinpoint the conditions under which such a metric exhibits isotropic projective Ricci curvature. Remarkably, on a closed Einstein Randers manifold, we establish that being PRic-flat is equivalent to being Ric-flat. This intriguing equivalence sheds light on the intricate interplay between projective and Riemannian geometry, offering valuable insights into the geometric structures underlying Finsler metrics.