Weighted Projective Ricci Curvature in Finsler Geometry

TL;DR

This paper introduces the weighted projective Ricci curvature in Finsler geometry, characterizes certain metrics with this curvature, and explores conditions for flatness and projective flatness in specific Finsler metrics.

Contribution

It extends the concept of projective Ricci curvature, providing new characterizations and conditions for Randers and Kropina metrics in Finsler geometry.

Findings

Characterization of weighted projective Ricci flat Randers metrics

Necessary and sufficient conditions for Kropina metrics to have flat weighted projective Ricci curvature

Classification of projectively flat metrics with isotropic weighted Ricci and S-curvature

Abstract

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.