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.

中文翻译：

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Finsler几何中的加权投影Ricci曲率

在本文中，我们介绍了加权投影Ricci曲率，作为Z. Shen引入的投影Ricci曲率的扩展。我们表征了加权投影Ricci平面曲率的Randers度量的类别。我们找到了Kropina度量加权加权Ricci平面曲率的必要和充分条件。最后，我们证明，具有各向同性加权投影Ricci和各向同性S曲率的每个射影平面度量都是Kropina度量或Randers度量。