射影Ricci平坦的Kropina度量

摘要

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.%In this paper, we study and characterize projective Ricci flat Kropina metrics. By using the formulas of S-curvature and Ricci curvature for Kropina metrics, we obtain the formula of the pro jective Ricci curvature for Kropina metrics. Based on this, we obtain the necessary and sufficient conditions for Kropina metrics to be projective Ricci flat metrics. Further, as a natural application, we study and characterize pro jective Ricci flat Kropina metrics defined by a Riemannian metric and a Killing 1-form of constant length. We also characterize pro jective Ricci flat Kropina metrics with isotropic S-curvature. In this case, the Kropina metrics are Ricci flat metrics.