射影Ricci平坦的Kropina度量
ON PROJECTIVE RICCI FLAT KROPINA METRICS

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 projective Ricci curvature for Kropina metrics. Based on this, we obtain the necessary and su-cient conditions for Kropina metrics to be projective Ricci flat metrics. Further, as a natural application, we study and characterize projective Ricci flat Kropina metrics deflned by a Riemannian metric and a Killing 1-form of constant length. We also characterize projective Ricci flat Kropina metrics with isotropic S-curvature. In this case, the Kropina metrics are Ricci flat metrics