Positive projectively flat manifolds are locally conformally flat-Kahler Hopf manifolds

We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kahler metrics on Hopf manifolds, explicitly characterized by Vaisman [23]. Finally, we review the known characterization and properties of zero projectively flat metrics. As applications, we make sharp a list of possible projectively flat metrics by Li, Yau, and Zheng [16, Theorem 1]; moreover we prove that projectively flat astheno-Kahler metrics are in fact Kahler and globally conformally flat.

Automated summary

In this study, the space of projectively flat metrics is divided into three classes based on the sign of the Chern scalar curvature. It is shown that the class of negative projectively flat metrics is empty, while the positive class consists of locally conformally flat-Kahler metrics. Additionally, the characterization and properties of zero projectively flat metrics are reviewed, and it is proven that projectively flat astheno-Kahler metrics are actually Kahler and globally conformally flat.