Positive projectively flat manifolds are locally conformally flat-Kähler 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-Kähler 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-Kähler metrics are in fact Kähler and globally conformally flat.

Keywords

projectively flat, locally conformally flat-Kähler, Boothby metric

2010 Mathematics Subject Classification

Primary 53C07. Secondary 53C55.

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This work was supported by GNSAGA of INdAM. During this research, the author was visiting assistant professor at University of Science and Technology, and funded by the Municipal Science and Technology Commission.

Received 12 June 2019

Accepted 24 June 2019

Published 14 June 2021