Positive projectively flat manifolds are locally conformally flat-K\"ahler Hopf manifolds

Simone Calamai

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arXiv · 1711.00929

Positive projectively flat manifolds are locally conformally flat-K\"ahler Hopf manifolds

Simone Calamai

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Abstract

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-K\"ahler metrics on Hopf manifolds, explicitly characterized by Vaisman. Finally, we review the properties of zero projectively flat metrics. As applications, we refine a list of possible projectively flat metrics by Li, Yau, and Zheng; moreover we prove that projectively flat astheno-K\"ahler metrics are in fact K\"ahler and globally conformally flat.

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BibTeX RIS

Simone Calamai. 2017-11-02. Positive projectively flat manifolds are locally conformally flat-K\"ahler Hopf manifolds. https://arxiv.org/abs/1711.00929

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