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

TL;DR

This paper classifies projectively flat metrics on complex manifolds, showing positive ones are locally conformally flat-K"ahler Hopf manifolds, and refines understanding of zero and negative cases with new characterizations.

Contribution

It introduces a classification of projectively flat metrics based on Chern scalar curvature and characterizes positive cases as locally conformally flat-K"ahler Hopf manifolds, expanding prior knowledge.

Findings

Negative projectively flat metrics do not exist.

Positive projectively flat metrics are exactly locally conformally flat-K"ahler Hopf manifolds.

Projectively flat astheno-K"ahler metrics are K"ahler and globally conformally flat.

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.