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We define a partition of the space of projectively flat metrics in threeclasses according to the sign of the Chern scalar curvature; we prove that theclass of negative projectively flat metrics is empty, and that the class ofpositive projectively flat metrics consists precisely of locally conformallyflat-K\"ahler metrics on Hopf manifolds, explicitly characterized by Vaisman.Finally, we review the properties of zero projectively flat metrics. Asapplications, we refine a list of possible projectively flat metrics by Li,Yau, and Zheng; moreover we prove that projectively flat astheno-K\"ahlermetrics are in fact K\"ahler and globally conformally flat.

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