  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.
