Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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SMI course "Conformal geometry, Cartan connection and locally conformally Kaehler structures"

A characterization of Projectively Flat manifolds

Simone Calamai ( Dipartimento di Matematica e Informatica "Ulisse Dini", Università di Firenze)

created on 27 Mar 2018
modified on 09 Apr 2018

17 may 2018 -- 15:00



We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we show 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-Kaehler metrics on Hopf manifolds, explicitly characterized by Vaisman. If time allows, 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; moreover we show that projectively flat astheno-Kaehler metrics are in fact Kaehler and globally conformally flat.

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