Chern-Ricci Curvatures,Holomorphic Sectional Curvature And Hermitian Metrics

Complex Kahler manifolds with constant holomorphic sectional curvature are called complex space forms.The classification of simply connected complex space forms has been given in the 1950s,but there is not much related work in the case of non-Kahler.In this paper,we mainly study Hermitian manifolds with constant(or pointwise constant)holomorphic sectional curvature.In 1985,Balas and Gauduchon proved that a compact Hermitian surface with constant nonpositive holomorphic sectional curvature is Kahler.Based on this conclu-sion,we consider:is it still true in higher dimensions(n?3)?However,their method was restricted to n=2,so when n?3 we consider a class of metrics called locally conformal Kahler metrics.The details are as follows:Firstly,based on Liu-Yang's study of Ricci curvatures on Hermitian manifolds in[11],we give some formulae about Ricci curvatures and scalar curvatures on locally conformal Kahler manifolds.Secondly,we prove that a compact locally conformal Kahler manifolds with con-stant nonpositive holomorphic sectional curvature is Kahler.In particular,its universal cover is complex hyperbolic space or complex Euclidean space.In addition,we prove that a compact locally conformal Kahler manifolds with pointwise constant nonposi-tive holomorphic sectional curvature(under such a more general condition)is globally conformal Kahler.Then,we give examples of complete non-Kahler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensor.Finally,we study k-Gauduchon metrics and obtain some propositions equivalent to k-Gaudouchon metrics.