A Note on Sectional Curvatures of Hermitian Manifolds

Abstract

First, we derive expression of the Chern sectional curvature of a Hermitian manifold in local complex coordinates. As an application, we find that a Hermitian metric is K\"ahler if the Riemann sectional curvature and the Chern sectional curvature coincide. Second, we prove that the sectional curvature restricted to orthogonal 2-planes of a G-K\"ahler-like manifold with non-negative (resp. non-positive) sectional curvature can take its maximum (resp. minimum) at a holomorphic plane section. And we also prove that the holomorphic bisectional curvature of a K\"ahler-like manifold with non-negative (resp. non-positive) Chern sectional curvature can take its maximum (resp. minimum) at the holomorphic sectional curvature.