The Chern Sectional Curvature of a Hermitian Manifold
On a Hermitian manifold, the Chern connection can induce a metric connection on the background Riemannian manifold. We call the sectional curvature of the metric connection induced by the Chern connection the Chern sectional curvature of this Hermitian manifold. First, we derive expression of the Chern sectional curvature 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. At last, we prove that the scalar curvature of the metric connection induced by the Lichnerowicz connection and that of the Levi-Civita connection coincide on a compact Hermitian manifold if and only if the Hermitian manifold is balanced.