The Chern Sectional Curvature of a Hermitian Manifold

TL;DR

This paper introduces the concept of Chern sectional curvature on Hermitian manifolds, deriving its local expression and exploring its implications for Kähler metrics and curvature properties.

Contribution

It defines and analyzes Chern sectional curvature, providing explicit formulas and linking it to Kähler conditions and classical curvature measures.

Findings

Chern sectional curvature formula derived in local coordinates

Hermitian metric is Kähler if Riemann and Chern sectional curvatures coincide

Ricci and scalar curvatures of the Chern-induced connection are obtained

Abstract

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. As subsequent results, Ricci curvature and scalar curvature of the metric connection induced by the Chern connection are obtained.