The Chern sectional curvature of a Hermitian manifold (Q6614549)

Let \((M,h)\) be a Hermitian manifold and let \(g=\Re (h)\) be the background Riemannian metric associated to the metric \(h\). As it is known, the Chern connection on the manifold \((M,h)\) induces the metric connection of the background Riemannian manifold.\N\NThe authors study the sectional curvature of the metric connection induced by the Chern connection under the name of Chern sectional curvature of this Hermitian manifold.\N\NThe authors obtain the expression of the Chern sectional curvature in local complex coordinates. Some formulae for Ricci curvature and scalar curvature of the metric connection induced by the Chern connection are also obtained.\N\NThe main result of the paper is the following: it is proved that if \((M,h)\) is a Hermitian manifold such that the Riemannian sectional curvature and the Chern sectional curvature coincide, then the metric \(h\) is Kähler.