On Curvature Tensors of Hermitian Manifolds

Abstract: In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a K\"ahler metric. We will call such metrics G-K\"ahler-like or K\"ahler-like, for lack of better terminologies. Such metrics are always balanced when the manifold is compact, so in a way they are more special than balanced metrics, which drew a lot of attention in the study of non-K\"ahler Calabi-Yau manifolds. In particular we derive various formulas on the difference between the Riemannian and Hermitian curvature tensors in terms of the torsion of the Hermitian connection. We believe that these formulas could lead to further applications in the study of Hermitian geometry with curvature assumptions.