In this article, we examine the behavior of the Riemannian and Hermitiancurvature tensors of a Hermitian metric, when one of the curvature tensorsobeys 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 betterterminologies. 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 ofattention in the study of non-K"ahler Calabi-Yau manifolds. In particular wederive various formulas on the difference between the Riemannian and Hermitiancurvature tensors in terms of the torsion of the Hermitian connection. Webelieve that these formulas could lead to further applications in the study ofHermitian geometry with curvature assumptions.
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