In recent years, the geometry of Hermitian manifolds has regenerated interests, with the intent of pushing analysis on Kaehler manifolds to general Hermitian ones, and also with the study of non-Kaehler Calabi-Yau manifolds from string theory.

For a given Hermitian metric on a complex manifold, there are two canonical connections associated with the metric, namely, the Hermitian (aka Chern) connection r and the Riemannian (aka Levi-Civita) connection. The former is the unique connection that is compatible with both the metric and the complex structure, while the latter is the unique torsion-free connection that is compatible with the metric. These two connection coincide precisely when the metric is Kaehler.

In this talk, we will explore properties and behaviors of the curvature tensors of the Hermitian and the Riemannian connection of a Hermitian manifold, and examine some conditions on the curvature that will lead to the Kaehlerness of the metric.