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 Kähler metric. We will call such metrics G-Kähler-like or Kähler-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-Kähler 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.

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature H, and when the metric is non-Kähler, it is slightly stronger than H. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.It is a natural question to ask when will a Hermitian manifold have constant real bisectional curvature. To this end, we have the following:Theorem 1.4. Let (M n , g) be a compact Hermitian manifold whose real bisectional curvature is constantly equal to c. Then c ≤ 0. Moreover, when c = 0, then (M, g) is a balanced manifold with vanishing first, second, and third Ricci tensors, and its Chern curvature satisfies the property R xyuv = −R uvxy for any type (1, 0) complex tangent vectors x, y, u, v.We would like to propose the following conjecture:Conjecture 1.5. Let M n (n ≥ 3) be a compact Hermitian manifold with vanishing real bisectional curvature c. Then c = 0, and its Chern curvature tensor R = 0.By Boothby's theorem, compact Hermitian manifolds with vanishing Chern curvature are precisely the compact quotients of complex Lie groups equipped with left invariant metrics.Besides the constant real bisectional curvature cases, more generally, it would certainly be very interesting to try to understand the class of all compact Hermitian

In this paper, we consider orthogonal Ricci curvature Ric ⊥ for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate their subtle relationship. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. The simply-connectedness is also shown when the complex dimension is smaller than five.