Pseudo-Bochner curvature tensor on Hermitian manifolds

Our main purpose of this paper is to introduce a natural generalization ${\displaystyle B_H}$ of the Bochner curvature tensor on a Hermitian manifold ${\displaystyle M}$ provided with the Hermitian connection. We will call ${\displaystyle B_H}$ the pseudo-Bochner curvature tensor. Firstly, we introduce a unique tensor P, called the (Hermitian) pseudo-curvature tensor, which has the same symmetries as the Riemannian curvature tensor on a Kähler manifold. By using P, we derive a necessary and sufficient condition for a Hermitian manifold to be of pointwise constant Hermitian holomorphic sectional curvature. Our pseudo-Bochner curvature tensor ${\displaystyle B_H}$ is naturally obtained from the conformal relation for the pseudo-curvature tensor P and it is conformally invariant. Moreover we show that ${\displaystyle B_H}$ is different from the Bochner conformal tensor in the sense of Tricerri and Vanhecke.