Pseudo-Bochner curvature tensor on Hermitian manifolds

Koji Matsuo¹

¹

Colloquium Mathematicum, Tome 80 (1999) no. 2, pp. 201-209.

Résumé

Our main purpose of this paper is to introduce a natural generalization $B_H$ of the Bochner curvature tensor on a Hermitian manifold $M$ provided with the Hermitian connection. We will call $B_H$ 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 $B_H$ is naturally obtained from the conformal relation for the pseudo-curvature tensor P and it is conformally invariant. Moreover we show that $B_H$ is different from the Bochner conformal tensor in the sense of Tricerri and Vanhecke.

Détail

DOI : 10.4064/cm-80-2-201-209

Keywords: Hermitian manifold, Hermitian connection, pseudo-Bochner curvature tensor, (Hermitian) pseudo-curvature tensor