THE BOCHNER CURVATURE TENSOR ON ALMOST HERMITIAN MANIFOLDS
Since S. Bochner introduced in [2] the Bochner curvature tensor on a K~ihler manifold, several papers have appeared concerning such manifolds with vanishing Bochner curvature tensor. The main purpose of this paper is to consider the same tensor for the more general class of almost Hermitian manifolds and to prove some theorems about these manifolds with vanishing Bochner curvature tensor. In particular, we consider holomorphic and antiholomorphic submanifolds and obtain properties analogous to those proved in [1],
{14-] for K~hler manifolds. We also give some results on conformally flat, almost Hermitian manifolds.
- Let (]Q, g, J) be a C Β°o manifold which is almost Hermitian, that is, the tangent bundle has an almost complex structure J and a Riemannian metric g such that g(JX, JY) = g(X, Y) for all X, Y~ 33(2Q). ~(37Β’) denotes the Lie algebra of C ~ vector fields on ~r. If ~ is the Riemannian connection, then the curvature tensor/~ is given by ~(x, Y) = [~x, ~Y] -~Ex,Y~-Curvature identities are a key to understanding the geometry of some classes of almost Hermitian manifolds [5], [6]. In this paper we shall only consider the following identities: