Kaehlerian manifolds with vanishing Bochner curvature tensor

Abstract

Let (M, F, g) be a Kaehlerian manifold of real dimension n with almost complex structure F and Kaehlerian metric g.We cover M by a system of coordinate neighborhoods {£/; x h }, where here and in the sequel the indices h, i, j, k ••• run over the range {1, 2, -•• , n} and denote by g jit P 19 K kji h , K jit K and Fj % local components of g, the operator of covariant differentiation with respect to the Levi-Civita connection, the curvature tensor, the Ricci tensor, the scalar curvature and F of M respectively.The Bochner curvature tensor of M is defined to be [6]whereIt is known that a Kaehlerian manifold with vanishing Bochner curvature tensor is a complex analogue to a conformally flat Riemannian manifold and that the Bochner curvature tensor has properties quite similar to those of Weyl conformal curvature tensor.Recently, S. I. Goldbreg [1] proved THEOREM A. Let M be an n-dimensional (n^3) compact conformally flat Riemannian manifold with constant scalar curvature.If the length of the Ricci tensor is less than K/ Vn-1 , then M is a space of constant curvature.Also, S. I. Goldberg and M. Okumura [2] proved THEOREM B. Let M be an n-dimenswnal (n^3) compact conformally flat