On locally conformal Kähler structures

Abstract

A locally conformal Kahler manifold is introduced in [7] as a Hermitian manifold whose metric is locally conformal to a Kahler metric. As a special case, a generalized Hopf manifold has been introduced, which is topologically different from a Kahler manifold if it is compact. In the first half of this paper, we will discuss the Riemannian curvature tensor of a generalized Hopf manifold in the case when holomorphic sectional curvature is constant except for a certain section. In the second half, we study a Riemannian manifold which admits more than one locally conformal Kahler structures with some relations.