Generalized Hopf manifolds

Abstract

The first example in the preceding section led to the study of locally conformal Kahler manifolds with a parallel Lee form. These are calledVaisman manifolds. I. Vaisman, to whom the notion is due (cf. [269] and [2751] adopts the terminologygeneralized Hopf (g.H.) manifolds.However, the term g.H. manifold is sometimes used to label a different generalization of complex Hopf manifolds (i.e. E. Brieskorn & A. Van de Ven’s g.H. manifolds H a n (b),[45], cf. our Section 3.7). In the following, we use both terminologies interchangeably. The example of an Inoue surface with the Tricerri metric shows that g.H. manifolds form a proper subset of the set of all l.c.K. manifolds.