Generalized Hopf manifolds

Let (M,J,g) be a Hermitian manifold with complex structure J, metric g, and Kähler form Ω. Then g is locally conformal Kähler iff dΩ=ω ∧ Ω for some closed and non-exact 1-form ω. Moreover, if ω is a parallel form, M is called a generalized Hopf manifold. The main results of this paper are: (a) the description of the geometric structure of the compact locally conformal Kähler-flat manifolds; (b) the description of the geometric structure of the compact generalized Hopf manifolds on which a certain canonically defined foliation is regular; (c) a description of the harmonic forms and Betti numbers of a general compact generalized Hopf manifold; (d) a method for studying analytic vector fields on generalized Hopf manifolds; (e) conditions for submanifolds of generalized Hopf manifolds to belong to the same class.