Conformally flat and locally conformal Kähler manifolds
The natural generalization of space forms, i.e., spaces of constant curvature, are conformally flat Riemannian manifolds equipped with certain structures, for example, locally equipped with a conformal Kähler structure. \textit{A. Gray} and \textit{L. Vanhecke} [Cas. Pestovani Mat. 104, 170-179 (1979; Zbl 0413.53011)] have constructed the examples of locally conformal Kähler manifolds with a pointwise constant holomorphic sectional curvature. In this paper the authors study conformally flat locally conformal Kähler manifolds of the most general form and they find conditions for the curvature as well as the holomorphic sectional curvature of a conformally flat locally conformal Kähler manifold to be constant. Moreover, they show that a compact conformally flat locally conformal Kähler manifold \(M^ n\), \(n > 4\), of non-positive scalar curvature is locally holomorphically isometric to the complex Euclidean space. A similar result is obtained for a conformally flat locally conformal Kähler manifold of nonpositive pointwise constant holomorphic sectional curvature. Also, they prove that a conformally flat generalized Hopf manifold \(M^{2m}\) is locally holomorphically isometric to a manifold of the form \(S^{2m - 1} \times \mathbb{R}\) equipped with a special structure.
scientific article; zbMATH DE number 525671
scientific article