Conformally flat and locally conformal Kähler manifolds
type: Item
instance of: scholarly article
title: Conformally flat and locally conformal Kähler manifolds
zbMATH Open document ID: 0809.53037
DOI: 10.1007/BF01262178
author: V. F. Kirichenko
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: 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.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 525671
zbMATH Keywords: conformally flat
zbMATH Keywords: conformal Kähler structure
zbMATH Keywords: locally conformal Kähler manifolds
zbMATH Keywords: holomorphic sectional curvature
zbMATH Keywords: generalized Hopf manifold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 525671