Conformally flat Lorentz hypersurfaces.
type: Item
instance of: scholarly article
title: Conformally flat Lorentz hypersurfaces.
zbMATH Open document ID: 1035.53097
DOI: 10.1016/S0393-0440(02)00177-8
author: Mohammed Guediri
published in: Journal of Geometry and Physics
publication date: 2003-08-13T00:00:00Z
review text: The conformally flat Lorentz hypersurfaces \(M^n\) of dimension \(n\geq 4\) in indefinite space forms \(\widetilde M^{n+1}(\widetilde c)\) are investigated. Here \(\widetilde M^{n+1}(\widetilde c)\) is a complete simply connected \((n+1)\)-dimensional Lorentz manifold of constant curvature \(\widetilde c\). In case \(\widetilde M^{n+1}(\widetilde c)= \mathbb{R}^{n+1}_1\) these hypersurfaces are locally classified by \textit{I. van de Woestijne} and \textit{L. Verstraelen} [Tôhoku Math. J., II. Ser. 39, 81--88 (1987; Zbl 0627.53042)]. In the present paper first a classification of shape operators \(A_r\) of conformally flat Lorentz hypersurfaces \(M^n\) for \(x\in M^n\) in space forms is given. Then the standard examples of Lorentz hypersurfaces are presented as models for the further classification. If \(A_x\) is nonsingular and has a simple eigenvalue, the point \(x\in M^n\) is called bad, all other points are called good. If all points are good, then \(M^n\) is said to be good. In the main result all conformally flat good Lorentz hypersurfaces \(M^n\) for \(n\geq 4\) in \(M^{n+1}(c)\) are explicitly described and classified. For the bad points it is established that if \(M^n\) of scalar curvature \(r\), \(n\geq 4\), is a conformally flat Lorentz hypersurface in \(\widetilde M^{n+1}(\widetilde c)\) with \(\widetilde c> r/n(n-1)\) everywhere, then all points are bad, and \(M^n\) is foliated by \((n-1)\)-dimensional spaces of constant curvature \(>\widetilde c\).
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1963589
zbMATH Keywords: conformally flat manifolds
zbMATH Keywords: isometric immersions
zbMATH Keywords: shape operators
zbMATH Keywords: relative nullities
zbMATH Keywords: \(B\)-scrolls
zbMATH Keywords: generalized umbilical hypersurfaces
zbMATH Keywords: Lorentz hypersurfaces
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1963589