Hypersurfaces of a Lorentz space form
type: Item
instance of: scholarly article
title: Hypersurfaces of a Lorentz space form
zbMATH Open document ID: 0806.53057
DOI: 10.1007/BF01189830
author: Qing-Ming Cheng
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: In this paper, we completely classify complete spacelike hypersurfaces in an anti-de Sitter space \(H^ 4_ 1(-c)\) with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature and give some characterizations on the complete spacelike hypersurfaces in a de Sitter space \(S^{n + 1}_ 1(c)\) with \(h^ 2 = 4(n - 1)c\). Namely we prove the following: Let \(M\) be a 3-dimensional complete spacelike hypersurface of \(H^ 4_ 1 (-c)\) with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature. Then \(M\) is totally umbilic or the hyperbolic cylinder \(H^ 1(c_ 1) \times H^ 2(c_ 2)\). Theorem 4.2. Let \(M\) be an \(n\)-dimensional complete and noncompact spacelike hypersurface of a de Sitter space \(S^{n + 1}_ 1 (c)\) with \(h^ 2 = 4(n - 1)c\). Then \(M\) is the hyperbolic cylinder \(H^ 1(c_ 1) \times S^{n - 1}(c_ 2)\) or \(M\) has unbounded volume and positive Ricci curvature, and the squared norm \(S\) of the second fundamental form is a subharmonic function with \(\sup S = nc\) and \(S < nc\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 687046
zbMATH Keywords: complete spacelike hypersurfaces
zbMATH Keywords: anti-de Sitter space
zbMATH Keywords: constant mean curvature
zbMATH Keywords: constant quasi-Gauss-Kronecker curvature
zbMATH Keywords: totally umbilic
zbMATH Keywords: hyperbolic cylinder
MaRDI profile type: Publication
cites work: Complete space-like hypersurfaces with constant mean curvature in a Lorentz space form of dimension 4
cites work: On spacelike hypersurfaces with constant mean curvature in the de Sitter space
cites work: Complete maximal spacelike hypersurfaces of \(H_ 1^ 4(c)\)
cites work: Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector
cites work: Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature
cites work: Isometric immersions of Riemannian manifolds
cites work: Q3029747
cites work: Q4102442
description: scientific article; zbMATH DE number 687046