Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms
type: Item
instance of: scholarly article
title: Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms
zbMATH Open document ID: 0906.53042
author: Francisco J. M. Estudillo
author: Alfonso Romero
author: Luis J. Alías
published in: Tsukuba Journal of Mathematics
publication date: 1999-02-09T00:00:00Z
review text: The study of constant mean curvature spacelike hypersurfaces in Lorentzian space forms, or, more generally, spacelike submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms, has been of substantial interest in the last years. Quite recently, the first and third authors developed some integral formulas for compact spacelike submanifolds in de Sitter spaces and applied them to the case of parallel mean curvature vector, obtaining some interesting characterizations of the totally geodesic and the totally umbilical round spheres [\textit{L. J. Alías} and \textit{A. Romero}, Manuscr. Math. 87, 405-416 (1995; Zbl 0838.53043)]. In the paper under review, the authors study compact spacelike \(n\)-submanifolds with (nonzero) parallel mean curvature vector in a pseudo-Riemannian space form \(N^{n+p+1}_{p}(c)\) with constant curvature \(c\) and Lorentzian normal bundle. They use the same approach as above, but the family of ambient spaces is extended in order to consider flat and negatively curved space forms. In particular, they prove that a compact spacelike submanifold \(M^n\) in \(N^{n+p+1}_{p}(c)\) is a totally umbilical and non-totally geodesic hypersurface in the corresponding Riemannian space form \(N^{n+1}(c)\subset N^{n+p+1}_p(c)\) if and only if \(M^n\) is pseudo-umbilical with nonzero spacelike parallel mean curvature vector. As a consequence of this, in the case where \(n=2\), they conclude that the only topological 2-spheres which are spacelike surfaces in \(N^{3+p}_p(c)\) with nonzero spacelike parallel mean curvature vector are the totally umbilical ones in \(N^3(c)\subset N^{3+p}_p(c)\). Finally, the authors also derive some other consequences for the case where the mean curvature vector has another causal character, namely, lightlike or timelike.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1102918
zbMATH Keywords: spacelike submanifold
zbMATH Keywords: totally umbilical submanifold
zbMATH Keywords: parallel mean curvature vector
zbMATH Keywords: pseudo-Riemannian space forms
Wikidata QID: Q115233546
reviewed by: Ángel Ferrández
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163169
OpenAlex ID: W1855915878
DOI: 10.21099/TKBJM/1496163169
description: scientific article; zbMATH DE number 1102918