Integral formulas for compact spacelike \(n\)-submanifolds in de Sitter spaces. Applications to the parallel mean curvature vector case
type: Item
instance of: scholarly article
title: Integral formulas for compact spacelike \(n\)-submanifolds in de Sitter spaces. Applications to the parallel mean curvature vector case
zbMATH Open document ID: 0838.53043
DOI: 10.1007/BF02570483
author: Luis J. Alías
author: Alfonso Romero
published in: Manuscripta Mathematica
publication date: 1996-01-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156096
review text: The authors introduce a new method to study compact spacelike \(n\)-submanifolds in de Sitter spaces \(S^{n+ p}_q\) by means of certain integral formulas. Several interesting results are then established. Theorem 3.1. Let \(M^n\) be a complete maximal submanifold in \(S^{n+ p}_q\), \(1\leq q\leq p\), such that \(\text{Ric }\geq n- 1\). Then \(M^n\) is totally geodesic in \(S^{n+ q}_q\). For surfaces: The only complete maximal surfaces in \(S^{2+ p}_q\), \(1\leq q\leq p\), with Gaussian curvature \(\geq 1\) are the totally geodesic ones. Several applications are given to the parallel mean curvature case; for instance Theorem 5.2: Let \(M^2\) be a compact spacelike surface with non-zero parallel mean curvature vector \(H\) in \(S^{2+ p}_q\), \(1\leq q\leq p\). If \(M^2\) is a topological sphere and its Gaussian curvature satisfies \(K\geq 1+ \langle H, H\rangle\), then \(M^2\) is totally umbilical.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 824168
zbMATH Keywords: integral formulas
zbMATH Keywords: maximal submanifold
zbMATH Keywords: de Sitter space
zbMATH Keywords: totally geodesic
zbMATH Keywords: parallel mean curvature vector
reviewed by: Bernard Rouxel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 824168