Compact space-like \(m\)-submanifolds in a pseudo-Riemannian sphere \(S_ p^{m+p} (c)\)
type: Item
instance of: scholarly article
title: Compact space-like \(m\)-submanifolds in a pseudo-Riemannian sphere \(S_ p^{m+p} (c)\)
zbMATH Open document ID: 0842.53038
author: Reiko Aiyama
published in: Tokyo Journal of Mathematics
publication date: 1996-07-28T00:00:00Z
review text: The author deals with the problem whether or not there exists a compact spacelike \(m\)-dimensional submanifold in an \((m+ p)\)-dimensional pseudo-Riemannian sphere \(S^{m+ p}_p(c)\) with index \(p\) and sectional curvature \(c\in \mathbb{R}_+\), with parallel mean curvature vector which is not totally umbilical. She proves the following theorem: Let \(M\) be a compact spacelike \(m\)-dimensional submanifold in \(S^{m+ p}_p(c)\) with parallel mean curvature vector. If the normal connection is flat (in particular, this holds when \(p\leq 2\)), then \(M\) is totally umbilical. For \(m= 2\), she obtains that only compact spacelike surfaces in \(S^{2+ p}_p(c)\) with parallel mean curvature vector are totally umbilical ones. So, the answer to the problem in the case \(m= 2\) is nonexistence. This last result was independently obtained, using another method, by \textit{L. J. Alias} and the reviewer [Manuscr. Math. 87, 405-416 (1995)]. Finally, she also proves, as a consequence of the previous theorem, that given a compact spacelike \(m\)-dimensional submanifold \(M\) in \(S^{m+ p}_p(c)\) with parallel mean curvature vector then \(M\) is totally umbilical, if the sectional curvature of \(M\) is nonnegative. Thus, it seems that the answer to the problem could be nonexistence, although in general it remains open.
reviewed by: Alfonso Romero
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 821711
zbMATH Keywords: spacelike submanifold
zbMATH Keywords: flat normal connection
zbMATH Keywords: pseudo-Riemannian sphere
zbMATH Keywords: parallel mean curvature vector
zbMATH Keywords: totally umbilical
Wikidata QID: Q115219801
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270043610
OpenAlex ID: W1999474771
DOI: 10.3836/TJM/1270043610
description: scientific article; zbMATH DE number 821711