Complete maximal spacelike submanifolds
type: Item
instance of: scholarly article
title: Complete maximal spacelike submanifolds
zbMATH Open document ID: 0908.53033
author: Qing-Ming Cheng
author: Susumu Ishikawa
published in: Kodai Mathematical Journal
publication date: 1998-10-18T00:00:00Z
review text: The authors obtain some interesting results on complete maximal spacelike \(n\)-submanifolds in de Sitter spaces \({\text S}^{n+p}_{q}(c)\), \(1\leq q\leq p\), where by maximal here it is meant that the mean curvature vector field vanishes everywhere. In a previous paper [\textit{L. J. Alías} and \textit{A. Romero}, Manuscr. Math. 87, 405-416 (1995; Zbl 0838.53043)], the reviewer and Romero obtained some integral formulas for compact spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\), with applications to the case of parallel mean curvature vector field. In particular, we characterized the totally geodesic submanifolds of \({\text S}^{n+p}_{q}(c)\) as the only complete maximal \(n\)-submanifolds satisfying \(\text{Ric}\geq (n-1)c\), and constructed elementary counterexamples for \(\text{Ric}\leq (n-2)c\). On the other hand, some other authors have recently studied the case of parallel mean curvature when \(p=q\) by using the classical Simons' formula for the Laplacian of the squared norm \(S\) of the second fundamental form. In the paper under review, the authors generalize Simons' method to complete spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\). As a consequence, they reprove and extend our previous result when \(q=1\), \(p=2\), including also the case \((n-2)c < \text{Ric} < (n-1)c\). In particular, they obtain that complete maximal spacelike \(n\)-submanifolds in \({\text S}^{n+2}_{1}(c)\) satisfying \(\text{Ric}\geq (n-2)c\) are either totally geodesic or Einstein with \(\text{Ric}= (n-2)c\) and parallel second fundamental form. For general \(p,q\), \(1\leq q\leq p\), they prove that the only compact maximal spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\) with \(S\leq \max\{ nc/(2-1/(p-q)), 2nc/3 \}\) are either totally geodesic or some concrete submanifolds (where the equality must be reached at some point). These concrete submanifolds appear just when \(p=q+1\) (a Clifford torus in the totally geodesic spacelike submanifold \({\…
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1180763
zbMATH Keywords: spacelike maximal submanifold
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: scalar curvature
zbMATH Keywords: Ricci curvature
zbMATH Keywords: sectional curvature
zbMATH Keywords: Simon's method
reviewed by: Luis J. Alías
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043791
OpenAlex ID: W2046577400
cites work: Integral formulas for compact spacelike \(n\)-submanifolds in de Sitter spaces. Applications to the parallel mean curvature vector case
cites work: Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
cites work: Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature
cites work: An intrinsic rigidity theorem for minimal submanifolds in a sphere
cites work: Minimal varieties in Riemannian manifolds
cites work: Submanifolds with Constant Mean Curvature
DOI: 10.2996/KMJ/1138043791
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description: scientific article; zbMATH DE number 1180763