Hypersurfaces of restricted type in Minkowski space
type: Item
instance of: scholarly article
title: Hypersurfaces of restricted type in Minkowski space
zbMATH Open document ID: 0874.53006
DOI: 10.1007/BF00181571
author: Christos Baikoussis
author: Filip Defever
author: David E. Blair
author: Bang-Yen Chen
published in: Geometriae Dedicata
publication date: 1997-10-14T00:00:00Z
review text: A submanifold \(M^n_r\), \(n\) being the dimension and \(r\) the index, of Minkowski space \(E^m_1\) is said to be of restricted type if its shape operator, with respect to the mean curvature vector, is the restriction of a fixed linear transformation of \(E^m_1\) to the tangent space of \(M^n_r\) at every point of \(M^n_r\). The study of this kind of submanifolds was initiated by \textit{B. Y. Chen, F. Dillen, L. Verstraelen} and \textit{L. Vranken} [J. Geom. 46, 20-32 (1993; Zbl 0778.53002)] where they stated the problem of classifying all submanifolds of restricted type of pseudo-Euclidean spaces. As for the classification of hypersurfaces of restricted type in \(E^{n+1}_1\), the authors show that it suffices to establish which of the parallel hypersurfaces of \(E^{n+1}_1\) are of restricted type. To do that the authors deal with the minimal of the shape operator, showing that the hypersurface has to be isoparametric. Then, following the papers by \textit{H. B. Lawson} [Ann. Math. 89, 187-197 (1969; Zbl 0174.24901)] and \textit{M. A. Magid} [Tsukuba J. Math. 8, 31-54 (1984; Zbl 0549.53052)], they obtain the main result: The only hypersurfaces of restricted type of \(E^{n+1}_1\) are (i) the minimal ones; (ii) an open part of \(S^k \times E^{n-k}_1\), \(S^k_1 \times E^{n-k}\), \(H^k \times E^{n-k}\), \(S^k_1\) and \(H^n\), with \(1\leq k\leq n-1\); and (iii) an open part of a cylinder on a plane curve of restricted type.
reviewed by: Ángel Ferrández
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 929921
zbMATH Keywords: Minkowski space
zbMATH Keywords: shape operator
zbMATH Keywords: classification of hypersurfaces of restricted type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181571
OpenAlex ID: W1992912986
description: scientific article; zbMATH DE number 929921