Semi-parallel hypersurfaces of a real space form
type: Item
instance of: scholarly article
title: Semi-parallel hypersurfaces of a real space form
zbMATH Open document ID: 0765.53012
DOI: 10.1007/BF02776024
published in: Israel Journal of Mathematics
publication date: 1992-12-16T00:00:00Z
review text: Let \(M\) be a hypersurface of an \((n+1)\)-dimensional standard space \(N\) of constant curvature \(c\). \(M\) is said to be semi-parallel, iff the second fundamental form \(h: TM\times_ M TM\to\mathbb{R}\) of \(M\) satisfies \(h(R(X,Y)U,V)+h(U,R(X,Y)V)=0\), where \(R\) denotes the curvature tensor of \(M\). The author shows that in case \(c\neq 0\) there are three (and no other) possibilities for semi-parallel hypersurfaces \(M\): (1) \(n=2\) and \(M\) is flat, (2) \(M\) is parallel, and (3) \(M\) is a rotation hypersurface whose radius function depends linearly on the arc length parameter of the profile curve. This classification follows from the fact that \(M\) has at most two principal curvatures \(\lambda\) and \(\mu\) everywhere, which satisfy \(\lambda\cdot\mu+c=0\) in case \(\lambda\neq\mu\). Semi-parallel hypersurfaces in Euclidean space were already classified by \textit{J. Deprez} [Rend. Semin. Mat., Torino 44, 303-316 (1986; Zbl 0616.53018)].
reviewed by: Helmut Reckziegel
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 78182
zbMATH Keywords: rotation hypersurface
zbMATH Keywords: principal curvatures
author: Franki Dillen
MaRDI profile type: Publication
Wikidata QID: Q124956039
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full work available at URL: https://doi.org/10.1007/bf02776024
OpenAlex ID: W2003942077
description: scientific article; zbMATH DE number 78182