A note on compact hypersurfaces in a Euclidean space
type: Item
instance of: scholarly article
title: A note on compact hypersurfaces in a Euclidean space
zbMATH Open document ID: 1260.53019
author: Sharief Deshmukh
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2012-12-13T00:00:00Z
review text: An affirmative answer to the question ``Does an orientable compact hypersurface in a Euclidean space that admits a unit Killing vector field \(\xi\) together with the condition that \(g(A\xi,\xi)\) is a constant necessary isometric to a sphere?'' is given in the form of the following theorem. Theorem. Let \(M\) be a compact orientable hypersurface of the Euclidean space \(\mathbb R^{n+1}\). \(M\) admits a unit Killing vector field \(\xi\) satisfying that, with respect to the induced metric \(g\) and the shape operator \(A\), \(g(A\xi,\xi)\) is a constant if and only if \(n=2m+1\) and \(M\) is isometric to the sphere \(S^{2m+1}(c)\) of constant curvature \(c\).
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 6115455
zbMATH Keywords: Killing vector field
zbMATH Keywords: compact hypersurface
zbMATH Keywords: shape operator
zbMATH Keywords: sphere
reviewed by: Huili Liu
MaRDI profile type: Publication
full work available at URL: j.crma.2012.10.027
OpenAlex ID: W1993343306
cites work: Q3545583
cites work: Q4720334
cites work: Contact manifolds in Riemannian geometry
cites work: Q4397392
cites work: Q3413014
cites work: REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM
cites work: Q3407102
cites work: Sufficient Conditions for Periodicity of a Killing Vector Field
cites work: Positive curvature, local and global symmetry, and fundamental groups
cites work: Killing Vector Fields on Complete Riemannian Manifolds
DOI: 10.1016/J.CRMA.2012.10.027
description: scientific article; zbMATH DE number 6115455