Stability of unit Hopf vector fields on quotients of spheres
type: Item
instance of: scholarly article
title: Stability of unit Hopf vector fields on quotients of spheres
zbMATH Open document ID: 1195.53051
author: Vincent Borrelli
author: Hanifi Zoubir
published in: Differential Geometry and its Applications
publication date: 2010-07-08T00:00:00Z
review text: The goal of this work is to investigate the problem of the existence of a critical radius for quotients of spheres. Any space form with positive sectional curvature is a quotient of \(S^{2m-1}(r)\) by a finite fixed point free isometry subgroup \(\Gamma\) of \(O(2m+2)\). It turns out that there is still (at last) a Hopf field \(H_r\) on the quotient \(S^{2m+1}(r)/\Gamma\) which remains critical for volume functional. But something a priori unexpected occurs: the field \(H_r\) is always stable, whatever the radius \(r> 0\). Theorem. Let \(M= S^{2m+1}(r)/\Gamma\) be a space form with \(\Gamma\neq(\text{id})\), then the Hopf field \(H_r\) is stable on \(S^{2m+1}(r)/\Gamma\).
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5731969
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Hopf vector field
zbMATH Keywords: Hopf fibration
zbMATH Keywords: Hessian
zbMATH Keywords: space form
zbMATH Keywords: volume functional
zbMATH Keywords: stability
reviewed by: Mihail B. Banaru
MaRDI profile type: Publication
full work available at URL: j.difgeo.2010.02.002
OpenAlex ID: W1978556360
Wikidata QID: Q115357109
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DOI: 10.1016/J.DIFGEO.2010.02.002
description: scientific article; zbMATH DE number 5731969