Killing vector fields of constant length on locally symmetric Riemannian manifolds
type: Item
instance of: scholarly article
title: Killing vector fields of constant length on locally symmetric Riemannian manifolds
zbMATH Open document ID: 1151.53046
published in: Transformation Groups
publication date: 2008-09-04T00:00:00Z
review text: Among others, the following results are obtained. {\parindent7mm \begin{itemize}\item[(1)] If \(M\) is a symmetric Riemannian manifold and \(X\) is a Killing field of constant length on \(M\), then the one-parameter isometry group generated by \(X\) consists of Clifford-Wolf translations. If, in particular, \(M\) is of positive sectional curvature, then the flow of \(X\) induces a free isometric \(S^1\)-action on \(M\). \item[(2)] If \(X\) is a Killing field of constant length on a symmetric Riemannian manifold, then \(X\) is either regular or it has no closed integral curves. (This is a corollary of (1).) \item[(3)] Sufficient conditions for the existence of quasi-regular Killing fields of constant length on a non simply connected Riemannian manifold are deduced. \item[(4)] A criterion is obtained for the existence of quasi-regular Killing fields of constant length on a locally Euclidean space. \item[(5)] A homogeneous Riemannian manifold of constant positive sectional curvature does not admit quasi-regular Killing fields of constant length, if and only if, \(M\) is either a Euclidean sphere or a real projective plane. \end{itemize}} Also, some open problems are listed.
reviewed by: József Szilasi
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 5320883
zbMATH Keywords: Killing field of constant length
zbMATH Keywords: Clifford-Wolf translation
zbMATH Keywords: Riemannian symmetric manifold
author: V. N. Berestovskij
author: Yurii G. Nikonorov
MaRDI profile type: Publication
full work available at URL: s00031-008-9000-6
OpenAlex ID: W2034125185
Wikidata QID: Q115389527
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DOI: 10.1007/S00031-008-9000-6
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description: scientific article; zbMATH DE number 5320883