Transformation Groups

, Volume 13, Issue 1, pp 25–45 | Cite as




In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S 1. Examples of unit Killing vector fields generated by almost free but not free actions of S 1 on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated.

Key words and phrases

Riemannian manifolds Killing vector fields Clifford-Wolf translations circle actions geodesics homogeneous spaces symmetric spaces 

AMS classification

53C20 (primary) 53C25 53C35 (secondary) 


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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Omsk Branch of the Sobolev Institute of Mathematics SD RASOmsk ul. PevtsovaRussia
  2. 2.Rubtsovsk Industrial Institute of Altay State Technical UniversityRubtsovsk ul. TraktornayaRussia

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