Mathematische Zeitschrift

, Volume 267, Issue 1–2, pp 221–233

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

  • José Luis Flores
  • Miguel Ángel Javaloyes
  • Paolo Piccione


We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Mathematics Subject Classification (2000)

53C22 53C50 53C12 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • José Luis Flores
    • 1
  • Miguel Ángel Javaloyes
    • 2
  • Paolo Piccione
    • 3
    • 4
  1. 1.Departamento de Álgebra, Geometría y Topología, Facultad de CienciasUniversidad de MálagaMálagaSpain
  2. 2.Departamento de Geometría y Topología, Facultad de CienciasUniversidad de GranadaGranadaSpain
  3. 3.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil
  4. 4.Department of MathematicsUniversity of MurciaEspinardoSpain

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