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

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Abstract

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.

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Author notes

  1. Paolo Piccione

    Present address: Department of Mathematics, University of Murcia, Campus de Espinardo, 30100, Espinardo, Murcia, Spain

Authors and Affiliations

  1. Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Teatinos s/n, 29071, Málaga, Spain

    José Luis Flores

  2. Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain

    Miguel Ángel Javaloyes

  3. Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, CEP 05508-900, Brazil

    Paolo Piccione

Authors
  1. José Luis Flores
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  2. Miguel Ángel Javaloyes
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  3. Paolo Piccione
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Corresponding author

Correspondence to José Luis Flores.

Additional information

The authors are grateful to Andrea Spiro for giving several interesting suggestions on isometric group actions on manifolds. J. L. Flores thanks the Departamento de Matemática, Universidade de São Paulo, where this work started, for its kind hospitality during his stay there. He is partially supported by Spanish MEC-FEDER Grant MTM2007-60731 and Regional J. Andalucía Grant P06-FQM-01951. M. Á. Javaloyes was partially supported by Regional J. Andalucía Grant P06-FQM-01951 and by Spanish MEC Grant MTM2007-64504. P. Piccione is sponsored by Capes, Brasil, Grant BEX 1509-08-0.

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Flores, J.L., Javaloyes, M.Á. & Piccione, P. Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field. Math. Z. 267, 221–233 (2011). https://doi.org/10.1007/s00209-009-0617-5

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  • DOI: https://doi.org/10.1007/s00209-009-0617-5

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