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
Article

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.

Mathematics Subject Classification (2000)

53C22 53C50 53C12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, S., Stuck, G.: The isometry group of a compact Lorentz manifold. I, II. Invent. Math. 129(2), 239–261, 263–287 (1997)Google Scholar
  2. 2.
    Baker A.: Matrix Groups. An Introduction to Lie Group Theory. Springer Undergraduate Mathematics Series. Springer, London (2002)Google Scholar
  3. 3.
    Beem J.K., Ehrlich P.E., Easley K.: Global Lorentzian Geometry. Marcel Dekker, New York (1996)MATHGoogle Scholar
  4. 4.
    Beem J.K., Ehrlich P.E., Markvorsen S.: Timelike isometries and Killing fields. Geom. Dedic. 26, 247–258 (1988)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Biliotti L., Mercuri F., Piccione P.: On a Gromoll–Meyer type theorem in globally hyperbolic stationary spacetimes. Comm. Anal. Geom. 16(2), 333–393 (2008)MATHMathSciNetGoogle Scholar
  6. 6.
    Bredon G.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972)Google Scholar
  7. 7.
    Cannas da Silva A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001)CrossRefGoogle Scholar
  8. 8.
    Caponio E., Masiello A., Piccione P.: Some global properties of static spacetimes. Math. Z. 244(3), 457–468 (2003)MATHMathSciNetGoogle Scholar
  9. 9.
    D’Ambra G.: Isometry groups of Lorentz manifolds. Invent. Math. 92, 555–565 (1988)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fintushel R.: Circle actions on simply connected 4-manifolds. Trans. Am. Math. Soc. 230, 147–171 (1977)MATHMathSciNetGoogle Scholar
  11. 11.
    Fintushel R.: Classification of circle actions on 4-manifolds. Trans. Am. Math. Soc. 242, 377–390 (1978)MATHMathSciNetGoogle Scholar
  12. 12.
    Galloway G.: Closed timelike geodesics. Trans. Am. Math. Soc. 285, 379–384 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Galloway G.: Compact Lorentzian manifolds without closed non spacelike geodesics. Proc. Am. Math. Soc. 98, 119–123 (1986)MATHMathSciNetGoogle Scholar
  14. 14.
    Giannoni F., Piccione P., Sampalmieri R.: On the geodesical connectedness for a class of semi- Riemannian manifolds. J. Math. Anal. Appl. 252(1), 444–476 (2000)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Guediri M.: On the existence of closed timelike geodesics in compact spacetimes. Math. Z. 239, 277–291 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Guediri M.: On the existence of closed timelike geodesics in compact spacetimes. II. Math. Z. 244, 577–585 (2003)MATHMathSciNetGoogle Scholar
  17. 17.
    Guediri M.: On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds. Trans. Am. Math. Soc. 355, 775–786 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Guediri M.: A new class of compact spacetimes without closed causal geodesics. Geom. Dedic. 126, 177–185 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Guediri M.: Closed timelike geodesics in compact spacetimes. Trans. Am. Math. Soc. 359(6), 2663–2673 (2007)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Knapp A.W.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140. Birkhõuser, Boston (1996)Google Scholar
  21. 21.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. I. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, New York (1996)Google Scholar
  22. 22.
    Kobayashi, S.: Transformation Groups in Differential geometry, Reprint of the 1972 edition. Classics in Mathematics. Springer, Berlin (1995)Google Scholar
  23. 23.
    Kollár J.: Circle actions on simply connected 5-manifolds. Topology 45, 643–671 (2006)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Marzantowicz W.: A G-Lusternik–Schnirelman category of space with an action of a compact Lie group. Topology 28, 403–412 (1989)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Masiello A.: On the existence of a closed geodesic in stationary Lorentzian manifolds. J. Differ. Equ. 104, 48–59 (1993)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Molino P.: Riemannian Foliations. Progress in Mathematics, vol. 73. Birkhõuser, Boston (1988)Google Scholar
  27. 27.
    O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)MATHGoogle Scholar
  28. 28.
    Piccione, P., Zeghib, A.: On the isometry group of a compact stationary Lorentzian manifold. Preprint (2009)Google Scholar
  29. 29.
    Romero A., Sánchez M.: Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field. Proc. Am. Math. Soc. 123(9), 2831–2833 (1995)MATHGoogle Scholar
  30. 30.
    Sachs R., Wu H.: General Relativity for Mathematicians. Graduate Texts in Mathematics, vol. 48. Springer, New York (1977)Google Scholar
  31. 31.
    Sánchez M.: Structure of Lorentzian tori with a Killing vector field. Trans. Am. Math. Soc. 349(3), 1063–1080 (1997)MATHCrossRefGoogle Scholar
  32. 32.
    Sánchez, M.: Lorentzian manifolds admitting a Killing vector field. In: Proceedings of the Second World Congress of Nonlinear Analysts, Part 1 (Athens, 1996). Nonlinear Anal. 30 (1), 643–654 (1997)Google Scholar
  33. 33.
    Sánchez M.: On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes. Differ. Geom. Appl. 24(1), 21–32 (2006)MATHCrossRefGoogle Scholar
  34. 34.
    Scott P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)MATHCrossRefGoogle Scholar
  35. 35.
    Seifert H.: Topologie dreidimensionaler gefaserte Räume. Acta Math. 60, 148–238 (1932)MathSciNetGoogle Scholar
  36. 36.
    Tipler F.J.: Existence of closed timelike geodesics in Lorentz spaces. Proc. Am. Math. Soc. 76, 145–147 (1979)MATHMathSciNetGoogle Scholar
  37. 37.
    Zeghib, A.: Sur les espaces-temps homogènes, The Epstein Birthday Schrift, pp. 551–576 (electronic). Geom. Topol. Monogr., 1, Geom. Topol. Publ. Coventry (1998)Google Scholar

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

Personalised recommendations