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The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

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Abstract.

We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the existence of conjugate points along timelike geodesics. We show for instance that a nonsingular timelike geodesic which is translated by an element of the group has a conjugate point. For those that we will call of GH-type, and for those which are globally hyperbolic with a timelike center and a one-dimensional derived subgroup, we show that if in addition the derived subgroup is spacelike, then nonsingular timelike geodesics maximize distance up to and including the first conjugate point. Other related results and corollaries are also obtained.

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References

  1. Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. Pure and Applied Mathematics, Marcel Dekker Inc., 1996

  2. Cheeger, J., Ebin, D.: Comparison theorems in Riemannian Geometry. North- Holland, Amsterdam, 1975

  3. Cordero, L.A., Parker, P.E.: Pseudo-Riemannian 2-step nilpotent Lie groups. Preprint (1999)

  4. Eberlein, P.: Geometry of 2-step nilpotent groups with a left invariant metric Ann. Scient. Éc. Norm. Sup. 27, 611–660 (1994)

    MATH  Google Scholar 

  5. Eberlein, P.: Geometry of 2-step nilpotent groups with a left invariant metric.II. Trans. Amer. Math. Soc. 343, 805–828 (1994)

    Google Scholar 

  6. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Springer-Verlag, 1990

  7. Gornet, R., Mast, M.: Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds. J. Geom. Analysis 13, 107–143 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Guediri, M.: Sur la complétude des pseudo-métriques invariantes à gauche sur les groupes de Lie nilpotents. Rend. Sem. Mat. Univ. Pol. Torino 52, 371–376 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Guediri, M., Lafontaine, J.: Sur la complétude des variétés pseudo-riemanniennes. J. Geom. Phys. 15, 150–158 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guediri, M.: On the geodesic connectedness of simply connected Lorentz surfaces. Ann. Fac. Sci. de Toulouse VI(3) 499–510 (1997)

    Google Scholar 

  11. Guediri, M.: On the existence of closed timelike geodesics in compact spacetimes. Math. Z. 239, 277–291 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guediri, M.: On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds. Trans. Amer. Math. Soc. 355, 775–786 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Guediri, M.: Lorentz geometry of 2-step nilpotent Lie groups. Geom. Dedicata 100, 11–51 (2003)

    Article  MATH  Google Scholar 

  14. Guediri, M.: On the global hyperbolicity of Lorentz 2-step nilpotent Lie groups. General Relativity and Gravitation 35(9), 1707–1714 (2003)

    Article  MATH  Google Scholar 

  15. Hawking, S.W., Ellis, G.F.R.: The large scale structure of spacetime. Cambridge Univ. Press, Cambridge, 1973

  16. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978

  17. Jang, C., Park, K.: Conjugate points on 2-step nilpotent groups. Geom. Dedicata 79, 65–80 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Amer. Math. Soc. 258, 147–153 (1980)

    MathSciNet  MATH  Google Scholar 

  19. Kaplan, A.: Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11, 127–136 (1981)

    MathSciNet  MATH  Google Scholar 

  20. Kaplan, A.: On the geometry of groups of Heisenberg type. Bull. Lond. Math. Soc. 15, 35–42 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Lee, K.B., Park, K.: Smoothly closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 45, 1–14 (1996)

    MathSciNet  Google Scholar 

  22. Mast, M.: Closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 43, 885–911 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Mast, M.: Low-dimensional 2-step nilpotent Lie groups in resonance. Algebras Groups Geom. 14, 321–337 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Nomizu, K.: Left invariant Lorentz metrics on Lie groups. Osaka J. Math. 16, 143–150 (1979)

    MathSciNet  MATH  Google Scholar 

  25. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York, 1983

  26. Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, New York, 1972

  27. Tipler, F.T.: Existence of closed timelike geodesics in Lorentz spaces. Proc. Amer. Math. Soc. 76, 145–147 (1979)

    MathSciNet  MATH  Google Scholar 

  28. Walschap, G.: Cut and conjugate loci in two-step nilpotent Lie groups. J. Geom. Analysis 7, 343–355 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Wolf, J.A.: Spaces of constant curvature. Publish or Perish, Boston, 1974

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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Mohammed Guediri

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  1. Mohammed Guediri
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Correspondence to Mohammed Guediri.

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Mathematics Subject Classification (2000): 53C50, 53C22, 22E25

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Guediri, M. The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups. manuscripta math. 114, 9–35 (2004). https://doi.org/10.1007/s00229-004-0443-3

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  • DOI: https://doi.org/10.1007/s00229-004-0443-3

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