Advertisement

Geometriae Dedicata

, Volume 179, Issue 1, pp 229–253 | Cite as

Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist

  • Sorin Dumitrescu
  • Karin Melnick
Original Paper
  • 78 Downloads

Abstract

We show that a germ of a real-analytic Lorentz metric on \({\mathbb R}^3\) which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously—meaning they act transitively on an open set admitting the origin in its closure, but not at the origin—and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneous action is semisimple, we provide a complete set of normal forms of the metric and the action.

Keywords

Real-analytic Lorentz metrics Transitive Killing Lie algebras  Local differential invariants 

Mathematics Subject Classification (2010)

53A55 53B30 53C50 

References

  1. 1.
    Amores, A.M.: Vector fields of a finite type \(G\)-structure. J. Differ. Geom. 14(1), 1–6 (1979)MATHMathSciNetGoogle Scholar
  2. 2.
    Benoist, Y.: Orbites des structures rigides (d’après M. Gromov). In: Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995), vol. 145 of Progress in Mathematics, pp. 1–17. Birkhäuser Boston, Boston (1997)Google Scholar
  3. 3.
    Benveniste, E.J., Fisher, D.: Nonexistence of invariant rigid structures and invariant almost rigid structures. Commun. Anal. Geom. 13(1), 89–111 (2005)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Benoist, Y., Foulon, P., Labourie, F.: Flots d’Anosov à distributions stable et instable différentiables. J. Am. Math. Soc. 5(1), 33–74 (1992)MATHMathSciNetGoogle Scholar
  5. 5.
    Calvaruso, G.: Einstein-like metrics on 3-dimensional homogeneous Lorentzian manifolds. Geom. Dedic. 127, 99–119 (2007)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Calvaruso, G., Kowalski, O.: On the Ricci operator of locally homogeneous Lorentzian 3-manifolds. CEJM 7(1), 124–139 (2009)MATHMathSciNetGoogle Scholar
  7. 7.
    Cordero, L.A., Parker, P.E.: Left invariant Lorentzian metrics on 3-dimensional Lie groups. Rend. Math. Appl. (7) 17(1), 129–155 (1997)MATHMathSciNetGoogle Scholar
  8. 8.
    D’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. In: Surveys in differential geometry (Cambridge. MA, 1990), pp. 19–111. Lehigh Univ, Bethlehem (1991)Google Scholar
  9. 9.
    Dumitrescu, S., Guillot, A.: Quasihomogeneous real analytic connections on surfaces. J. Topol. Anal. 5(4), 491–532 (2013)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dumitrescu, S.: Dynamique du pseudo-groupe des isométries locales sur une variété lorentzienne analytique de dimension 3. Ergod. Theory Dyn. Syst. 28(4), 1091–1116 (2008)MATHCrossRefGoogle Scholar
  11. 11.
    Dumitrescu, S., Zeghib, A.: Géométries lorentziennes de dimension trois: classification et complétude. Geom. Dedic. 149, 243–273 (2010)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Feres, R.: Rigid geometric structures and actions of semisimple Lie groups. In Rigidité, groupe fondamental et dynamique, vol. 13 of Panor. Synthèses, pp. 121–167. Soc Math. France, Paris (2002)Google Scholar
  13. 13.
    Gromov, M.: Rigid transformations groups. In: Géométrie différentielle (Paris. 1986), vol. 33 of Travaux en Cours, pp. 65–139. Hermann, Paris (1988)Google Scholar
  14. 14.
    Kirilov, A.: Eléments de la théorie des représentations. M.I.R, Moscou (1974)Google Scholar
  15. 15.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I. Wiley Classics Library, Wiley, New York (1996). Reprint of the 1963 originalGoogle Scholar
  16. 16.
    Melnick, K.: Compact Lorentz manifolds with local symmetry. J. Differ. Geom. 81(2), 355–390 (2009)MATHMathSciNetGoogle Scholar
  17. 17.
    Melnick, K.: A Frobenius theorem for Cartan geometries, with applications. Enseign. Math. (2) 57(1–2), 57–89 (2011)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Nomizu, K.: On local and global existence of Killing vector fields. Ann. Math. 2(72), 105–120 (1960)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nomizu, K.: Left invariant Lorentz metrics on Lie groups. Osaka J. Math. 16, 143–150 (1979)MATHMathSciNetGoogle Scholar
  20. 20.
    Pecastaing, V.: On two theorems about local automorphisms of geometric structures. Annales de l’Institut Fourier. arXiv:1402.5048 (to appear) (2014)
  21. 21.
    Prüfer, F., Tricerri, F., Vanhecke, L.: Curvature invariants, differential operators and local homogeneity. Trans. Am. Math. Soc. 348(11), 4643–4652 (1996)MATHCrossRefGoogle Scholar
  22. 22.
    Rosenlicht, M.: On quotient varieties and the affine embedding of certain homogeneous spaces. Trans. Am. Math. Soc. 101, 211–223 (1961)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Sharpe, R.W.: Differential geometry, volume 166 of Graduate Texts in Mathematics. Springer, New York, 1997. Cartan’s generalization of Klein’s Erlangen program. With a foreword by S. S. Chern (1997)Google Scholar
  24. 24.
    Singer, I.: Infinitesimally homogeneous spaces. Commun. Pure Appl. Math. 13, 685–697 (1960)MATHCrossRefGoogle Scholar
  25. 25.
    Thurston, W.P.: Three-dimensional geometry and topology, vol. 1, volume 35 of Princeton mathematical series. Princeton University Press, Princeton (1997). Edited by Silvio LevyGoogle Scholar
  26. 26.
    Wolf, J.A.: Spaces of constant curvature. McGraw-Hill Book Co., New York (1967)MATHGoogle Scholar
  27. 27.
    Zeghib, A.: Killing fields in compact Lorentz \(3\)-manifolds. J. Differ. Geom. 43(4), 859–894 (1996)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Laboratoire J.-A. Dieudonné, UMR 7351Université Côte d’Azur, Université Nice Sophia Antipolis, CNRSNice Cedex 2France
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations