Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 088, 13 pages      arXiv:0806.1632      https://doi.org/10.3842/SIGMA.2008.088

Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds

Shirley Bromberg a and Alberto Medina b
a) Departameto de Matemáticas, UAM-Iztapalapa, México
b) Département des Mathématiques, Université de Montpellier II, UMR, CNRS, 5149, Montpellier, France

Received June 24, 2008, in final form December 10, 2008; Published online December 18, 2008

Abstract
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.

Key words: Lorentzian metrics; complete geodesics; 3-dimensional Lie groups; Euler equation.

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References

  1. Bromberg S., Medina A., Complétude de l'équation d'Euler, in Proceedings of the Colloquium in Tashkent "Algebra and Operator Theory" (September 29 - October 5, 1997, Tashkent), Editors Y. Khakimdjanov, M. Goze and S.A. Ayupov, Kluwer Acad. Publ., Dordrecht, 1998, 127-144.
  2. Bromberg S., Medina A., Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst. 6 (2005), 181-185.
  3. Dumitrescu S., Zeghib A., Géométries Lorentziennes de dimension 3: classification et complétude, math.DG/0703846.
  4. Guediri M., Lafontaine J., Sur la complétude des varietés pseudo-Rimanniennes, J. Geom. Phys. 15 (1995), 150-158.
  5. Guediri M., Sur la complétude des pseudo-métriques invariantes a gauche sur les groupes de Lie nilpotents, Rend. Sem. Math. Univ. Politec. Torino 52 (1994), 371-376.
  6. Guediri M., On completeness of left-invariant Lorentz metrics on solvable Lie groups, Rev. Mat. Univ. Complut. Madrid 9 (1996), 337-350.
  7. Kaplan J.L., Yorke J.A., Non associative real algebras and quadratic differential equations, Nonlinear Anal. 3 (1979), 49-51.
  8. Milnor J., Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), 293-329.

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