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.

pdf (217 kb)   ps (160 kb)   tex (15 kb)

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.

Previous article   Next article   Contents of Volume 4 (2008)