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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
- 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.
- Bromberg S., Medina A., Completeness of homogeneous quadratic
vector fields, Qual. Theory Dyn. Syst. 6 (2005),
181-185.
- Dumitrescu S., Zeghib A.,
Géométries Lorentziennes de dimension 3: classification et
complétude, math.DG/0703846.
- Guediri M., Lafontaine J., Sur la
complétude des varietés pseudo-Rimanniennes, J. Geom.
Phys. 15 (1995), 150-158.
- 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.
- Guediri M., On completeness of left-invariant
Lorentz metrics on solvable Lie groups, Rev. Mat. Univ.
Complut. Madrid 9 (1996), 337-350.
- Kaplan J.L., Yorke J.A., Non associative real algebras and
quadratic differential equations, Nonlinear Anal. 3 (1979), 49-51.
- Milnor J., Curvatures of left invariant metrics on Lie groups,
Advances in Math. 21 (1976), 293-329.
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