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

SIGMA 4 (2008), 088, 13 pages      arXiv:0806.1632

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

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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