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Homogeneous structures on three-dimensional Lorentzian manifolds. (English) Zbl 1112.53051

Summary: We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
22E15 General properties and structure of real Lie groups
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[1] Ambrose, W.; Singer, I.M., On homogeneous Riemannian manifolds, Duke math. J., 25, 647-669, (1958) · Zbl 0134.17802
[2] Bueken, P., Three-dimensional Lorentzian manifolds with constant principal Ricci curvatures \(\rho_1 = \rho_2 \ne \rho_3\), J. math. phys., 38, 1000-1013, (1997) · Zbl 0876.53042
[3] Bueken, P.; Djorić, M., Three-dimensional Lorentz metrics and curvature homogeneity of order one, Ann. global anal. geom., 18, 85-103, (2000) · Zbl 0947.53037
[4] Bueken, P.; Vanhecke, L., Examples of curvature homogeneous Lorentz metrics, Classical quantum gravity, 14, (1997), L93-96
[5] G. Calvaruso, Einstein-like Lorentzian metrics on three-dimensional homogeneous manifolds, 2006 (preprint) · Zbl 1126.53044
[6] Cahen, M.; Leroy, J.; Parker, M.; Tricerri, F.; Vanhecke, L., Lorentz manifolds modelled on a Lorentz symmetric space, J. geom. phys., 7, 571-591, (1990) · Zbl 0736.53056
[7] Chaichi, M.; García-Río, E.; Vázquez-Abal, M.E., Three-dimensional Lorentz manifolds admitting a parallel null vector field, J. phys. A, 38, 841-850, (2005) · Zbl 1068.53049
[8] Cordero, L.A.; Parker, P.E., Left-invariant Lorentzian metrics on 3-dimensional Lie groups, Rend. mat. appl. (7), 17, 129-155, (1997) · Zbl 0948.53027
[9] Gadea, P.M.; Oubiña, J.A., Homogeneous pseudo-riemannian structures and homogeneous almost para-Hermitian structures, Houston J. math., 18, 3, 449-465, (1992) · Zbl 0760.53029
[10] Gromov, M., ()
[11] Milnor, J., Curvature of left invariant metrics on Lie groups, Adv. math., 21, 293-329, (1976) · Zbl 0341.53030
[12] O’Neill, B., Semi-Riemannian geometry, (1983), Academic Press New York · Zbl 0531.53051
[13] Nomizu, K., Left-invariant Lorentz metrics on Lie groups, Osaka J. math., 16, 143-150, (1979) · Zbl 0397.53047
[14] Patrangenaru, V., Locally homogeneous pseudo-Riemannian manifolds, J. geom. phys., 17, 59-72, (1995) · Zbl 0832.53017
[15] Rahmani, S., Métriques de Lorentz sur LES groupes de Lie unimodulaires de dimension trois, J. geom. phys., 9, 295-302, (1992) · Zbl 0752.53036
[16] Sekigawa, K., On some three-dimensional curvature homogeneous spaces, Tensor (N.S.), 31, 87-97, (1977) · Zbl 0356.53016
[17] Singer, I.M., Infinitesimally homogeneous spaces, Comm. pure appl. math., 13, 685-697, (1960) · Zbl 0171.42503
[18] Sternberg, S., Lectures on differential geometry, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0129.13102
[19] Tricerri, F.; Vanhecke, L., Homogeneous structures on Riemannian manifolds, () · Zbl 0509.53043
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