Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds
Tóm tắt
We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise.
Tài liệu tham khảo
Abbena E., Garbiero S. (1992). Curvature forms and Einstein-like metrics on Sasakian manifoilds. Math. J. Okayama Univ. 34: 241–248
Abbena E., Garbiero S., Vanhecke L. (1992). Einstein-like metrics on three-dimensional Riemannian homogeneous manifolds. Simon Stevin Quart. J. Pure Appl. Math. 66: 173–182
Boeckx E. (1997). Einstein-like semi-symmetric spaces. Arch. Math. (Brno) 16: 789–800
Bueken P., Djorić M. (2000). Three-dimensional Lorentz metrics and curvature homogeneity of order one. Ann. Glob. Anal. Geom. 18: 85–103
Bueken P., Vanhecke L. (1999). Three- and four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75: 123–136
Calvaruso G. (2000). Einstein-like and conformally flat contact metric three-manifolds. Balkan J. Geom. 5(2): 17–36
Calvaruso G. (2007). Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57: 1279–1291
Calvaruso, G.: Einstein-like Lorentz metrics and three-dimensional curvature homogeneity of order one. Preprint (2007)
Calvaruso G., Vanhecke L. (1997). Special ball-homogeneous spaces. Z. Anal. Anwendungen 16: 789–800
Chaichi M., García-Río E., Vázquez-Abal M.E. (2005). Three-dimensional Lorentz manifolds admitting a parallel null vector field. J. Phys. A: Math. Gen. 38: 841–850
Cordero L.A., Parker P.E. (1997). Left-invariant Lorentzian metrics on three-dimensional Lie groups. Rend. Mat., Serie VII 17: 129–155
Gray A. (1978). Einstein-like manifolds which are not Einstein. Geom. Dedicata 7: 259–280
Milnor J. (1976). Curvature of left invariant metrics on Lie groups. Adv. Math 21: 293–329
O’Neill B. (1983). Semi-Riemannian Geometry. Academic Press, New York
Nomizu K. (1979). Left-invariant Lorentz metrics on Lie groups. Osaka J. Math. 16: 143–150
Rahmani S. (1992). Métriques de Lorentz sur les groupes de Lie unimodulaires de dimension trois. J. Geom. Phys. 9: 295–302
Sekigawa K. (1977). On some three-dimensional curvature homogeneous spaces. Tensor N.S. 31: 87–97
Takagi H. (1975). Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tohôku Math. J. 27: 103–110
Tricerri, F., Vanhecke, L.: Homogeneous structures on Riemannian manifolds. London Math. Soc. Lect. Notes 83. Cambridge Univ. Press (1983)