×

On curvature homogeneous three-dimensional Lorentzian manifolds. (English) Zbl 0881.53055

A pseudo-Riemannian manifold \((M,g)\) is said to be curvature homogeneous if, for every \(p,q\in M\), there exists a linear isometry \(F:T_pM \to T_qM\) such that \(F^*R_q = R_p\) where \(R\) denotes the Riemannian curvature tensor. All locally homogeneous spaces \((M,g)\) are trivial examples. Moreover, a curvature homogeneous space \((M,g)\) is said to have the same curvature tensor as a homogeneous space \((\overline M, \overline g)\) if, for any pair of points \(m\in M\) and \(\overline m\in\overline M\), there exists a linear isometry \(\varphi: T_m M \to T_{\overline m} \overline M\) such that \(\varphi^* \overline R_{\overline m} =R_m\). In this case, \((\overline M, \overline g)\) is said to be a (homogeneous) model space for \((M,g)\).
Curvature homogeneous Riemannian spaces have been studied in detail by several people, and a lot of examples, with or without a homogeneous model, have been discovered. Note that, in dimension three, \((M,g)\) is curvature homogeneous if and only if the Ricci operator has constant eigenvalues. We refer to [E. Boeckx, O. Kowalski and L. Vanhecke, ‘Riemannian manifolds of conullity two’ (World Scientific, River Edge, NJ) (1996)] for a survey about the results and for further references.
A similar study has already been started for pseudo-Riemannian manifolds, in particular for Lorentz spaces. It turns out that there are remarkable differences mainly due to the fact that the Ricci operator is not always diagonalizable. In another paper, the author studied in detail the diagonalizable case for three-dimensional Lorentz spaces when the constant eigenvalues satisfy \(\rho_1= \rho_2\neq \rho_3\) and determined a complete local classification. In this paper he continues this study and treats the case when the non-diagonalizable Ricci operator has three equal constant eigenvalues whose associated eigenspace is two-dimensional. He proves the existence of such metrics and studies when they are locally isometric, obtaining in this way a classification result. In particular, he studies the case when \((M^3,g)\) has an indecomposable, non-irreducible Lorentzian symmetric space as homogeneous model (the case of vanishing Ricci eigenvalues) and provides several new examples in this class.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C35 Differential geometry of symmetric spaces
53C30 Differential geometry of homogeneous manifolds
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Boeckx, E., Foliated semi-symmetric spaces, () · Zbl 0807.53041
[2] Boeckx, E.; Kowalski, O.; Vanhecke, L., Non-homogeneous relatives of symmetric spaces, Diff. geom. appl., 4, 45-69, (1994) · Zbl 0796.53046
[3] Boeckx, E.; Kowalski, O.; Vanhecke, L., Reimannian manifolds of conullity two, (1996), World Scientific Singapore · Zbl 0904.53006
[4] Bueken, P., Three-dimensional Riemannian manifolds with constant principal Ricci curvatures ϱ1 = ϱ2 ≠ ϱ3, J. math. phys., 37, 4062-4075, (1996) · Zbl 0866.53026
[5] P. Bueken, Three-dimensional Lorentzian manifolds with constant principal Ricci curvatures ϱ1 = ϱ2 ≠ ϱ3, J. Math. Phys., to appear. · Zbl 0876.53042
[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] Hall, G.; Morgan, T.; Perjes, Z., Three-dimensional space-times, Gen. relativity gravitation, 19, 1137-1147, (1987) · Zbl 0629.53022
[8] Kowalski, O., A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ϱ1 = ϱ2 ≠ ϱ3, Nagoya math. J., 132, 1-36, (1993) · Zbl 0788.53038
[9] O. Kowalski, An explicit classification of 3-dimensional Riemannian spaces satisfying R(X, Y)·R = 0, Czech. Math. J., to appear. · Zbl 0879.53014
[10] Kowalski, O.; Prüfer, F., On Riemannian three-manifolds with distinct constant Ricci eigenvalues,, Math. ann., 300, 17-28, (1994) · Zbl 0813.53020
[11] Kowalski, O.; Prüfer, F., A classification of special Riemannian three-manifolds with distinct constant Ricci eigenvalues, Z. anal. anwendungen, 14, 43-58, (1995) · Zbl 0821.53036
[12] Kowalski, O.; Sekizawa, M., Local isometry classes of Riemannian three-manifolds with constant Ricci eigenvalues ϱ1 = ϱ2 ≠ ϱ3 > 0, Arch. math., 32, 137-145, (1996) · Zbl 0903.53015
[13] Kowalski, O.; Tricerri, F.; Vanhecke, L., New examples of nonhomogeneous riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space, C. R. acad. sci. Paris, 311, 355-360, (1990) · Zbl 0713.53028
[14] Kowalski, O.; Tricerri, F.; Vanhecke, L., Curvature homogeneous Riemannian manifolds, J. math. pures appl., 71, 471-501, (1992) · Zbl 0836.53029
[15] Kowalski, O.; Tricerri, F.; Vanhecke, L., Curvature homogeneous spaces with a solvable Lie group as homogeneous model, J. math. soc. Japan, 44, 461-484, (1992) · Zbl 0762.53031
[16] O. Kowalski and Z. Vlášek, Classification of Riemannian three-manifolds with distinct constant principal Ricci curvatures, preprint.
[17] McManus, D., Riemannian three-metrics with degenerate Ricci tensors, J. math. phys., 36, 362-369, (1995) · Zbl 0824.53067
[18] McManus, D., Lorentzian three-metrics with degenerate Ricci tensors, J. math. phys., 36, 1353-1364, (1995) · Zbl 0824.53068
[19] O’Neill, B., Semi-Riemannian geometry (with applications to relativity), (1983), Academic Press New York, London · Zbl 0531.53051
[20] V. Patrangenaru, Some curvature homogeneous cosmological models, preprint. · Zbl 1046.53033
[21] Sekigawa, K., On the Riemannian manifolds of the form \(B × ƒ F\), Kodai math. sem. rep., 26, 343-347, (1975) · Zbl 0304.53019
[22] Sekigawa, K., On some 3-dimensional curvature homogeneous spaces, Tensor (N.S.), 31, 87-97, (1977) · Zbl 0356.53016
[23] Singer, I.M., Infinitesimally homogeneous spaces, Comm. pure appl. math., 13, 685-697, (1960) · Zbl 0171.42503
[24] Takagi, H., On curvature homogeneity of Riemannian manifolds, Tôhoku math. J., 26, 581-585, (1974) · Zbl 0302.53022
[25] Tricerri, F.; Vanhecke, L., Curvature homogeneous Riemannian manifolds, Ann. sci. ecole norm. sup., 22, 535-554, (1989) · Zbl 0698.53033
[26] Vanhecke, L., Curvature homogeneity and related problems, (), 103-122 · Zbl 0727.53049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.