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**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.

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.

Reviewer: L.Vanhecke (Leuven)

### MSC:

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

53C35 | Differential geometry of symmetric spaces |

53C30 | Differential geometry of homogeneous manifolds |

### Keywords:

curvature homogeneous Lorentz 3-manifolds; classification up to local isometry; non-diagonalizable Ricci operator; homogeneous model
Full Text:
DOI

### References:

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