S.V. Pastukhova, O.P. Khromova
On the Signature of Ricci Curvature Tensor Operator of Three-Dimensional Lie Groups with Left-Invariant Lorentzian Metrics
Curvatures of left-invariant Riemannian metrics on Lie groups were studied by J. Milnor. Possible signatures of the Ricci operator were found for the case of three-dimensional Lie groups with leftinvariant Riemannian metric. Further, O. Kowalski and S. Nikcevic found three-dimensional metric Lie groups and three-dimensional Riemannian locally homogeneous spaces with prescribed values of the Ricci operator. Y.G. Nikonorov and A.G. Kremlev identified possible signatures of the Ricci operator on four-dimensional Lie groups with left-invariant Riemannian metric. Similar results for the onedimensional curvature operator and the sectional curvature operator were obtained by D.N. Oskorbin, E.D. Rodionov, O.P. Khromova. In the case of left-invariant Lorentz metrics on Lie groups, the situation is less clear. In this paper, we define the possible signatures of the Ricci tensor operator on three-dimensional Lie group with left-invariant Lorenzian metric. Results of G. Calvaruso, E.D. Rodionov, V.V. Slavskii, L.N. Chibrikova studies on the structure of 3-dimensional homogeneous Lorentzian manifolds are used in our study.
Key words: Lie algebras and Lie groups, leftinvariant Lorentzian metrics, Ricci tensor
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