On the Sectional Curvature Operator of Three-Dimensional Lie Groups with Left-Invariant Lorentzian Metrics
Abstract
The study of curvature operator properties is interesting for understanding of geometrical and topological structure of a homogeneous (pseudo)Riemannian manifold. Some results of J. Milnor, V.N. Berestovskii, E.D. Rodionov, V.V. Slavskii on the connection between the Ricci curvature, one-dimensional curvature and topology of the homogeneous Riemannian space are well known in the homogeneous case. J. Milnor investigated the curvatures of left-invariant Riemannian metrics on Lie groups. The problem of prescribed values of the Ricci operator on three-dimensional Riemannian locally homogeneous spaces and three-dimensional metric Lie groups was solved by O. Kowalski and S. Nikcevic. Similar results were obtained by D.N. Oskorbin, E.D. Rodionov, O.P. Khromova for the one-dimensional curvature operator and the sectional curvature operator. The situation is less clear in the case of left-invariant Lorentzian metrics on Lie groups. The problem of existence of a Lie group with left-invariant Lorentzian metrics and prescribed values of Ricci operator spectrum for left-invariant Lorentzian metrics on three-dimensional Lie groups is studied in the paper of G. Calvaruso and O. Kowalski. In this paper, we consider the problem of prescribed values for the operator of the sectional curvature on three-dimensional metric Lie groups.Downloads
References
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