On the Spectrum of One-Dimensional Curvature operators on Three-Dimensional Lie Groups with Left-Invariant Lorentzian Metrics
Abstract
The problem of the establishing of connections between topology and curvature of a Riemannian manifold is one of the important problems of Riemannian geometry. J. Milnor, V.N. Berestovskii, E.D. Rodionov, V.V. Slavskii studies on the connection among the Ricci curvature, onedimensional curvature and topology of the homogeneous Riemannian space are well known in the homogeneous case . The curvatures of left-invariant Riemannian metrics on Lie groups were studied by J. Milnor. Namely, possible signatures of the Ricci operator were found in the case of three-dimensional Lie groups with a left-invariant Riemannian metric. Futher, O. Kowalski and S. Nikcevic found three-dimensional metric Lie groups and threedimensional Riemannian locally homogeneous spaces with prescribed values of the Ricci operator. Similar results were obtained by D.N. Oskorbin, E.D. Rodionov, O.P. Khromova for the onedimensional curvature operator and the sectional curvature operator. The situation is less clear in the case of leftinvariant Lorentzian metrics on Lie groups. In this paper, we consider the problem of the prescribed values for the operator of one-dimensional curvature. Besides, we define the possible signatures of the form of one-dimensional curvature on three-dimensional Lie groups with a left-invariant Lorenzian metric.
DOI 10.14258/izvasu(2016)1-21
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