InvestigationofCurvatureOperatorson Three-Dimensional Locally Homogeneous Lorentzian Manifolds with Application of Symbolic Computations Packages
DOI:
https://doi.org/10.14258/izvasu(2017)4-20Keywords:
symbolic computation packages, locally homogeneous Lorentzian manifolds, curvature operatorsAbstract
The study of the properties of curvature operators is interesting for understanding the geometrical and topological structure of a homogeneous (pseudo)Riemannian manifold. One of the actual problems in this area is the problem of restoring (pseudo)Riemannian manifolds with respect to a prescribed curvature operator. The problem of prescribed values of the Ricci operator on 3–dimensional locally homogeneous Riemannian manifolds have been solved by O. Kowalski and S. Nikcevic. Similar results for the one-dimen-sional and sectional curvature operators have been obtained by D.N. Oskorbin, E.D. Rodionov and O.P. Khromova. The research of G. Calvaruso and O. Kowalski is known fo the case of a three-dimensional locally homogeneous Lorentzian manifold. There, the problem of existence of a three–dimensional locally homogeneous Lorentzian manifold with a prescribed Ricci operator is studied. The problem of existence of a three–dimensional Lie group with a left-invariant Lorentzian metric and prescribed one-dimensional or sectional curvature operator has been previously solved by the authors. This paper continues the authors’ investigations for the case of three-dimensional locally homogeneous Lorentzian manifolds. With the help of symbolic computation packages, the problem of the existence of a three-dimensional locally homogeneous Lorentzian manifold with the prescribed one-dimen-sional or sectional curvature operator is solved.
DOI DOI 10.14258/izvasu(2017)4-20
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