InvestigationofCurvatureOperatorson Three-Dimensional Locally Homogeneous Lorentzian Manifolds with Application of Symbolic Computations Packages
Abstract
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
Downloads
Metrics
References
Kowalski O., Nikcevic S. On Ricci eigenvalues of locally homogeneous Riemann 3-manifolds // Geom. Dedicata. — 1996. — No. 1. DOI: 10.1007/BF00240002.
Calvaruso G., Kowalski O. On the Ricci operator of locally homogeneous Lorentzian 3-manifolds // Cent. Eur. J. Math. — 2009. -V. 7(1). DOI: 10.2478/s11533-008-0061-5.
Kowalski O. Nonhomogeneous Riemannian 3-manifolds with distinct constant Ricci eigenvalues // Nagoya Math. J. — 1993. — V. 132.
Bueken P. On curvature homogeneous three-dimensional Lorentzian manifolds // J. Geom. Phys. — 1997. — V. 22.
Calvaruso G. Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues // Diff. Geom. Appl. — 2008. — V. 26. DOI: 10.1016/j.difgeo.2007.11.031.
Оскорбин Д.Н., Родионов Е.Д. О спектре оператора кривизны трехмерных групп Ли с левоинвариантной римановой метрикой // ДАН. — 2013. — Т. 450. DOI: 10.7868/S0869565213140077.
Оскорбин Д.Н., Родионов Е.Д., Хромова О.П. О вычислении спектра оператора кривизны конформно (полу)плоских римановых метрик // Известия Алтайского гос. ун-та. — 2013. — № 1-2(77). DOI: 10.14258/izvasu(2013)1.2-04.
Bueken P., Djoric M. Three-dimensional Lorentz metrics and curvature homogeneity of order one // Ann. Glob. Anal. Geom. — 2000. — V. 18. DOI: 10.1023/A:1006612120550.
Calvaruso G. Homogeneous structures on three-dimensional Lorentzian manifolds // J. Geom. Phys. — 2007. — V. 57. DOi:
1016/j.geomphys.2006.10.005.
Клепиков П.Н. О допустимых значениях спектра оператора одномерной кривизны трехмерных групп Ли с левоинвариантной лоренцевой метрикой // Математика и ее приложения: фундаментальные проблемы науки и техники : сб. тр. Всеросс. конф., Барнаул, 24-26 ноября, 2015. — Барнаул, 2015.
Клепиков П.Н., Клепикова С.В., Хромова О.П. О спектре операторов одномерной кривизны левоинвариантных лоренцевых метрик трехмерных групп Ли // Известия Алтайского гос. ун-та. — 2016. — № 1(89). DOI: 10.14258/izvasu(2016)1-21.
Клепикова С.В., Хромова О.П. Об операторе секционной кривизны на трехмерных группах Ли с левоинвариантной лоренцевой метрикой // Известия Алтайского гос. ун-та. — 2017. — № 1(93). DOI: 10.14258/izvasu(2017)1-17.
Copyright (c) 2017 С.В. Клепикова, О.П. Хромова
This work is licensed under a Creative Commons Attribution 4.0 International License.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
