Application of Computer Mathematics Systems to the Study of Homogeneous (pseudo)Riemannian Manifolds with the Trivial Schouten — Weyl Tensor
Abstract
Papers of many mathematicians are devoted to the investigation of (pseudo)Riemannian Einstein manifolds, locally symmetric, Ricci parallel and conformally flat manifolds. All these manifolds (as special cases) are contained in the class of (pseudo)Riemannian manifolds with trivial Schouten — Weyl tensor.
In the case of low dimension, it is possible to apply computer mathematics systems for studying the homogeneous (pseudo)Riemannian manifolds with trivial Schouten — Weyl tensor, since all invariant tensor fields are expressed in terms of Lie algebra structure constants of isometry group and the components of the metric tensor. A key step to solving the problem of classifying homogeneous (pseudo)Riemannian manifolds with the zero Schouten — Weyl tensor is a sequential consideration of all possible Segre types of the Ricci operator.
This study is aimed at developing a mathematical model, as well as a computer program for studying and classifying the homogeneous (pseudo)Riemannian manifolds with zero Schouten — Weyl tensors of finite dimensions. Also, this paper contains an example that shows the basic steps of the developed algorithm.
DOI 10.14258/izvasu(2018)1-18
Downloads
Metrics
References
Besse A. Einstein manifolds. — Springer-Ver-lag, Berlin-Heidelberg, 1987. DOI: 10.1007/978-3540-74311-8
Gray A. Einstein-like manifolds which are not Einstein // Geom. Dedicata. — 1978. — Vol. 7. DOI: 10.1007/BF00151525
Calvaruso G., Zaeim A. Conformally flat homogeneous pseudo-riemannian four-manifolds // Tohoku Math. J. — 2014. — Vol. 66. DOI: 10.2748/tmj/1396875661
Calvaruso G., Zaeim A. Four-dimensional Lorentzian Lie groups // Differential Geometry and its Applications. — 2013. — Vol. 31. DOI: 10.1016/j.difgeo.2013.04.006
Calvaruso G., Zaeim A. Neutral Metrics on Four-Dimensional Lie Groups // Journal of Lie Theory. — 2015. — Vol. 25.
Zaeim A., Haji-Badali A. Einstein-like Pseudo-Riemannian Homogeneous Manifolds of Dimension Four // Mediterranean Journal of Mathematics. — 2016. — Vol. 13, No 5. DOI: 10.1007/s00009-016-0696-6
Клепиков П.Н. Левоинвариантные псев-доримановы метрики на четырехмерных группах Ли с нулевым тензором Схоутена — Вейля // Известия вузов. Математика. — 2017. — No 8.
Gladunova O.P., Rodionov E.D., Slavskii V.V. Harmonic Tensors on Three-Dimensional Lie Groups with Left-Invariant Lorentz Metric // Journal of mathematical sciences. — 2014. — Vol. 198, No 5. DOI: 10.1007/s10958-014-1806-2
Voronov D.S., Rodionov E.D. Left-invariant Riemannian metrics on four-dimensional nonunim-odular Lie groups with zero-divergence Weyl tensor // Doklady Mathematics. — 2010. — Vol. 81, No 3. DOI: 10.1134/S1064562410030154
Gladunova O.P., Slavskii V.V. Harmonicity of the Weyl tensor of left-invariant Riemannian metrics on four-dimensional unimodular Lie groups // Siberian Advances in math. — 2013. — Vol. 23, No 1. DOI: 10.3103/S1055134413010033
Law P.R. Algebraic classification of the Ricci curvature tensor and spinor for neutral signature in four dimensions // arXiv:1008.0444, 2010.
Гладунова О.П. Применение математических пакетов к вычислению инвариантных тензорных полей на трехмерных группах Ли с левоинвариантной (псевдо)римановой метрикой // Вестник Алт. гос. пед. ун-та. — 2006. — № 6-2.
Гладунова О.П., Оскорбин Д.Н. Применение пакетов символьных вычислений к исследованию спектра оператора кривизны на метрических группах Ли // Изв. Алт. гос. ун-та. — 2013. — № 1-1 (77).
Хромова О.П. Применение пакетов символьных вычислений к исследованию оператора одномерной кривизны на нередуктивных однородных псевдоримановых многообразиях // Изв. Алт. гос. ун-та. — 2017. — № 1 (93). DOI: 10.14258/izvasu(2017)1-28
Milnor J. Curvatures of left invariant metrics on Lie groups // Adv. Math. — 1976. — Vol. 21, No 3. DOI: 10.1016/S0001-8708(76)80002-3
Родионов Е.Д., Славский В.В., Чибрико-ва Л.Н. Левоинвариантные лоренцевы метрики на трехмерных группах Ли с нулевым квадратом тензора Схоутена — Вейля // ДАН. — 2005. -Т. 401, № 4.
Kodama H., Takahara A., Tamaru H. The space of left-invariant metrics on a Lie group up to isometry and scaling // Manuscripta math. — 2011. — Vol. 135. DOI: 10.1007/s00229-010-0419-4
Kubo A., Onda K., Taketomi Y., Tamaru H. On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups // arXiv:1509.08336, 2015.
Клепиков П.Н., Оскорбин Д.Н. Построение обобщенных базисов Милнора некоторых четырехмерных метрических алгебр Ли // Изв. Алт. гос. ун-та. — 2015. — № 1/1 (85). DOI: 10.14258/izvasu(2015)1.1-13
O’Neill B. Semi-Riemannian Geometry With Applications to Relativity — Academic Press, 1983
Copyright (c) 2018 П.Н. Клепиков
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).
