On Degenerated Singular Points of Dynamical Systems Related to the Normalized Ricci Flow on Generalized Wallach Spaces

  • Н.А. Абиев M.Kh. Dulaty Taraz State University (Taraz, Kazakhstan) Email: abievn@mail.ru
DOI_disc_logo https://doi.org/10.14258/izvasu(2019)1-09
Keywords: generalized Wallach space, Riemannian metric, Einstein metric, normalized Ricci flow, Ricci curvature, dynamical system, singular point

Abstract

In this paper, we study degenerated singular points (equilibrium points) of a dynamical system obtained by reduction of the normalized Ricci flow on generalized Wallach spaces. It is known that every generalized Wallach space is characterized by a triple of positive numbers satisfying well-defined inequalities. Therefore, the corresponding system of differential equations also depends on three real parameters. In the works of N.A. Abiev, A. Arvanitoyeorgos, Yu.G. Nikonorov, and P. Siasos a new approach was developed for studying of singular points. This approach is based on the idea of building a surface of parameters providing the normalized Ricci flow degenerate singular points. At natural (geometric) values of parameters, we established that for the normalized Ricci flow the nilpotent case never occurs, and the linearly zero case can occur only at a unique special combination of parameters. As a consequence, any other degenerate singular point of the system may be only semi-hyperbolic. In this paper, we remove the previous restrictions and study an abstract dynamical system abstracted from geometric essence. It is proved that some results of the mentioned works also hold for arbitrary values of the real parameters.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Author Biography

Н.А. Абиев, M.Kh. Dulaty Taraz State University (Taraz, Kazakhstan)
кандидат физико-математических наук, доцент, заведующий кафедрой математики

References

Abiev N.A., Arvanitoyeorgos A., Nikonorov Yu.G., Siasos P. The dynamics of the Ricci flow on generalized Wallach spaces // Differ. Geom. Appl. 2014. V. 35.

Abiev N.A., Arvanitoyeorgos A., Nikonorov Yu.G., Siasos P. The Ricci flow on some generalized Wallach spaces // Geometry and its Applications. Springer Proceedings in Mathematics and Statistics (Switzerland, Cham). 2014. V. 72.

Абиев Н.А., Арванитойоргос А., Никоноров Ю.Г., Сиасос П. Нормализованный поток Риччи на обобщенных пpостранствах Уоллаха // Математический форум. 2014. Т. 8, ч. 1.

Ломшаков А.М., Никоноров Ю.Г., Фирсов Е.В. Инвариантные метрики Эйнштейна на три-локально-симметрических пространствах // Математические труды. 2003. Т. 6, № 2.

Никоноров Ю.Г. Об одном классе однородных компактных многообразий Эйнштейна // Сибирский математический журнал. 2000. Т. 41, № 1.

Nikonorov Yu.G., Rodionov E.D., Slavskii V.V. Geometry of homogeneous Riemannian manifolds // Journ. Math. Sciences (New York). 2007. V. 146, № 7.

Nikonorov Yu. G. Classification of generalized Wallach spaces // Geom. Dedicata. 2016. V. 181, № 1.

Abiev N.A. On topological structure of some sets related to the normalized Ricci flow on generalized Wallach spaces // Владикавказский математический журнал. 2015. Т.17, № 3.

Batkhin A.B., Bruno A.D. Investigation of a real algebraic surface // Prog. Comp. Soft. 2015. V. 41, № 2.

Batkhin A.B. A real variety with boundary and its global parameterization // Prog. Comp. Soft. 2017. V. 43, № 2.

Abiev N.A. Two-parametric bifurcations of singular points of the normalized Ricci flow on generalized Wallach spaces // AIP Conference Proceedings (Turkey, Antalya). 2015. V. 1676.
Published
2019-03-06
How to Cite
Абиев Н. On Degenerated Singular Points of Dynamical Systems Related to the Normalized Ricci Flow on Generalized Wallach Spaces // Izvestiya of Altai State University, 2019, № 1(105). P. 60-63 DOI: 10.14258/izvasu(2019)1-09. URL: http://izvestiya.asu.ru/article/view/%282019%291-09.
Issue
No 1(105) (2019): Izvestiya of Altai State University
Section
Математика и механика

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:

  1. 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.
  2. 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.
  3. 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).