Killing Fields on 2-Symmetric Lorentzian Manifolds

  • Д.Н. Оскорбин Altai State University (Barnaul, Russia) Email: oskorbin@yandex.ru
  • Е.Д. Родионов Altai State University (Barnaul, Russia) Email: edr2002@mail.ru
  • И.В. Эрнст Altai State University (Barnaul, Russia) Email: igeh@ya.ru
DOI_disc_logo https://doi.org/10.14258/izvasu(2019)1-16
Keywords: Killing field, Walker manifold, Lorentzian manifold, k-symmetric manifold, coordinate system

Abstract

  In the following paper, we describe Killing fields on 2-symmetric Lorentzian manifolds of dimension four. Killing fields play an important role in the study of Ricci solitons which were introduced by R. Hamilton. Ricci solitons are the generalization of the Einstein metrics on (pseudo)Riemannian manifolds. Ricci soliton equation was studied by many mathematicians on different classes of manifolds. In particular, in the recent author’s papers, solvability of the Ricci soliton equation on 3-symmetric Lorentzian manifolds was proved, and the general solution of the Ricci soliton equation on 2-symmetric Lorentzian manifolds was described. We describe Killing fields using Brinkmann normal coordinates which exist on the class of Lorentzian manifolds, the so-called pp-waves. The system of differential equations that corresponds to the Killing equation can be reduced to a simpler form. This was done by W. Globke and T. Leistner. By applying their result, the general solution of the system was found, the dimension of an algebra of Killing fields was calculated. The results stated in this paper continue author’s research on Ricci solitons on Lorentzian manifolds.

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Author Biographies

Д.Н. Оскорбин, Altai State University (Barnaul, Russia)
старший преподаватель кафедры математического анализа  
Е.Д. Родионов, Altai State University (Barnaul, Russia)
доктор физико-математических наук, профессор кафедры математического анализа  
И.В. Эрнст, Altai State University (Barnaul, Russia)

студент факультета математики и информационных технологий

References

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Alekseevsky D.V., Galaev A.S. Two-symmetric Lorentzian manifolds // Journal of Geometry and Physics. 2011. V. 61, N. 12.

Галаев A.C. Группы голономии лоренцевых многообразий и супермногообразий // Математический сборник. 2013. T. 204, 9.

Brozos-Vazquez M., Garcia-Rio E., Gavino-Fernandez S. Locally conformally flat lorentzian gradient Ricci soliton // Journal of Geometric Analysis. 2013. V. 23, N 3.

Walker A.G. Canonical form for a Riemannian space with a parallel field of null planes // Quart. J. Math. Oxford, 1950. V. 1, N 2.

Globke W., Leistner T. Locally homogeneous pp-waves // Journal of Geometry and Physics. 2016. V. 108.

Оскорбин Д.Н., Родионов Е.Д., Эрнст И.В. О солитонах Риччи на 2-симметрических четырёхмерных лоренцевых многообразиях // Известия Алтайского гос. ун-та 2017. N 4 (96).
Published
2019-03-06
How to Cite
Оскорбин Д., Родионов Е., Эрнст И. Killing Fields on 2-Symmetric Lorentzian Manifolds // Izvestiya of Altai State University, 2019, № 1(105). P. 95-98 DOI: 10.14258/izvasu(2019)1-16. URL: http://izvestiya.asu.ru/article/view/%282019%291-16.
Issue
No 1(105) (2019): Izvestiya of Altai State University
Section
Математика и механика

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