Geodesic
On conformal factor in the conformal Killing equation on the $2$-symmetric five-dimensional indecomposable Lorentzian manifold
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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Conformally Killing vector fields are a natural generalization of Killing vector fields and play an important role in the study of the group of conformal transformations of a manifold, Ricci flows on a manifold, and the theory of Ricci solitons. Pseudo-Riemannian symmetric spaces of order $k$, where $k \geq 2$, arise in the study of pseudo-Riemannian geometry and in physics. At present, they have been investigated in cases $k=2, 3$ by D. V. Alekseevsky, A. S. Galaev and others. In the case of low dimensions, these spaces and Killing vector fields on them were studied by D. N. Oskorbin, E. D. Rodionov, and I. V. Ernst. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds, and their equation has been studied on various classes of manifolds by many mathematicians. In particular, D. N. Oskorbin and E. D. Rodionov found a general solution of the Ricci soliton equation on $2$-symmetric Lorentzian manifolds of low dimension, and proved the local solvability of this equation in the class of $3$-symmetric Lorentzian manifolds. For a single Einstein constant in the Ricci soliton equation the Killing vector fields make it possible to find the general solution of the Ricci soliton equation corresponding to the given constant. However, for different values of the Einstein constant, conformally Killing vector fields play the role of Killing fields. Therefore, there is a need to study them. In this paper, we investigate the conformal analogue of the Killing equation on five-dimensional $2$-symmetric indecomposable Lorentzian manifolds, and investigate the properties of the conformal factor of the conformal analogue of the Killing equation on them. Nontrivial examples of conformally Killing vector fields with a variable conformal factor are constructed. 
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T. A. Andreeva; D. N. Oskorbin; E. D. Rodionov. On conformal factor in the conformal Killing equation on the $2$-symmetric five-dimensional indecomposable Lorentzian manifold. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 5-14. http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a0/

[1] Berestovskii V. N., Nikonorov Yu. G., Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, Springer, Cham, 2020, xxii+482 pp. | DOI | MR | Zbl

[2] Oskorbin, D. N. and Rodionov, E. D., “Ricci Solitons and Killing Fields on Generalized Cahen–Wallach Manifolds”, Siberian Mathematical Journal, 60 (2019), 911–915 | DOI | DOI | MR | Zbl

[3] Andreeva, T. A., Balashchenko, V. V. and Oskorbin, D. N., “Conformal Killing Fields on Symmetric Lorentzian Manifolds of Low Dimension”, Proceedings of the Seminar on Geometry and Mathematical Modeling, 6 (2020), 19–25 (in Russian)

[4] Hall G. S., Symmetries and Curvature Structure in General Relativity, World Sci. Lect. Notes Phys., 46, 2004, 440 pp. | DOI | MR

[5] Andreeva, T. A., Balashchenko, V. V., Oskorbin D. N. and Rodionov E. D., “Conformally Killing Fields on 2-Symmetric Five-Dimensional Lorentzian Manifolds”, Izvestiya of Altai State University, 117:1 (2021), 68–71 (in Russian) | DOI

[6] Cahen M., Wallach N., “Lorentzian symmetric spaces”, Bull. Amer. Math. Soc., 76:3 (1970), 585–592 | DOI | MR

[7] Galaev A. S., Alexeevskii D. V., “Two-symmetric Lorentzian manifolds”, J. Geometry Phys., 61:12 (2011), 2331–2340 | DOI | MR | Zbl

[8] Blanco O. F., Sanchez M., Senovilla J. M., “Structure of second-order symmetric Lorentzian manifold”, J. Eur. Math. Soc, 15:2 (2013), 595–634 | DOI | MR | Zbl

[9] Galaev A. S., Leistner T., “Holonomy groups of Lorentzian manifolds: classification, examples, and applications”, Recent Developments in Pseudo-Riemannian Geometry, 2008, 53–96 | DOI | MR | Zbl

[10] Walker A. G., “On parallel fields of partially null vector spaces”, Quart. J. Math., os-20:1 (1949), 135–145 | DOI | MR | Zbl

[11] Brozos-Vázquez M., García-Río E., Gilkey P., Nikčević S., Vázquez-Lorenzo R., The geometry of Walker manifolds, Synthesis Lect. Math. Statistics, Morgan Claypool Publ, 2009, 179 pp. | DOI | MR | Zbl

[12] Wu H., “On the de Rham decomposition theorem”, Illinois J. Math., 8:2 (1964), 291–311 | DOI | MR | Zbl

[13] Hall G. S., “Conformal symmetries and fixed points in spacetime”, J. Math. Phys., 31:5 (1989), 1198–1207 | DOI | MR

[14] Blau M., O'Loughlin M., “Homogeneous plane waves”, Nuclear Phys., 654:1–2 (2003), 135–176 | DOI | MR | Zbl

[15] Fedoryuk, M. V., “Eyri Functions”, Encyclopaedia of Mathematics, v. 5, ed. I. M. Vinogradov, Sovetskaya Entsiklopediya Publisher, M., 1985, 939–941 | MR