Geodesic
On the Einstein equation on Lorentzian manifolds with parallel distributions of isotropic lines
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 3, pp. 165-174 Cet article a éte moissonné depuis la source Math-Net.Ru

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Some recent results about Einstein Lorentzian manifolds that admit parallel distributions of isotropic lines are reviewed. We find all holonomy algebras of such manifolds and describe special coordinates that allow us to simplify the Einstein equation. Examples in dimension 4 are considered.
Keywords: Lorentzian manifold, Einstein equation, Walker metric, holonomy algebra, recurrent light-like vector field, Petrov classification. 
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A. S. Galaev. On the Einstein equation on Lorentzian manifolds with parallel distributions of isotropic lines. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 3, pp. 165-174. http://geodesic.mathdoc.fr/item/UZKU_2011_153_3_a14/

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

[2] Brannlund J., Coley A., Hervik S., “Supersymmetry, holonomy and Kundt spacetimes”, Class. Quantum Grav., 25 (2008), 195007-1–195007-10 | DOI | MR

[3] Bryant R., “Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor”, Sémin. Congr., 4 (2000), 53–94 | MR | Zbl

[4] Figueroa-O'Farrill J. M., “Breaking the M-waves”, Class. Quantum Grav., 17:15 (2000), 2925–2947 | DOI | MR | Zbl

[5] Gibbons G. W., Pope C. N., “Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy $\mathrm{Sim}(n-2)$”, Class. Quantum Grav., 25 (2008), 125015-1–125015-21 | DOI | MR

[6] Schimming R., “Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie”, Math. Nachr., 59 (1974), 129–162 | DOI | MR | Zbl

[7] Galaev A. S., Leistner T., “On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines”, Class. Quantum Grav., 27 (2010), 225003-1–225003-16 | DOI | MR

[8] Lewandowski J., “Reduced holonomy group and Einstein equations with a cosmological constant”, Class. Quantum Grav., 9:10 (1992), L147–L151 | DOI | MR | Zbl

[9] Hall G. S., Lonie D. P., “Holonomy groups and spacetimes”, Class. Quantum Grav., 17:6 (2000), 1369–1382 | DOI | MR | Zbl

[10] Galaev A. S., “Examples of Einstein spacetimes with recurrent null vector fields”, Class. Quantum Grav., 28 (2011), 175022-1–175022-6 | DOI | MR

[11] Goldberg J. N., Kerr R. P., “Some applications of the infinitesimal-holonomy group to the Petrov classification of Einstein spaces”, J. Math. Phys., 2 (1961), 327–332 | DOI | MR | Zbl

[12] Goldberg J. N., Kerr R. P., “Einstein spaces with four-parameter holonomy groups”, J. Math. Phys., 2 (1961), 332–336 | DOI | MR | Zbl

[13] Schell J. F., “Classification of four-dimensional Riemannian spaces”, J. Math. Phys., 2 (1961), 202–206 | DOI | MR | Zbl

[14] Besse A. L., Einstein manifolds, Springer-Verlag, Berlin–Heidelberg–New York, 1987, 510 pp. | MR | Zbl

[15] Galaev A. S., “Holonomy of Einstein Lorentzian manifolds”, Class. Quantum Grav., 27 (2010), 075008-1–075008-13 | DOI | MR

[16] Galaev A. S., Leistner T., “Holonomy groups of Lorentzian manifolds: classification, examples, and applications”, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, 53–96 | MR | Zbl

[17] Astrahancev V. V., “The holonomy groups of four-dimensional pseudo-Riemannian spaces”, Mat. Zametki, 9:1 (1971), 59–66 | MR | Zbl