Geodesic
Conformally flat Lorentzian manifolds with special holonomy groups
Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1264-1284

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a local classification of conformally flat Lorentzian manifolds with special holonomy groups. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics. Bibliography: 28 titles.
Keywords: Lorentzian manifold, special holonomy group, Walker manifold, conformally flat manifold, Nordström's gravity. 
A. S. Galaev. Conformally flat Lorentzian manifolds with special holonomy groups. Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1264-1284. http://geodesic.mathdoc.fr/item/SM_2013_204_9_a1/
@article{SM_2013_204_9_a1,
     author = {A. S. Galaev},
     title = {Conformally flat {Lorentzian} manifolds with special holonomy groups},
     journal = {Sbornik. Mathematics},
     pages = {1264--1284},
     year = {2013},
     volume = {204},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_9_a1/}
}
TY  - JOUR
AU  - A. S. Galaev
TI  - Conformally flat Lorentzian manifolds with special holonomy groups
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 1264
EP  - 1284
VL  - 204
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_9_a1/
LA  - en
ID  - SM_2013_204_9_a1
ER  - 
%0 Journal Article
%A A. S. Galaev
%T Conformally flat Lorentzian manifolds with special holonomy groups
%J Sbornik. Mathematics
%D 2013
%P 1264-1284
%V 204
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2013_204_9_a1/
%G en
%F SM_2013_204_9_a1

[1] M. Kurita, “On the holonomy groups of the conformally flat Riemannian manifold”, Nagoya Math. J., 9 (1955), 161–171 | MR | Zbl

[2] R. Bryant, “Classical, exceptional, and exotic holonomies: a status report”, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Semin. Congr., 1, Soc. Math. France, Paris, 1996, 93–165 | MR | Zbl

[3] A. Galaev, Th. Leistner, “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

[4] M. Brozos-Vázquez, E. Garcia-Rio, P. Gilkey, S. Nikčević, R. Vázquez-Lorenzo, The geometry of Walker manifolds (San Rafael, CA), Synth. Lect. Math. Stat., 5, Morgan Claypool Publ., Williston, VT, 2009 | MR | Zbl

[5] Th. Leistner, P. Nurowski, “Ambient metrics for $n$-dimensional $pp$-waves”, Comm. Math. Phys., 296:3 (2010), 881–898 | DOI | MR | Zbl

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

[7] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact solutions of Einstein's field equations, Cambridge Monogr. Math. Phys., Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[8] R. Ghanam, G. Thompson, “Two special metrics with $R_{14}$-type holonomy”, Classical Quantum Gravity, 18:11 (2001), 2007–2014 | DOI | MR | Zbl

[9] J. D. Norton, “Einstein, Nordström and the early demise of scalar, Lorentz-covariant theories of gravitation”, Arch. Hist. Exact Sci., 45:1 (1992), 17–94 | DOI | MR | Zbl

[10] N. Ravndal, “Scalar gravitation and extra dimensions”, Proceedings of the Gunnar Nordström Symposium on Theoretical Physics, Comment. Phys.-Math., 166, Finn. Soc. Sci. Lett., Helsinki, 2004, 151–164 | MR

[11] V. P. Vizgin, Relyativistskaya teoriya tyagoteniya (istoki i formirovanie, 1900–1915), Nauka, M., 1981 | MR | Zbl

[12] G. W. Gibbons, C. N. Pope, “Time-dependent multi-centre solutions from new metrics with holonomy $\operatorname{Sim}(n-2)$”, Classical Quantum Gravity, 25:12 (2008), ID 125015 | DOI | MR | Zbl

[13] H. Baum, “Conformal Killing spinors and the holonomy problem in Lorentzian geometry – a survey of new results”, Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 251–264 | DOI | MR | Zbl

[14] Ya. V. Bazaikin, “Globally hyperbolic Lorentzian spaces with special holonomy groups”, Siberian Math. J., 50:4 (2009), 567–579 | DOI | MR | Zbl

[15] G. Hall, “Projective relatedness and conformal flatness”, Cent. Eur. J. Math., 10:5 (2012), 1763–1770 | DOI | MR | Zbl

[16] D. V. Alekseevskii, B. N. Kimel'fel'd, “Classification of homogeneous conformally flat Riemannian manifolds”, Math. Notes, 24:1 (1978), 559–562 | DOI | MR | Zbl | Zbl

[17] V. F. Kirichenko, “Conformally flat and locally conformal Kähler manifolds”, Math. Notes, 51:5 (1992), 462–468 | DOI | MR | Zbl

[18] V. V. Slavskii, “Conformally-flat metrics and pseudo-Euclidean geometry”, Siberian Math. J., 35:3 (1994), 605–613 | DOI | MR | Zbl

[19] M. Erdogan, T. Ikawa, “On conformally flat Lorentzian spaces satisfying a certain condition on the Ricci tensor”, Indian J. Pure Appl. Math., 26:5 (1995), 417–424 | MR | Zbl

[20] K. Honda, “Conformally flat semi-riemannian manifolds with commuting curvature and Ricci operators”, Tokyo J. Math., 26:1 (2003), 241–260 | DOI | MR | Zbl

[21] K. Honda, K. Tsukada, “Conformally flat semi-Riemannian manifolds with nilpotent Ricci operators and affine differential geometry”, Ann. Global Anal. Geom., 25:3 (2004), 253–275 | DOI | MR | Zbl

[22] K. Honda, K. Tsukada, “Three-dimensional conformally flat homogeneous Lorentzian manifolds”, J. Phys. A, 40:4 (2007), 831–851 | MR | Zbl

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

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

[25] K. Yano, “On pseudo-Hermitian and pseudo-Kählerian manifolds”, Proceedings of the International Congress of Mathematicians (Amsterdam, 1954), v. III, North-Holland, Amsterdam, 1954, 190–197 | MR | Zbl

[26] A. S. Galaev, “The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds”, Differential Geom. Appl., 22:1 (2005), 1–18 | DOI | MR | Zbl

[27] A. S. Galaev, “One component of the curvature tensor of a Lorentzian manifold”, J. Geom. Phys., 60:6–8 (2010), 962–971 | DOI | MR | Zbl

[28] A. S. Galaev, T. Leistner, “On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines”, Classical Quantum Gravity, 27 (2010), ID 225003 | DOI | MR | Zbl