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Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter collineations, curvature and Weyl collineations. Several results are given for the broader class of three-dimensional Walker manifolds.
Keywords: Walker manifolds; Killing vector fields; affine vector fields; Ricci collineations; curvature and Weyl collineations; matter collineations. 
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     author = {Giovanni Calvaruso and Amirhesam Zaeim},
     title = {Symmetries of {Lorentzian} {Three-Manifolds} with {Recurrent} {Curvature}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a62/}
}
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Giovanni Calvaruso; Amirhesam Zaeim. Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a62/

[1] Aichelburg P. C., “Curvature collineations for gravitational ${\rm pp}$ waves”, J. Math. Phys., 11 (1970), 2458–2462 | DOI | MR

[2] Alekseevski D., “Self-similar Lorentzian manifolds”, Ann. Global Anal. Geom., 3 (1985), 59–84 | DOI | MR | Zbl

[3] Batat W., Calvaruso G., De Leo B., “Homogeneous Lorentzian 3-manifolds with a parallel null vector field”, Balkan J. Geom. Appl., 14 (2009), 11–20 | MR | Zbl

[4] Beem J. K., “Proper homothetic maps and fixed points”, Lett. Math. Phys., 2 (1978), 317–320 | DOI | MR | Zbl

[5] Brozos-Vázquez M., García-Río E., Gilkey P., Nikčević S., Vázquez-Lorenzo R., “The geometry of Walker manifolds”, Synthesis Lectures on Mathematics and Statistics, 5, Morgan Claypool Publishers, Williston, VT, 2009 | MR | Zbl

[6] Calvaruso G., Zaeim A., “Invariant symmetries on non-reductive homogeneous pseudo-Riemannian four-manifolds”, Rev. Mat. Complut., 28 (2015), 599–622 | DOI | MR | Zbl

[7] Calvaruso G., Zaeim A., “Geometric structures over four-dimensional generalized symmetric spaces”, Collect. Math. (to appear) | DOI

[8] Calviño-Louzao E., Seoane-Bascoy J., Vázquez-Abal M. E., Vázquez-Lorenzo R., “Invariant Ricci collineations on three-dimensional Lie groups”, J. Geom. Phys., 96 (2015), 59–71 | DOI | MR | Zbl

[9] Camci U., Hussain I., Kucukakca Y., “Curvature and Weyl collineations of Bianchi type V spacetimes”, J. Geom. Phys., 59 (2009), 1476–1484 | DOI | MR | Zbl

[10] Camci U., Sharif M., “Matter collineations of spacetime homogeneous Gödel-type metrics”, Classical Quantum Gravity, 20 (2003), 2169–2179, arXiv: gr-qc/0306129 | DOI | MR | Zbl

[11] Carot J., da Costa J., Vaz E. G. L. R., “Matter collineations: the inverse “symmetry inheritance” problem”, J. Math. Phys., 35 (1994), 4832–4838 | DOI | MR | Zbl

[12] Chaichi M., García-Río E., Vázquez-Abal M. E., “Three-dimensional Lorentz manifolds admitting a parallel null vector field”, J. Phys. A: Math. Gen., 38 (2005), 841–850 | DOI | MR | Zbl

[13] Flores J. L., Parra Y., Percoco U., “On the general structure of Ricci collineations for type B warped space-times”, J. Math. Phys., 45 (2004), 3546–3557, arXiv: gr-qc/0405133 | DOI | MR | Zbl

[14] García-Río E., Gilkey P. B., Nikčević S., “Homogeneity of Lorentzian three-manifolds with recurrent curvature”, Math. Nachr., 287 (2014), 32–47, arXiv: 1210.7764 | DOI | MR | Zbl

[15] Hall G., “Symmetries of the curvature, Weyl conformal and Weyl projective tensors on 4-dimensional Lorentz manifolds”, Proceedings of the International Conference “Differential Geometry – Dynamical Systems”, DGDS-2007, BSG Proc., 15, Geom. Balkan Press, Bucharest, 2008, 89–98 | MR | Zbl

[16] Hall G. S., Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004 | DOI | MR

[17] Hall G. S., Capocci M. S., “Classification and conformal symmetry in three-dimensional space-times”, J. Math. Phys., 40 (1999), 1466–1478 | DOI | MR | Zbl

[18] Hall G. S., Low D. J., Pulham J. R., “Affine collineations in general relativity and their fixed point structure”, J. Math. Phys., 35 (1994), 5930–5944 | DOI | MR | Zbl

[19] Hall G. S., Roy I., Vaz E. G. L. R., “Ricci and matter collineations in space-time”, Gen. Relativity Gravitation, 28 (1996), 299–310 | DOI | MR | Zbl

[20] Kühnel W., Rademacher H.-B., “Conformal Ricci collineations of space-times”, Gen. Relativity Gravitation, 33 (2001), 1905–1914 | DOI | MR

[21] Levichev A. V., “Methods for studying the causal structure of homogeneous Lorentz manifolds”, Sib. Math. J., 31 (1990), 395–408 | DOI | MR | Zbl

[22] Tsamparlis M., Apostolopoulos P. S., “Ricci and matter collineations of locally rotationally symmetric space-times”, Gen. Relativity Gravitation, 36 (2004), 47–69, arXiv: gr-qc/0309034 | DOI | MR | Zbl

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