### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 063, 12 pages      arXiv:1602.03693      https://doi.org/10.3842/SIGMA.2016.063

### Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature

Giovanni Calvaruso a and Amirhesam Zaeim b
a) Dipartimento di Matematica e Fisica ''E. De Giorgi'', Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
b) Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Received February 12, 2016, in final form June 17, 2016; Published online June 26, 2016

Abstract
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.

Key words: Walker manifolds; Killing vector fields; affine vector fields; Ricci collineations; curvature and Weyl collineations; matter collineations.

pdf (320 kb)   tex (17 kb)

References

1. Aichelburg P.C., Curvature collineations for gravitational ${\rm pp}$ waves, J. Math. Phys. 11 (1970), 2458-2462.
2. Alekseevski D., Self-similar Lorentzian manifolds, Ann. Global Anal. Geom. 3 (1985), 59-84.
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.
4. Beem J.K., Proper homothetic maps and fixed points, Lett. Math. Phys. 2 (1978), 317-320.
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, Vol. 5, Morgan & Claypool Publishers, Williston, VT, 2009.
6. Calvaruso G., Zaeim A., Invariant symmetries on non-reductive homogeneous pseudo-Riemannian four-manifolds, Rev. Mat. Complut. 28 (2015), 599-622.
7. Calvaruso G., Zaeim A., Geometric structures over four-dimensional generalized symmetric spaces, Collect. Math., to appear.
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.
9. Camci U., Hussain I., Kucukakca Y., Curvature and Weyl collineations of Bianchi type V spacetimes, J. Geom. Phys. 59 (2009), 1476-1484.
10. Camci U., Sharif M., Matter collineations of spacetime homogeneous Gödel-type metrics, Classical Quantum Gravity 20 (2003), 2169-2179, gr-qc/0306129.
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.
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.
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, gr-qc/0405133.
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.
15. Hall G., Symmetries of the curvature, Weyl conformal and Weyl projective tensors on 4-dimensional Lorentz manifolds, in Proceedings of the International Conference ''Differential Geometry - Dynamical Systems'' (DGDS-2007), BSG Proc., Vol. 15, Geom. Balkan Press, Bucharest, 2008, 89-98.
16. Hall G.S., Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, Vol. 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
17. Hall G.S., Capocci M.S., Classification and conformal symmetry in three-dimensional space-times, J. Math. Phys. 40 (1999), 1466-1478.
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.
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.
20. Kühnel W., Rademacher H.-B., Conformal Ricci collineations of space-times, Gen. Relativity Gravitation 33 (2001), 1905-1914.
21. Levichev A.V., Methods for studying the causal structure of homogeneous Lorentz manifolds, Sib. Math. J. 31 (1990), 395-408.
22. Tsamparlis M., Apostolopoulos P.S., Ricci and matter collineations of locally rotationally symmetric space-times, Gen. Relativity Gravitation 36 (2004), 47-69, gr-qc/0309034.
23. Walker A.G., On parallel fields of partially null vector spaces, Quart. J. Math. Oxford Ser. 20 (1949), 135-145.