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Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Published online by Cambridge University Press:  20 November 2018

Giovanni Calvaruso  and
Anna Fino
Giovanni Calvaruso
Affiliation:
Dipartimento di Matematica “E. De Giorgi”, Universitá del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy email: giovanni.calvaruso@unisalento.it
Anna Fino
Affiliation:
Dipartimento di Matematica, Universitá di Torino, Via Carlo Alberto 10, 10123 Torino, Italy email: annamaria.fino@unito.it
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Abstract

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We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

Keywords

53C2153C5053C25non-reductive homogeneous spacespseudo-RiemannianmetricsRicci solitonsEinsteinlike metrics
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 778 - 804
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Akbar, M. M. and E.Woolgar, Ricci solitons and Einstein-scalar field theory. Classical Quantum Gravity 26(2009), no. 5, 055015, 14 pp. http://dx.doi.org/10.1088/0264-9381/26/5/055015 Google Scholar
[2] Baird, P. and Danielo, L., Three-dimensional Ricci solitons which project to surfaces. J. Reine Angew. Math. 608(2007), 65–91. http://dx.doi.org/10.1515/CRELLE.2007.053 Google Scholar
[3] Brozos-Vazquez, M., Calvaruso, G., Garcia-Rio, E., and Gavino-Fernandez, S., Three-dimensional Lorentzian homogeneous Ricci solitons. Israel J. Math., to appear.Google Scholar
[4] Bueken, P. and Vanhecke, L., Three- and four-Dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75(1999), no. 2, 123–136. http://dx.doi.org/10.1023/A:1005060208823 Google Scholar
[5] Calvaruso, G., Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57(2007), no. 4, 1279–1291; Addendum: J. Geom. Phys. 58(2008), no. 2, 291–292. http://dx.doi.org/10.1016/j.geomphys.2006.10.005 Google Scholar
[6] Calvaruso, G., Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geom. Dedicata 127 (2007), 99–119. http://dx.doi.org/10.1007/s10711-007-9163-7 Google Scholar
[7] Calvaruso, G., Three-dimensional semi-symmetric homogeneous Lorentzian manifolds. Acta Math. Hungar. 121(2008), no. 1–2, 157–170. http://dx.doi.org/10.1007/s10474-008-7194-7 Google Scholar
[8] Calvaruso, G. and De Leo, B., Ricci solitons on Lorentzian Walker three-manifolds. Acta Math. Hungar. 132(2011), no. 3, 269–293.Google Scholar
[9] Calvaruso, G. and Fino, A., Four-dimensional pseudo-Riemannian homogeneous Ricci solitons. arxiv:1111.6384.Google Scholar
[10] H.-D. Cao, , Recent progress on Ricci solitons. In: Recent advances in geometric analysis, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010, pp. 1–38.Google Scholar
[11] Case, J. S., Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor. J. Geom. Phys. 60(2010), no. 3, 477–490. http://dx.doi.org/10.1016/j.geomphys.2009.11.001 Google Scholar
[12] di Cerbo, L. F., Generic properties of homogeneous Ricci solitons. arxiv:0711.0465v1.Google Scholar
[14] Fels, M. E. and Renner, A. G., Non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Canad. J. Math. 58(2006), no. 2, 282–311. http://dx.doi.org/10.4153/CJM-2006-012-1 Google Scholar
[13] Dušek, Z. and Kowalski, O., Light-like homogeneous geodesics and the geodesic lemma for any signature. Publ. Math. Debrecen, 71(2007), no. 1–2, 245–252.Google Scholar
[15] Friedan, D. H., Nonlinear models in 2 + “ dimensions. Ann. Physics 163(1985), no. 2, 318–419. http://dx.doi.org/10.1016/0003-4916(85)90384-7 Google Scholar
[16] Gray, A., Einstein-like manifolds which are not Einstein. Geom. Dedicata 7 (1978), no. 3, 259–280.Google Scholar
[17] Gadea, P. M. and Oubi˜na, J. A., Homogeneous pseudo-Riemannian structures and homogeneous almost para-Hermitian structures. Houston J. Math., 18(1992), no. 3, 449–465.Google Scholar
[18] Hervik, S., Ricci nilsoliton black holes. J. Geom. Phys. 58(2008), no. 9, 1253–1264. http://dx.doi.org/10.1016/j.geomphys.2008.05.001 Google Scholar
[19] Kholodenko, A. L., Towards physically motivated proofs of the Poincaré and the geometrization conjectures. J. Geom. Phys. 58(2008), no. 2, 259–290. http://dx.doi.org/10.1016/j.geomphys.2007.11.003 Google Scholar
[20] Komrakov, B. Jr., Einstein-Maxwell equation on four-dimensional homogeneous spaces. Lobachevskii J. Math. 8(2001), 33–165.Google Scholar
[21] Lauret, J., Ricci solitons solvmanifolds. J. Reine Angew. Math. 650(2011), 1–21. http://dx.doi.org/10.1515/CRELLE.2011.001 Google Scholar
[22] O’Neill, B., Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103, Academic Press, New York, 1983.Google Scholar
[23] Patera, J., Sharp, R. T., P.Winternitz, and Zassenhaus, H., Invariants of real low dimension Lie algebras. J. Mathematical Phys. 17(1976), no. 6, 986–994. http://dx.doi.org/10.1063/1.522992 Google Scholar
[24] Payne, T. L., The existence of soliton metrics for nilpotent Lie groups. Geom. Dedicata 145(2010), 71–88. http://dx.doi.org/10.1007/s10711-009-9404-z Google Scholar
[25] Pina, R. and Tenenblat, K., On solutions of the Ricci curvature equation and the Einstein equation. Israel J. Math. 171(2009), 61–76. http://dx.doi.org/10.1007/s11856-009-0040-yGoogle Scholar