Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T19:43:23.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One

Published online by Cambridge University Press:  20 November 2018

G. Calvaruso
G. Calvaruso*
Affiliation:
Dip. di Matematica “E. De Giorgi”, Università di Lecce, Prov. Lecce-Arnesano, Lecce, Italy e-mail: giovanni.calvaruso@unile.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

Keywords

53C5053C2053C30Lorentz manifoldscurvature homogeneityEinstein-like metrics
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 412 - 424
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Abbena, E. and Garbiero, S., Curvature forms and Einstein-like metrics on Sasakian manifolds. Math. J. Okayama Univ. 34(1992), 241–248.Google Scholar
[2] Abbena, E., Garbiero, S., and Vanhecke, L., Einstein-like metrics on three-dimensional Riemannian homogeneous manifolds. Simon Stevin 66(1992), no. 1–2, 173–183.Google Scholar
[3] Bueken, P. and Djorić, M., Three-dimensional Lorentz metrics and curvature homogeneity of order one. Ann. Glob. Anal. Geom. 18(2000), no. 1, 85–103. doi:10.1023/A:1006612120550Google Scholar
[4] Bueken, P. and Vanhecke, L., Three- and four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75(1999), no. 2, 123–136. doi:10.1023/A:1005060208823Google Scholar
[5] Calvaruso, G., Einstein-like and conformally flat contact metric three-manifolds. Balkan J. Geom. Appl. 5(2000), no. 2, 17–36.Google Scholar
[6] Calvaruso, G., Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57(2007), no. 4, 1279–1291. doi:10.1016/j.geomphys.2006.10.005Google Scholar
[7] Calvaruso, G., Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geom. Dedicata 127(2007), 99–119. doi:10.1007/s10711-007-9163-7Google Scholar
[8] Calvaruso, G. and Marinosci, R. A., Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three. Adv. Geom. 8(2008), no. 4, 473–489. doi:10.1515/ADVGEOM.2008.030Google Scholar
[9] Calvaruso, G. and Vanhecke, L., Special ball-homogeneous spaces. Z. Anal. Anwendungen 16(1997), no. 4, 789–800.Google Scholar
[10] Chaichi, M., García-Río, E., and Vázquez-Abal, M. E., Three-dimensional Lorentz manifolds admitting a parallel null vector field. J. Phys. A 38(2005), no. 4, 841–850. doi:10.1088/0305-4470/38/4/005Google Scholar
[11] Gray, A., Einstein-like manifolds which are not Einstein. Geom. Dedicata 7(1978), no. 3, 259–280.Google Scholar
[12] O’Neill, B., Semi-Riemannian geometry. Pure and Applied Mathematics, 103, Academic Press Inc., New York, 1983.Google Scholar
[13] Pedersen, H. and Tod, P., The Ledger curvature conditions and D’Atri geometry. Differential Geom. Appl. 11(1999), 155–162. doi:10.1016/S0926-2245(99)00026-1Google Scholar
[14] Sekigawa, K., On some 3-dimensional curvature homogeneous spaces. Tensor (N.S.) 31(1977), no. 1, 87–97.Google Scholar
[15] Singer, I. M., Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13(1960), 685–697. doi:10.1002/cpa.3160130408Google Scholar
[16] Sternberg, S., Lectures on differential geometry. Prentice Hall, Englewood Cliffs, NJ, 1964.Google Scholar
[17] Takagi, H., Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tôhoku Math. J. 27(1975), no. 1, 103–110. doi:10.2748/tmj/1178241040Google Scholar