Three-dimensional homogeneous Lorentzian Yamabe solitons

  • E. Calviño-Louzao
  • J. Seoane-Bascoy
  • M. E. Vázquez-Abal
  • R. Vázquez-Lorenzo
Article
First Online:
Received:

Abstract

A geometric characterization of Yamabe solitons on homogeneous Lorentzian manifolds of dimension three is given. As a consequence, Lorentzian Yamabe solitons and left-invariant Lorentzian Yamabe solitons are classified in this setting, showing the existence of Yamabe solitons which are not left-invariant.

Keywords

Homogeneous spaces Lie groups Yamabe solitons 

Mathematics Subject Classification

53C50 53B30 
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Notes

Acknowledgement

We would like to thank the Referee for his comments which resulted in an improved version of our original manuscript.

References

  1. 1.
    Batat, W., Brozos-Vázquez, M., García-Río, E., Gavino-Fernández, S.: Ricci solitons on Lorentzian manifolds with large isometry groups. Bull. Lond. Math. Soc. 43, 1219–1227 (2011) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Blau, M., O’Loughlin, M.: Homogeneous plane waves. Nucl. Phys. B 654, 135–176 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brozos-Vázquez, M., Calvaruso, G., García-Río, E., Gavino-Fernández, S.: Three-dimensional Lorentzian homogeneous Ricci solitons. Isr. J. Math. 188, 385–403 (2012) CrossRefGoogle Scholar
  4. 4.
    Brozos-Vázquez, M., García-Río, E., Gilkey, P., Nikčevic, S., Vázquez-Lorenzo, R.: The Geometry of Walker Manifolds. Synthesis Lect. Mathematics and Statistics, vol. 5. Morgan & Claypool, Williston (2009) MATHGoogle Scholar
  5. 5.
    Calviño-Louzao, E., García-Río, E., Vázquez-Abal, M.E., Vázquez-Lorenzo, R.: Curvature operators and generalizations of symmetric spaces in Lorentzian geometry. Adv. Geom. 12, 83–100 (2012) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Calvaruso, G.: Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geom. Dedic. 127, 99–119 (2007) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Calvaruso, G.: Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57, 1279–1291 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cordero, L.A., Parker, Ph.: Left-invariant Lorentzian metrics on 3-dimensional Lie groups. Rend. Mat. VII 17, 129–155 (1997) MathSciNetMATHGoogle Scholar
  9. 9.
    García-Río, E., Haji-Badali, A., Vázquez-Abal, M.E., Vázquez-Lorenzo, R.: Lorentzian 3-manifolds with commuting curvature operators. Int. J. Geom. Methods Mod. Phys. 5, 557–572 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hall, G.S.: The global extension of local symmetries in general relativity. Class. Quantum Gravity 6, 157–161 (1989) MATHCrossRefGoogle Scholar
  11. 11.
    Hall, G.S., Capocci, M.S.: Classification and conformal symmetry in three-dimensional space-times. J. Math. Phys. 40, 1466–1478 (1999) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kühnel, W., Rademacher, H.B.: Conformal vector fields on pseudo-Riemannian spaces. Differ. Geom. Appl. 7, 237–250 (1997) MATHCrossRefGoogle Scholar
  13. 13.
    Onda, K.: Lorentz Ricci solitons on 3-dimensional Lie groups. Geom. Dedic. 147, 313–322 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Walker, A.G.: On Ruse’s spaces of recurrent curvature. Proc. Lond. Math. Soc. (2) 52, 36–64 (1950) MATHCrossRefGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • E. Calviño-Louzao
    • 1
  • J. Seoane-Bascoy
    • 1
  • M. E. Vázquez-Abal
    • 1
  • R. Vázquez-Lorenzo
    • 1
  1. 1.Department of Geometry and Topology, Faculty of MathematicsUniversity of Santiago de CompostelaSantiago de CompostelaSpain

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