Journal of Geometric Analysis

, Volume 23, Issue 3, pp 1196–1212 | Cite as

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

  • M. Brozos-VázquezEmail author
  • E. García-Río
  • S. Gavino-Fernández


It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.


Ricci solitons Gradient Ricci solitons Lorentzian locally conformally flat manifolds 

Mathematics Subject Classification (2000)

53C21 53C50 53C25 


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  1. 1.
    Alekseevsky, D.V., Galaev, A.S.: Two-symmetric Lorentzian manifolds. J. Geom. Phys. 61, 2331–2340 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Batat, W., Brozos-Vázquez, M., García-Río, E., Gavino-Fernández, S.: Ricci solitons on Lorentzian manifolds with large isometry groups. Bull. London Math. Soc. doi: 10.1112/blms/bdr057, to appear
  3. 3.
    Blanco, O.F., Sánchez, M., Senovilla, J.M.: Complete classification of second-order symmetric spacetimes. J. Phys.: Conf. Ser. 229 (2010), 5pp. Google Scholar
  4. 4.
    Brinkmann, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94, 119–145 (1925) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brozos-Vázquez, M., Calvaruso, G., García-Río, E., Gavino-Fernández, S.: Three-dimensional Lorentzian homogeneous Ricci solitons. Israel J. Math. doi: 10.1007/s11856-011-0124-3, to appear
  6. 6.
    Brozos-Vázquez, M., García-Río, E., Vázquez-Lorenzo, R.: Some remarks on locally conformally flat static space-times. J. Math. Phys. 46 (2005), 11pp. Google Scholar
  7. 7.
    Calvaruso, G., García-Río, E.: Algebraic properties of curvature operators in Lorentzian manifolds with large isometry groups. SIGMA Symmetry Integrab. Geom. Methods Appl. 6 (2010), 8pp. Google Scholar
  8. 8.
    Candela, A.M., Flores, J.L., Sánchez, M.: On general plane fronted waves. Geodesics Gen. Relativ. Gravit. 35, 631–649 (2003) zbMATHCrossRefGoogle Scholar
  9. 9.
    Candela, A.M., Sánchez, M.: Geodesics in semi-Riemannian manifolds: geometric properties and variational tools. In: Recent developments in pseudo-Riemannian geometry. ESI Lect. Math. Phys., pp. 359–418. Eur. Math. Soc., Zürich (2008) CrossRefGoogle Scholar
  10. 10.
    Cao, H.D., Chen, Q.: On locally conformally flat steady gradient Ricci solitons. Trans. Am. Math. Soc., to appear Google Scholar
  11. 11.
    Chow, B., Chu, S.-Ch., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects. Mathematical Surveys and Monographs, vol. 135. Am. Math. Soc., Providence (2007) Google Scholar
  12. 12.
    Derdzinski, A., Roter, W.: Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math. J. 59, 565–602 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Derdzinski, A., Roter, W.: Some theories of conformally symmetric manifolds. Tensor (N.S.) 32, 11–23 (1978) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fernández-López, M., García-Río, E.: Rigidity of shrinking Ricci solitons. Math. Z. 269, 461–466 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fernández-López, M., García-Río, E., Kupeli, D., Ünal, B.: A curvature condition for a twisted product to be a warped product. Manuscr. Math. 106, 213–217 (2001) zbMATHCrossRefGoogle Scholar
  16. 16.
    Galaev, A.S.: Lorentzian manifolds with recurrent curvature tensor. arXiv:1011.6541v1
  17. 17.
    Hamilton, R.S.: The formation of singularities in the Ricci flow. In: Surveys in Differential Geometry, Cambridge, MA, 1993, vol. II, pp. 7–136. International Press, Cambridge (1995) Google Scholar
  18. 18.
    Kobayashi, S.: A theorem on the affine transformation group of a Riemannian manifold. Nagoya Math. J. 9, 39–41 (1955) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kühnel, W., Rademacher, H.-B.: Einstein spaces with a conformal group. Results Math. 56, 421–444 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Leistner, T.: Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds. Differ. Geom. Appl. 24, 458–478 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Munteanu, O., Sesum, N.: On gradient Ricci solitons. J. Geom. Anal. doi: 10.1007/s12220-011-9252-6, to appear
  22. 22.
    Onda, K.: Lorentz Ricci solitons on 3-dimensional Lie groups. Geom. Dedic. 147, 313–322 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Patrangenaru, V.: Lorentz manifolds with the three largest degrees of symmetry. Geom. Dedic. 102, 25–33 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Petersen, P., Wylie, W.: On gradient Ricci solitons with symmetry. Proc. Am. Math. Soc. 137, 2085–2092 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pac. J. Math. 241, 329–345 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ponge, R., Reckziegel, H.: Twisted products in pseudo-Riemannian geometry. Geom. Dedic. 48, 15–25 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Senovilla, J.M.: Second-order symmetric Lorentzian manifolds. I. Characterization and general results. Class. Quantum Grav. 25 (2008), 25pp. Google Scholar
  28. 28.
    Walker, A.G.: On Ruse’s spaces of recurrent curvature. Proc. Lond. Math. Soc. 52, 36–64 (1950) zbMATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2011

Authors and Affiliations

  • M. Brozos-Vázquez
    • 1
    Email author
  • E. García-Río
    • 2
  • S. Gavino-Fernández
    • 2
  1. 1.Department of MathematicsUniversity of A CoruñaA CoruñaSpain
  2. 2.Faculty of MathematicsUniversity of Santiago de CompostelaSantiago de CompostelaSpain

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