Advertisement

Monatshefte für Mathematik

, Volume 173, Issue 2, pp 175–186 | Cite as

Locally conformally flat Lorentzian quasi-Einstein manifolds

  • M. Brozos-Vázquez
  • E. García-RíoEmail author
  • S. Gavino-Fernández
Article

Abstract

We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a Robertson-Walker spacetime or a \(pp\)-wave.

Keywords

Quasi-Einstein Lorentzian metrics Locally conformally flat manifolds 

Mathematics Subject Classification (2010)

53C21 53C50 53C25 

References

  1. 1.
    Alekseevsky, D.V., Galaev, A.S.: Two-symmetric Lorentzian manifolds. J. Geom. Phys. 61, 2331–2340 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Anderson, M.T.: Topics in conformally compact Einstein metrics. In: Perspectives in Riemannian geometry. CRM Proceedings and Lecture Notes, vol. 40, pp. 1–26. American Mathematical Society, Providence (2006)Google Scholar
  3. 3.
    Blanco, O.F., Sánchez, M., Senovilla, J.M.: Structure of second-order symmetric Lorentzian manifolds. J. Eur. Math. Soc. (JEMS) 15, 595–634 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brozos-Vázquez, M., García-Río, E., Gavino-Fernández, S.: Locally conformally flat Lorentzian gradient Ricci solitons. J. Geom. Anal. 23, 1196–1212 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brozos-Vázquez, M., García-Río, E., Gavino-Fernández, S.: Quasi-Einstein metrics and plane waves. In: XX International Fall Workshop on Geometry and Physics. AIP Conference Proceedings, vol. 1460, pp. 174–179. American Institute of Physics, Melville (2012)Google Scholar
  6. 6.
    Brozos-Vázquez, M., García-Río, E., Vázquez-Lorenzo, R.: Some remarks on locally conformally flat static space-times. J. Math. Phys. 46, 022501 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cao, H.-D.: Recent progress on Ricci solitons. In: Recent Advances in Geometric Analysis. Advanced Lectures in Mathematics (ALM), vol. 11, pp. 1–38. International Press, Somerville (2010)Google Scholar
  8. 8.
    Case, J., Shu, Y.-J., Wei, G.: Rigidity of quasi-Einstein metrics. Diff. Geom. Appl. 29, 93–100 (2010)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Case, J.S.: Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor. J. Geom. Phys. 60, 477–490 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Case, J.S.: The nonexistence of quasi-Einstein metrics. Pacific J. Math. 248, 277–284 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Catino, G., Mantegazza, C., Mazzieri, L., Rimoldi, M.: Locally conformally flat quasi-Einstein manifolds. J. Reine Angew. Math. 675, 181–189 (2013)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fernández-López, M., García-Río, E.: Rigidity of shrinking Ricci solitons. Math. Z. 269, 461–466 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fernández-López, M., García-Río, E., Kupeli, D., Ünal, B.: A curvature condition for a twisted product to be a warped product. Manuscripta Math. 106, 213–217 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kim, D.-S., Kim, Y.H.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Am. Math. Soc. 131, 2573–2576 (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Leistner, T.: Conformal holonomy of \(C\)-spaces, Ricci-flat, and Lorentzian manifolds. Differ. Geom. Appl. 24, 458–478 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ponge, R., Reckziegel, H.: Twisted products in pseudo-Riemannian geometry. Geom. Dedicata 48, 15–25 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Sakai, T.: Riemannian geometry. 149 Translations of Mathematical Monographs American Mathematical Society. Providence, RI (1996)Google Scholar
  18. 18.
    Wang, L.F.: On noncompact \(\tau \)-quasi-Einstein metrics. Pacific J. Math. 254, 449–464 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

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

Personalised recommendations