Skip to main content
SpringerLink
Log in

On solutions of the Ricci curvature equation and the Einstein equation

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider the pseudo-Euclidean space (R n, g), with n ≥ 3 and g ij = δ ij ε i , ε i = ±1, where at least one ε i = 1 and nondiagonal tensors of the form T = Σ ij f ij dx i dx j such that, for i ≠ j, f ij (x i , x j ) depends on x i and x j . We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R n, on the n-dimensional torus T n and on cylinders T k×R n-k, that solve the Ricci equation or the Einstein equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Cao and D. DeTurck, The Ricci curvature equation with rotational symmetry, American Journal of Mathematics 116 (1994), 219–241.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. DeTurck, Existence of metrics with prescribed Ricci Curvature: Local Theory, Inventiones Mathematicae 65 (1981), 179–207.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. DeTurck, Metrics with prescribed Ricci curvature, Seminar on Differential Geometry, in Annals of Mathematics Studies, Vol. 102, Princeton University Press, Princeton, NJ, 1982, pp. 525–537.

    Google Scholar 

  4. D. DeTurck, The Cauchy problem for Lorentz metrics with Prescribed Ricci curvature, Compositio Mathematica 48 (1983), 327–349.

    MATH  MathSciNet  Google Scholar 

  5. D. DeTurck and H. Goldschmidt, Metrics with Prescribed Ricci Curvature of Constant Rank, Advances in Mathematics 145 (1999), 1–97.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. DeTurck and W. Koiso, Uniqueness and non-existence of metrics with prescribed Ricci curvature, Annales de l’Institut Henry Poincaré. Analyse Non Linéaire 1 (1984), 351–359.

    MATH  MathSciNet  Google Scholar 

  7. R. S. Hamilton, The Ricci curvature equation, in Seminar on nonlinear partial differential equations (Berkeley, California, 1983), Mathematical Science Research Institute Publications, Springer, New York, 1984, pp. 47–72.

    Google Scholar 

  8. J. Lohkamp, Metrics of negative Ricci curvature, Annals of Mathematics 140 (1994), 655–683.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Pina, Conformal Metrics and Ricci Tensors in the Hyperbolic space, Matemática Contemporâ nea 17 (1999), 254–262.

    MathSciNet  Google Scholar 

  10. R. Pina and K. Tenenblat, Conformal Metrics and Ricci Tensors in the pseudo-Euclidean space, Proceedings of the American Mathematical Society 129 (2001), 1149–1160.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Pina and K. Tenenblat, On metrics satisfying equation R ij -Kg ij /2 = T ij for constant tensors T, Journal of Geometry and Physics 40 (2002), 379–383.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Pina and K. Tenenblat, Conformal Metrics and Ricci Tensors on the Sphere Proceedings of the American Mathematical Society 132 (2004), 3715–3724.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Pina and K. Tenenblat, On the Ricci and Einstein equations on the pseudo-Euclidean and hyperbolic spaces, Differential Geometry and its Applications 24 (2006), 101–107.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Pina and K. Tenenblat, A class of solutions of the Ricci and Einstein equations, Journal of Geometry and Physics 57 (2007), 881–888.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein field equations, Cambridge University Press, Cambridge, MA, 2003.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IME, Universidade Federal de Goiás, 74001-970, Goiânia GO, Brazil

    Romildo Pina & Keti Tenenblat

Authors
  1. Romildo Pina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keti Tenenblat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Romildo Pina.

Additional information

Partially supported by CAPES/PROCAD.

Partially Supported By Cnpq, Capes/Procad.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pina, R., Tenenblat, K. On solutions of the Ricci curvature equation and the Einstein equation. Isr. J. Math. 171, 61–76 (2009). https://doi.org/10.1007/s11856-009-0040-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-009-0040-y

Keywords

Search

Navigation