Skip to main content
SpringerLink
Log in

Some potentials for the curvature tensor on three-dimensional manifolds

  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

We study equations of Riemann–Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.

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. Rendall, A.D., Weaver, M.: Class. Quantum Gravit. 18, 2959–2975 (2001)

    Article  Google Scholar 

  2. Ringström, H.: Lecture at the Miami Waves conference. (January 2004)

  3. Lanczos, C.: Rev. Mod. Phys. 34, 379–389 (1962)

    Article  Google Scholar 

  4. Taub, A.H.: Computers and Mathematics with Applications. Oxford, Pergamon pp. 377–380 (1976)

    Google Scholar 

  5. Zund, J.D.: Ann. Mat. Pura Appl. 104(4), 239–268 (1975)

    Google Scholar 

  6. Bampi, F., Caviglia, G.: Gen. Relativ. Gravit. 15, 375–386 (1983)

    Article  Google Scholar 

  7. Illge, R.: Gen. Relativ. Gravit. 20, 551–564 (1988)

    Article  Google Scholar 

  8. Andersson, F., Edgar, S.B.: Local existence of spinor potentials. (1999) [gr-qc/9902080]

  9. O’Donnell, P.: Gen. Relativ. Gravit. 36, 1415–1422 (2004)

    Article  Google Scholar 

  10. Dolan, P., Muratori, B.D.: J. Math. Phys. 39, 5406–5420 (1998)

    Article  Google Scholar 

  11. Edgar, S.B., Höglund, A.: Gen. Relativ. Gravit. 34, 2149–2153 (2002)

    Article  Google Scholar 

  12. Edgar, S.B., Senovilla, J.M.M.: A local potential for the Weyl tensor in all dimensions. (2004) [gr-qc/0408071v1]

  13. Udeschini Brinis, E.: Istit. Lombardo Accad. Sci. Lett. Rend. A 111 (1977), 466–475 (1978)

  14. Lanczos, C.: Rev. Mod. Phys. 21, 497–502 (1949)

    Article  Google Scholar 

  15. Bampi, F., Caviglia, G.: Gen. Relativ. Gravit. 16, 423–433 (1984)

    Article  Google Scholar 

  16. Gerber, A.: The Weyl–Lanczos and the Riemann–Lanczos problems as exterior differential systems, with applications to space-times. Ph.D. thesis, Imperial College, London (2001)

    Google Scholar 

  17. Dolan, P., Gerber, A.: J. Math. Phys. 44, 3013–3034 (2003)

    Article  Google Scholar 

  18. Berger, M., Ebin, D.: J. Differ. Geom. 3, 379–392 (1969)

    Google Scholar 

  19. Jezierski, J.: Private Communication

  20. Christodoulou, D., Choquet-Bruhat, Y.: Acta. Math. 146, 129–150 (1981)

    Google Scholar 

  21. Thomas, T.Y.: The Differential Invariants of Generalized Spaces. Cambridge University Press (1934)

  22. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York, London, Sydney (1962)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, F37200 Tours, France

    Piotr T. Chruściel

  2. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, Site Colbert, 76821, Mont-Saint-Aignan, France

    Annelies Gerber

Authors
  1. Piotr T. Chruściel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Annelies Gerber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annelies Gerber.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chruściel, P.T., Gerber, A. Some potentials for the curvature tensor on three-dimensional manifolds. Gen Relativ Gravit 37, 891–905 (2005). https://doi.org/10.1007/s10714-005-0074-3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-005-0074-3

Keywords

Search

Navigation