Skip to main content
SpringerLink
Log in

On rigidity of generalized conformal structures

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension \({\ge }3\) . Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined by its 2-jet at any point. We prove here a similar rigidity for generalized conformal structures defined by giving a one parameter family of metrics (instead of scalar multiples of a given one) on each tangent space.

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. Ballmann, W.: Geometric structures. http://people.mpim-bonn.mpg.de/hwbllmnn/notes.html. Lecture notes page

  2. Bekkara, E., Frances, C., Zeghib, A.: On lightlike geometry: isometric actions, and rigidity aspects. C. R. Math. Acad. Sci. Paris 343(5), 317–321 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bekkara, S., Zeghib, A.: Singular Riemannian metrics, sub-rigidity versus rigidity. Math. Res. Lett. 18(6), 1203–1214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bekkara, S., Zeghib, A.: On automorphism groups of generalized conformal structures (in preparation)

  5. Benoist, Y.: Orbites des structures rigides (d’après M. Gromov). Feuilletages et systèmes intégrables (Montpellier, 1995), pp. 1–17, Progr. Math., 145. Birkhäuser Boston, Boston, MA (1997)

  6. Berger, M.: Geometry I, Corrected Fourth Printing. Springer, Berlin (2009)

    Google Scholar 

  7. Candel, A., Quiroga-Barranco, R.: Rigid and finite type geometric structures. Geom. Dedicata 106, 123–143 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. D’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. Surveys in differential geometry (Cambridge, MA, 1990), pp. 19–111. Lehigh University, Bethlehem (1991)

  9. Frances, C.: Une démonstration du théorème de Liouville en géométrie conforme. Enseignement Mathématique (2) 49(1–2), 95–100 (2003)

    MATH  Google Scholar 

  10. Frances, C.: Sur le groupe d’automorphismes des géométries paraboliques de rang 1. Ann. Sci. École Norm. Sup. (4) 40(5), 741–764 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gromov, M.: Partial Differential Relations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  12. Gromov, M.: Rigid, transformations groups. Géométrie différentielle (Paris, 1986), pp. 65–139. Travaux en Cours, 33, Hermann, Paris (1988)

  13. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  14. Kobayashi, S.: Transformation groups in differential geometry. Reprint of the 1972 edition. Classics in mathematics. Springer, Berlin (1995)

  15. Spivak, M.A.: comprehensive introduction to differential geometry. Publish or Perish (1999)

  16. Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall Inc., Englewood Cliffs (1964)

    MATH  Google Scholar 

  17. Zeghib, A.: On Gromov’s theory of rigid transformation groups: a dual approach. Ergod. Theory Dyn. Syst. 20(3), 935–946 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UST, Oran, Algeria

    Samir Bekkara

  2. CNRS, UMPA, ENS, Lyon, France

    Abdelghani Zeghib

Authors
  1. Samir Bekkara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelghani Zeghib
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samir Bekkara.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bekkara, S., Zeghib, A. On rigidity of generalized conformal structures. Geom Dedicata 189, 59–78 (2017). https://doi.org/10.1007/s10711-017-0217-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-017-0217-1

Keywords

Mathematics Subject Classification

Search

Navigation