Skip to main content
SpringerLink
Log in

Geodesic rigidity of conformal connections on surfaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We show that a conformal connection on a closed oriented surface \(\Sigma \) of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on \(\Sigma \) determine the metric up to constant rescaling. It is also shown that every conformal connection on the \(2\)-sphere lies in a complex \(5\)-manifold of conformal connections, all of which share the same unparametrised geodesics.

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

Notes

  1. As usual, by an affine torsion-free connection on \(\Sigma \) we mean a torsion-free connection on \(T\Sigma \).

  2. We define \(\varepsilon _{ij}=-\varepsilon _{ji}\) with \(\varepsilon _{12}=1\).

  3. In order to keep notation uncluttered we omit writing \(\lambda ^*\) for pull-backs by \(\lambda \).

References

  1. Beltrami, E.: Resoluzione del problema: riportari i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette. Annali di Mat 1, 185–204 (1865). doi:10.1007/BF03198517. (Italian)

    Article  Google Scholar 

  2. Bryant, R.L.: Two exotic holonomies in dimension four, path geometries, and twistor theory, in complex geometry and Lie theory. Proceedings Symposium Pure Mathematics 53 American Mathematical Society, Providence, RI, (1991), pp. 33–88. doi:10.1090/pspum/053/1141197

  3. Bryant, R.L., Dunajski, M., Eastwood, M.: Metrisability of two-dimensional projective structures. J. Differ. Geom. 83, 465–499 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Cartan, E.: Sur les variétés à connexion projective. Bull. Soc. Math. Fr. 52, 205–241 (1924)

    MATH  MathSciNet  Google Scholar 

  5. Dubois-Violette, M.: Structures complexes au-dessus des variétés, applications, in Mathematics and physics. Progr. Math. 37, Birkhäuser Boston, Boston, MA, (1983), pp. 1–42. doi:10.1007/BF02591680

  6. Dunajski, M., Tod, P.: Paraconformal geometry of \(n\)th-order ODEs, and exotic holonomy in dimension four. J. Geom. Phys. 56, 1790–1809 (2006). doi:10.1016/j.geomphys.2005.10.007

    Article  MATH  MathSciNet  Google Scholar 

  7. Hitchin, N.J.: Complex manifolds and Einstein’s equations, in Twistor geometry and nonlinear systems, Lecture Notes in Math, vol. 970. Springer, Berlin (1982)

    Google Scholar 

  8. Kobayashi, S., Nagano, T.: On projective connections. J. Math. Mech. 13, 215–235 (1964)

    MATH  MathSciNet  Google Scholar 

  9. Kodaira, K.: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds. Ann. of Math. (2) 75, 146–162 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  10. Matveev, V.S.: Projectively equivalent metrics on the torus. Differ. Geom. Appl. 20, 251–265 (2004). doi:10.1016/j.difgeo.2003.10.009

    Article  MATH  Google Scholar 

  11. Matveev, V.S., Topalov, P.J.: Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electron. Res. Announc. Amer. Math. Soc. 6, 98–104 (2000). doi:10.1090/S1079-6762-00-00086-X

    Article  MATH  MathSciNet  Google Scholar 

  12. Mettler, T.: Weyl metrisability of two-dimensional projective structures. Math. Proc. Cambridge Philos. Soc. 156, 99–113 (2014). doi:10.1017/S0305004113000522

    Article  MATH  MathSciNet  Google Scholar 

  13. O’Brian, N.R., Rawnsley, J.H.: Twistor spaces. Ann. Global Anal. Geom. 3, 29–58 (1985). doi:10.1007/BF00054490

    Article  MATH  MathSciNet  Google Scholar 

  14. Weyl, H.: Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 1921, 99–112 (1921)

    MATH  Google Scholar 

Download references

Acknowledgments

This paper would not have come into existence without several very helpful discussions with Nigel Hitchin. I would like to warmly thank him here. I also wish to thank Vladimir Matveev for references and the anonymous referee for her/his careful reading and useful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, 8092, Zürich, Switzerland

    Thomas Mettler

  2. Department of Mathematics, University of Fribourg, 1700, Fribourg, Switzerland

    Thomas Mettler

Authors
  1. Thomas Mettler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Mettler.

Additional information

Research for this article was carried out while the author was visiting the Mathematical Institute at the University of Oxford as a postdoctoral fellow of the Swiss NSF, PA00P2_142053. The author would like to thank the Mathematical Institute for its hospitality.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mettler, T. Geodesic rigidity of conformal connections on surfaces. Math. Z. 281, 379–393 (2015). https://doi.org/10.1007/s00209-015-1489-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1489-5

Keywords

Mathematics Subject Classification

Search

Navigation