Advertisement

Mathematische Zeitschrift

, Volume 281, Issue 1–2, pp 379–393 | Cite as

Geodesic rigidity of conformal connections on surfaces

  • Thomas Mettler
Article
  • 155 Downloads

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.

Keywords

Projective structures Conformal connections Geodesic rigidity  Twistor space 

Mathematics Subject Classification

Primary 53A20 Secondary 53C24 53C28 

Notes

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.

References

  1. 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)CrossRefGoogle Scholar
  2. 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. 3.
    Bryant, R.L., Dunajski, M., Eastwood, M.: Metrisability of two-dimensional projective structures. J. Differ. Geom. 83, 465–499 (2009)MATHMathSciNetGoogle Scholar
  4. 4.
    Cartan, E.: Sur les variétés à connexion projective. Bull. Soc. Math. Fr. 52, 205–241 (1924)MATHMathSciNetGoogle Scholar
  5. 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. 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 MATHMathSciNetCrossRefGoogle Scholar
  7. 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. 8.
    Kobayashi, S., Nagano, T.: On projective connections. J. Math. Mech. 13, 215–235 (1964)MATHMathSciNetGoogle Scholar
  9. 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)MATHMathSciNetCrossRefGoogle Scholar
  10. 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 MATHCrossRefGoogle Scholar
  11. 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 MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mettler, T.: Weyl metrisability of two-dimensional projective structures. Math. Proc. Cambridge Philos. Soc. 156, 99–113 (2014). doi: 10.1017/S0305004113000522 MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    O’Brian, N.R., Rawnsley, J.H.: Twistor spaces. Ann. Global Anal. Geom. 3, 29–58 (1985). doi: 10.1007/BF00054490 MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Weyl, H.: Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung. Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. 1921, 99–112 (1921)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZürichSwitzerland
  2. 2.Department of MathematicsUniversity of FribourgFribourgSwitzerland

Personalised recommendations