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Mathematische Zeitschrift

, Volume 241, Issue 1, pp 181–226 | Cite as

Diophantine approximation for negatively curved manifolds

  • Sa'ar Hersonsky
  • Frédéric Paulin
Original article

Abstract.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space.

Mathematics Subject Classification (2002): 53 C 22, 11 J 06, 30 F 40, 11 J 70. 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Sa'ar Hersonsky
    • 1
  • Frédéric Paulin
    • 2
  1. 1.Caltech, Department of Mathematics, Pasadena CA 91125, USA (e-mail: saar@cco.caltech.edu) USA
  2. 2.Laboratoire de Mathématiques, UMR 8628 CNRS (Bât. 425), Equipe de Topologie et Dynamique, Université Paris-Sud, 91405 Orsay Cedex, France (e-mail: Frederic.Paulin@math.u-psud.fr) France

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