Inventiones mathematicae

, Volume 151, Issue 3, pp 579–609

Hyperbolic manifolds are geodesically rigid

  • Vladimir S. Matveev

DOI: 10.1007/s00222-002-0263-6

Cite this article as:
Matveev, V. Invent. math. (2003) 151: 579. doi:10.1007/s00222-002-0263-6


We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are proportional.

Mathematical Subject Classification (2000): 53C24, 53C15, 37J35, 37J30, 53A20, 53D25

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Vladimir S. Matveev
    • 1
  1. 1.Mathematisches Institut, Universität Freiburg, 79104 Freiburg, Germany (e-mail: