The theorem of Hadamard-Cartan and complete simply connected manifolds of nonpositive curvature

  • Werner Ballmann
  • Mikhael Gromov
  • Viktor Schroeder
Part of the Progress in Mathematics book series (PM, volume 61)


Let V be a complete Riemannian manifold of nonpositive curvature and p ∈ V. Then expp is defined on the entire tangent space TpV. The convexity of the norm of every Jacobi field (see 1.3) implies that no (non-zero) Jacobi field along a geodesic c: [0,1] → V vanishes at c(0) and c(1). Hence there are no conjugate points, and the differential of expp is everywhere nonsingular. Thus we can define a new metric on TpV, such that expp is a local isometry. This metric is complete by the Hopf-Rinow theorem, because the lines through the origin in TpV are geodesics in this metric. Now it is easy to prove that a local isometry φ: V1 → V2, where V1 is complete, is a covering map (comp. [Cheeger-Ebin, 1975], p. 35).


Convex Subset Fundamental Group Universal Covering Complete Riemannian Manifold Nonpositive Curvature 


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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Werner Ballmann
    • 1
    • 2
  • Mikhael Gromov
    • 3
  • Viktor Schroeder
    • 4
    • 5
  1. 1.Dept. of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Math. Institut der UniversitätBonnWest Germany
  3. 3.Inst. des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  4. 4.Math. Institut der UniversitätMünsterGermany
  5. 5.Math. Institut der UniversitätBaselSwitzerland

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