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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)

Abstract

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).

Keywords

Convex Subset Fundamental Group Universal Covering Complete Riemannian Manifold Nonpositive Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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