Manifolds of bounded negative curvature

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


In this lecture we use the Margulis lemma more systematically to study manifolds of nonpositive curvature. As usual we represent our manifold as V = X/Γ where π: X → V is the canonical projection. Let us assume that V satisfies the curvature assumption -1 ≤ K ≤ 0 and let p ∈ V be a point with Inj Rad(p) < μ/2. We choose x ∈ X with π(x) = p, then dΓ(x) < μ. By the Margulis lemma, Γμ (x) is a nontrivial almost nilpotent subgroup and we can use our knowledge of these groups (§7) to get information about the geometry of V in a neighborhood of p. Thus we will divide V into the subsets {Inj Rad ≥ μ/2} and {Inj Rad < μ/2}. The local geometry of the first part is under control because we have a lower bound on the injectivity radius, and for the second part we use the Margulis lemma.


Common Fixed Point Closed Geodesic Complete Riemannian Manifold Nonpositive Curvature Injectivity Radius 
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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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