Local geometry and convexity

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


We start with an intuitive description of what the sign of the sectional curvature in a Riemannian manifold describes locally. Let us consider two geodesic rays starting from the same point p in a Riemannian manifold V and let α be the angle between these rays. If the sectional curvature K of V is everywhere nonnegative (K ⩾ 0), then the geodesics tend to come together compared with two corresponding rays (also with angle α) in the Euclidean plane, while K ⩽ 0 forces the geodesies to diverge faster than in the Euclidean situation:
To be more precise: let V be an n-dimensional Riemannian manifold, p ∈ V and r > 0 small enough such that expp: Br(0) → Bp(p) is a diffeomorphism, where Br(0) is the open ball of radius r in the tangent space TpV and Br(p) the corresponding distance ball in V.


Riemannian Manifold Convex Subset Sectional Curvature Variation Formula 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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