Manifolds of Nonpositive Curvature pp 1-14 | Cite as

# Local geometry and convexity

Chapter

## Abstract

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 exp

_{p}: B_{r}(0) → B_{p}(p) is a diffeomorphism, where B_{r}(0) is the open ball of radius r in the tangent space T_{p}V and B_{r}(p) the corresponding distance ball in V.## Keywords

Riemannian Manifold Convex Subset Sectional Curvature Variation Formula 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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© Springer Science+Business Media New York 1985