Local geometry and convexity
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.
KeywordsRiemannian Manifold Convex Subset Sectional Curvature Variation Formula Nonpositive Curvature
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© Springer Science+Business Media New York 1985