Local geometry and convexity

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_pV and B_r(p) the corresponding distance ball in V.