Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature

Abstract

We want to study the metric of a Riemannian manifold. The first tasks to address are: 1. to compute the metric d as defined by equation 4.13 on page 174 (namely itd (p, q) is the infimum of the lengths of curves connecting p to q) 2. to determine if there are curves realizing this distance (called segments or shortest paths or minimal geodesics according to your taste) and 3. to study them.