Geodesics and Distance

Abstract

In this chapter, we study in detail the relationships among geodesies, lengths, and distances on a Riemannian manifold. A primary goal is to show that all length-minimizing curves are geodesies, and that all geodesies are length minimizing, at least locally. A key ingredient in the proofs is the symmetry of the Riemannian connection. Later in the chapter, we study the property of geodesic completeness, which means that all maximal geodesies are defined for all time, and prove the Hopf-Rinow theorem, which states that a Riemannian manifold is geodesically complete if and only if it is complete as a metric space.