Geodesics and Distance
In this chapter, we explore the relationships among geodesics, lengths, and distances on a Riemannian manifold. One of the main goals is to show that all length-minimizing curves are geodesics, and all geodesics are locally length minimizing. Later, we prove the Hopf–Rinow theorem, which states that a connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space. At the end of the chapter, we study distance functions (which express the distance to a point or other subset) and show how they can be used to construct coordinates that put a metric in a particularly simple form.