Introduction to Riemannian Manifolds pp 151-191 | Cite as

# Geodesics and Distance

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## Abstract

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.

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