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Geodesics and Distance

  • John M. LeeEmail author
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 176)

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.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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