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Geodesics and Distance

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.

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  • DOI: 10.1007/978-3-319-91755-9_6
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Correspondence to John M. Lee .

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Lee, J.M. (2018). Geodesics and Distance. In: Introduction to Riemannian Manifolds. Graduate Texts in Mathematics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-91755-9_6

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