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Differential GeometryBundles, Connections, Metrics and Curvature$
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Clifford Henry Taubes

Print publication date: 2011

Print ISBN-13: 9780199605880

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199605880.001.0001

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(p.78) 8 Geodesics
Differential Geometry

Clifford Henry Taubes

Oxford University Press

Let M denote a smooth manifold. A metric on TM can be used to define a notion of the distance between any two points in M and the distance travelled along any given path in M. This chapter first explains how this is done then considers the distance minimizing paths. The discussions cover Riemannian metrics and distance; length minimizing curves; the existence of geodesics; examples of metrics with their corresponding geodesics; geodesics on SO(n); geodesics on U(n) and SU(n); and geodesics and matrix groups.

Keywords:   metrics, distance, Riemannian metrics, length minimizing curves, matrix groups

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