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Jacobi Fields

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

Abstract

To generalize to higher dimensions some of the geometric and topological consequences of the Gauss–Bonnet theorem, we need to develop a new approach: instead of using Stokes’s theorem and differential forms to relate the curvature to global topology as in the proof of the Gauss–Bonnet theorem, we analyze the way that curvature affects the behavior of nearby geodesics. In this chapter, we study Jacobi fields, which encode the first-order behavior of families of geodesics. We then introduce conjugate points, which are pairs of points along a geodesic where some nontrivial Jacobi field vanishes, and show that no geodesic is minimizing past its first conjugate point.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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