Advertisement

Jacobi Fields

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

Abstract

Our goal for the remainder of this book is 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 study how curvature affects the behavior of nearby geodesies. Roughly speaking, positive curvature causes nearby geodesies to converge (Figure 10.1), while negative curvature causes them to spread out (Figure 10.2). In order to draw topological consequences from this fact, we need a quantitative way to measure the effect of curvature on a one-parameter family of geodesies.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • John M. Lee
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations