Comparison Theory

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


The purpose of this chapter is to show how upper or lower bounds on curvature can be used to derive bounds on other geometric quantities such as lengths of tangent vectors, distances, and volumes. In the first section of the chapter, we show how the growth of Jacobi fields in a normal neighborhood is controlled by the Hessian of the radial distance function, which satisfies a first-order differential equation called a Riccati equation. We then state and prove a fundamental comparison theorem for Riccati equations. Then we derive some of the most important geometric comparison theorems that follow from the Riccati comparison theorem.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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