The Comparison Geometry of Ricci Curvature

Summary

We survey comparison results that assume a bound on the manifold's Ricci curvature.

1. Introduction

This is an extended version of the talk I gave at the Comparison Geometry Workshop at MSRI in the fall of 1993, giving a relatively up-to-date account of the results and techniques in the comparison geometry of Ricci curvature, an area that has experienced tremendous progress in the past five years. The term “comparison geometry” had its origin in connection with the success of the Rauch comparison theorem and its more powerful global version, the Toponogov comparison theorem. The comparison geometry of sectional curvature represents many ingenious applications of these theorems and produced many beautiful results, such as the -pinched Sphere Theorem [Berger 1960; Klingenberg 1961], the Soul Theorem [Cheeger and Gromoll 1972], the Generalized Sphere Theorem [Grove and Shiohama 1977], The Compactness Theorem [Cheeger 1967; Gromov 1981c], the Betti Number Theorem [Gromov 1981a], and the Homotopy Finiteness Theorem [Grove and Petersen 1988], just to name a few. The comparison geometry of Ricci curvature started as isolated attempts to generalize results about sectional curvature to the much weaker condition on Ricci curvature. Starting around 1987, many examples were constructed to demonstrate the difference between sectional curvature and Ricci curvature; in particular, Toponogov's theorem was shown not to hold for Ricci curvature. At the same time, many new tools and techniques were developed to generalize results about sectional curvature to Ricci curvature. We will attempt to present the highlights of this progress.