This is a survey on the convergence theory developed first by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, including Anderson's generalizations to the case where all one has is bounded Ricci curvature. The exposition is streamlined by the introduction of a norm for riemannian manifolds, which makes the theory more like that of Holder and Sobolev spaces.

1. Introduction

This paper is an outgrowth of a talk given in October 1993 at MSRI and a graduate course offered in the Spring of 1994 at UCLA. The purpose is to introduce readers to the convergence theory of riemannian manifolds not so much through a traditional survey article, but by rigorously proving most of the key theorems in the subject. For a broader survey of this subject, and how it can be applied to various problems, we refer the reader to [Anderson 1993]. The prerequisites for this paper are some basic knowledge of riemannian geometry, Gromov-Hausdorff convergence and elliptic regularity theory. In particular, the reader should be familiar with the comparison geometry found in [Karcher 1989], for example. For Gromov-Hausdorff convergence, it suffices to read Section 6 in [Gromov 1981a] or Section 1 in [Petersen 1993]. In regard to elliptic theory, we have an appendix that contains all the results we need, together with proofs of those theorems that are not explicitly stated in [Gilbarg and Trudinger 1983].