Convergence

Abstract

In this chapter we will give an introduction to several of the convergence ideas for Riemannian manifolds. The goal is to understand what it means for a sequence of Riemannian manifolds, or more generally metric spaces, to converge to a space. In the first section we develop the weakest convergence concept: Gromov-Hausdorff convergence. We then go on to explain some of the elliptic regularity theory we need for some of the later developments. We have confined ourselves to the simpler Hölder and Schauder theories. In Section 3 we develop the idea of norms of Riemannian manifolds. This is a concept developed by the author in the hope that it will make it easier to understand convergence theory as a parallel to the easier Hölder theory for functions (as is explained in Section 2). At the same time, we also feel that it has made some parts of the theory more concise. In this section we examine some stronger convergence ideas that were developed by Cheeger and Gromov and study their relation to the norms of manifolds. These preliminary discussions will enable us in subsequent sections to establish the convergence theorem of Riemannian geometry and its generalizations by Anderson and others. These convergence theorems contain the Cheeger finiteness theorem, which states that certain classes of Riemannian manifolds contain only finitely many diffeomorphism types.