文档介绍：Comparison Geometry
MSRI Publications
V olume 30 , 1997
Con v ergence Theorems in Riemannian Geometry
PETER PETERSEN
Abstra ct . This is a surv ey on the con v ergence theory dev elop ed rst b y
Cheeger and Gromo v. In their theory one is concerned with pact-
ness of the class of riemannian manifolds with b ounded curv ature and lo w er
b ound on the injectivit y radius. W e explain and giv e pro ofs of almost all
the ma jor results, including Anderson's generalizations to the case where
all one has is b ounded i curv ature. The exp osition is streamlined b y the
in tro duction of a norm for riemannian manifolds, whic h mak es the theory
more lik e that of H older and Sob olev spaces.
1. In tro duction
This pap er is an outgro wth of a talk giv en in Octob er 1993 at MSRI and
a graduate course o ered in the Spring of 1994 at UCLA. The purp ose is to
in tro duce readers to the con v ergence theory of riemannian manifolds not so m uc h
through a traditional surv ey article, but b y rigorously pro ving most of the k ey
theorems in the sub ject. F or a broader surv ey of this sub ject, and ho w it can b e
applied to v arious problems, w e refer the reader to [Anderson 1993].
The prerequisites for this pap er are some basic kno wledge of riemannian geom-
etry , Gromo v{Hausdor con v ergence and elliptic regularit y theory . In particular,
the reader should b e familiar with parison geometry found in [Karc her
1989], for example. F or Gromo v{Hausdor con v ergence, it suces to read Sec-
tion 6in [Gromo v 1981a] or Section 1 in [P etersen 1993]. In regard to elliptic
theory ,w eha v e an app endix that con tains all the results w e need, together with
pro ofs of those theorems that are not explicitly stated in [Gilbarg and T rudinger
1983].
k +
In Section 2 w ein tro duce the concept of p oin ted C con v ergence of rie-
mannian manifolds. This in tro duces a natural top ology on p oin ted riemannian
manifolds and imme