Summary: Manifolds with A Lower Ricci Curvature Bound
This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.
The purpose of this paper is to give a survey on the structure of manifolds with a lower Ricci
curvature bound. A Ricci curvature bound is weaker than a sectional curvature bound but stronger
than a scalar curvature bound. Ricci curvature is also special that it occurs in the Einstein equation
and in the Ricci flow. The study of manifolds with lower Ricci curvature bound has experienced
tremendous progress in the past fifteen years. Our focus in this article is strictly restricted to results
with only Ricci curvature bound, and no result with sectional curvature bound is presented unless
for straight comparison. The reader is referred to John Lott's article in this volume for the recent
important development concerning Ricci curvature for metric measure spaces by Lott-Villani and
Sturm. We start by introducing the basic tools for studying manifolds with lower Ricci curvature
bound (Sections 2-4), then discuss the structures of these manifolds (Sections 5-9), with examples
in Section 10.
The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula.
From there one can derive powerful comparison tools like the mean curvature comparison, the
Laplacian comparison, and the relative volume comparison. For the Laplacian comparison (Section
3) we discuss the global version in three weak senses (barrier, distribution, viscosity) and clarify