388 G. ELEMENTARY ASPECTS OF METRIC GEOMETRYbooks on metric and related Riemannian geometry are Gromov's [78] and
Burago, Burago, and Ivanov's [18]. We shall refer to these books freely
throughout this appendix.
In §1 we discuss length spaces, the Gromov-Hausdorff distance and con-
vergence, the Gromov precompactness theorem (I), and the tangent and
asymptotic cones.
In §2 we recall Aleksandrov spaces and their basic properties and asso-
ciated notions, the Gromov precompactness theorem (II), the Aleksandrov
space splitting theorem, and the existence of the tangent cone.1. Metric spaces and length spaces
In this section we recall some basic facts about metric and length spaces.
This serves two purposes:
(1) One of Perelman's many contributions to Ricci flow is the intro-
duction of a space-time length-type geometry (see [152] or the
expository Chapter 7 in Part I). We hope that the discussion of
background material on metric and length spaces may facilitate
the reader's study of the foundations of Perelman's theory.
(2) Metric and length spaces prepare us for the study of Aleksandrov
spaces, which are natural spaces for discussing compactness theo-
rems for Riemannian manifolds in the absence of injectivity radius
bounds.1.1. (Quasi-)metric and (quasi-)length spaces.
In this subsection we present a few basic notions related to metric spaces
and length spaces. At its end, we also indicate a relevance of metric and
length spaces to Ricci flow.
1.1.1. Metric-type spaces.
Recall that a metric d on a set X is a nonnegative symmetric function
defined on the Cartesian product X x X which vanishes only on the diagonal
and satisfies the triangle inequality, i.e.,d (x, z) :S d (x, y) + d (y, z)
for all x, y, z EX. We also refer to d as the distance function. A metric
space is the pair (X, d) of a topological space and a metric, where the
topology on Xis the coarsest topology for which B (x, r) is an open set for
all x E X and r > 0. A metric space is said to be complete if every Cauchy
sequence converges.
If Sis a subset of a metric space (X, dx), it has a subspace metric ds
simply defined by
ds (x, y)-==;= dx (x, y)