1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

406 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

DEFINITION G.41 (Strained points and strainers). Let X be an Alek-
sandrov space with curvature bounded from below. Given m EN and f > 0,
a point p E X is an ( m, f )-strained point if there exist m pairs of points
{(ai, bi)};: 1 in X such that

2ai,Pbi > 7r - f, 2aipaj > ~ - lOf,

2aipbj > ~ - lOf, 2biPbj > ~ - lOf,

where i f j E {1, 2, ... , m }. The collection {(ai, bi)};: 1 is called an (m, c)-

strainer for p.

The following result, which is Theorem 10.8.3 of [18], is proved using
strainers.

THEOREM G .42 (Regularity of Aleksandrov spaces). Let X be a complete

length space with curvature bounded from below. If dimH X = n < oo, then

X is locally compact and an open dense subset of X is homeomorphic to an

n-manifold.

Besides the notion of Hausdorff dimension given above, there are other

definitions of dimension (rough dimension, covering dimension, and strainer

number) on Aleksandrov spaces with curvature bounded from below, but

they are more or less equivalent (see Proposition 155 of [159]).

A key property of Aleksandrov spaces is the following (see Proposition

10.7.1 of [18]); we do not discuss the pointed version.

THEOREM G.43 (Closedness under Gromov-Hausdorfflimit). If a metric

space (X, d) is the Gromov-Hausdorff limit (pointed or not) of a sequence

of Aleksandrov spaces of curvature 2: k, then (X, Cd) is a complete length

space of curvature 2: k.

Let 9J1 ( n, k, D) denote the space of Aleksandrov spaces of curvature 2: k

with Hausdorff dimension ::; n and diameter ::; D. The following is Theorem

10.7.2 of [18].

THEOREM G.44 (Gromov compactness theorem, II). For any sequence

in 9J1 (n, k, D) there exists a subsequence which converges in the Gromov-

Hausdorff topology to an element of 9J1 (n, k, D).

Since Riemannian manifolds with sectional curvatures bounded from
below are Aleksandrov spaces of curvature bounded from below, the above
theorem implies that any sequence of n-dimensional Riemannian manifolds
with sectional curvature 2: k and diameter::; D subconverges in 9J1 (n, k, D).
Hence Aleksandrov spaces of curvature bounded from below are nice gener-
alizations of Riemannian manifolds with sectional curvatures bounded from
below. There are other nice aspects of this generalization: some of the the-
orems and tools in Riemannian geometry can be generalized to Aleksandrov
spaces. The following splitting theorem of Toponogov and of Milka [128] is
an example (see Theorem 10.5.1 of [18]).