1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 405

We say that a path "( emanating from p has a direction at p if its

angle with itself exists and is zero, i.e., Lp ("!, "!) = 0.^18

LEMMA G.38 (Comparison angle monotonicity). Let (X, d) be an Alek-

sandrov space with curvature bounded from below. Suppose a is a shortest

path between a (0) and a (E) and suppose (3 is a shortest path between (3 (0)

and (3 (E). Then

La ( u) P f3 ( v)

is a nonincreasing function of u and v.

For a proof of the nonnegative curvature case, see Proposition 4.3.2 of
[18] or, for a proof of the general case, see §2.6(B) of [19]. In particular, the
angle Lp (a, (3) exists.

PROPOSITION G.39 (Existence of angle between shortest paths). If X is

an Aleksandrov space of curvature 2: k and if a and (3 are shortest paths em-

anating from a common point p, then Lp (a, (3) is well defined. In particular,

any shortest path emanating from a point has a direction.

Another consequence of the monotonicity of La ( u) p (3 ( v) is that a

shortest path does not bifurcate in an Aleksandrov space with curvature

bounded from below. That is, let a (t) and (3 (t), t E [O, b], be two (unit

speed) minimal paths; if there is a 8 > 0 such that a (t) = (3 (t) fort E [O, 8],
then a (t) = (3 (t) for all t E [O, b].

2.3. Some basic notions and properties of Aleksandrov spaces

with curvature bounded from below.

Given d E [O, oo), there is the standard notion of d-dimensional Haus-
dorff measure μd (X) of a metric space X (see p. 19 of [18] for the def-
inition). The Hausdorff dimension of a metric space X is defined to

be the unique number dimHaus (X) E [O, oo] such that μd (X) = 0 for all

d > dimHaus (X) and μd (X) = oo for all d < dimHaus (X). By Theorem

1.7.16 of [18], such a number always exists.
For the proof of the following result, see Theorems 10.8.1 and 10.8.2 of
[18].19

THEOREM G.40 (Aleksandrov spaces have integer dimension). The Haus-

dorff dimension of an Aleksandrov space X with curvature bounded from

below is a finite nonnegative integer.

A basic tool used to study the local topological structure of Aleksandrov
spaces is strainers, which we now define.

(^18) Note that if Lp ("!, '!') is well defined, then it must be equal to zero (by taking u = v
in the limit on the RHS of (G.18)).
(^19) Since X is locally compact by definition, we have that dimHaus (X) is not equal to
00.