1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

404 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY

there exists an open neighborhood U of x such that for any triangle 6.pqr

contained in U and point s E rp we have

de (q, s) ?. d (q, s)'

where 6.pqf is the k-comparison triangle of 6.pqr and s E fp is the point

with d (p, s) =de (p, s). In this definition, if we drop the local compactness

assumption, we then call X a complete length space with curvature bounded

from below.

If k = 0, then we say that (X, ..C) is an Aleksandrov space of non-

negative curvature. By Theorem G.33(3), if (Mn, g) is a complete Rie-

mannian manifold with sectional curvatures bounded below by k E IR, then

(Mn,Lg) is an Aleksandrov space of curvature?. k.

The global version of the definition of Aleksandrov space is as follows.

DEFINITION G.36 (Aleksandrov space-global version). Given k E IR,

we say that a complete locally compact length space (X, ..C) has curvature

?. k in the large if for any triangle 6.pqr contained in X and point s E rp

we have

de (q, s) ?. d (q, s)

whenever the k-comparison triangle 6.pqf of 6.pqr exists and s E f p satisfies

d (p, s) =de (p, s).

The local. and global versions of the definition of Aleksandrov space are

the same (see Theorem 10.3.1 of [18]).

THEOREM G.37 (Toponogov's globalization theorem). If (X, ..C) is an

Aleksandrov space of curvature ?. k, then (X, ..C) has curvature ?. k in the

large.

The following notions are used in Aleksandrov space theory.

Let (X, d) be a metric space. Given x, y, z EX distinct, the Euclidean

comparison angle of (x, y, z) at y is

(G.17) /. _, _^1 d(x,y)2+d(y,z)

(^2) -d(x,z) (^2) [ ]
4...XYZ-;-cos 2d (x, y) d (y, z) E 0, 1f •
There is an equivalent definition of Aleksandrov spaces with curvature
bounded.from below given by using comparison angles (see Definition 2.3 of
[19]).
If (X, d) is a length space and if a and f3 are paths emanating from
p EX, i.e., a, /3: [O, c] -+ X for some e: > 0 and a (0) = f3 (0) = p, then the
angle between a and f3 is defined by

(G.18) Lp(a,{3)~ lim La(u)pf3(v),

u,v-tO+

provided the limit exists.