1547671870-The_Ricci_Flow__Chow

2. LOCAL VERSUS GLOBAL GEOMETRY 295

COROLLARY B.22. Let (Mn,g) be a complete Riemannian manifold with
sectional curvatures bounded above by K > 0. Let f3 be a geodesic path from
p to q in Mn with L ({3) < 7f / VK. Then for any points p' and q' sufficiently
near p and q respectively, there is a unique geodesic path {3' from p' to q'
which is close to f3.

PROOF. By the previous corollary, the map

expPIB(O;rr/VK) : B(O, 7f /VK) c TpMn -7 Mn

is an immersion. We may assume that the geodesic f3 : [O, 1] __, Mn is

parameterized so that f3 (s) = expP (sV) for some V E B(O, 7f /VK) with

expP (V) = q. Since expPIB(O,rr/VK) is a local diffeomorphism, there is for all

q' sufficiently close to q a unique V' E B ( 0, 7f / VK) such that s f---? expP ( s V')

is the unique geodesic path from p to q' lying close to {3. Since for all

p' = expP W sufficiently close to p, the map

(expP)* : TpMn ~ Tw (TpMn) __, Tp'Mn

is an isomorphism, there exists a unique vector V" E Tp' Mn such that

expp' (V") = expP (V') = q'.

The path {3' : [O, 1] __, Mn defined by {3' (s) ~ expp' (sV") is the unique

geodesic near f3 joining p' to q'. D

2. Distinguishing between local geometry and global geometry

In Section 1, we observed that the exponential map is always an embed-

ding of a sufficiently small ball. We now study larger length scales, in order

to investigate some of the ways that the topology of a complete Riemannian

manifold (Mn, g) affects its geometry. In this section, interpret geodesic to

mean unit-speed geodesic.

2.1. Cut points and the injectivity radius. Given p E Mn and a

geodesic "f : [O, oo) __,Mn with "f (0) = p, define

S'Y ~ {t E [O,oo): d("t(O) ,1 (t)) = t}.

A sufficiently short geodesic always minimizes distance. Its behavior at

larger length scales is described by the following elementary observation.

LEMMA B.23. Either d("t(0),1(t)) = t for all t E [O,oo), or else

there exists a unique t'Y such that db (0), 'Y (t)) = t for all t E [O, t'Y] and

db (0), 'Y (t)) < t for all t > t"I' In short,

either S'Y = [O, oo) or else S'Y = [O, t'Y].

PROOF. If db (0), 'Y (l)) < f for some f > 0, then for all t 2 f,

db (0) '"'( (t)) :::; db (0) '"'( (l)) +db (l) '"'( (t)) < f + (t - l) = t.

Take t'Y ~ inf { t 2 0 : d b ( 0) , 'Y ( t)) < t}. D