1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. SHARAFUTDINOV RETRACTION THEOREM 459

The following property concerning approximating long geodesics by rays

is useful (for a proof see Lemma B.49 of Volume One).

LEMMA I.4 (Angle closeness of long geodesics to rays). Given p E Mn,

define e : [O, oo) -+ [O, 7r] by

O(r)= sup inf Lp(0-(0),i'(O)),

uES ( r) ')'ERay M (p)

where S (r) is the set of all minimal geodesic segments <Y of length L (<Y) 2 r

emanating from p. Then

r-too lim e (r) = 0.

1.1.2. Half-spaces in Riemannian manifolds.

In this subsection we do not assume the condition of nonnegative sec-

tional curvature. Now we use the distance function to define the Riemannian

analogue of a half-space in Euclidean space.

DEFINITION I.5 (Half-spaces and geodesic balls). Let I : [O, oo) --+ Mn

be a ray emanating from a point p E M.

(1) The open right geodesic half-space is

JIB'Y =i= LJ B (! ( s) , s).
sE(O,oo)
(2) The closed left geodesic half-space Ry is
Ry =i= M - JIB')'.
The following lemma, which implies the equivalence of the Busemann
function formulation and the half-space formulation, is easy to verify.

LEMMA I.6. The closed left geodesic half-space is a sublevel set of the

Busemann function, i.e., b'Y (x) ::::; 0 if and only if x E JHI'Y.

Given a ray I: [O, oo) -+Mn and s E [O, oo ), define Is : [O, oo) -+ M by

Is ( s') =i= I ( s' + s)

for all s' E [O, oo). An elementary property is that if s ::::; t, then JIB'Yt C JIB'Ys.
We also have JIB'Ys is the open (t - s)-neighborhood of JIB'Yti for a proof, seep.
138 of [30].
LEMMA I.7 (e-neighborhood of a half-space is a half-space). Whenever

0 < s < t,

JIB'Ys = {x E Mn: d(x,JIB'Yt) < t-s} =Nt-s (JIB'Yt).

Given a point p E Mn and s E [O, oo ), the sublevel set of the Buse-

mann function associated to p is defined by

Cs (p) =i= n Rys = M - LJ JIB'Ys'

')'ERay M (p) ')'ERay M (p)

The terminology above is justified by the following (for a proof, see Lemma
B.59 of Volume One).