492 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION
is distance nonincreasing (in particular, the map <Ps,t is continu-
ous);
(2) (composition rule) for any u ::'.'.: t ::'.'.: s ::'.'.: so (p),
c/Jt,u o c/Js,t = c/Js,u;
(3) (preserves rays emanating from p) for any a E Ray M (p) and t ::'.'.:
s ::'.'.: so (p),.
<Ps,t (a (s)) =a (t).6.2.2. Equivalence of rays.
Two rays a, f3 E Ray M are said to be equivalent if(I.61) lim d (a (s) 'f3 (s)) = O;
s-+oo Sin this case we write a rv f3. Clearly rv is an equivalence relation.^20 Let
[a]= {/3 E Ray M : f3 rv a} denote the equivalence class of a ray a.
Fix a point p EM and define the distance dR on Ray Ml rv by
(I.62) dR ([a], [/3]) = s-+oo lim ds(p,s) (an S (p, S s), f3 n S (p, s)).
Regarding the issue of the existence of the limit on the RHS of (I.62), see
Lemma I.53(iii) below.
EXAMPLE I. 51 (Euclidean space). If (Mn, g) = IEn is Euclidean space,
then (Ray Ml rv, da) = 5n-^1 , .the unit (n - 1)-sphere. Note that here a rv f3
if and only if a and f3 are parallel and pointing in the same direction. That
is, Ray M = IEn x sn-l (a basepoint and a direction) and (x, v) rv (y, w) if
and only if v = w; so the quotient identifies the factor IEn to a point.
Let Ray M (p) be the set of rays emanating from a fixed point p.^21
The equivalence relation rv on Ray M induces an equivalence relation ~ on
RayM (p).
LEMMA I.52. For any a E Ray M and p E M there exists ap E Ray M (p)
such that ap rv a.PROOF. Given t E (O,oo), let at : [O,d(p,a(t))] -+ M be a unit
speed minimal geodesic joining p to a (t). For the sequence of unit vectors
{ai (O)}iEN' there exists a subsequence such that limi-+oo ai (0) ~VE TpM
exists. Let ap : [O, oo) -+ M be the unit speed geodesic with ap (0) = V.
Then ap E Ray M (p) and ap rv a. (Exercise: Prove that ap is a ray and
ap rv a.) 0
The following lemma justifies the definition (I.62).(^20) Refiexivity and symmetry are obvious, whereas transitivity follows from the triangle
inequality.
(^21) Note that there is an injection j: Ray M (p)-+ s;- (^1) c TpM, where s;- (^1) denotes
the unit sphere. ·