1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 493

LEMMA I. 53. Let (Mn, g) be a complete noncom pact Riemannian mani-

fold with nonnegative sectional curvature. For any a1, a2 E Ray M, we have

the following. ·

(i) dR ([a1], [a2]) = 0 is equivalent to lims-+oo d(ai(s)t"^2 (s)) = 0.

(ii) dR is well defined independent of the choice of representatives of

[a1] and [a2].

(iii) Assuming Presumed Theorem I.50 is true, we have dR ([a1], [a2])

exists.

PROOF. (i) (==?). By the triangle inequality, we have

(I.63) u - d (ai (0) ,p) :::; d (ai (u) ,p) :::; u + d (ai (0) ,p) ..

Thus, if C¥i (si) E C¥i n S (p, s), then Si E [s - d (ai (0) ,p), s + d (ai (0) ,p)].

Hence

1 lill. ---'-------d ( C¥1 ( s) ' C¥2 ( s))

s-+oo S

1

. d (a1 (s), a1 (s1)) + d (a1 (s1), a2 (s2)) + d (a2 (s2), a2 (s))

< -s-+oo Ill --------------------~ S

1

. d (ar(O) ,p) + d (a 2 (0) ,p)
1
. ds(p,s) (a1 n S (p, s), a2 n S (p, s))
< lill + lll------------

• s-+oo S s-+oo S

=0.

( {==). Using the modified distance function rs and following the idea
of the proof of Proposition I.41, one can show that there is a constant C
independent of s such that (exercise)

(I.64) ds(p,s) (x, y) :::; C · d (x, y) for x, y ES (p, s).

Hence, for any s let Si be defined by C¥i (;i) E C¥i n S (p, s),

. ds(p s) ( C¥1 n s (p, s) ' C¥2 n s (p, s))

hm '

S-700 S

.C

1

. · d (a1 (s1), a2 (s2))
< lll-------

• s-+oo S ,

C 1

. d (a1 (s1), a1 (s)) + d (a1 (s), a2 (s)) + d (a2 (s), a2 (s2))
< - s-too lill ----'---'--------'-----------------· S

C 1

. d (ar(O) ,p) + d (a2 (0) ,p) C
1
. d (ar(s), a2 (s))
< lill + lill ------

• s-+oo · S s-+oo S
  =0.
  (ii) If C¥i rv /3i, then by the triangle inequality

ds(p,s) (a1 n S (p, s), a2 n S (p, s)) ds(p,s) (/31 n S (p, s), /32 n S (p, s))

s s

< ds(p,s)(a1nS(p,s),/31,ns(;,s)) + ds(p,s)(a2nS(p,s),/32nS(p,s)).

• s s. '