1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

6. EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 487

(note that [rs, rt]= 0 and 'Vrsrs = 0), equation (I.52) implies

a2

1rtl 8s2 1rtl 2:: (R (rs, rt) rs, rt)

2:: -r;,^2 lrtl^2 lrsl^2

2:: -E2r;,2 Jrtl2

since, given any t, we have

Inequality (ii) follows immediately. Thus the claim has been proven.

Now we may apply the Sturmian comparison theorem to (I.50)-(I.51) to

obtain

Jrtl (s, t) 2:: (cos (Er;,s) - ~sin (Er;,s)) L (7r o 'Y) ~ 'ljJ (s);

note that 'ljJ satisfies

In particular,

Thus

L ('Y) = fo

1

Jrtl (1, t) dt

2:: (cos (Er;,) - ~sin (Er;,)) L (7r o 'Y).

0

Finally, we leave it as an exercise for the reader to formulate local ver-

sions of Lemma I.42 and Theorem I.44.

6. Equivalence classes of rays and points at infinity

The monotonicity property of distance spheres in nonnegatively curved

manifolds given by Proposition I.29 suggests another approach to the exis-

tence of asymptotic cones in Theorem I.26, originally suggested by Gromov

[9] and later considered by Kasue [107]. On pp. 593-594 of [107] it is

written: