- EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 489
We begin with an example on a compact manifold which contrasts to
the following discussion regarding a positive lower bound of IV' re j.
EXAMPLE I.46 (Mollifying distance on the sphere). Consider the unitsphere sn. Let p E sn and let r ( x) ~ d ( x, p). The mollified distance
functions re, where c E (0, 7r /2), are rotationally symmetric with respect to
the axis spanned by p. Moreover, re attains its maximum at the antipodal
point -p, so that
. IY'rel (-p) = 0
for all c E (0, 1f /2). From the rotational symmetry of re, it is not difficult to
show that
V're (x)
IY're (x)I = \i'r (x)
for all x E 5n - {p, -p} and c E (0, 1f /2).
Now we return to the general case of complete noncompact manifolds
with nonnegative sectional curvature. Define B1 : (0, oo) -i-[O, oo) by
B1 ( s) ~ max { ang p ( x) : r ( x) 2 s} ,where angp (x) is defined in (I.27). This is the maximum angle of r in the
complement of the ball ofradius s. By (I.28), we have B1 is well defined and
finite and -
(I.55) s-+oo lim B1 (s) = 0.
On manifolds with nonnegative sectional curvature, re has the following
properties, which may be used to approach Presumed Theorem I.50.
LEMMA I.47 (Properties of the derivatives of re). Let (Mn, g) be a com-
plete noncompact Riemannian manifold with nonnegative sectional curva-
ture. For any p E M and s1 > 0 there exist an co > 0 and functions ""i ( c),
defined for c E (0, co) and i = 1, 2, 3, with
(I.56) e-+0 lim ""i (c) = 0
such that for any c E (0, co) we have the following:
(a) (gradient estimate) for x E f3 (p, s1),
(I.57) 1 - ""1 (c) - e1 (r (x)) ::::; IY'rel (x) ::::; 1 + ""1 (c),
(b) (improved gradient estimate outside cut locus) ford (x, Cut (p)) 2
""2 (c) and x E f3 (p, s1) 1
(I.58) 1 - ""2 (c) S IY'rel (x),
and
(c) (Hessian comparison) for x E f3 (p, s 1 ),^19
1 + ""3 (c)
(I.59) V'V're (x) S re ( ). x(^19) Compare with (I.20).