- MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS 471
so that
(I.21) dv di UI^2 (ao ( v)) ::=:: r (ao^2 (v)) IUI^2 (ao (v)).
Since r (ao (v)) = v, we have
(I.22) :v log cu1 (:o (v))) ::::: o.
We conclude that
(I. 23 ) (<ps,t)* (Uo) = IUI (ao (t)) < IUI (ao (s)) = Uo
t t - s s·
This implies the distance-nonincreasing property of <ps,t· D
An example where one can see the proposition directly is when (Mn, g)
is a rotationally symmetric complete Riemannian manifold with nonnegative
sectional curvature, Mn is diffeomorphic to IRn, and p is the origin; in this
case, inj (p) = oo. In particular, writing the metric as
g = dr^2 + w^2 (r) g5n-1
for 0 ::; r < oo and where w (0) = 0 and w' (0) = 1,^5 we have that the
sectional curvatures are given by (see (20.53) and (20.54))
1 - ( w' ( r))^2 w" ( r)
v1= w(r) 2 and v2=-w(r).Since the metric is complete and the sectional curvatures are nonnegative,
we have 0 ::; w' (r) ::; 1 and w" (r) ::; 0. This implies (rw' (r) - w (r))' =
rw" (r) ::; 0 and hence rw' (r) - w (r) ::; 0, so that(w ( r) )1
= rw' ( r) - w ( r) ::; 0
r r^2
for all r E [O, oo). Since(s(p,s),~ds(p,s)) = (sn-l, w;s)dsn-1)
(isometric), the distance-nonincreasing property of <p 8 ,t in (I.16) for s ::; t
follows.
One expects that, on manifolds with nonnegative sectional curvatures,
there are maps analogous to (I.16) when the radii of the spheres are allowed
to be larger than the injectivity radii at the centers. Such maps would be
useful for the study of the geometry at infinity. Working toward this end,
we shall develop some techniques regarding the distance function and its
mollifications.(^5) This guarantees that the metric extends smoothly over the origin even though the
coordinates are singular there.