- MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS 4(39
endowed with the subspace metric, which we denote by ds(p,s)· The metric
ds(p,s) gives us the induced intrinsic metric defined by (see (G.6))
~
(I.15) ds(p ' s) ~ dcd S(p,s).We consider S (p, s) with the metric ds(p,s). It is easy to see that ds(p,s) (x, y)
is finite if and only if x and y lie in the same component of S (p, s). When
S (p, s) is ,smooth, ds(p,s) is the same a~ the distance induced by the Rie-
mannian metric gJs(p,s) on S (p, s).
For intuition we note the (trivial) model case.REMARK I.28 (Euclidean space). Given a point 0 (considered as the
origin) in Euclidean space, for any s, t E (0, oo) there are natural maps <p~t:
S ( 0, s) --+ S ( 0, t) given by identifying points which lie on the same ray e~-
anating from O. Themaps<p~t: (s(O,s),~ds(o,s))-+ (s(O,t),fds(O,t))
are isometries.
The following is the main result of this section.PROPOSITION I.29 (Distance nonincreasing maps between small geodesic
spheres). Let (Mn, g) be a complete Riemannian manifold with nonnegativesectional curvature. Given p E M, for any s :::; t :::; u < inj (p),
(1) the map(I.16) 'Ps,t : ( S (p, s) , ~ds(p,s) )--+ (s (p, t) , tds(p,t)),
defined by (I.14), is distance nonincreasing and
(2)
(I.17) 'Pt,u o 'Ps,t = 'Ps,u·FIRST PROOF. The composition property (I.17) of <p 8 ,t follows directly
from (I.14). Below we shall argue using Jacobi fields and the Rauch compar-
ison theorem that 'Ps,t in (I.16) is distance nonincreasing for s:::; t <-inj (p).
Let x, y ES (p, s) be arbitrary points and let
f3: [O, p].-+ S (p, s),
where p ~ ds(p,s) (x, y), be a minimal geodesic, with respect to the (smooth)
induced Rierhannian ·metric on S (p, s), satisfying (3 (0) = x and (3 (p) = y.
For every u E [O, p], let · ,
au : [O, t] --+ M
be the unique unit speed minimal geodesic with au ( 0) = p and au ( s)
f3 ( u). Consider the path
'Y : [O, p] --+ S (p, t)
defined by
ry(u)=~(t).