312 B. SOME RESULTS IN COMPARISON GEOMETRY
PROPOSITION B.64. For any choice of origin 0 E Mn and alls< t,
(B.10) Cs= {x E Ct: d(x,8Ct) 2: t-s}.
In particular, 0 E 8Co and
8Cs = { x E Ct : d (x, 8Ct) = t - s}.PROOF. By Lemma B.59, we have 0 E 8Co = { x E Mn : b (x) = 0 },
because b ( 0) = 0. And since the distance function is continuous, the char-
acterization of 8Cs given here follows from (B.10). So it will suffice to prove
both inclusions<;;;; and 2 in (B.10).
(<;;;;) Suppose x E Ct and d (x, 8Ct) < t - s. We want to show that
x tJ. Cs. Because
d (x, u,ER(O)JIB,t) = d (x, Mn\ Ct) = d (x, 8Ct) < t - s,
there exists a ray j3 emanating from 0 such that d (x, JIB.at) < t - s. By
Lemma B.57, this proves that x E JIB.es, hence that x tJ. Cs.
(2) Suppose x E Ct \Cs. We want to show that d (x, 8Ct) < t - s. Since
x tJ. Cs = n,ER(O)lHils, there is a ray /3 emanating from 0 such that x E JIB.as.
By Lemma B.57, this proves that d (x, JIB.at) < t - s. Hence
d (x, 8Ct) = d (x, Mn\ Ct) = d (x, U 1 En(o)JIB 1 t) < t - s.
- Estimating injectivity radius in positive curvature
The objective of this section is to prove the following result.
D
THEOREM B.65. Let (Mn,g) be a complete noncompact Riemannian
manifold of positive sectional curvatures bounded above by some K E (0, oo).
Then its injectivity radius satisfies
.. mJ (Mn ,g ) :::: VR. 7rBefore proving the theorem, we will establish some preliminary results.
Assume for now only that the sectional curvatures of (Mn, g) are bounded
above by K > 0.
DEFINITION B.66. If k E N*, a proper geodesic k-gon is a collection
r = bi : [O, Ri] _____, Mn : i = 1, ... , k}
of unit-speed geodesic paths between k pairwise-distinct vertices Pi E Mnsuch that Pi = /i (0) = /i-1 (Ri-1) for each i, where all indices are interpreted
modulo k. The total length of a proper geodesic k-gon is
k
L(I') ~ LL(ri).
i=l
r is a nondegenerate proper geodesic k -gon if Lp; ( -i'i-l, )'i) # 0 for