- CONSTRUCTION OF GOOD COVERINGS BY BALLS 159
Bk C (Mk, gk), and functions A (r), K (r), I (n, Co), where Co is the cur-
vature bound in Theorem 3. 9 and A and K are nondecreasing in r, so that
the following hold.
(1) B'f:, Bk, Bk, B'f:, and B'f: are concentric, i.e., they have the same
center, with center denoted as xk such that x£ =Ok.
(2) Bk and B~ are disjoint for a-=!= (3.
(3) The exponential map expxk' oLk : Ba -+ Bk is a difjeomorphism,
where Ba is a ball in JR.n and Lk : JR.n-+ Txk'Mk is a linear isometry
defined using an orthonormal frame at xk. Moreover, for each a the
ball Bk is geodesically convex fork large enough (depending on a).
( 4) We have the containmentif k 2:: K (r).
B (Ok,r) C LJ Bk
a:=:;A(r)(5) The number of (3 such that B~ n Bk-=!= 0 is fewer than I (n, Co).
(6) If a, (3 < A (r), then Bk n B~ is either empty for all k 2:: K (r) or
nonempty for all k 2:: K ( r).(7) If Bk n B~-=!= 0, where a, (3 ~A (r) and k 2:: K (r), then Bk c B~
- .... (3
and Bk c Bk.
The proof of the above lemma will occupy the rest of the section. In
order to prove the lemma, we will need the following result on how the in-
jectivity radius can decay in relation to distance. On a complete manifold
with bounded curvature the injectivity radius at a point can decay at most
exponentially in distance. A strong partial result in this direction was first
obtained by Cheng, Li, and Yau [93]. Later, Cheeger, Gromov, and Tay-
lor [75] obtained the following stronger estimate using different techniques.
Recall that inj (x) denotes the injectivity radius at x.
PROPOSITION 4.19 (Injectivity radius decay estimate). Let (Mn,g) be
a complete Riemannian manifold with sectional curvatures IKI ~ Co and
injectivity radius inj (0) 2:: lo> 0. Then there exist constants a= a (n, Co) >
0 and C = C (n, Co)< oo such that for any x EM
inj (x) 2:: μ [d (x, 0), lo],
where
(4.6) μ [ r, lo l ::;=. a· min · { lo, l}n · e -Cr.
REMARK 4.20. The reason for the unnatural looking exponent n in the
estimate is that the result is proved by using relative volume comparisons,
and the injectivity radius inj bounds the volume comparable to inr whereas
the volume V bounds the injectivity radius comparable only to V. That is,
the exponent n arises from the conversion from injectivity radius to volume
and then back again to injectivity radius.