166 4. PROOF OF THE COMPACTNESS THEOREM
as restrictions of the map
expx"' k oLf.,
where Ea, JJJa, and iJa are the appropriately sized Euclidean balls centered
at the origin (the three maps are the same, but they are defined on different
domains).^1
We have the following standard result (for the proof see Corollary 4.12
in [187]).
PROPOSITION 4.32 (The IV'e Rmj ::::; Ce imply l8mgl ::::; Cm in normal
coordinates). Let (Mn, g) be a Riemannian manifold. Let p E M and ro E
(0, ~ inj (p)). Assume that for all£ ;:::: 0 there are constants Ce < oo such
that
l\i'e Rm I ::::; Ce in B(p, ro).Then in the normal coordinates {xi} on B(p, ro) there are constants Ce
depending on n, inj (p), Co, ... , Ce and a constant ci depending only on n
such that for any multi-index a with lal 2:: 1
1
2 ( 5ij) ::::; (9ij) ::::; 2 ( 5ij) andin B (p, min { ci/ vlCQ, ro}).
I
aagij I< c
oxa - fa[The { (Bk, ( Hf:)-^1 ) } form coordinate charts covering B (Ok, r) c
a~A(r)
Mk· Since
gk d3 7. ( Hk -/3) * 9kare Riemannian metrics on jjj/3 in normal coordinates with uniformly bounded
curvatures, by Proposition 4.32, all partial derivatives of the metrics are uni-
formly bounded. Hence by the Arzela-Ascoli theorem there is a subsequence
so that the gf converge uniformly in C^00 on compact sets to a limit Rie-
mannian metric ~ defined locally on EfJ. We use ou,r convention that
the subsequence is still indexed by k.
We have transition maps on. Mk defined as follows. Recall that for
a, f3 ::::; A (r) , if Bk n B~ -=/= 0 for some k, then it is true for all k ;:::: K (r) by
Lemma 4.18. If Bk n B~-=/= 0, then it makes sense to define the maps
,r:/3 : Ea -+ jjJ/3'by
JafJ k ='=. (fI/3)-1 k o Ha kl J-a{:J k ='=. (H-/3)-1 k o H-a k•
The maps Jf /3 and Jf/3 are embeddings; since for all "/, gr c E'Y c E'Y, Jf /3
is a restriction of Jf f3.
(^1) The balls Bf, Bf, and Bf are given by Definition 4.26; consequently the radii of
E"', E"', and E"' are equal to 5>.."', 45e^10 cG >.."', and 205e^20 cG >.."', respectively.