- APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 155
- 7, we have for 0 :::; r :::; p
IV'() ((k^0 k-1 ° · · ·^0 o)* gk+I - go) lg 0
= IY'o ((<I>k-1 ° · · ·^0 <I>o)* <I>kgk+i - go) lg 0
k-1 k
:::; Gp LE:i + E:kCp:::; Gp LCi·
i=O i=O
Similarly, by induction, <I>k o · · · o <I>1 is a (Gp 'Ef= 1 E:i,P )-approximate isom-
etry. By Proposition 4.7,\ \7k+l ( ( ( k o ... o 1 o o )-
1
) *go - gk+l) \gk+l= \\7k+l ( [ ( ( <I>k 0 ..• 0 <1>1)-^1 )*] ( <I>()^1 )*go - gk+l) \gk+l
k k:::; Gp LEi + coCp:::; Gp LE:i·
i=l i=O'Thus k o · · · o o is a (Gp 'E~=O E:i,P )-approximate isometry. D
2.2. Compactness of maps. We shall need a version of the Arzela-
Ascoli theorem which applies to maps. We define C^00 -convergence on com-
pact sets for maps in Euclidean space as follows.
DEFINITION 4.9 (GP-convergence of maps). Let U and V be two open
sets in ]Rn and let K C U be a compact set. We say that a sequence of maps
k : U --+ V converges to a map <1> 00 : U --+ V in GP on K if for every
c > 0 there exists ko = ko ( c, p) such that
sup sup IV'r (<I>k (x) - <I>oo (x))I:::; c fork 2: ko.
O:Sr:Sp xEKNote that the norm given is the Euclidean norm and \7 is the gradient with
respect to the Euclidean metric.
DEFINITION 4.10 ( C^00 -convergence of maps uniformly on compact sets).
Let U and V are two open sets in ]Rn. A sequence of maps k : U --+ V
converges to a map 00 : U--+ Vin C^00 uniformly on compact sets if
for any compact set K c U and any p > 0 there exists k1 = k1 (K, p) such
that { kh>k - 1 converges to <1> 00 in GP on K.
The following is a corollary to the Arzela-Ascoli theorem, Lemma 3.14.COROLLARY 4.11 (Compactness of sequence of isometries). Let U and
V be two bounded open sets in JRn. Let {gkhEN and {hk}kEN be Riemannian
metrics on U and V, respectively, such that the gk and hk are all uniformly
equivalent to the Euclidean metric and all of their derivatives (covariant
derivatives with respect to the Euclidean metric) are bounded. If the k :
(U, gk) --+ (V, hk) are isometries, then there is a subsequence of k which