152 4. PROOF OF THE COMPACTNESS THEOREMPROOF. Suppose x E B 9 (xo, r). Then
dh («I> (x), cl> (xo)) s inf lb h («I>* (a), cl>* (a))^1 l^2 dt
a:[a,b]-+M as inf lb ((l+s)g(a,a))^112 dt
a:[a,b]-+M a
S (1 + s)^1 /^2 d 9 (x, xo),where a (a)= x and a (b) = xo. D
We shall need the following propositions about how approximate isome-
tries affect tensors and how to compose approximate isometries.
LEMMA 4.5 (Norms of covariant derivatives of tensors, I). Given p EN
and q1, q2 E N U { 0} , there exists a positive constant Cp,qi,qz < oo such that
if cl>: (Mn, g)--+ (Nn, h) is an (c:,p)-approximate isometry withs< 1, then
for any (q1, q2)-tensor field T on M we have
r-1
(4.2) IV~Tl 9 s 1v~.hTl 9 + c:Cp,qim I: [vi.hr[
k=O gfor all 0 < r S p.
PROOF. We begin by proving (4.2) for p = 1. Note that I' 9 - I'<P*h is
a global (1, 2)-tensor field since it is the difference of two connections. By
(3.8) and (4.1) with C = 1 + s, we find
1r g - I'q,·hlg s (1 + s)^312 1r g - r<P·hli[>h s ~ (1 + c )^3 /^2 IV gel> hliP*h
(4.3) S ~ (1 +s)^3 IV 9 cI>*hl 9 S 12s
since s < 1. We have
The above expression has one term of the form V q,• hT and q1 + q2 terms of
the form (I' 9 - I'q,•h) * T. Hence we get the estimate
using (4.3).
IVgTl 9 s IV<P·hTl 9 + (q1+q2)1rg -r<P·hl 9 ITl 9
S IV <P*hTl 9 + 12s (q1 + q2) ITl 9 ,We now induct on p. Suppose (4.2) is true for any tensor and p s p.
Certainly we may choose Cp+1,q 1 ,q 2 to be greater than Cp,q 1 ,q 2 and so we