- APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 153
need only prove inequality (4.2) for r = p + 1. Then
l'\JP+lTI g g = l'JP'\J g g g Tl
p-1
:S l'\J~*h '\J gTlg + cCp,qi,q2+1 L l'\Ji*h '\J gTI
k=O g
by the inductive hypothesis. For the first term on the RHS,I '\J~h '\J gTlg :S l'\J~h ('\Jg - '\J h) Tlg + I ('\J h)P+i rig·
In turn, a sum of terms of the formI ('\J h)k (r g - rh) lg I ('\J *h)p-k rig'
where 0 :S k < p, bounds the first term. As was shown in the proof of
Lemma 3 .11,
k+l
l'\Ji*h (rg -r<I>*h)I :S ck L l'\J~*hgl
g j=l g
k+l:S ck~ ~ (1 + c)(2+j)/^2 1'\Jj q, h gl
j=l
2(k+4)/2::::; ck v'2 2-1 10
for some constant Ck depending on k, where we have used that ~ is an
(10, p + 1)-approximate isometry and 0 :S k :Sp. Hence we have bounded the
term l'\J~h '\J gTlg· Similarly, we can bound the terms l'Jih '\J gTI, where
O:Sk:Sp-1.
Putting it all together, we see that in the estimate of l'\Jg+irlg, the
coefficients of l'Ji*hT\g are all bounded and all have an c in them except
l'\J~-t~Tlg. Thus we get
p
\'\J~+ir\g :S l'\J~-t~rl + 10Cp+i,q 1 ,q2 L l'\Ji*hrl.
g k=O gThis completes the proof of the induction. D
COROLLARY 4.6 (Norms of covariant derivatives of tensors, II). There
exist Cp,O,q 2 for p, q2 E N, such that if~ : (Mn, g) --+ (Nn, h) is an (10,p)-approximate isometry, then for any (0, q2)-tensor T field on N
1v; (
for each 1 :S r :S p.