- APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 151
i.e., qi-l : (N, h) -t (M, g) is also an (c,p)-pre-approximate isometry.
Note the condition I ( qi-l ) g - hih Sc is equivalent to lg - hliP*h Sc
and l\7~ [(-^1 ) g] lh Sc is equivalent to l\7~hgliP*h Sc. Another way to
express the condition supxEM l* h - gl 9 Sc is
Iaqia aqib 12( [ ik aqia J ab k) (^0 qic [ ·e^0 qid J j)
hab oxi oxj - 9ij g = hab9 oxi oxj - Ij oxk hcd9J axe - Ik
= I ( d? ( d) - id 1
2
,where id is the identity map on TM, the transpose comes from the two met-
rics hand g, and IFl^2 =trace (F^2 ). Approximate isometries allow pointwise
bounding of metric tensors as follows.
PROPOSITION 4.2 (Approximate isometries and norms). Let c E (0, 1).
(i) If<I>: (Mn,g) -t (Nn,h) is an (c,0)-pre-approximate isometry andX is a vector field on M, then
IXl~*h S (1 + c) IXI~.
(ii) If: (Mn,g) -t (Nn,h) is an (c,0)-approximate isometry and X
is a vector field on M, then
1 2 2 ( 2
1 + c IXliPh S IXl^9 S^1 + c) IXliPh.
PROOF. (i) Using the Cauchy-Schwarz inequality on the tensor space,
we find that
( h )ij xi xj = ( ( h )ij - 9ij) xi xj + 9ijxi xj
s l<I>*h-gl 9 • 9ijxixj + 9ijxixj
s (1+c)9ijxixj.(ii) Inequality IXI~ :::; (1 + c) IXl~·h follows from (i) and the fact that
qi-l : (N, h) -t (M, g) is also an (c,p)-pre-approximate isometry. D
The following now immediately follows from Lemma 3.13.
COROLLARY 4.3 (Norms of tensors). If : (Mn, g) -t (Nn, h) is an
(c, 0)-approximate isometry, then for any (p, q)-tensor field T on M we
have
(4.1) (1 + c)-(p+q)/^2 ITliP*h S ITl 9 S (1 + c)(p+q)/^2 ITlq,•h·
The following proposition shows how (c, 0)-approximate isometries de-
form distances by small amounts.
PROPOSITION 4.4 (Distances). If : (Mn, g) -t (Nn, h) is an (c, 0)-
pre-approximate isometry, then
(B 9 (xo, r)) C Bh ( (xo), (1 + c)^1 /^2 r).