- CENTER OF MASS AND NONLINEAR AVERAGES 181
Using lzl::; 2c/JCQ, we can choose co~ sn 4 c2 (1). We have proved
( 4.22)Now we can apply the implicit function theorem to F (x, y, z) ~ f (1, y, z)
-x. From
oz
Dyf (1, y, z) + Dzf (1, y, z) oy = 0,
Dzf(l,y,z) ~=-id=O,
we can take higher-order derivatives of the equations above to get formulas£ or (By)cx(ax)/3 acx+/3 z m. t erms o f th e part1a.^1 d. envatives. o f (By)a(az)/3 acx+/3 fk (1 , y, z ) an d
(Dzf (1, y, z))-^1. From (4.21) and (4.22) we can estimate
fp+!3z -
(8yr' (8x)/3 <C - lal+l/3 I +1by induction on the order of derivatives. From (4.18) we know that the
bounds of the covariant derivatives I v~^1 v;^2 z I follow from the bounds ofI
aa+f3z I
(By)°'(Bx)/3.(ii) Let w be normal coordinates on B (p, r) for sufficiently small r*
and let 9ij ~ g (a~i, a~j). From Theorem 4.10 in [187] we have (9ij)---+ (<5ij)
on B (p, r) as r ---+ 0. Hence the geodesic equation ( 4.20) in the coordinate
chart w has a solution J (r, y, z) which converges in C^1 tof00 (r, y, z) ~ y+rz
as r ---+ 0. So for x, y E B (p, r), expy-^1 x converges in C^1 to exp~~ x ~
x - y as r ---+ 0. The estimate ( 4.1 7) follows from (gij) ---+ ( <5ij) on B (p, r *)
and
oxi^8 exp-1 d^8 -1
00 y x = ei an (Jyi exp 00 Y x = -ei,where ei = (0, · · · , 1, · · · , 0) is the unit vector in i-th direction. The lemma
now is proved. D
REMARK 4.49. (i) Under the assumptions of Proposition 4.48, let p E
B (p, ri) and let w be normal coordinates on B (p, r) for sufficiently small
r. Then when x, y---+ p*, we have that expy-^1 x converges in C^1 to the map
x - y, where x = { xk} stands both for a point in M and its coordinates in
the coordinate system w (and the same for y = { yk}).
(ii) Suppose h is another metric on M. From the proof of Proposition
4.48 it is not difficult to see that the map (exp9)-^1 : M x M---+ TM is close
to (exphr^1 : M x M ---+ TM on any compact set in Ck when g is very
close to h on any compact set in Ck, for any k E N.