1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. CENTER OF MASS AND NONLINEAR AVERAGES 179

which implies that the maximum of f ( "( ( t)) occurs at the endpoints. Hence

d(O,"((t)) :S max{d(O,x) ,d(O,y)} < r.

D

The next lemma will be used in proving the smooth dependence of the

center of mass in the next subsection.

PROPOSITION 4.48 (On the derivatives of exp-^1 ). Let (Mn, g) be a Rie-

mannian manifold such that all derivatives of the curvature are bounded:

IV'eRml :SCe forf=0,1,2, ....

There is a constant c (n) > 0 such that for any p EM and x, y E B (p, rl),

where rl ::; min { ~ inj (p) , c/ VCO} , if x is not in the cut locus of y, then

(i) we have

(4.16)

(ii)

(4.17)

1\7~^1 \7;^2 exp;^1 xi :S Ce 1 +e 2 +i for f1, f2 = 0, 1, 2, ... ,

where Ce= Ce (n, inj (p) ,f, Co, ... , Ce) > 0 are constants indepen-

dent of x and y, and \7 y and \7 x are the covariant derivatives with

respect to y and x, respectively;

when x, y ~ p* EB (p, rl), we have

(Y' x exp;^1 x : TxM ~ TyM) ~ (id : Tp*M ~ Tp*M) ,

(Y' y exp;^1 x: TyM ~ TyM) ~(-id: Tp*M ~ Tp*M),

where we use parallel translation to identify TxM and TyM with

Tp* M and to define the convergences above.

PROOF. (i) Let w = {wk} be normal coordinates on B (p, rl). By
Proposition 4.32 (Corollary 4.12 in [187]) we have in the coordinates w,

(4.18) 2 1 ( <5ij) :S (9ij) :S 2 ( bij) and 18°' (ow) a 9ij 1-:S Gia!,

where a is a multi-index. In particular the Christoffel symbols r~j satisfy

(4.19)

I

8°' k I -

(ow)°'rij :S Ciat+i·

Now we consider the exponential map exp : TM ~ M with expy z = x

for z ET Musing the coordinates w; here we abuse notation in that x = (xk)
stands for both a point in M and its coordinates in the coordinate system

w (and the same for y). Define f (r, y, z), 0::; r::; 1, by

d^2 fk k dfidfj -

(4.20) dr2 + rij (f) dr dr - 0,

fk (0,y,z) = yk,

dfk - k

dr (O,y,z) - z.