- CENTER OF MASS AND NONLINEAR AVERAGES 183
PROPOSITION 4.51 (Dependence of cm on weights and points). Suppose
(Mn, g) is a Riemannian manifold such that all of the derivatives of the
curvature are bounded:
Iv£ Rml 'S. Cc fore= 0, 1, 2, ....Let μ1, ... , μk be nonnegative weights with μ1 + · · · + μk > 0. There exists
a constant c (n) E (0, ~) such that for any p E M, if inj (q) > 3r for all
q EB (p, r), where r <. ~· Then we have the following.
(i) (Bounds on the derivatives of cm) The unique center of mass
Cm(μ 1 , ... ,μk) {q1, .. ·, qk}
is a smooth function of ql, ... , qk E B (p, r) and μ1, ... , μk. TheV~V~-covariant derivatives of cm(μi,.. .,μk) {q1, ... , qk}, with respect
to ql, ... , qk and μ1, ... , μk, satisfy
(4.24) jv~v~ cm(μi, ... ,μk) {q1, ... , qk}j s. C1a<1+1,s1+i'
where V q = (V qu ... , V qk) and V μ = ( Bμi, a ... , Bμk a ) and Cial+l,81+1 -
are constants depending onn, inj (p), lal+l,81, and Co, ... ,Clal+l,61+1·
(ii) For ql, ... , qk E B (p, r) such that ql, ... , qk --+ q* E B (p, r) (i.e.,
the points tend to each other), we have
(a) (change in a weight has negligible effect on cm)IV μi cm(μ1, ... ,μk) {q1, ... 'qk}I --+ 0,
(b) (effect of the change in a point on cm)(V qi Cm(μi ' ... ' μk) {q1, · · ·, qk} : TqiM --+Tern( μli···if.Lk ){qi, ... ,qk}M)--+ (z:/i j=l μJ. id : TqM --+ TqM) ,
(c) (effect of the change in a weight and point on cm)( V qj O~j Cm(μ 1 , .. .,μk) {q1, ... , qk} : TqiM --+ Tcm(μi,. .. ,μk)fo, .. .,qk}M)
--+(I:/ i=l μi. id: TqM--+ TqM).
The convergences above are defined using parallel translationto identify Tqi M with Tq* M.
PROOF. (i) We apply the implicit function theorem to the family of
maps
defined by
k
Gq1, ... ,qk>μ,1, ... ,μk (q) ~ L μi exp;;-^1 qi.
i=l