- NOTES AND COMMENTARY 187
4.49(ii) that by choosing k large and x to be in a very small neighborhood
of xo, we can make cm^9 (^00 μ,k 1 ( x ) , ... ,μ,k A( x l) {Fl (x), ... , Ff (x)} arbitrarily close
in the C^1 -topology to the Euclidean center of mass1 ()
1A() (μk(x)Fl(x)+ .. ·+μt(x)Ff(x)),
μk x + ... + μk x
where we have identified the point Ff (x) E B1 with its coordinates in the
coordinate chart w. Since μ"fc (x) + · · · + μt (x) = 1, we have
1 ()
1A() (μk(x)Fl(x)+ .. ·+μt(x)Ff(x))-x
μk x + ... + μk x
= μ"fc (x) (Fl (x) - x) + · · · + μt (x) (Ff (x) - x),
which clearly converges to 0 on any compact set within the coordinate chart
win C^1 when k-+ oo. Now the proposition is proved. D
6. Notes and commentary
For some additional references on compactness theorems not cited in
the previous chapter, see Cheeger and Gromov [73], [74], Gao [152], Yang
[374], [375], Anderson [4] and Anderson and Cheeger [6].