# Download Journal of Differential Geometry-Volume 12 by Lehigh University; International Press PDF

By Lehigh University; International Press

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Additional info for Journal of Differential Geometry-Volume 12

Example text

Is a strictly increasing function of. the arguments x and X and a strictly decreasing function of the arguments x and k, in the region >0, ks> -K 2, 0

Let us put p(p, l') = 2p > 0. Then, for a sufficiently large v, l') > By Lemma 3 108 B. V. DEKSTER cos a - cos , < C- p(p, q) when p(p, q) < E. 5) passing to the limit when v, oo. Remark. 3. Originally, Lemmas 2 and 3 were stated in much more generality, but their proof was very long. The idea of the present short proof was suggested by the referee whom the author thanks very much. This proof can still be easily generalized for the case when I' is an arbitrary compact set in a complete space and p, q I F.

IZnllk2 -s In each case Xn IIl < C(x0)tn, II Zn IIZ-s <_ C(xo)tns, Ilxn+1III < C(x0)tnS+l < C(x0)tns C(xo)tn2e-1os (or any power of tn), so surely each term above is and IIZnIII-c < bounded by C(x0)tn 2c. Thus IZn+IIIZ-8 1, but again it is trivial for n Thus the lemma holds. Corollary. For every i and every c This proves lz,,11 8 II dxn IIZ < C(xo)tn c , 0 II Xn - X,, IIZ < C(x0)tn Proof. 5. Derivatives of the projection. We now wish to show that on the entire set Vb the map Pb is smooth and all its derivatives are continuous and tame.