The phrase "deformation (through isometric embeddings)" includes the word "through". But to expand on that definition: Given any eps > 0, there is a family of isometric embeddings H_t : S^2 -> R^3 given by H_t(x) := H(x,t), where H: S^2 x [0,1] -> R^3 is a continuous mapping, and such that a) H_0 is the standard inclusion of the unit sphere S^2 into R^3 and b) H_1(S^2) lies inside an open ball in R^3 of radius eps. --Dan << I think I can guess what deformation of a surface "through" a set of surfaces might mean; but can you give a pointer to where these matters are defined in more detail? In particular, why is there no mention of "through" in your and Gerver's result? [I] wrote: << Joe Gerver and I have found a way to perform a C^0-isometric deformation (through isometric embeddings) of the round S^2 in R^3 that reduces it to fitting into a small ball of radius eps for arbitrarily small eps > 0.
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