I somehow managed completely to fail to say what I meant, which was that though there were restrictions on the continuity of the family in the other statements, there seemed to be none in this. But of course, any embedding of the sphere must be C^0 by definition; and that's all you are claiming anyway (crawls back underneath stone). WFL On 6/19/12, Dan Asimov <dasimov@earthlink.net> wrote:
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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