Re: [math-fun] "C^1" isometric deformations of S^2 in R^3
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. --Dan On 13/11/2009 09:32, Gerard Westendorp wrote: << . . . Daniel Asimov wrote: << It's known that the standard sphere S^2 in R^3 is rigid with respect to C^2 deformations through C^2 surfaces. (See, e.g., Edgar Kann, "A new method for infinitesimal rigidity of surfaces with K > 0," J. Diff. Geom., 1970 pp. 5-12.) But have there been results about isometric deformation of the standard S^2 in R^3, through surfaces that need be only C^1 embeddings? The deformation itself need be only C^0. And what about the same question but with C^0 replacing C^1 ?
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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? Fred Lunnon On 6/19/12, Dan Asimov <dasimov@earthlink.net> 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.
--Dan
On 13/11/2009 09:32, Gerard Westendorp wrote: << . . .
Daniel Asimov wrote: << It's known that the standard sphere S^2 in R^3 is rigid with respect to C^2 deformations through C^2 surfaces. (See, e.g., Edgar Kann, "A new method for infinitesimal rigidity of surfaces with K > 0," J. Diff. Geom., 1970 pp. 5-12.)
But have there been results about isometric deformation of the standard S^2 in R^3, through surfaces that need be only C^1 embeddings? The deformation itself need be only C^0.
And what about the same question but with C^0 replacing C^1 ?
. . .
________________________________________________________________________________________ It goes without saying that .
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Fred lunnon