On Oct 12, 2007, at 11:59 AM, Dan Asimov wrote:
I had been originally just thinking C^0. But what is a C^1 isometric embedding of S^2 -> R^3 that approximates the inclusion map, yet whose image is strictly in the interior of it ?
It has to be pretty weird. I think it's qualitatively like fingers that have soaked a long time in a bathtub, or like the skin of an animal that has just molted. Since the embedding can't be C^2, I think it's probably hard to give an explicit concrete description. It could be interesting to try to do computer simulations--this should be feasible for someone with enough time and motivation. A good first step might be to shrink a square to an isometric embedding in E^3 that approximates the map say (x,y,z) -> .8 (x,y,z), by alternately rippling in the x direction and the y direction. Bill