12 Oct
2007
12 Oct
'07
10 a.m.
(Sorry about the garbled first line.) Bill wrote: << . . . The Nash-Kuiper embedding theorem can be done in a parametrized way, i.e. any C^1 continuous family of short C^1 embeddings parametrized say by a cell complex, i.e. P -> {C^1 short embeddings S^2 -> R^3}, can be deformed through short maps to a family of isometric embeddings, by the same proof. I.e. there would be a homotopy P x [0,1] -> {C^1 short embeddings S^2 -> R} that ends up in {C^1 isometric embeddings S^2 -> R}. . . .
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 ? --Dan