If a subset of the plane has a continuous shrinking motion of itself, then every smooth isometric embedding of that subset into 3d can be smoothly flattened. However, there exist subsets of the plane with holes, for which some smooth embeddings that are topologically equivalent to a flat embedding cannot be smoothly flattened.
(Slides)
Journals – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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