Locked and unlocked smooth embeddings of surfaces.
D. Eppstein.
arXiv:2206.12989.
34th Canadian Conference on Computational Geometry, 2022, pp. 135–142.
Computing in Geometry and Topology 2 (2): 5.1–5.20, 2023, doi:10.57717/cgt.v2i2.28.
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.
(CCCG'22 surface-unfolding slides – Blog post: The motion of bend lines on smooth surfaces)