TITLE: Is a smooth closed surface in Euclidean 3-space rigid?
QUESTION [14 upvotes]: Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically.
Robert Connelly found an example of a polyhedral surface that can be deformed isometrically. A metal hinged model of it can be found at IHES.
But what about an arbitrary not-necessarily-convex smooth closed surface? Is it necessarily rigid? Or maybe it might be possible to make a smooth version of Connelly's example? It's easy to make smooth "hinges". The real challenge is finding a smooth model of the vertices, which is where two or more hinges meet.

REPLY [3 votes]: Apparently the question is still open for smooth enough surfaces and deformations (that is, at least $C^2$). 
Mike Anderson wrote a preprint claiming to prove local rigidity of smooth enough surfaces, but it was later withdrawn. 
Idjad Sabitov and his collaborators have been working on this question, developing for instance a theory of higher-order isometric deformations, see e.g. Sabitov, I. Kh. Local theory of bendings of surfaces [MR1039820 (91c:53004)]. Geometry, III, 179–256, Encyclopaedia Math. Sci., 48, Springer, Berlin, 1992. He conjectures that local rigidity holds for analytic surfaces.