Let $S$ be an immersed surface in $\mathbb{R}^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s\_t: S\to \mathbb{R}^3$, such that each $s\_t$ induces the same metric on $S$ and no $s\_{t\_1}$ and $s\_{t\_2}$ are related by an isometry of $\mathbb{R}^3$ (i.e. we rule out trivial deformations by one-parameter subgroups of ambient isometries).


1) Do there exist flexible smooth closed surfaces in $\mathbb{R}^3$? In the polyhedral world the answer is yes, with a well-known example of flexible polyhedron by Connelly.


Convex surfaces seem to be rigid, which should be a theorem of Alexandrov or Cauchy. Are there any other global criterions for rigidity/flexibility?


2) Is locally any surface flexible?


3) Can anybody give me an example of a rigid surface with boundary?



Bending surfaces in Riemannian manifolds
The standard reference, where the state of the art concerning all of your questions is found:



> 
> Q. Han, J.-X. Hong, **Isometric Embedding of Riemannian Manifolds in Euclidean Spaces,** Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
> 
> 
> 


There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson recently [produced](https://arxiv.org/abs/0709.3202) a proof, but a flaw emerged.


The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.


Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.


Another very important work, *[Open problems in geometry of curves and surfaces](http://people.math.gatech.edu/%7Eghomi/Papers/op.pdf)* (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.


Rigidity of the top half of a torus is discussed [here](https://mathoverflow.net/questions/77760/why-is-the-half-torus-rigid?rq=1), but perhaps without a complete proof having become clear yet.


