Isometric but differently shaped closed surfaces in $\mathbb{R}^3$ Starting from the following inclusions for surfaces $M_1,M_2$ in $\mathbb{R}^3$:

     $M_1,M_2$ have the same shape, i.e. are related by an ambient isometry
→ $M_1,M_2$ have the same metric
→ $M_1,M_2$ have the same Gaussian curvature

the only examples of isometric but differently shaped surfaces in $\mathbb{R}^3$ I have seen so far do have boundaries:


*

*cone and cylinder

*catenoid and helicoid 

*other associate families of isometric minimal surfaces. 


I wonder if there are no (examples of) isometric but differently shaped closed surfaces, and why that could be. (I am particularly interested in smooth surfaces.)
And I am still looking for (closed) surfaces with the same Gaussian curvature but different metrics.
A picture gallery would be highly welcome, because I really would like to see two such (non-)isometric surfaces.
 A: Here is the standard explict example of 2-spheres $S_1, S_2$ embedded in $R^3$ with identical curvature functions (under suitable parameterization), so that $S_1$ is not isometric (as a Riemannian manifold) to $S_2$. Both surfaces will be surfaces of revolution. Take two cylinders of revolution $C_i$, $i=1, 2$ with unit radius and different heights. Now, attach isometric rotationally-symmetric caps $C_i^\pm, i=1,2$ at the top and the bottom of the cylinders $C_i$. One can easily write explicit functions for the cups (similarly to writing equations for the bump-functions) so that the resulting surfaces $S_i$ are smooth. Now, it is clear that, under suitable parameterization, surfaces $S_1, S_2$ have the same curvature functions, but will not be isometric to each other (since their regions of zero curvature are not isometric as they have different heights). I think, you can easily draw pictures of such surfaces yourself.   
A: Let $D$ be the unit disc and take a smooth function $f:D\to \mathbb{R}$ such that $f$ and all its derivatives are zero near the boundary of $D$.  Then the graph of $f$ and the graph of $-f$ are isometric, so if you construct a smooth surface which contains the graph of $f$, you can replace it with the graph of $-f$.  This is a bit of an unsatisfying example, though -- I'm not sure whether there is, say, a closed smooth manifold with a continuous family of isometric deformations, like the flexihedra that Lee mentioned.
Pogorelov's monograph (in Russian) on "Unique determination of general convex surfaces" proves the theorem that the boundary of any convex body in $\mathbb{R}^3$ is rigid (i.e., its embedding in $\mathbb{R}^3$ is determined up to translation and rotation by its metric), so any examples would have to be nonconvex.
A: Flexihedra
