TITLE: Isometric but differently shaped closed surfaces in $\mathbb{R}^3$
QUESTION [5 upvotes]: 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.

REPLY [7 votes]: 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.