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.**