Is there a "non-curved" manifold?

Question

I've seen some visualizations of manifolds. It seems that they are all "curved" shapes. Is there a "non-curved" manifold?

Answer 1

Yes, you can get a flat, compact manifold embedded in $\Bbb R^3$. This is a consequence of the Nash embedding theorem. In particular, a couple of years ago, this was done in practice with a torus, and the results are quite visually interesting. Here are the first three steps in the construction:

It may look curved, but it does actually give a $C^1$ isometric embedding of the flat torus $S^1\times S^1$ in three-dimensional Euclidean space. ($C^1$, i.e. continuously differentiable, is important because otherwise distances along the surface are not necessarily well-defined.)