Can we deform a curvy 2-manifold (surface embedded in 3D) so that the resulting homeomorphic surface consists of flat faces only. Like taking a sphere and deforming it to a cube. If that's true, is there a way of knowing when the resulting shape is going to be convex or non-convex?
I don't know if this is what you're after, but the torus is homeomorphic to this:
(not a perfect representation, but you get the idea)
and you can do the same thing for any closed orientable surface - basically, just add more holes and more pieces as needed.