Until recently, I did not believe that it would be possible to have a flat (in the Riemannian sense) torus or doughnut. But then I figured it is possible:
$$(x^1, x^2, x^3, x^4)=\Big(\cos\phi, \sin\phi, \cos\psi, \sin\psi\Big)$$
which is a 2D manifold embedded in the 4D Euclidean space ($\mathbb{E}^4$) with the induced, flat metric
$$ds^2=d\phi^2+d\psi^2$$
It also is a torus, clearly! So my question is: Is it possible to have a flat sphere? A manifold with vanishing Riemann tensor everywhere (Not almost everywhere, like a cube) which has a spherical topology?
If yes, are there any topologies (high genus, etc.), for which this is not possible? Thanks!
Best Answer
Great question!
It is not possible to have a flat sphere. In fact, it follows from the Gauss-Bonnet theorem that any (edit: connected compact orientable) surface which is flat is topologically a torus.
See https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem