TITLE: Why no flat metric on a sphere?
QUESTION [8 upvotes]: What is a simple proof that there is no flat metric on a sphere, but there is a flat metric on a torus?

Ideally such a proof would clearly distinguish between the sphere and the torus. Naively one might think neither can be
endowed with a flat metric. I'm hoping for some insight that
reveals this naive view to be incorrect.

REPLY [4 votes]: As the other answer also states, a simple proof that a flat metric on the torus is possible can be seen by pushing forward the flat metric on $\mathbb{R}^2$ by the cover $\pi: \mathbb{R}^2 \to S^1 \times S^1$, which can be done since the metric in $\mathbb{R}^2$ is invariant by translations.
An alternative to Gauss-Bonnet to see that the sphere admits no flat metric is by using that a flat metric on a simply connected manifold has trivial holonomy, which allows us to define a global non-vanishing section of the tangent bundle by taking the parallel transports of some chosen non-zero tangent vector, a contradiction with the hairy-ball theorem.