A map from the torus to the real line which is not a trivial fiber bundle

Question

I want to understand the map $\pi: \mathbb R^2/\mathbb Z^2\to \mathbb R$, by $(x,y)\mod \mathbb Z^2\mapsto y-\sqrt 2x $. Why it can not be a local trivial fiber bundle? Is $\pi$ surjective? If not, is $\pi( \mathbb R^2/\mathbb Z^2)$ a manifold?