No Smooth Onto Map from Circle to Torus

This follows from Sard's theorem, which implies that the image of a smooth map $S^1 \to S^1 \times S^1$ has Lebesgue measure zero. More generally, Sard's theorem implies that there are no smooth onto maps from a smooth manifold of dimension $n$ to a smooth manifold of dimension $m > n$.

The obvious statement is that there are no diffeomorphisms between such manifolds, which just follows by computing differentials (the differential of a diffeomorphism is an isomorphism). The differential of a surjective smooth map is not surjective in general (consider the surjection $\mathbb{R} \ni x \mapsto x^3 \in \mathbb{R}$) so the obvious argument doesn't work (or at least it doesn't obviously work).

Note also that this statement is false if "smooth" is replaced by "continuous" due to the existence of space-filling curves.

A smooth map $S^1 \rightarrow S^1 \times S^1$ is a differentiable curve; by differentiability it must have finite arc length. But a curve of finite length can't cover the 2-dimensional torus.