Is there a smooth area-preserving (i.e. unit jacobian) mapping from the 2-torus to the 2-sphere, or can we prove it's not possible?
While we can glue together two maps of the form $$f(x,y) = (\sqrt{1-y^2}\cos x, \sqrt{1-y^2}\sin x, y)$$ to define a continuous such mapping, with non-differentiability in the $y$ coordinate, I suspect the answer is no for a smooth mapping. Intuitively I think such a smooth mapping would be in contradiction with the Gauss-Bonnet theorem, but I can't make this precise.
If the Jacobian of a smooth map $T^2 \to S^2$ were nonzero everywhere then that map would be a local diffeomorphism and hence (using compactness) a covering map. Since $S^2$ is simply connected, every smooth covering map to $S^2$ with connected domain is a diffeomorphism. But $T^2$ is not diffeomorphic to $S^2$.