Greenfield–Wallach conjecture on globally hypoelliptic vector fields

Let $M$ be a closed, connected, orientable manifold, and let $X$ be a vector field on $M$. The vector field $X$ is globally hypoelliptic when every distributional solution of $Xu=f$ is smooth whenever $f$ is smooth. A constant vector field on a torus is Diophantine when its frequency vector satisfies a Diophantine lower-bound condition. Greenfield–Wallach conjecture. If $M$ admits a globally hypoelliptic vector field $X$, then $M$ is diffeomorphic to a torus and $X$ is smoothly conjugate to a constant Diophantine vector field. This conjecture predicts that globally hypoelliptic vector fields occur only in the toral, Diophantine setting; its general status is open.