2 problems

Let $V:\mathbb R^3\to S^2$ be a unit vector field defining a fibration of $\mathbb R^3$ by oriented lines. Put $U=V(\mathbb R^3)$, and for $u\in U$ let $u^\perp$ be the plane throu…

Great-circle conjecture. If there exist multiple $u\in U$ for which $S_u$ is noncompact, then the set of such $u$ is contained in some great circle in $S^2$.