Central figure-8 cross-cuts make surfaces cylindrical

Abstract

Let M be a complete connected $C^2$-surface in ℝ$^3$ in general position, intersecting some plane along a clean figure-8 (a loop with total curvature zero) and such that all compact intersections with planes have central symmetry. We prove that $M$ is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and $k$ = 0.