Surfaces with central convex cross-sections

Abstract

Say that a surface in $S\subset {\mathbb{R}}^3$ has the central plane oval property, or cpo, if

• $S$ meets some affine plane transversally along an oval, and
• Every such transverse plane oval on $S$ has central symmetry.

We show that a complete, connected $C^2$ surface with cpo must either be a cylinder over a central oval, or else quadric. We apply this to deduce that a complete $C^2$ surface containing a transverse plane oval but no skewloop must be cylindrical or quadric.