Surfaces with central cross-sections

TL;DR

This paper characterizes complete, connected C^2 surfaces in R^3 with the central plane oval property, showing they are either generalized cylinders or quadrics, and explores implications for surfaces with transverse plane ovals but no skewloops.

Contribution

It provides a complete classification of surfaces with the central plane oval property and links this to the absence of skewloops in such surfaces.

Findings

Surfaces with cpo are either generalized cylinders or quadrics.

Complete surfaces with a transverse plane oval and no skewloop are cylinders or quadrics.

The central plane oval property imposes strong geometric constraints on surfaces.

Abstract

A surface S in R^3 has the central plane oval property (cpo) if (i) S meets at least one affine plane transversally along a strictly convex oval, and (ii) Every such transverse oval on S has central symmetry. We show that a complete, connected C^2 surface with cpo must be either a generalized cylinder, or quadric. Applying this, we deduce that a complete C^2 surface containing a transverse plane oval but no skewloop, must be a cylinder or a quadric.