Shapes of surfaces that contain a great and a small circle through each point

TL;DR

This paper classifies certain surfaces in the 3-sphere that contain both a great and a small circle through each point, revealing five normal forms and their topological and intersection properties.

Contribution

It provides a complete topological classification of these surfaces, identifying their homeomorphism types and geometric configurations in the 3-sphere.

Findings

Surfaces are homeomorphic to five normal forms.

Surfaces are either products of circles in quaternions or have five concurrent circles.

Classified the real singular loci and circle intersection behaviors.

Abstract

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the pointwise product of circles in the unit quaternions or contain five concurrent circles. We classify the real singular loci of such surfaces and characterize how circles in the surface meet the self-intersection locus.