Quadric surfaces of revolution in the 3-sphere as Weingarten surfaces

TL;DR

This paper classifies and characterizes quadric surfaces of revolution in the 3-sphere, revealing they form a special class of Weingarten surfaces defined by a cubic relation between principal curvatures.

Contribution

It extends the classical study of quadric surfaces of revolution from Euclidean space to the 3-sphere, introducing a new classification and showing these surfaces satisfy a unique cubic Weingarten relation.

Findings

Surfaces are characterized by a specific cubic relation between principal curvatures.

They form a class of Weingarten surfaces in the 3-sphere.

The results unify Euclidean, Lorentzian, and spherical cases.

Abstract

The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere $\mathbb{S}^3$ has not been sufficiently explored. This article addresses this important gap by providing a rigorous classification and characterization of non-degenerate quadric surfaces of revolution in $\mathbb{S}^3$, namely spherical ellipsoids, hyperboloids and paraboloids, generated by the rotation of spherical conics around a geodesic axis containing their foci or is orthogonal to them. Using the concept of spherical angular momentum as a prominent geometric invariant, we discover that these surfaces constitute a remarkable class of Weingarten surfaces and prove that they are uniquely characterised by a specific cubic functional relation between their principal curvatures. This result not only provides a unified description of spherical quadric surfaces of revolution, but also highlights a profound geometric universality, reflecting exactly the same cubic Weingarten relations observed in their Euclidean and Lorentzian counterparts.