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

Ildefonso Castro, Daniel López-López

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.

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

Abstract

The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere 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 , 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.
Paper Structure (12 sections, 10 theorems, 42 equations, 10 figures)

This paper contains 12 sections, 10 theorems, 42 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Rotational surfaces in the 3-sphere generated by spherical curves
  3. Spherical curves
  4. Rotational surfaces in $\mathbb{S}^3$
  5. The spherical angular momentum
  6. The spherical angular momentum of a spherical curve
  7. The spherical angular momentum of a rotational surface in $\mathbb{S}^3$
  8. Spherical conics
  9. Spherical conics as geometric loci
  10. Spherical conics as intersection with elliptic cylinders
  11. Quadric surfaces of revolution in $\mathbb{S}^3$
  12. Cubic rotational Weingarten surfaces in $\mathbb{S}^3$

Key Result

Theorem 3.2

Any spherical curve $\xi=(x,y,z):I\subseteq \mathbb{R} \rightarrow \mathbb{S}^2$, with $z$ non-constant, is uniquely determined by its spherical angular momentum $\mathcal{K}$ as a function of its distance to the equator $\xi_0$, that is, by $\mathcal{K}= \mathcal{K}(z)$. The uniqueness is modulo ro

Figures (10)

  • Figure 1: Generating curves $\eta_{\delta}$, $\delta \geq 0$, and open sights of a stereographic projection of totally umbilical spheres $S_{\eta_{\delta}}$.
  • Figure 2: Generating curve $\xi_{\varphi_0}$, $\varphi_0 \in (0,\pi/2)$, and stereographic projection of a standard torus $S_{\xi_{\varphi_0}}$.
  • Figure 3: Generating curve $\mu_\theta$, $\theta \in (0,\pi)$, $\theta \neq \pi/2$, and stereographic projection of a spherical moon $S_{\mu_{\theta}}$.
  • Figure 4: Spherical conics with foci on the equator in terms of the parameters $d$ and $e$.
  • Figure 5: Spherical conic with foci on the equator.
  • ...and 5 more figures

Theorems & Definitions (40)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 3.1
  • Theorem 3.2
  • ...and 30 more