Quadric surfaces of revolution in the 3-sphere as Weingarten surfaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918356288798720 |
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| author | Castro, Ildefonso López-López, Daniel |
| author_facet | Castro, Ildefonso López-López, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadric surfaces of revolution in the 3-sphere as Weingarten surfaces Castro, Ildefonso López-López, Daniel Differential Geometry 53A05 (Primary), 53A04 (Secondary) 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. |
| title | Quadric surfaces of revolution in the 3-sphere as Weingarten surfaces |
| topic | Differential Geometry 53A05 (Primary), 53A04 (Secondary) |
| url | https://arxiv.org/abs/2602.21785 |