Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909313367277568 |
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| author | Manfio, Fernando Santos, João Batista Marques dos Santos, João Paulo dos Van der Veken, Joeri |
| author_facet | Manfio, Fernando Santos, João Batista Marques dos Santos, João Paulo dos Van der Veken, Joeri |
| contents | We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it denotes the hyperbolic space $\mathbb{H}^3$ if $\varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $\mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $\mathbb{Q}^3\times\mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $\mathbb{Q}^3\times\mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07978 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures Manfio, Fernando Santos, João Batista Marques dos Santos, João Paulo dos Van der Veken, Joeri Differential Geometry 53C40 (Primary), 53C42 (Secondary) We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it denotes the hyperbolic space $\mathbb{H}^3$ if $\varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $\mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $\mathbb{Q}^3\times\mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $\mathbb{Q}^3\times\mathbb{R}$. |
| title | Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures |
| topic | Differential Geometry 53C40 (Primary), 53C42 (Secondary) |
| url | https://arxiv.org/abs/2409.07978 |