Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ and $\mathbb{H}^{n}\times \mathbb{R}^{m}$
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| Format: | Preprint |
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2025
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| _version_ | 1866910252878790656 |
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| author | Tan, Huixin Xie, Yuquan Yan, Wenjiao |
| author_facet | Tan, Huixin Xie, Yuquan Yan, Wenjiao |
| contents | We first show that every isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, we prove that an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ also has constant principal curvatures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ and $\mathbb{H}^{n}\times \mathbb{R}^{m}$ Tan, Huixin Xie, Yuquan Yan, Wenjiao Differential Geometry 53C42 (Primary) 53B25, 53C40 (Secondary) We first show that every isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, we prove that an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ also has constant principal curvatures. |
| title | Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ and $\mathbb{H}^{n}\times \mathbb{R}^{m}$ |
| topic | Differential Geometry 53C42 (Primary) 53B25, 53C40 (Secondary) |
| url | https://arxiv.org/abs/2511.07782 |