Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910253021396992 |
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| author | Tan, Huixin Xie, Yuquan Yan, Wenjiao |
| author_facet | Tan, Huixin Xie, Yuquan Yan, Wenjiao |
| contents | We prove that the angle function associated with the canonical product structure is constant for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{S}^{m}$, $\mathbb{S}^{n}\times \mathbb{H}^{m}$, or $\mathbb{H}^{n}\times \mathbb{H}^{m}$. This rigidity result enables us to provide a complete classification of isoparametric and homogeneous hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$. Furthermore, we establish a geometric characterization in these two spaces: a hypersurface is isoparametric if and only if it has constant principal curvatures and a constant angle function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00447 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$ Tan, Huixin Xie, Yuquan Yan, Wenjiao Differential Geometry 53C42 (Primary) 53B25, 53C40 (Secondary) We prove that the angle function associated with the canonical product structure is constant for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{S}^{m}$, $\mathbb{S}^{n}\times \mathbb{H}^{m}$, or $\mathbb{H}^{n}\times \mathbb{H}^{m}$. This rigidity result enables us to provide a complete classification of isoparametric and homogeneous hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$. Furthermore, we establish a geometric characterization in these two spaces: a hypersurface is isoparametric if and only if it has constant principal curvatures and a constant angle function. |
| title | Isoparametric hypersurfaces in $\mathbb{S}^{n}\times \mathbb{S}^{m}$ and $\mathbb{S}^{n}\times \mathbb{H}^{m}$ |
| topic | Differential Geometry 53C42 (Primary) 53B25, 53C40 (Secondary) |
| url | https://arxiv.org/abs/2603.00447 |