Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916340835549184 |
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| author | Dimitric, Ivko |
| author_facet | Dimitric, Ivko |
| contents | We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic space forms $\mathbb HP^m$ and $\mathbb HH^m$ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius $r \in (0, π/2)$ except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a canonically embedded $\mathbb CP^m \subset \mathbb HP^m $. On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in $\mathbb HH^m$ is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane $\mathbb HH^{m-1}.$ Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere $H_3$ in $\mathbb HH^m $ is not of finite type but satisfies $Δ^2\widetilde x =$ const. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms Dimitric, Ivko Differential Geometry C40 We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic space forms $\mathbb HP^m$ and $\mathbb HH^m$ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius $r \in (0, π/2)$ except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a canonically embedded $\mathbb CP^m \subset \mathbb HP^m $. On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in $\mathbb HH^m$ is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane $\mathbb HH^{m-1}.$ Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere $H_3$ in $\mathbb HH^m $ is not of finite type but satisfies $Δ^2\widetilde x =$ const. |
| title | Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms |
| topic | Differential Geometry C40 |
| url | https://arxiv.org/abs/2407.21158 |