On Umbilical Real Hypersufaces of Products of Complex Space Forms
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866911586475573248 |
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| author | Domingos, Iury da Silva, Ranilze de Sousa, Alexandre Vitório, Feliciano |
| author_facet | Domingos, Iury da Silva, Ranilze de Sousa, Alexandre Vitório, Feliciano |
| contents | Tashiro and Tachibana proved that there exist no totally umbilical hypersurfaces in complex space forms with nonzero constant holomorphic sectional curvature, and it is also known that the shape operator of such hypersurfaces cannot be parallel. Motivated by these results, we study real hypersurfaces in products of complex space forms. We establish rigidity and nonexistence results for totally umbilical real hypersurfaces in this setting. In particular, we show that if a real hypersurface in a product of complex space forms does not admit a local product structure, then its shape operator cannot be parallel. Moreover, we provide a classification of totally umbilical real hypersurfaces, showing that those admitting a local almost product structure are necessarily totally geodesic or extrinsic hyperspheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24352 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Umbilical Real Hypersufaces of Products of Complex Space Forms Domingos, Iury da Silva, Ranilze de Sousa, Alexandre Vitório, Feliciano Differential Geometry 53C42, 53C40 Tashiro and Tachibana proved that there exist no totally umbilical hypersurfaces in complex space forms with nonzero constant holomorphic sectional curvature, and it is also known that the shape operator of such hypersurfaces cannot be parallel. Motivated by these results, we study real hypersurfaces in products of complex space forms. We establish rigidity and nonexistence results for totally umbilical real hypersurfaces in this setting. In particular, we show that if a real hypersurface in a product of complex space forms does not admit a local product structure, then its shape operator cannot be parallel. Moreover, we provide a classification of totally umbilical real hypersurfaces, showing that those admitting a local almost product structure are necessarily totally geodesic or extrinsic hyperspheres. |
| title | On Umbilical Real Hypersufaces of Products of Complex Space Forms |
| topic | Differential Geometry 53C42, 53C40 |
| url | https://arxiv.org/abs/2603.24352 |