A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929260486197248 |
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| author | Pinamonti, Andrea Verzellesi, Simone |
| author_facet | Pinamonti, Andrea Verzellesi, Simone |
| contents | In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^n$, with $n\geq 2$, we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_13148 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups Pinamonti, Andrea Verzellesi, Simone Differential Geometry 53C17, 49Q05 In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^n$, with $n\geq 2$, we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces. |
| title | A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups |
| topic | Differential Geometry 53C17, 49Q05 |
| url | https://arxiv.org/abs/2305.13148 |