Engel CR submanifolds of $\mathbb{C}^3$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909791591333888 |
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| author | Fernández, Eduardo del Pino, Álvaro Zhou, Wei |
| author_facet | Fernández, Eduardo del Pino, Álvaro Zhou, Wei |
| contents | We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Engel CR submanifolds of $\mathbb{C}^3$ Fernández, Eduardo del Pino, Álvaro Zhou, Wei Differential Geometry Geometric Topology Symplectic Geometry We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg. |
| title | Engel CR submanifolds of $\mathbb{C}^3$ |
| topic | Differential Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2509.11488 |