Concurrent normals of immersed manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916122749566976 |
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| author | Panina, Gaiane Siersma, Dirk |
| author_facet | Panina, Gaiane Siersma, Dirk |
| contents | It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$.
Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$ vanishes. Then under mild conditions, almost every normal line to $M^m$ contains an intersection point of at least $β+4$ normals from different points of $M^m$, where $β$ is the sum of Betti numbers of $M^m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_07742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concurrent normals of immersed manifolds Panina, Gaiane Siersma, Dirk Geometric Topology Differential Geometry It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$. Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$ vanishes. Then under mild conditions, almost every normal line to $M^m$ contains an intersection point of at least $β+4$ normals from different points of $M^m$, where $β$ is the sum of Betti numbers of $M^m$. |
| title | Concurrent normals of immersed manifolds |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2301.07742 |