Gaiane Panina ; Dirk Siersma - Concurrent normals of immersed manifolds

cm:10840 - Communications in Mathematics, May 9, 2023, Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin) - https://doi.org/10.46298/cm.10840
Concurrent normals of immersed manifoldsArticle

Authors: Gaiane Panina ; Dirk Siersma

    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 $\beta +4$ normals from different points of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.


    Volume: Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin)
    Published on: May 9, 2023
    Accepted on: April 16, 2023
    Submitted on: January 21, 2023
    Keywords: Mathematics - Geometric Topology,Mathematics - Differential Geometry

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