Critical points of the distance function to a generic submanifold
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929353106915328 |
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| author | Arnal, Charles Cohen-Steiner, David Divol, Vincent |
| author_facet | Arnal, Charles Cohen-Steiner, David Divol, Vincent |
| contents | In general, the critical points of the distance function $d_{\mathsf{M}}$ to a compact submanifold $\mathsf{M} \subset \mathbb{R}^D$ can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and $μ$-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to small $C^2$ perturbations. More specifically, for any compact abstract manifold $M$, the set of embeddings $i:M\rightarrow \mathbb{R}^D$ such that the submanifold $i(M)$ satisfies those conditions is open and dense in the Whitney $C^2$-topology. When those regularity conditions are fulfilled, we prove that the distance function to $i(M)$ satisfies Morse-like conditions and that the critical points of the distance function to an $\varepsilon$-dense subset of the submanifold (e.g., obtained via some sampling process) are well-behaved. We also provide many examples that showcase how the absence of these conditions allows for pathological situations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13147 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Critical points of the distance function to a generic submanifold Arnal, Charles Cohen-Steiner, David Divol, Vincent Differential Geometry General Topology In general, the critical points of the distance function $d_{\mathsf{M}}$ to a compact submanifold $\mathsf{M} \subset \mathbb{R}^D$ can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and $μ$-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to small $C^2$ perturbations. More specifically, for any compact abstract manifold $M$, the set of embeddings $i:M\rightarrow \mathbb{R}^D$ such that the submanifold $i(M)$ satisfies those conditions is open and dense in the Whitney $C^2$-topology. When those regularity conditions are fulfilled, we prove that the distance function to $i(M)$ satisfies Morse-like conditions and that the critical points of the distance function to an $\varepsilon$-dense subset of the submanifold (e.g., obtained via some sampling process) are well-behaved. We also provide many examples that showcase how the absence of these conditions allows for pathological situations. |
| title | Critical points of the distance function to a generic submanifold |
| topic | Differential Geometry General Topology |
| url | https://arxiv.org/abs/2312.13147 |