A Whitney Extension Problem for Manifolds
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910289531764736 |
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| author | O'Neill, Kevin |
| author_facet | O'Neill, Kevin |
| contents | The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set $E\subset\mathbb{R}^n$, how can we tell if there exists a $d$-dimensional, $C^m$-smooth manifold $\mathcal{M}\supset E$? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when $d=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13115 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Whitney Extension Problem for Manifolds O'Neill, Kevin Functional Analysis The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set $E\subset\mathbb{R}^n$, how can we tell if there exists a $d$-dimensional, $C^m$-smooth manifold $\mathcal{M}\supset E$? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when $d=1$. |
| title | A Whitney Extension Problem for Manifolds |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2310.13115 |