The geometric Cauchy problem for rank-one submanifolds
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arXiv
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| Format: | Preprint |
| Published: |
2018
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| _version_ | 1866910593480392704 |
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| author | Raffaelli, Matteo |
| author_facet | Raffaelli, Matteo |
| contents | Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $γ$ in $\mathbb{R}^{m+n}$, we consider the following problem: to find an $m$-dimensional rank-one submanifold of $\mathbb{R}^{m+n}$, that is, an $(m-1)$-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $γ$ coincides with $\mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_08114 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The geometric Cauchy problem for rank-one submanifolds Raffaelli, Matteo Differential Geometry 53A07 (Primary) 53B20, 53B25 (Secondary) Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $γ$ in $\mathbb{R}^{m+n}$, we consider the following problem: to find an $m$-dimensional rank-one submanifold of $\mathbb{R}^{m+n}$, that is, an $(m-1)$-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $γ$ coincides with $\mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution. |
| title | The geometric Cauchy problem for rank-one submanifolds |
| topic | Differential Geometry 53A07 (Primary) 53B20, 53B25 (Secondary) |
| url | https://arxiv.org/abs/1811.08114 |