$h$-Principles for smooth curves and knots with prescribed curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916992963837952 |
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| author | Ghomi, Mohammad Raffaelli, Matteo |
| author_facet | Ghomi, Mohammad Raffaelli, Matteo |
| contents | We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{α\geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be $C^1$-approximated by $C^α$ curves of any larger curvature, prescribed as a function of arclength. It follows that there exist $C^\infty$ knots of prescribed curvature in every isotopy class of closed curves embedded in $R^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $h$-Principles for smooth curves and knots with prescribed curvature Ghomi, Mohammad Raffaelli, Matteo Differential Geometry Geometric Topology 53A04, 57K10 We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{α\geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be $C^1$-approximated by $C^α$ curves of any larger curvature, prescribed as a function of arclength. It follows that there exist $C^\infty$ knots of prescribed curvature in every isotopy class of closed curves embedded in $R^3$. |
| title | $h$-Principles for smooth curves and knots with prescribed curvature |
| topic | Differential Geometry Geometric Topology 53A04, 57K10 |
| url | https://arxiv.org/abs/2510.05275 |