On a Complete Riemannian Metric on the Space of Embedded Curves
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912766035492864 |
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| author | Döhrer, Elias Reiter, Philipp Schumacher, Henrik |
| author_facet | Döhrer, Elias Reiter, Philipp Schumacher, Henrik |
| contents | We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this metric is complete in the following senses: (i) bounded sets are relatively compact with respect to the weak $H^s$ topology; (ii) every Cauchy sequence with respect to the induced geodesic distance converges; (iii) solutions of the geodesic initial-value problem exist for all times; and (iv) there are length-minimizing geodesics between every pair of curves in the same path component (i.e., in the same knot class). As a by-product, we show $C^\infty$-smoothness of the tangent-point energies in the Hilbert case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Complete Riemannian Metric on the Space of Embedded Curves Döhrer, Elias Reiter, Philipp Schumacher, Henrik Differential Geometry 58B20, 58D10, 58E10 We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this metric is complete in the following senses: (i) bounded sets are relatively compact with respect to the weak $H^s$ topology; (ii) every Cauchy sequence with respect to the induced geodesic distance converges; (iii) solutions of the geodesic initial-value problem exist for all times; and (iv) there are length-minimizing geodesics between every pair of curves in the same path component (i.e., in the same knot class). As a by-product, we show $C^\infty$-smoothness of the tangent-point energies in the Hilbert case. |
| title | On a Complete Riemannian Metric on the Space of Embedded Curves |
| topic | Differential Geometry 58B20, 58D10, 58E10 |
| url | https://arxiv.org/abs/2501.16647 |