Homotopy type of spaces of curves with constrained curvature on flat surfaces
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866912670826889216 |
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| author | Saldanha, Nicolau C. Zühlke, Pedro |
| author_facet | Saldanha, Nicolau C. Zühlke, Pedro |
| contents | Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1410_8590 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Homotopy type of spaces of curves with constrained curvature on flat surfaces Saldanha, Nicolau C. Zühlke, Pedro Geometric Topology Differential Geometry Primary: 58D10, Secondary: 53C42 Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed. |
| title | Homotopy type of spaces of curves with constrained curvature on flat surfaces |
| topic | Geometric Topology Differential Geometry Primary: 58D10, Secondary: 53C42 |
| url | https://arxiv.org/abs/1410.8590 |