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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2405.13789 |
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| _version_ | 1866914807291052032 |
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| author | Espinosa-García, Manuel A. González, Ahtziri Villicaña-Molina, Yesenia |
| author_facet | Espinosa-García, Manuel A. González, Ahtziri Villicaña-Molina, Yesenia |
| contents | In this paper we study the space $\mathbb{L}(n)$ of $n$-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of $\mathbb{C}^n$, and describe its topology in terms of the manifold $\mathbb{M}(n)$ of $n$-gons degenerated to segments and with the first vertex at 0. We show that $\mathbb{M}(n)$ and $\mathbb{L}(n)$ contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of $\mathbb{L}(n)$ by the diagonal action of the affine complex group and the re-enumeration of the vertices is described. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The manifold of polygons degenerated to segments Espinosa-García, Manuel A. González, Ahtziri Villicaña-Molina, Yesenia Differential Geometry 53A07, 53C22 (Primary) 53C15, 53C40 (Secondary) In this paper we study the space $\mathbb{L}(n)$ of $n$-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of $\mathbb{C}^n$, and describe its topology in terms of the manifold $\mathbb{M}(n)$ of $n$-gons degenerated to segments and with the first vertex at 0. We show that $\mathbb{M}(n)$ and $\mathbb{L}(n)$ contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of $\mathbb{L}(n)$ by the diagonal action of the affine complex group and the re-enumeration of the vertices is described. |
| title | The manifold of polygons degenerated to segments |
| topic | Differential Geometry 53A07, 53C22 (Primary) 53C15, 53C40 (Secondary) |
| url | https://arxiv.org/abs/2405.13789 |