Topology of closed asymptotic curves on negatively curved surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912580949245952 |
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| author | Ghomi, Mohammad Raffaelli, Matteo |
| author_facet | Ghomi, Mohammad Raffaelli, Matteo |
| contents | Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $Γ$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using Călugăreanu's theorem, we obtain a formula for the linking number $Lk(Γ,n)$ of $Γ$ with the normal $n$ of $M$. It follows that when $Lk(Γ, n)=0$, $Γ$ cannot have any locally star-shaped planar projections with vanishing crossing number, which extends observations of Kovaleva, Panov and Arnold. These results hold also for curves with nonvanishing torsion and their binormal vector field. Furthermore we construct an example where $n$ is injective but $Lk(Γ, n)\neq 0$, and discuss various restrictions on $Γ$ when $n$ is injective. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology of closed asymptotic curves on negatively curved surfaces Ghomi, Mohammad Raffaelli, Matteo Differential Geometry Analysis of PDEs Geometric Topology Primary 53A04, 53A05, Secondary 57K10, 35L10 Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $Γ$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using Călugăreanu's theorem, we obtain a formula for the linking number $Lk(Γ,n)$ of $Γ$ with the normal $n$ of $M$. It follows that when $Lk(Γ, n)=0$, $Γ$ cannot have any locally star-shaped planar projections with vanishing crossing number, which extends observations of Kovaleva, Panov and Arnold. These results hold also for curves with nonvanishing torsion and their binormal vector field. Furthermore we construct an example where $n$ is injective but $Lk(Γ, n)\neq 0$, and discuss various restrictions on $Γ$ when $n$ is injective. |
| title | Topology of closed asymptotic curves on negatively curved surfaces |
| topic | Differential Geometry Analysis of PDEs Geometric Topology Primary 53A04, 53A05, Secondary 57K10, 35L10 |
| url | https://arxiv.org/abs/2412.19266 |