The total absolute curvature of closed curves with singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911787227545600 |
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| author | Honda, Atsufumi Tanaka, Chisa Yamauchi, Yuta |
| author_facet | Honda, Atsufumi Tanaka, Chisa Yamauchi, Yuta |
| contents | In this paper, we give a generalization of Fenchel's theorem for closed curves as frontals in Euclidean space $\mathbb{R}^n$. We prove that, for a non-co-orientable closed frontal in $\mathbb{R}^n$, its total absolute curvature is greater than or equal to $π$. It is equal to $π$ if and only if the curve is a planar locally $L$-convex closed frontal whose rotation index is $1/2$ or $-1/2$. Furthermore, if the equality holds and if every singular point is a cusp, then the number $N$ of cusps is an odd integer greater than or equal to $3$, and $N=3$ holds if and only if the curve is simple. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00487 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The total absolute curvature of closed curves with singularities Honda, Atsufumi Tanaka, Chisa Yamauchi, Yuta Differential Geometry Primary 53A04, Secondary 57R45, 53C65, 53C42 In this paper, we give a generalization of Fenchel's theorem for closed curves as frontals in Euclidean space $\mathbb{R}^n$. We prove that, for a non-co-orientable closed frontal in $\mathbb{R}^n$, its total absolute curvature is greater than or equal to $π$. It is equal to $π$ if and only if the curve is a planar locally $L$-convex closed frontal whose rotation index is $1/2$ or $-1/2$. Furthermore, if the equality holds and if every singular point is a cusp, then the number $N$ of cusps is an odd integer greater than or equal to $3$, and $N=3$ holds if and only if the curve is simple. |
| title | The total absolute curvature of closed curves with singularities |
| topic | Differential Geometry Primary 53A04, Secondary 57R45, 53C65, 53C42 |
| url | https://arxiv.org/abs/2403.00487 |