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arXiv · 2403.00487

The total absolute curvature of closed curves with singularities

Atsufumi Honda·Chisa Tanaka·Yuta Yamauchi
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Abstract

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.

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Atsufumi Honda, Chisa Tanaka, Yuta Yamauchi. 2024-03-01. The total absolute curvature of closed curves with singularities. https://arxiv.org/abs/2403.00487

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