The total absolute curvature of closed curves with singularities

TL;DR

This paper extends Fenchel's theorem to non-co-orientable closed curves with singularities in Euclidean space, establishing lower bounds on total absolute curvature and characterizing cases of equality involving cusps.

Contribution

It generalizes Fenchel's theorem for singular closed curves in Euclidean space, providing conditions for minimal total absolute curvature and cusp configurations.

Findings

Total absolute curvature ≥ π for non-co-orientable closed frontals.

Equality holds for planar locally L-convex frontals with rotation index ±1/2.

Number of cusps is an odd integer ≥ 3, with 3 cusps characterizing simple curves.

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 $\pi$. It is equal to $\pi$ 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.