The total absolute curvature of closed curves with singularities

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.

Figures (9)

• Figure 1: A co-orientable closed front having arbitrary small total absolute curvature (Example \ref{['ex:megata']}).
• Figure 2: Closed non-co-orientable fronts with total absolute curvature $K(\gamma)=\pi$ in $\mathbb{R}^2$. All curves (a), (b) and (c) are given by hypocycloids (Example \ref{['ex:hypo']}). Although the leftmost (a) is a simple closed curve, the others, (b) and (c), are not simple. All the singular points of these curves are cusp.
• Figure 3: A frontal having the same endpoints but with no singular point in the interior.
• Figure 4: A frontal having the same endpoints but with two singular points in the interior.
• Figure 5: Frontals having the same endpoints but with one singular point in the interior, the case (I).

Theorems & Definitions (29)

• Theorem A
• Theorem B
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.4
• proof
• Definition 2.5
• Lemma 2.6
• proof