Tangent lines, inflections, and vertices of closed curves

TL;DR

This paper establishes new inequalities relating parallel tangent lines, inflections, and vertices of smooth closed curves in 3D space, extending classical curve theory results using curve shortening flow techniques.

Contribution

It introduces novel inequalities connecting geometric features of closed curves in 3D, employing curve shortening flow with surgery, and generalizes classical theorems like Arnold's tennis ball theorem.

Findings

Proves 2(P+I)+V > 5 for smooth closed curves in 3D.

Establishes 2(P'+I)+V > 3 for such curves.

Extends classical results in curve theory to higher dimensions.

Abstract

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem".