Tangent lines, inflections, and vertices of closed curves

Abstract

We show that every smooth closed curve $\Gamma$ immersed in Euclidean space ${\mathbf{R}}^3$ satisfies the sharp inequality $2(\mathcal{P}+\mathcal{I})+\mathcal{V}\ge 6$ which relates the numbers $\mathcal{P}$ of pairs of parallel tangent lines, $\mathcal{I}$ of inflections (or points of vanishing curvature), and $\mathcal{V}$ of vertices (or points of vanishing torsion) of $\Gamma$. We also show that $2({\mathcal{P}}^{+}+\mathcal{I})+\mathcal{V}\ge 4$, where ${\mathcal{P}}^{+}$ 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 ${\mathbf{RP}}^2$ and the sphere ${\mathbf{S}}^2$ which intersect every closed geodesic. These findings extend some classical results in curve theory from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”