Tangent lines, inflections, and vertices of closed curves

Abstract

We show that every smooth closed curve Γ immersed in Euclidean space R 3 satisfies the sharp inequality 2 ( P + I ) + V ≥ 6 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 Γ . We also show that 2 ( P + + I ) + V ≥ 4 , 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 RP 2 and the sphere 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.”