Tangent lines, inflections, and vertices of closed curves

Abstract. We show that every smooth closed curve Γ immersed in Euclidean space R3 satisfies the sharp inequality 2(P + I) + V ≥ 6 which relates the num-bers 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, singular-ities, and inflections of closed curves in the real projective plane RP2 and the sphere S2 which intersect every closed geodesic. These findings extend some clas-sical results in curve theory including works of Möbius, Fenchel, and Segre, whic