Abstract
We show that every smooth closed curve immersed in Euclidean space satisfies the sharp inequality which relates the numbers of pairs of parallel tangent lines, of inflections (or points of vanishing curvature), and of vertices (or points of vanishing torsion) of . We also show that , where 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 from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”
Citation
Mohammad Ghomi. "Tangent lines, inflections, and vertices of closed curves." Duke Math. J. 162 (14) 2691 - 2730, November 2013. https://doi.org/10.1215/00127094-2381038
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