November 2013 Tangent lines, inflections, and vertices of closed curves
Mohammad Ghomi
Duke Math. J. 162(14): 2691-2730 (November 2013). DOI: 10.1215/00127094-2381038


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.”


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Mohammad Ghomi. "Tangent lines, inflections, and vertices of closed curves." Duke Math. J. 162 (14) 2691 - 2730, November 2013.


Received: 16 January 2012; Revised: 16 February 2013; Published: November 2013
First available in Project Euclid: 6 November 2013

zbMATH: 1295.53002
MathSciNet: MR3127811
Digital Object Identifier: 10.1215/00127094-2381038

Primary: 53A04 , 53C44
Secondary: 57R45 , 58E10

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 14 • November 2013
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