Duke Mathematical Journal

Tangent lines, inflections, and vertices of closed curves

Mohammad Ghomi

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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.”

Article information

Source
Duke Math. J., Volume 162, Number 14 (2013), 2691-2730.

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1383760702

Digital Object Identifier
doi:10.1215/00127094-2381038

Mathematical Reviews number (MathSciNet)
MR3127811

Zentralblatt MATH identifier
1295.53002

Subjects
Primary: 53A04: Curves in Euclidean space 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 57R45: Singularities of differentiable mappings 58E10: Applications to the theory of geodesics (problems in one independent variable)

Citation

Ghomi, Mohammad. Tangent lines, inflections, and vertices of closed curves. Duke Math. J. 162 (2013), no. 14, 2691--2730. doi:10.1215/00127094-2381038. https://projecteuclid.org/euclid.dmj/1383760702


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