Duke Mathematical Journal
- Duke Math. J.
- Volume 162, Number 14 (2013), 2691-2730.
Tangent lines, inflections, and vertices of closed curves
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.”
Duke Math. J., Volume 162, Number 14 (2013), 2691-2730.
Received: 16 January 2012
Revised: 16 February 2013
First available in Project Euclid: 6 November 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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)
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