Tangent lines, inflection points, and vertices of closed space curves

Geometry Topology Working Seminar
Friday, October 14, 2011 - 2:00pm
2 hours
Skiles 006
Ga Tech
We show that every smooth closed curve C immersed in Euclidean 3-space  satisfies the sharp inequality 2(P+I)+V>5 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 C. The proof, which employs curve shortening flow, is based on a corresponding inequality  for the numbers of double points, singularites, and inflections of closed contractible curves in the real projective plane which intersect every closed geodesic. In the process we will also obtain some generalizations of classical theorems due to Mobius, Fenchel, and Segre (which includes Arnold's ``tennis ball theorem'').