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Graduate Student Geometry-Topology Seminar

Monday, April 17, 2017 - 3:15pm

Jingye Yang

University of Pennsylvania


University of Pennsylvania


The total curvature of a smooth simple closed curve in Euclidean 3-space  R3  is always  ≥  2 pi ,  with equality only for plane convex curves.


By contrast, the Fary-Milnor theorem states that if the curve is knotted, then its total curvature must be more than double this, thus > 4 pi .


We will outline a proof of this result, and then make a few remarks about its generalization to knotted surfaces in Euclidean 4-space R4 .