Monday, September 18, 2017 - 4:00pm
University of Pennsylvania
The curvature k of a smooth curve in 3-space is by definition non-negative, and its integral with respect to arc length is called the total curvature of the curve. According to Fenchel's Theorem, the total curvature of any simple closed curve in Euclidean 3-space is at least 2π , with equality if and only if it is a plane convex curve. According to the Fary–Milnor Theorem, if the simple closed curve is knotted, then its total curvature is strictly greater than 4π . In this talk, I will say a few words about Fenchel's Theorem, indicate one proof of the Fary–Milnor Theorem, and then discuss generalizations about knots in 3-space, knotted surfaces in 4-space, and knotted n-manifolds in N-space. At the end, I will state what I regard as the fundamental direction for further study.