2026-09-18T18:05:38Zhttps://wakespace.lib.wfu.edu/server/oai/request

oai:null:10339/146992026-09-02T12:46:46Zcom_10339_14934col_10339_38132

Evans, Charles 2010-05-05T16:11:11Z 2010-06-18T18:57:32Z 2010-05-05T16:11:11Z 2010-06-18T18:57:32Z 2010-05-05T16:11:11Z https://wakespace.lib.wfu.edu/handle/10339/14699 We present an exposition of various results dealing with the total curvature of curves in Euclidean 3-space. There are two primary results: Fenchel's theorem and the theorem of Fary and Milnor. Fenchel's theorem states that the total curvature of a simple closed curve is greater than or equal to $2\pi$, with equality if and only if the curve is planar convex. The Fary-Milnor theorem states that the total curvature of a simple closed knotted curve is strictly greater than $4\pi$. Several methods of proof are supplied, utilizing both curve-theoretic and surface-theoretic techniques, surveying methods from both differential and integral geometry. Related results are considered: the connection between total curvature and bridge number; an analysis of total curvature plus total torsion; a lower bound on the length of the normal indicatrix. en_US Wake Forest University total curvature knots Curves, Knots, and Total Curvature Thesis Release the entire work immediately for access worldwide. Kuzmanovich, James Howards, Hugh Mathematics open.access