On the total absolute curvature of closed curves in spheres
The total absolute curvature (tac) of a closed curve in a Euclidean space satisfies Fenchel's resp. Fary-Milnor's inequality. This article treats tac of closed curves in spheres. We derive similar inequalities depending on the length of the curve resp. the area of surfaces of disk-type bounded by the curve. The proof starts from the computation of tac as mean value of the number of critical points of certain level functions [see the author's article, ibid. 31, 119-147 (1980; Zbl 0429.53038)] uses some topological considerations and Poincaré's integral geometric formula for the computation of length resp. area.
scientific article; zbMATH DE number 3947068
scientific article