Total curvatures of a closed curve in Euclidean n-space

摘要

A classical result by J. W. Milnor states that the total curvature of a closed curve C in the Euclidean n-space is the limit of the total curvatures of polygons inscribed in C. In the present paper a similar geometric interpretation is given for all total curvatures integral(C)k(r)ds, r = 1,..., n - 1.