Total curvatures of a closed curve in Euclidean 𝑛-space

Abstract

A classical result by J. W. Milnor states that the total curvature of a closed curve C C in the Euclidean n n -space is the limit of the total curvatures of polygons inscribed in C C . In the present paper a similar geometric interpretation is given for all total curvatures ∫ C | κ r | d s \int _{C}|\kappa _{r}|\mathrm {d}s , r = 1 , … , n − 1 r=1,\ldots ,n-1 .