Some Additional Remarks

Abstract

For a curve in an n-dimensional Euclidean space E n which obeys certain conditions of regularity, to each point of the curve there is put in correspondence (n − 1) numbers of k1, k2, ... , kn − 1 of the curvatures at the point of the curve. Let us set $$ {\kappa _i}\left( s \right) = \int_0^s {{k_i}\left( \sigma \right)d\sigma } $$ where integration is carried out with respect to the arc length. The functions obtained in this way will be called integral curvatures of the curve. There arises a problem — to give such a definition of the ith integral curvature which could be applied to an essentially wider class of curves than the class considered in differential geometry. Accordingly, the contents of Chapter V can be considered as a theory of the curves of a limited integral first curvature. Chapter IV presents a theory of curves of a limited integral first curvature in a space S n .