On the curvatures of a curve in \(n\)-dimensional Euclidean space (Q2066428)

Consider a smooth curve in \(n\)-dimensional Euclidean space \(\mathbb E^n\), parametrized by \(r(t)\). Assuming the system of vectors \(r',r'',\dots, r^{(n)}\) is linearly independent at each point, the Frenet equations lead to the curvatures \(k_1,\dots, k_{n-1}\). The first result of the present paper is the calculation of the curvatures \(k_p, p=1,\dots, n-1\) in case of a curve in \(\mathbb E^n\), given implicitly by equations\\ \(F_1(x_1, x_2,\dots, x_n)=0, F_2(x_1, x_2,\dots, x_n)=0,\dots, F_{n-1}(x_1, x_2,\dots, x_n)=0\).\\ The given formula shows, that the curvatures \(k_p\) are expressed in terms of partial derivatives of \(F_i\) only. Even in simple cases the calculations are complicated. This is illustrated by the example of the well-known Viviani curve (the intersection of a sphere and a cylinder in a suitable position).\\ The second goal is the generalization of Beltrami's theorem: Let \(V\) be a developable surface formed by the tangents of a smooth space curve \(l\) in \(\mathbb E^3\), \(\pi_0\) the osculating plane of this curve at a point \(M_0\), and let \(l'\) be the intersection curve of \(\pi_0\) and \(V\). Then \(k'=\frac{3}{4} k\), where \(k\) and \(k'\) are the curvatures of \(l\) and \(l'\) respectively at their common point \(M_0\). Generalizing the above theorem, let \(l\) be a smooth curve in \(\mathbb E^n\), \(M_0\) a point on \(l\), and \(\pi_0\) the osculating \(m\)-plane at \(M_0\). Denote by \(l_m\) the curve cut in the \(m\)-plane \(\pi_0\) by \((n-m)\)-dimensional osculating planes of \(l\). Then the curvatures \(k_p\) and \(k_p^m\) of \(l\) and \(l_m\) at \(M_0\) are related by \(k_p^m=\frac{n}{n-p}\frac{m-p}{m}k_p\).