On the principal angles between the osculating \(K\)-planes of a curve (Q1586363)

Two \(k\)-dimensional subspaces \(V\) and \(W\) of an \(n\)-dimensional Euclidean vector space \(E^n\) determine \(k\) so-called principal angles; their squared cosines are the eigenvalues of that selfadjoint linear mapping \(V\to V\) which is the product of the orthogonal projection \(V\to W\) and the orthogonal projection \(W\to V\). The squared cosine of the angle between \(V\) and \(W\) is defined as the determinant of the above mentioned endomorphism of \(V\). By translation, these concepts carry over to affine subspaces in a natural way. Let \(\gamma(I)\) be a curve in \(E^n\) (parameterized by arc length) of general type, i.e., the first \(n-1\) deriviatives are linearly independent in \(I\). By \textit{H. Gluck} [Am. Math. Mon. 74, 1049-1056 (1967; Zbl 0171.19801)], the angle between the osculating \(k\)-subspaces at points \(\gamma(0)\) and \(\gamma(s)\) equals -- up to terms of higher order -- the product of the \(k\)-th curvature \(\kappa(0)\) and the absolute value of the arc-length \(s\) between the two points. In the present paper the author establishes a similar asymptotic formula for the principal angles of the osculating \(k\)-subspaces at points \(\gamma(0)\) and \(\gamma(s)\).