The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

Abstract

The well known formulas express the curvature and the torsion of a curve in $R^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in $R^n$. It follows that a curve in $R^n$ is determined up to an isometry by the norms of its n derivatives. We extend these observations to curves in arbitrary riemannian manifolds.