Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds

The well-known formulas express the curvature and the torsion of a curve in R3R3 in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in arbitrary Riemannian manifolds. Our motivation comes from physics. It follows that regular curves in RnRn are determined up to isometry by the norms of their nn consecutive derivatives. We extend this fact to two-point homogeneous spaces.

► We study regular curves in Riemannian manifolds.
► We express their curvatures via Euclidean invariants of their derivatives.
► Our results extend standard formulas for Euclidean curves in three dimensions.