The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

TL;DR

This paper generalizes formulas for curvature and torsion from 3D to n-dimensional Euclidean spaces and extends these concepts to curves in arbitrary Riemannian manifolds, showing that derivatives' norms determine the curve up to isometry.

Contribution

It provides new expressions for all curvatures of curves in R^n and extends the invariance results to Riemannian manifolds, broadening the understanding of curve invariants.

Findings

Curvatures in R^n can be expressed via derivatives' invariants.

A curve in R^n is determined by the norms of its derivatives up to isometry.

Extension of curvature invariants to curves in Riemannian manifolds.

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.