The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

Details The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

2010-07-17

arxiv.org/abs/1007.2960v1

Mathematics

Differential Geometry

27 pages

Scientific paper

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.

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