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

2010/07/17 by Eugene Gutkin, Eugène Gutkin
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53A04 #msc:53C30 #msc:53C44 #msc:53Z05

paper · pdf · doi:10.1016/j.geomphys.2011.06.013

published as Journal of geometry and physics 61 (2011), 2147 -- 2161 · 27 pages

arxiv created 2010/07/17 · openalex publication_date 2011/07/05 · arxiv updated 2012/12/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01

Abstract

The well known formulas express the curvature and the torsion of a curve in R³ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in Rⁿ. It follows that a curve in Rⁿ is determined up to an isometry by the norms of its n derivatives. We extend these observations to curves in arbitrary riemannian manifolds.

Citations

• There is More than One Way to Frame a Curve
• Capillary Floating and the Billiard Ball Problem
• There is More than One Way to Frame a Curve