Curves with constant curvature ratios

TL;DR

This paper characterizes curves in Euclidean space with constant ratios of consecutive curvatures, linking them to geodesics on flat tori, and explores their spherical counterparts and relation to intrinsic helices.

Contribution

It provides a geometric characterization of these special curves and compares their properties in Euclidean space and on spheres, extending understanding of their structure.

Findings

Curves with constant curvature ratios are geodesics on flat tori.

Spherical analogs of these curves are studied and compared with intrinsic helices.

The geometric properties of these curves are characterized in Euclidean and spherical contexts.

Abstract

Curves in ${\mathbb R}^n$ for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For $n= 3,4$, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere.