Curves orthogonal to a vector field in Euclidean spaces

TL;DR

This paper characterizes rectifying curves in Euclidean spaces as geodesics on hypercones, extends previous results to higher dimensions, and explores their relationships with other special curves like slant helices and spherical curves.

Contribution

It generalizes the characterization of rectifying curves as geodesics on hypercones to any space dimension and links them to spherical curves in higher dimensions.

Findings

Rectifying curves are geodesics on hypercones.

Characterization of rectifying slant helices as geodesics of circular cones.

Establishment of a mapping between rectifying curves and spherical curves in higher dimensions.

Abstract

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an $(m + 2)$-dimensional space and spherical curves in an $(m + 1)$-dimensional space. A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector.