Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

TL;DR

This paper investigates the geometric properties of rotation minimizing vector fields along curves in various Riemannian manifolds, including Euclidean, Hyperbolic, and Kähler spaces, expanding understanding beyond classical Euclidean cases.

Contribution

It extends the study of rotation minimizing vector fields to new ambient spaces, analyzing their properties in Euclidean, Hyperbolic, and Kähler manifolds.

Findings

Characterization of RM vector fields in Euclidean space

Analysis of RM fields in Hyperbolic space

Properties of RM vector fields in Kähler manifolds

Abstract

Rotation minimizing vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed into a Riemannian manifold are studied when the ambient manifold is the Euclidean 3-space, the Hyperbolic 3-space and a K\"{a}hler manifold.