Rotation Minimizing vector fields and frames in Riemannian manifolds

TL;DR

This paper characterizes rotation minimizing vector fields along curves in Riemannian manifolds, extending classical Euclidean results to more general geometric settings.

Contribution

It establishes the equivalence between RM vector fields and parallel normal vector fields, generalizing RM frames to Riemannian manifolds.

Findings

RM vector fields are equivalent to parallel normal vector fields.

The results extend classical Euclidean RM frames to Riemannian manifolds.

Provides a foundation for further geometric analysis in curved spaces.

Abstract

We prove that a normal vector field along a curve in R3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.