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Rotation Minimizing vector fields and frames in Riemannian manifolds

Fernando Etayo

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.

Rotation Minimizing vector fields and frames in Riemannian manifolds

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.

Paper Structure

This paper contains 5 sections, 3 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. RM vector fields and frames of a curve in $\mathbb{R}^{3}$
  3. Curves immersed in a Riemmanian manifold
  4. RM vector fields over a curve immersed in a Riemmanian manifold
  5. RM frames and transformations

Key Result

Theorem 6

A normal vector field $v$ over a curve $\gamma$ immersed in $\mathbb{R}^{3}$ is a RM vector field iff ir is parallel respect to the normal connection of $\gamma$.

Theorems & Definitions (9)

  • Definition 1
  • Remark 2
  • Example 3
  • Definition 4
  • Example 5
  • Theorem 6
  • Corollary 7
  • Definition 8
  • Theorem 9