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

Fernando Etayo

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 Kähler manifold.

Geometric properties of rotation minimizing vector fields along curves in Riemannian 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 Kähler manifold.

Paper Structure

This paper contains 4 sections, 9 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. RM vector fields along curves in $\mathbb{R}^{3}$
  3. RM vector fields along curves in $\mathbb{H}^{3}$
  4. RM vector fields along curves in Kähler manifolds

Key Result

Theorem 2

Et A normal vector field $N$ over a curve $\alpha$ immersed in $\mathbb{R}^{3}$ is an RM vector field in the sense of Bishop if and only if it is parallel with respect to the normal connection of $\alpha$.

Theorems & Definitions (18)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • Proposition 4: B
  • Proposition 5
  • Proposition 6
  • Definition 7
  • Theorem 8
  • Corollary 9
  • Proposition 10
  • ...and 8 more