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Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

Luiz C. B. da Silva, Gilson S. Ferreira, José D. da Silva

Abstract

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are consequences of the specificities of the ambient manifold and what could be generic. In this work, we propose a theoretical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices; we prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux vector field of the directrix. We characterize rectifying surfaces of constant angle; in other words, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, then the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction.

Curves and surfaces making a constant angle with a parallel transported direction in Riemannian spaces

Abstract

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are consequences of the specificities of the ambient manifold and what could be generic. In this work, we propose a theoretical framework to unify parts of these findings. We study curves and surfaces by prescribing the angle they make with a parallel transported vector field. We show that the characterization of Euclidean helices in terms of their curvature and torsion is also valid in any Riemannian manifold. Among other properties, we prove that surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces. We also investigate the relation between their geodesics and the so-called slant helices; we prove that surfaces of constant angle are the rectifying surface of a slant helix, i.e., the ruled surface with rulings given by the Darboux vector field of the directrix. We characterize rectifying surfaces of constant angle; in other words, when their geodesics are slant helices. As a corollary, we show that if every geodesic of a surface of constant angle is a slant helix, then the ambient manifold is flat. Finally, we characterize surfaces in the product of a Riemannian surface with the real line making a constant angle with the vertical real direction.
Paper Structure (11 sections, 23 theorems, 78 equations)

This paper contains 11 sections, 23 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Curves of constant angle
  3. Generalized parallel helices
  4. Parallel slant helices
  5. Helices and geodesics of cylinders
  6. Surfaces of constant parallel angle
  7. Slant helices and geodesics of surfaces of constant parallel angle
  8. Rectifying surfaces
  9. Rectifying surfaces and constant parallel angle
  10. Surfaces of constant parallel angle in product manifolds
  11. Concluding remarks

Key Result

Theorem A

A curve $\gamma: I\rightarrow M^{3}$ with curvature $\kappa>0$ makes a constant angle with a parallel transported field $V$ if, and only if, the ratio $\tau / \kappa$ of torsion to curvature is constant. In addition, $\tau / \kappa=\cot(\theta)$ and $V=\cos(\theta) T+\sin(\theta) B$, where $\{T,N,B\

Theorems & Definitions (51)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Definition 1
  • Theorem 1
  • proof
  • Example 1: Helices in Product Manifolds
  • ...and 41 more