Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds

Ji-Eun Lee

doi:10.3390/sym11060784

Abstract

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of κ and τ − 1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that γ is a slant curve if and only if M is Sasakian for a contact magnetic curve γ in contact Lorentzian three-manifold M. As an example, we find contact magnetic curves in Lorentzian Heisenberg three-space.

Highlights

We find contact magnetic curves in Lorentzian Heisenberg three-space
As a generalization of Legendre curve, we defined the notion of slant curves in [1,2]
For a contact Riemannian manifold, we proved that a slant curve in a Sasakian three-manifold is that its ratio of κ and τ − 1 is constant

Summary

Introduction

As a generalization of Legendre curve, we defined the notion of slant curves in [1,2]. We define the magnetic curve γ with contact magnetic field Fξ,q of the length q in three-dimensional Sasakian Lorentzian manifold M3. We call it the contact magnetic curve or trajectories of Fξ,q. Using the Lorentzian cross product, we prove that the ratio of κ and τ − 1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold.

Preliminaries

Slant Curves in Contact Lorentzian Three-Manifolds

Lorentzian Cross Product

Frenet Slant Curves

Null Slant Curves

Contact Magnetic Curves