Pseudo-Hermitian magnetic curves in normal almost contact metric 3-manifolds

Commun. Korean Math. Soc. 2020 Vol. 35, No. 4, 1269-1281 https://doi.org/10.4134/CKMS.c200122 Published online August 31, 2020 Printed October 31, 2020

Ji-Eun Lee Chonnam National University

Abstract : In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric $3$-manifold equipped with the canonical affine connection $\hat\nabla^t$ is a slant helix with pseudo-Hermitian curvature $\hat{\kappa}=|q|\sin\theta$ and pseudo-Hermitian torsion $\hat\tau=q\cos\theta$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric $3$-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve $\gamma$ in a quasi-Sasakian 3-manifold $M$ is a slant curve with curvature $\kappa=|(t-\alpha)\cos\theta\>+q|\sin\theta$ and torsion $\tau=\alpha+\{(t-\alpha)\cos\theta+q\}\cos\theta$. These curves are not helices, in general. Note that if the ambient space $M$ is an $\alpha$-Sasakian $3$-manifold, then $\gamma$ is a slant helix.

Keywords : Magnetic curve, slant curve, almost contact metric 3-manifold, CR-structure

MSC numbers : Primary 53B25, 53C25

Supported by : The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1l1A1A01043457).