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

Commun. Korean Math. Soc. Published online August 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 $\hat{\kappa}=\mid q\mid\sin\theta$
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
$\kappa=\mid (t-\alpha)\cos\theta\>+q\mid\sin\theta$ 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 a $\alpha$-Sasakian $3$-manifold
then $\gamma$ is a slant helix.

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