Communications of the
Korean Mathematical Society CKMS

In this paper, we prove that a diffeomorphism $f$ on a normal almost contact $3$-manifold $M$ is a CRL-{\it transformation} if and only if $M$ is an $\alpha$-Sasakian manifold. Moreover, we show that a $CR$-loxodrome in an $\alpha$-Sasakian $3$-manifold is a pseudo-Hermitian magnetic curve with a strength $q=\widetilde{r}\eta(\gamma')=(r+\alpha-t)\eta(\gamma')$ for constant $\eta(\gamma')$. A non-geodesic $CR$-loxodrome is a non-Legendre slant helix. Next, we prove that let $M$ be an $\alpha$-Sasakian $3$-manifold such that $(\nabla_Y S)X=0$ for vector fields $Y$ to be orthogonal to $\xi$, then the Ricci tensor $\rho$ satisfies $\rho=2\alpha^2 g$. Moreover, using the CRL-{\it transformation} $\widetilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\widetilde{R}$, the pseudo-Ricci tensor $\widetilde{\rho}$ and the torsion tensor field $ \widetilde{ \mathfrak{T}}^{t} (\widetilde{S}X,Y)$.

Keywords: Loxodrome, magnetic curves, normal almost contact manifold, pseudo-Hermitian geometry

MSC numbers: Primary 58E20, 53B25

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).