Characterization of 3D Sasakian manifold from magnetic Hopf surfaces

In a three-dimensional Riemannian manifold M that admits a unit Killing vector field $\xi$, we regard $\xi$ as a magnetic vector field. A magnetic Hopf surface is a surface obtained by Lie dragging the magnetic curve with $\xi$. Then we characterize Sasakian structure on M from magnetic Hopf surfaces. That is, we show that if an arbitrary magnetic Hopf surface is a constant mean curvature surface then M is a Sasakian manifold.