Characterization of 3D Sasakian manifold from magnetic Hopf surfaces

TL;DR

This paper characterizes three-dimensional Sasakian manifolds by analyzing magnetic Hopf surfaces, showing that constant mean curvature magnetic Hopf surfaces imply the manifold's Sasakian structure.

Contribution

It establishes a novel link between magnetic Hopf surfaces and Sasakian structures in 3D Riemannian manifolds, providing a new characterization criterion.

Findings

Magnetic Hopf surfaces with constant mean curvature indicate a Sasakian manifold.

The paper links magnetic curves and Sasakian geometry in three dimensions.

It offers a geometric condition for identifying Sasakian structures via magnetic surfaces.

Abstract

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.