= Magnetic flow on a Riemannian surface
{title2=$F=X+(f\circ\pi)V$}
= Magnetic flow
{synonym}
A unit-speed curve with $D_t\dot\gamma=f(\gamma)i\dot\gamma$ defines the magnetic flow. The vertical term rotates the velocity while the horizontal part transports it along the base. Since $V(f\circ\pi)=0$, the divergence of $X+fV$ in Liouville volume vanishes. The associated two-form on the unit tangent bundle is $d\alpha-f\pi^*\Omega_a=-\iota_F\mu$, whose kernel is the magnetic flow direction.
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