Cross sections to flows via intrinsically harmonic forms

We establish a new criterion for the existence of a global cross section to a non-singular volume-preserving flow on a compact manifold. Namely, if $\Phi$ is a non-singular smooth flow on a compact, connected manifold $M$ with a smooth invariant volume form $\Omega$, then $\Phi$ admits a global cross section if and only if the $(n-1)$-form $i_X \Omega$ is intrinsically harmonic, that is, harmonic with respect to some Riemannian metric on $M$.