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 \(M\).