On the existence of global cross sections to volume-preserving flows

We establish a new criterion for the existence of a global cross section to a non-singular volume-preserving flow $\mathrm{\Phi }$ on a closed smooth manifold $M$. Namely, if $X$ is the infinitesimal generator of the flow and $\mathrm{\Phi }$ preserves a smooth volume form $\mathrm{\Omega }$, then $\mathrm{\Phi }$ admits a global cross section if there exists a smooth Riemannian metric $g$ on $M$ with Riemannian volume $\mathrm{\Omega }$ and $g(X,X)=1$ such that $\parallel {\delta }_g(i_X\mathrm{\Omega }){\parallel }_g<1$, where ${\delta }_g$ denotes the codifferential relative to $g$; (equivalently, $\parallel d{X}^{\mathrm{♭}}{\parallel }_g<1$). In that case, there in fact exists another smooth Riemannian metric on $M$ with respect to which the canonical form $i_X\mathrm{\Omega }$ is co-closed and therefore harmonic.