  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 $M$.
