Let $M$ be an orientable closed smooth manifold of dimension n. Let $\Omega$ be a volume form for $M$, i.e., a nowhere-zero smooth n-form. A smooth $\Phi_t$ flow defined on $M$ is volume-preserving with respect to $\Omega$ if $\mathcal{L}_X\Omega=0,$ where $X$ is the vector field generating the flow. In this case, the flux form, which is defined by $\omega=i_X\Omega$, is a closed $(n-1)$-form (where $\mathcal{L}_X$ denotes the Lie derivative with respect to $X$, and $i_X$ denotes the interior product with $X$). In fact, by Cartan's formula we have $\mathcal{L}_X\Omega=di_X\Omega+i_Xd\Omega$. A cross section for a flow is a closed codimension-one smooth submanifold of $M$ that intersects every trajectory of the flow transversely. If the flux form of a non-singular volume-preserving flow is zero in cohomology, this flow can't admit a cross-section (for if $N$ is a cross-section, then $([\omega],[N])=\int _N\omega\neq 0$, then $[\omega]\neq 0$ in $H^{n-1}(M;\mathbb{R})$). The question is:
Does there exist a non-singular volume-preserving flow whose flux form is not exact and which has no cross-section?
