Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say that a smooth $1$-form $\theta$ on $M$ is *adapted* to this flow if

- $\theta(X)$ is everywhere positive; and
- The Lie derivative ${\mathcal L}_X \theta$ is an exact $1$-form.

(By the way, I'd be happy to take suggestions for a better name than "adapted". Most adjectives such as "calibrated", "polarised", etc. are unfortunately already taken.)

Question.Is it true that every smooth flow with no fixed points has at least one $1$-form adapted to it?

At first I was sure that there must be counterexamples (perhaps many such), but every construction of a smooth flow I tried to make ended up having at least one adapted $1$-form. Some examples:

- If the flow is isometric (that is, it preserves some Riemannian metric $g$), one can take $\theta$ to be the $1$-form dual to $X$ with respect to the metric $g$.
- If the flow is an Anosov flow, one can take $\theta$ to be the canonical $1$-form.
- If $M$ is the cosphere bundle of some compact Riemannian manifold $N$ and $(M,X)$ is the geodesic flow, then one can again take $\theta$ to be the canonical $1$-form. (This example can be extended to a number of other Hamiltonian flows, such as flows that describe a particle in a potential well, which was in fact the original context in which this question arose for me.)
- If the flow is a suspension, one can take $\theta$ to be $dt$, where $t$ is the time variable (in local coordinates).
- If there is a morphism $\phi: M \to M'$ from the flow $(M,X)$ to another flow $(M',X')$ (thus $\phi$ maps trajectories to trajectories), and the latter flow has an adapted $1$-form $\theta'$, then the pullback $\phi^* \theta'$ of that form will be adapted to $(M,X)$.

Some simple remarks:

- If $\theta$ is adapted to a flow $(M,X)$, then so is $(e^{tX})^* \theta$ for any time $t$, where $e^{tX}: M \to M$ denotes the time evolution map along $X$ by $t$. In many cases this allows one to average along the flow and restrict attention to cases where $\theta$ is $X$-invariant. In the case when the flow is ergodic, this would imply in particular that we could restrict attention to the case when $\theta(X)$ is constant. Conversely, in the ergodic case one can almost (but not quite) use the ergodic theorem to relax the requirement that $\theta(X)$ be positive to the requirement that $\theta(X)$ have positive mean with respect to the invariant measure.
- The condition that ${\mathcal L}_X \theta$ be exact implies that $d\theta$ is $X$-invariant, and is in turn implied by $\theta$ being closed. For many vector fields $X$ it is already easy to find a closed $1$-form $\theta$ with $\theta(X) > 0$, but this is not always possible in general, in particular if $X$ is the divergence of a $2$-vector field with respect to some volume form, in which case the integral of $\theta(X)$ along this form must vanish when $\theta$ is closed. However, in all the cases in which this occurs, I was able to locate a non-closed example of $\theta$ that was adapted to the flow. (But perhaps if the flow is sufficiently "chaotic" then one can rule out non-closed examples also?)