TITLE: closed dual of vector fields
QUESTION [5 upvotes]: Given a closed manifold and a nowhere zero vector field $X$, can one always find a closed 1-form $\alpha$ such that $\alpha(X)=1$ as the constant function? Or under what kind of conditions on $X$ can this be true? Thanks!

REPLY [4 votes]: On the other hand, there are situations where the answer is positive. Actually Schwartzman and Fried gave a very satisfactory criterion for the existence of forms which are positive along the flow (see for example Fried's The geometry of cross sections to flows).
For $X$ a vector field on a compact manifold $M$, denote by $\phi^t$ the associated flow. For $p$ a point in $M$, let $k(p,t)$ be a loop obtained by following $X$ from $p$ to $\phi^t(p)$, and closed by an arbitrary (short) segment. Consider the classes $[k(p,t)]/t$ in $H_1(M, \mathbb R)$, denote by $C_\phi$ the set of such classes, and by $S_\phi$ the set of accumulation points of $C_\phi$. As long as the length of the closing segments is bounded, $S_\phi$ depends on $X$ only. Schwartzman-Fried's theorem states that $X$ admits a closed $1$-form $\alpha$ with $\alpha(X)>0$ everywhere if and only if $S_\phi$ lies in an open half-space in $H_1(M, \mathbb R)$.