**Edit:** According to the comment of Prof. Bryant, I revise my question. 


Is there a reasonable and non trivial (geometric) interpretation for the following quantity on a compact Riemannian manifold $(M,g)$ of dimension $3$ 


$$q=\sup\_{\alpha}\int\_M \alpha \wedge d\alpha $$


where $\sup$ is taken over all $1$-forms $\alpha$ of $g$-length $1$.



Is there a geometric interpretation for this quantity?
There is no reason to believe that there is a supremum of this functional. For example, consider the $3$-torus $M = \mathbb{R}^3/\mathbb{Z}^3$
with the quotient metric and the unit $1$-forms
$$
\alpha\_n = \cos(2\pi n z)\,\mathrm{d} x - \sin(2\pi n z)\,\mathrm{d} y,
$$
where $n$ is an integer, which are well-defined on $M$. 


One finds by calculation that
$$
\alpha\_n\wedge\mathrm{d}\alpha\_n = 2\pi n\, \mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z\,
$$
Hence 
$$
\int\_M \alpha\_n\wedge\mathrm{d}\alpha\_n = 2\pi n,
$$
so that the integral you describe can be made arbitrarily large (positive or negative) by appropriate choice of $\alpha$. 


I expect that some similar construction could be made for any compact oriented $3$-manifold, showing that the functional is always unbounded (in either direction).


