A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, $\mathit{\text{a}}\mathit{\text{priori}}$ non-integrable, distribution (plane field) $\mathcal{D}$ and a vector field $T$ transverse to $\mathcal{D}$. Using a 1-form $\omega$ such that $\mathcal{D}=\ker \omega$ and $\omega (T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $\omega$ and~$T$. For a compatible Riemannian metric on $M$, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of $\mathcal{D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $(\mathcal{D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,\mathcal{D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$iscritical,characterizecriticalpairswhen${\cal D}$ is integrable, and prove that these critical pairs are not extrema.