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, a priori non-integrable, distribution (plane field) D and a vector field T transverse to D. Using a 1-form ω such that D=ker⁡ω and ω(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 ω 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 D. We deduce Euler-Lagrange equations of associated functionals: for variable (D,T) on M, and for variable Riemannian metric on (M,D). We show that for a geodesic field T (e.g., for a contact structure) such (D,T) is critical, characterize critical pairs when D is integrable (with examples among twisted products) and prove that these critical pairs are not extrema.