On the Godbillon-Vey invariant of transversely parallelizable foliations

We consider a (2q+1)-dimensional smooth manifold M equipped with a (q+1)-dimensional, a priori non-integrable, distribution D and a q-vector field T=T₁∧…∧T_q, where {T_i} are linearly independent vector fields transverse to D. Using a q-form ω such that D=ker⁡ω and ω(T)=1, we construct a (2q+1)-form analogous to that defining the generalized Godbillon-Vey class of a foliation of codimension q, and show how does this form depend on ω and T. For a compatible Riemannian metric g on M, we express this (2q+1)-form in terms of T and extrinsic geometry of D and normal distribution D^⊥. We find Euler-Lagrange equations of associated functionals on a closed manifold: for a variable pair (ω,T) on (M,g) and for a variable metric on (M,D). We show that for a harmonic distribution D^⊥ such (ω,T) is critical, characterize critical pairs when D is integrable and find sufficient conditions for critical pairs when variations are among foliations. We also calculate the index form and consider examples of critical foliations among twisted products.