Variations of 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 $\mathcal{D}$ and a $q$-vector field $\mathbf{T}=T_1\wedge \dots \wedge T_q$, where $\{T_i\}$ are linearly independent vector fields transverse to~$\mathcal{D}$. Using a $q$-form $\omega$ such that $\mathcal{D}=\ker \omega$ and $\omega (\mathbf{T})=1$, we construct a $(2q+1)$-form analogous to that defining the Godbillon–Vey class of a $(q+1)$-dimensional foliation, and show how does this form depend on $\omega$ and $\mathbf{T}$. For a compatible Riemannian metric $g$ on $M$, we express this $(2q+1)$-form in terms of $\mathbf{T}$ and extrinsic geometry of~$\mathcal{D}$ and normal distribution ${\mathcal{D}}^{\mathrm{\perp }}$. We find Euler-Lagrange equations of associated functionals: for variable $(\omega ,\mathbf{T})$ on $(M,g)$, and for variable metric on $(M,\mathcal{D})$, when distributions/foliations and forms are defined outside a “singularity set” under additional assumption of convergence of certain integrals. We show that for a harmonic distribution ${\mathcal{D}}^{\mathrm{\perp }}$ such $(\omega ,\mathbf{T})$ is critical, characterize critical pairs when $\mathcal{D}$ is integrable and find sufficient conditions for critical pairs when variations are among foliations, calculate the index form and consider examples of critical foliations among twisted products, Reeb foliations and transversely holomorphic flows.