Transverse structures and first integrals. Remarks (Q1319094)

A foliation \(\mathcal F\) of codimension one on a manifold \(V\) given by a 1- form \(\tau_ 0\) satisfying the Frobenius integrability requirement \(\tau_ 0 \wedge d\tau_ 0 = 0\) is said to be of the \(p\)th kind (at most) if there exists a sequence of 1-forms (the Godbillon-Vey cocycle) \(\tau_ 0,\dots,\tau_ p\) (\(\tau_ p = 0\)) such that \[ d\tau_ n = \sum a^{ij}_ n \tau_ i \wedge \tau_ j\quad (i < j,\quad i + j = n + 1) \] with appropriate \(a^{ij}_ n\). Assuming \(V\) a simply connected real or complex Stein manifold, \(SL(2,\mathbb{R})\)- or \(SL(2,\mathbb{C})\)- transverse structures are studied for 3rd kind foliations, and examples of 4th kind are mentioned.