Variations of the Godbillon–Vey Invariant of Foliated 3-Manifolds

Abstract

Let M be a smooth three-dimensional manifold equipped with a vector field T transverse to a plane field ${\mathcal {D}}$—the kernel of a one form $\omega$ such that $\omega (T)=1$. Recently, in Rovenski and Walczak (A Godbillon–Vey type invariant for a 3-dimensional manifold with a plane field, 2017. arXiv:1707.04847), we constructed a three-form analogous to that defining the Godbillon–Vey class of a foliation, showed how does this form depend on $\omega$ and T, and deduced Euler–Lagrange equations of the associated functional. In this paper, we continue our study when distributions/foliations and forms are defined outside a “singularity set” (a finite union of pairwise disjoint closed submanifolds of codimension $\ge 2$) under additional assumption of convergence of certain integrals. We characterize critical pairs $(\omega ,T)$ and foliations for different types of variations, find sufficient conditions for critical pairs when variations are among foliations, consider applications to transversely holomorphic flows, calculate the index form and present examples of critical foliations among Reeb foliations and twisted products.