A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

TL;DR

This paper introduces a new invariant for 3D manifolds with a plane field, generalizing the Godbillon-Vey class, and explores its geometric properties and critical configurations under various metric conditions.

Contribution

It constructs a Godbillon-Vey type invariant for non-integrable plane fields on 3-manifolds and analyzes its dependence on geometric structures and variational principles.

Findings

Defined a 3-form analogous to the Godbillon-Vey class for plane fields.

Derived Euler-Lagrange equations for associated functionals.

Characterized critical pairs, including contact structures, and showed they are not extrema.

Abstract

We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $\omega$ such that ${\cal D} = \ker\,\omega$ and $\omega(T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $\omega$ 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 ${\cal D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable, and prove that these critical pairs are not extrema.