Godbillon-Vey type functional for almost contact manifolds

TL;DR

This paper introduces a Godbillon-Vey type functional for almost contact manifolds, derives its Euler-Lagrange equations, and constructs critical examples with specific geometric structures, expanding variational methods in contact geometry.

Contribution

It extends the Godbillon-Vey functional to almost contact manifolds and finds critical points with specific geometric structures, a novel approach in contact geometry.

Findings

Derived Euler-Lagrange equations for the functional.

Constructed critical almost contact manifolds with double-twisted product structure.

Identified solutions within the $C_{5}igoplus C_{12}$ class.

Abstract

Many contact metric manifolds are critical points of curvature functionals restricted to spaces of associated metrics. The Godbillon-Vey functional has never been considered in a variational context in contact geometry. Recently we extended this functional from foliations to arbitrary plane fields on a 3-dimensional manifold, so, the following question arises: can one use the Godbillon-Vey functional to find optimal almost contact manifolds? In the paper, we introduce a Godbillon-Vey type functional for a 3-dimensional almost contact manifold and find its Euler-Lagrange equations for all variations preserving the Reeb vector field. We construct critical (for our functional) 3-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class $C_{5}\oplus C_{12}$ according to Chinea-Gonzalez classification.