Godbillon–Vey type functional for 3-dimensional almost contact manifolds

Many contact metric manifolds are critical points of curvature functionals restricted to spaces of associated metrics. The Godbillon–Vey functional was never considered in a variational context in Contact Geometry. Recently we extended this functional from foliations to arbitrary plane fields on a three-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 three-dimensional almost contact manifold, present it in Reinhart–Wood form and find its Euler–Lagrange equations for all variations preserving the Reeb vector field. We construct critical (for our functional) three-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class \(C_{5}\oplus C_{12}\) $C_5\oplus C_{12}$ according to Chinea-Gonzalez classification.