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

Abstract

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 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, 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) 3-dimensional almost contact manifolds having a double-twisted product structure, these solutions belong to the class C 5 ⊕ C 12 according to Chinea-Gonzalez classification.

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

“…A straightforward calculation gives In the special case where (C 16) simplifies to If , i.e. then and the space is then foliated (Reinhart & Wood 1973; Rovenski & Walczak 2019 a , b ).…”

“…If , i.e. then and the space is then foliated (Reinhart & Wood 1973; Rovenski & Walczak 2019 a , b ).…”

“…The Godbillon–Vey one-form is given by (e.g. Rovenski & Walczak 2019 a , b ). Taking the exterior derivative of (E 15) gives The Godbillon–Vey three-form is given by where the superscript in (E 17) refers to Reinhart & Wood (1973), and is the volume element for the three-form (E 17).…”

“…Taking the exterior derivative of (E 15) gives The Godbillon–Vey three-form is given by where the superscript in (E 17) refers to Reinhart & Wood (1973), and is the volume element for the three-form (E 17). The Godbillon–Vey three-form (E 17) is the formula given by Reinhart & Wood (1973) and Rovenski & Walczak (2019 a , b ) (note in Rovenski & Walczak (2019 a , b )).The Godbillon–Vey three-form (E 17) is equivalent to the Godbillon–Vey three-form used in the present paper in the sense that the helicity density where is the Godbillon–Vey helicity density used in the present paper, modulo a pure divergence term, i.e. (see below).…”

“…The differences of these 2 forms are described below. Following Reinhart & Wood (1973) and Rovenski & Walczak (2019 a , b ) we first identify a one form that is dual to the normal to the foliation, such that The analogue of the Serret–Frenet equation for in (E 2) using the dual one-form is given by Cartan’s magic formula because .There is some freedom in the choice of and in (E 19). For example if we choose Here .…”