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

Abstract

We consider a 3-dimensional smooth manifold M equipped with an arbitrary, a priori nonintegrable, distribution (plane field) D and a vector field T transverse to D. Using a 1-form ω such that D = ker ω and ω(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 ω 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 D. We deduce Euler-Lagrange equations of associated functionals: for variable (D, T ) on M , and for variable Riemannian or Randers metric on (M, D). We show that for a geodesic field T (e.g., for a contact structure) such (D, T ) is critical, characterize critical pairs when D is integrable, and prove that these critical pairs are not extrema.

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 .…”