On 2 - Framed Riemannian Manifolds with Godbillon - Vey Structure Form

In the last decade, contact, almost contact, paracontact cosymplectic, and conformal cosymplectic manifolds carrying κ > 1 structure vector fields ξ have been studied by many authors, e.g. [2], [7], [11], [15]. In the present paper we consider a (2m + 2)-dimensional Riemannian manifold carrying two structure vector fields ξ (r ∈ {2m+1, 2m+2}), a (1, 1)-tensor field Φ, and a structure 2 form Ω of rank 2m, such that for η := (ξr) [ Φ = −Id + η ⊗ ξr Φ ξr = 0, η (ξs) = δ s Ω(Z, Z′) = g(Φ Z, Z′), Ω ∧ η ∧ η 6= 0 (0.1) holds. Here the (2m)-dimensional subspace ImΦ of the tangent space is supposed to be Kählerian (see eq. (2.12) below). If the 3-forms γ = η ∧ dη (0.2) satisfy dγ = 0 , (0.3) they are called Godbillon-Vey forms [6]. On the other hand, if ∇X ξr = fr X r = 2m + 1, 2m + 2 (0.4) holds for all X orthogonal to ξr and for some fr ∈ ΛM , the structure vector fields define a concircular pairing [1]. It will turn out that (0.3) follows from (0.1) and (0.4). Therefore we call such manifolds M(Φ, Ω, η, ξr) 2-framed GodbillonVey manifold (abbreviated 2FG-V). We shall prove that they have the following properties: Any 2FG-V manifold is equipped with a conformal symplectic structure CSp(m + 1, IR) with ξ := ∑ fr ξr as vector of Lee, i.e. dΩ = 2ξ ∧ Ω (0.5) and M is the local Riemannian product M = M⊥ ×M>