On the Curvature Groups of a Cr Manifold

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups H(C(M),Γ) of the Fefferman space C(M) of a strictly pseudoconvex real hypersurface M ⊂ C. 1. Statement of results Let M be a strictly pseudoconvex CR manifold of CR dimension n and θ a contact form on M such that the Levi form Lθ is positive definite. Let S → C(M) → M be the canonical circle bundle and Fθ the Fefferman metric on C(M), cf. [6]. Let GL(2n + 2,R) → L(C(M)) → C(M) be the principal bundle of linear frames tangent to C(M) and Γ : u ∈ L(C(M)) 7→ Γu ⊂ Tu(L(C(M))) the LeviCivita connection of Fθ. Let H (C(M),Γ) be the curvature groups of (C(M),Γ), cf. [3] and our Section 2. Our main result is Theorem 1. If (M, θ) is a pseudo-Einstein manifold of constant pseudohermitian scalar curvature ρ then the curvature groups H(C(M),Γ) are isomorphic to the de Rham cohomology groups of C(M). Otherwise (that is if either θ is not pseudo-Einstein or ρ is nonconstant) H(C(M),Γ) = 0, 1 ≤ k ≤ 2n+ 2. The key ingredient in the proof of Theorem 1 is the explicit calculation of the infinitesimal conformal transformations of the Lorentz manifold (C(M), Fθ). Corollary 1. Let Ω ⊂ C be a smoothly bounded strictly pseudoconvex domain. There is a defining function φ of Ω such that θ = i 2 (∂−∂)φ is a pseudo-Einstein contact form on ∂Ω. If (∂Ω, θ) has constant pseudohermitian scalar curvature then H(C(∂Ω),Γ) ≈ H(∂Ω,R)⊕H(∂Ω,R), for any 1 ≤ k ≤ 2n+ 2.