A gauged Fefferman construction for partially integrable CR geometry

A non-degenerate almost CR structure T10⊂TM⊗CT10⊂TM⊗C of hypersurface type on a smooth manifold MnMn of odd dimension n=2m+1≥3n=2m+1≥3 is called partially integrable iff [Γ(T10),Γ(T10)]⊂Γ(T10⊕T10¯). For any choice of pseudo-Hermitian form θθ there exists a canonical linear connection ∇W∇W on a partially integrable CR manifold (M,T10)(M,T10). This connection is a natural generalisation of the Tanaka–Webster connection of pseudo-Hermitian geometry. We use the generalised connection for the construction of a Fefferman metric on the total space of the canonical circle bundle of any (strictly pseudoconvex) partially integrable CR manifold. The construction is a CR invariant. In fact, we invent here a slightly more general version of this construction by the use of a gauge form ℓℓ. This is called the gauged Fefferman construction of partially integrable CR geometry. The scalar curvature of the gauged Fefferman metric and the Laplacian of the given pseudo-Hermitian structure θθ are calculated in explicit form.