Pseudohermitian geometry on contact Riemannian manifolds

Abstract

Starting from work by S. Tanno, [39], and E. Barletta et al., [3], we study the geometry of (possibly non integrable) almost CR structures on contact Riemannian manifolds. We characterize CR-pluriharmonic functions in terms of differential operators naturally attached to the given contact Riemannian structure. We show that the almost CR structure of a contact Riemannian manifold (M,η) admitting global nonzero closed sections (with respect to which η is volume normalized) in the canonical bundle is integrable and η is a pseudo-Einstein contact form. The pseudohermitian holonomy of a Sasakian manifold M^{2n+1} is shown to be contained in SU(n) × 1 if and only if the Tanaka-Webster connection is Ricci flat. Also, for any quaternionic Sasakian manifold (M^{4m+1}, (F,T,θ,g)) either the Tanaka-Webster connection of (M^{4m+1},θ) is Ricci flat or m = 1 and then (M^5,θ) is pseudo-Einstein if and only if 4p + ρ∗ θ is closed, where p is a local 1-form on M^5 such that ∇G = p ⊗ H and ∇H = −p ⊗ G for some frame {F, G, H}, and ρ∗ is the pseudohermitian scalar curvature of (M^5,θ). On any Sasakian manifold M there is a smooth integrable Pfaffian system, invariant by Ψ(x) (the pseudohermitian holonomy group at x ∈ M) containing the contact flow as a subfoliation. We build canonical connections (reminiscent of the Tanaka connection, [38]) on complex vector bundles over contact Riemannian manifolds, carrying a pre-∂-operator and a Hermitian metric. As an application, we compute the first structure function of the underlying almost CR structure of a contact Riemannian manifold. The restricted conformal class [G_η] of the (generalized) Fefferman metric and certain canonical connections D (with trace Λ_gR^D = 0) are shown to be gauge invariants.

Keywords

• riemannian manifold
• functions pluriharmonic
• pseudohermitian holonomy
• sasakian manifold