Pseudohermitian geometry on contact Riemannian manifolds (original) (raw)

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 276 DAVID E. BLAIR – SORIN DRAGOMIR [2] 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 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, (F, T, θ, g)) either the Tanaka-Webster connection of (M, θ) is Ricci flat or m = 1 and then (M, θ) is pseudo-Einstein if and only if 4p + ρ∗ θ is closed, where p is a local 1-form on M such that ∇G = ...