On the Ricci tensor and the generalized Tanaka--Webster connection of real hypersurfaces in a complex space form

Abstract

We prove that the Ricci tensor Ŝ with respect to the generalized Tanaka-Webster connection of a real hypersurface with the almost contact structure (η, φ, ξ, g) in a complex space form of complex dimension n ≥ 3 satisfies Ŝ(X,φY ) = λg(X,φY ) for any vector field X and Y , λ being a function, if and only if the real hypersurface is locally congruent to some type (A) hypersurface.