On differential geometry of non-degenerate CR manifolds

Abstract

In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form θ, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{θ}. This induces a natural Riemannian metric g_{h,θ} on M compatible with the structure. The synthetic object (M,θ,J,h) will be called a pseudo-Hermitian manifold, which generalizes the usual notion of pseudo-Hermitian manifold (M,θ,J,L_{θ}) in the literature. Our purpose is to investigate the differential-geometric aspect of pseudo-Hermitian manifolds. By imitating Hermitian geometry, we find a canonical connection on (M,θ,J,h), which generalizes the Tanaka-Webster connection on (M,θ,J,L_{θ}). We define the pseudo-Kähler 2-form by g_{h,θ} and J; and introduce the notion of a pseudo-Kähler manifold, which is an analogue of a Kähler manifold. It turns out that (M,θ,J,L_{θ}) is pseudo-Kählerian. Using the structure equations of the canonical connection, we derive some curvature and torsion properties of a pseudo-Hermitian manifold, in particular of a pseudo-Kähler manifold. Then some known results in Riemannian geometry are generalized to the pseudo-Hermitian case. These results include some Cartan type results. As an application, we give a new proof for the classification of Sasakian space forms.