2024/08/20 by Yuxin Dong, Yibin Ren, Dong, Yuxin +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2408.10591
openalex publication_date 2024/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Lθ. This induces a natural Riemannian metric gh,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,Lθ) 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,Lθ). We define the pseudo-Kähler 2-form by gh,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,Lθ) 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.