On Cauchy-Riemann circle bundles (Q2386865)

Recall that a Sasakian manifold \(M\) is endowed with: i) a contact distribution \(\mathcal D\); ii) a smooth family \(J\) of complex structures on \(J_x: {\mathcal D}_x \to {\mathcal D}_x\), \(x \in M\), which makes \(({\mathcal D}, J)\) a CR structure; iii) a 1-form \(\eta\) so that \(\text{ker}\;\eta_x = {\mathcal D}_x\) for any \(x\in M\); iv) a vector field \(\xi\) so that \(\eta(\xi) \equiv 1\); v) a Riemannian metric \(g\) so that \(g(\xi, \cdot) = \eta\) and \(g| _{{\mathcal D} \times {\mathcal D}}\) is the Levi form of \(({\mathcal D}, J)\). The vector field \(\xi\) is called contact vector field. The author considers real orientable hypersurfaces in a Sasakian manifold \(M\), which are tangent to the contact vector field \(\xi\) at all points. Such submanifolds are naturally endowed with a CR structure of codimension two and, for the manifolds of this kind, the author constructs a canonical connection, which is an analogue of the Tanaka-Webster connection of Levi non-degenerate CR manifolds of codimension one with distinguished contact form. He also considers the class of submanifolds \(M \subset S^{2 n+ 1}\), which are tangent to the vertical distribution of the Hopf fibration \(\pi: S^{2 n+ 1} \to {\mathbb C}P^n\) and which project onto a CR submanifold \(M' = \pi(M) \subset {\mathbb C}P^n\). He calls such manifolds Cauchy-Riemann circle bundles. The circle bundles of codimension one are examples of the hypersurfaces of Sasakian manifolds considered above. For any Cauchy-Riemann circle bundle \(M\subset S^{2n +1}\) that projects onto a generic, real analytic, compact CR submanifold \(M' \subset {\mathbb C}P^n\), it is shown that any CR function, which is constant along the fibers of \(\pi\), extends to CR function of an open subset of \(S^{2n +1}\). Also, for the Cauchy-Riemann circle bundles of codimension one in \(S^{2n+1}\), some sufficient conditions for the vanishing of the first Khon-Rossi cohomology group in terms of the curvature of their canonical connection is determined.