On Cauchy-Riemann circle bundles

Abstract

Building on ideas of R. Mizner, [17] - [18], and C. Laurent-Thiébaut, [14], we study the CR geometry of real orientable hypersurfaces of a Sasakian manifold. These are shown to be CR manifolds of CR codimension two and to possess a canonical connection D (parallelizing the maximally complex distribution) similar to the Tanaka-Webster connection (cf. [21]) in pseudohermitian geometry. Examples arise as circle subbundles $S^1 \to N \stackrel{\pi}{\rightarrow} M$, of the Hopf fibration, over a real hypersurface M in the complex projective space. Exploiting the relationship between the second fundamental forms of the immersions N → S²ⁿ⁺¹ and M → CPⁿ and a horizontal lifting technique we prove a CR extension theorem for CR functions on N. Under suitable assumptions [$\mathrm{Ric}_D(Z,\overline{Z})+2g(Z,(I-a)\overline{Z})\geq 0$, $Z \in T_{1,0}(N)$, where a is the Weingarten operator of the immersion N → S²ⁿ⁺¹] on the Ricci curvature Ric_D of D, we show that the first Kohn-Rossi cohomology group of M vanishes. We show that whenever $\mathrm{Ric}_D(Z,\overline{W})-2g(Z,\overline{W})=(\mu \circ \pi)g(Z,\overline{W})$ for some $\mu \in C^\infty (M)$, M is a pseudo-Einstein manifold.