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 S1→N\stackrel{π}{→}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 [RicD(Z, \bar{Z})+2g(Z, (I−a)\bar{Z})≥0, Z∈T1, 0(N), where a is the Weingarten operator of the immersion N→S2n+1] on the Ricci curvature RicD of D, we show that the first Kohn-Rossi cohomology group of M vanishes. We show that whenever RicD(Z, \bar{W})−2g(Z, \bar{W})=(μ{\circ}π)g(Z, \bar{W}) for some μ∈C∞(M), M is a pseudo-Einstein manifold.
