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 RicD 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.