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 [Ric
D(
Z, \bar{
Z})+2
g(
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 Ric
D of
D, we show that the first Kohn-Rossi cohomology group of
M vanishes. We show that whenever Ric
D(
Z, \bar{
W})−2
g(
Z, \bar{
W})=(μ{\circ}π)
g(
Z, \bar{
W}) for some μ∈
C∞(
M),
M is a pseudo-Einstein manifold.
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