  We prove some Liouville type results for generalized holomorphic maps in
three classes: maps from pseudo-Hermitian manifolds to almost Hermitian
manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds
and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds,
assuming that the domains are compact. For instance, we show that any $(J,J^N)$
holomorphic map from a compact pseudo-Hermitian manifold $M$ with nonnegative
(resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian
manifold $N$ with negative (resp. nonpositive) holomorphic sectional curvature
is constant. We also construct explicit almost CR structures on a complex
vector bundle over an almost CR manifold.
