Given an almost contact structure, we select a fixed almost
complex structure on it and some additional geometric data. We show how
to construct sections of suitable sequences of complex bundles that are
"approximately C-R" in a suitable sense. We use these sections to
provide a number of geometric constructions in the manifold: existence
of submanifolds with special properties, existence of open book
decompositions, embeddings in the projective space preserving the
geometric structure, etc. As a corollary we show the existence of smooth
open book decompositions in a large class of 5-manifolds.
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