Abstract: Complex
contact manifolds generalise real notion, which originates
from classical mechanics. However in the complex case the
known examples are very rare, and according to
LeBrun-Salamon conjecture, there is no more projective
examples. Projective contact manifolds are studied in
relation with a classification of Riemannian manifolds with
respect to their holonomy groups. We pose the question if
there are any other examples of non-projective contact
manifolds. I will present a simple result obtained
with Thomas Peternell, which proves the classification in
dimension 3 and with first Betti number 1. Further I will
explain how the non-projective classification or examples
(in any dimension) can potentially affect the projective
classification.