  The real projective plane is a compact, non-orientable orbifold of Euler
characteristic 1 without boundaries, which can be described as a twisted Klein
bottle. We shortly review the motivations for choosing such a geometry among
all possible two-dimensional orbifolds, while the main part of the study will
be devoted to dark matter study and limits in Universal Extra Dimensional (UED)
models based on this peculiar geometry. In the following we consider such a UED
construction based on the direct product of the real projective plane with the
standard four-dimensional Minkowski space-time and discuss its relevance as a
model of a weakly interacting Dark Matter candidate. One important difference
with other typical UED models is the origin of the symmetry leading to the
stability of the dark matter particle. This symmetry in our case is a remnant
of the six-dimensional Minkowski space-time symmetry partially broken by the
compactification. Another important difference is the very small mass splitting
between the particles of a given Kaluza-Klein tier, which gives a very
important role to co-annihilation effects. Finally the role of higher
Kaluza-Klein tiers is also important and is discussed together with a detailed
numerical description of the influence of the resonances.
