  We study a class of six-dimensional models based on positive curvature
surfaces (spherical 2-orbifolds) as extra-spaces. Using the Newman-Penrose
formalism, we discuss the particle spectrum in this class of models. The
fermion spectrum problem, which has been addressed with flux compactifications
in the past, can be avoided using localised fermions. In this framework, we
find that there are four types of geometry compatible with the existence of a
stable dark matter candidate and we study the simplest case in detail. Using
the complementarity between collider resonance searches and relic density
constraints, we show that this class of models is under tension, unless the
model lies in a funnel region characterised by a resonant Higgs s-channel in
the dark matter annihilation.
