  In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped
with a homogeneous spin structure can be viewed as the simplest manifolds after
Riemannian symmetric spin spaces. In this paper, we give some characterizations
and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds
and we obtain the classification of simply-connected cyclic homogeneous
Riemannian manifolds of dimension less than or equal to four. We also present a
wide list of examples of non-compact irreducible Riemannian $3$-symmetric
spaces admitting cyclic metrics and give the expression of these metrics.
