  Consider the nonstandard embedding of SO(3) into SO(5) given by the
5-dimensional irreducible representation of SO(3), henceforth called SO(3)_\ir.
In this note, we study the topology and the differential geometry of
5-dimensional Riemannian manifolds carrying such an SO(3)_\ir structure, i.\,e.
with a reduction of the frame bundle to SO(3)_\ir.
