On the topology and the geometry of SO(3)-manifolds

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.