A note on the topology of irreducible SO(3)-manifolds

A smooth, connected, oriented 5-manifold M is an irreducibleSO(3)-manifold if there exists a rank 3 vector bundle η over M such that the tangent bundle is isomorphic to the bundle of symmetric trace-free endomorphisms of η. In this article we give necessary and sufficient conditions for the existence of irreducible SO(3)-manifolds. We also provide some new examples of irreducible SO(3)-manifolds.