Hard Lefschetz theorem for Sasakian manifolds

We prove that on a compact Sasakian manifold (M, \eta, g) of dimension 2n+1, for any 0 \le p \le n the wedge product with \eta \wedge (d\eta)^p defines an isomorphism between the spaces of harmonic forms \Omega^{n-p}_\Delta (M) and \Omega^{n+p+1}_\Delta (M). Therefore it induces an isomorphism between the de Rham cohomology spaces H^{n-p}(M) and H^{n+p+1}(M). Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.