Hard Lefschetz Theorem for Sasakian manifolds

We prove that on a compact Sasakian manifold (M,n, g) of dimension 2n + 1, for any 0 ≤ p ≤ n the wedge product with n (dn)p defines an isomorphism between the spaces of harmonic forms Ωn-p Δ (M) and Ωn+p+1 Δ (M). Therefore it induces an isomorphism between the de Rham cohomology spaces Hn-p(M) and Hn+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.