Hard Lefschetz theorem for Sasakian manifolds

We prove that on a compact Sasakian manifold (M, eta, g) of dimension 2n + 1, for any 0 <= p <= n the wedge product with eta Lambda (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 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.