On the space of Riemannian metrics on symplectic and contact manifolds

Let $\mathscr A\mathscr M_\omega$ be the space of all almost Kahlerian smooth metrics on a symplectic manifold $M^2n,\omega$ such that the fundamental form of each metric coincides with $\omega$. It is well known that $\mathscr A\mathscr M_\omega$ is a retractor of the space $\mathscr M$ of all smooth metrics on $M$. We show that $\mathscr M$ is a smooth trivial bundle over $\mathscr A\mathscr M_\omega$. A similar fact holds also in the case of a contact manifold.