Bi-invariant metric on symplectic diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the symplectic diffeomorphisms group $\mathcal{D}_\omega$ of a symplectic Riemannian manifold $(M,g,\omega)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_\omega$ and calculate the sectional curvature of $\mathcal{D}_\omega (T^n)$.