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 ${\mathcal{D}}_{\omega }(T^n)$.