Bi-invariant metric on contact diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group ${\mathcal{D}}_{\theta }$ of a contact Riemannian manifold $(M,g,\theta )$ and study its properties. We describe the Euler’s equation on a Lie algebra of group ${\mathcal{D}}_{\theta }$ and calculate the sectional curvature of ${\mathcal{D}}_{\theta }$. In a case $\dim M=3$ connection between the bi-invariant metric on ${\mathcal{D}}_{\theta }$ and the bi-invariant metric on volume-preserving diffeomorphisms group ${\mathcal{D}}_{\mu }$ of $M^3$ is discover.