  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.
