Bi-invariant metric on contact diffeomorphisms group

Abstract: 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.