Riemannian geometry of the contactomorphism group

Abstract

We define a right-invariant Riemannian metric on the group of contactomorphisms and study its Euler-Arnold equation. If the metric is associated to the contact form, the Euler-Arnold equation reduces to $m_t + u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$, the contact vector field $u$ defined by $f$, and the momentum $m = f - \Delta f$. Here the equation is considered on a compact manifold $M$ of dimension $2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize the analogy with the higher-order equation. We use the usual momentum conservation law for Euler-Arnold equations to rewrite the geodesic equation as a smooth first-order equation on the contactomorphism group of Sobolev class $H^s$, and thus obtain local existence in time of solutions which depend smoothly on initial data. In addition we prove a global existence criterion analogous to the Beale-Kato-Majda criterion in fluid mechanics, and show how this criterion is automatically satisfied on the totally geodesic subgroup of quantomorphisms. Finally we briefly discuss singular solutions and conservation laws of the Euler-Arnold equation.