Riemannian geometry on the quantomorphism group

TL;DR

This paper explores the Riemannian geometry of the quantomorphism group of contact-preserving diffeomorphisms, deriving geodesic equations, proving global existence, and connecting to geophysical models like the quasigeostrophic equation.

Contribution

It introduces a Riemannian metric on the quantomorphism group, analyzes geodesic equations, and links these to geophysical fluid dynamics, extending results to related diffeomorphism groups.

Findings

Solutions to geodesic equations exist globally and depend smoothly on initial conditions.

The quasigeostrophic equation is interpreted as a geodesic equation on a central extension.

The geometric structures are smooth in Sobolev topology.

Abstract

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $\theta$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and show that solutions exist for all time and depend smoothly on initial conditions. In certain special cases (such as on the 3-sphere), the geodesic equation is a simplified version of the quasigeostrophic equation, so we obtain a new geodesic interpretation of this geophysical system. We also show that the genuine quasigeostrophic equation on $S^2$ can be obtained as an Euler-Arnold equation on a one-dimensional central extension of $T_{\id}\mathcal{D}_q(M)$, and that our global existence result extends to this case. If $E$ is the Reeb field of $\theta$ and $\mu$ is the volume form, assumed compatible in the sense that $\text{div} E=0$, we show that $\mathcal{D}_q(M)$ is a smooth submanifold of $\mathcal{D}_{E,\mu}(M)$, the space of diffeomorphisms preserving the vector field $E$ and the volume form $\mu$, in the sense of $H^s$ Sobolev completions. The latter manifold is related to symmetric motion of ideal fluids. We further prove that the corresponding geodesic equations and projections are $C^{\infty}$ objects in the Sobolev topology.