Contactomorphisms with $L^2$ metric on stream functions

TL;DR

This paper explores the geometric structure of the contactomorphism group on compact contact manifolds with an $L^2$ metric, revealing non-negative curvature and properties of the exponential map, with implications for related groups.

Contribution

It demonstrates that the contactomorphism group has non-negative sectional curvature, the exponential map is not locally $C^1$, and the quantomorphism group forms a totally geodesic submanifold.

Findings

Sectional curvature is always non-negative.

The Riemannian exponential map is not locally $C^1$.

The quantomorphism group is a totally geodesic submanifold.

Abstract

Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_\theta(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as a generalization to the $\mathcal{D}(S^1)$, the diffeomorphism group of the circle, we show that its sectional curvature is always non-negative and that the the Riemannian exponential map is not locally $C^1$. Lastly, we show that the quantomorphism group is a totally geodesic submanifold of $\mathcal{D}_\theta(M)$ and talk about its Riemannian submersion onto the symplectomorphism group of the Boothby-Wang quotient of $M$.