A new metric on the contactomorphism group of orderable contact manifolds

TL;DR

This paper introduces a new pseudo-metric on the contactomorphism group of contact manifolds, linking orderability with metric properties and answering key questions about the topology of these groups.

Contribution

It defines a contact analogue of Hofer's metric, establishes its non-degeneracy criterion, and relates it to the interval topology, providing new insights into contactomorphism groups.

Findings

The pseudo-metric is non-degenerate iff the manifold is strongly orderable.

The metric topology coincides with the interval topology.

The interval topology is Hausdorff when non-trivial.

Abstract

We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,\xi)$ with a cooriented contact structure $\xi$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if $(M,\xi)$ is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers.