Bi-invariant metrics on contactomorphism groups

TL;DR

This paper reviews recent developments in bi-invariant metrics on contactomorphism groups, exploring their construction and relation to rigidity phenomena like orderability and non-squeezing in contact topology.

Contribution

It provides an overview of how bi-invariant metrics are constructed on contactomorphism groups and connects these metrics to recent rigidity results in contact topology.

Findings

Construction methods for bi-invariant metrics on contactomorphism groups

Relationship between these metrics and contact rigidity phenomena

Connection to orderability and non-squeezing theorems in contact topology

Abstract

Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the Hofer metric, has been studied since more than 20 years ago, it is only recently that some somehow analogous bi-invariant metrics have been discovered on the group of diffeomorphisms that preserve a contact structure. In this expository article I will review some constructions of bi-invariant metrics on the contactomorphism group, and how these metrics are related to some other global rigidity phenomena in contact topology which have been discovered in the last few years, in particular the notion of orderability (due to Eliashberg and Polterovich) and an analogue in contact topology (due to Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem.