Lorentzian distance functions in contact geometry

TL;DR

This paper introduces a Lorentzian distance function on contactomorphism groups, linking contact geometry with Lorentzian metrics and orderability, and explores its properties and implications for Legendrian isotopy classes.

Contribution

It defines a Lorentzian distance on contactomorphisms, proves its continuity and finiteness conditions, and relates it to orderability and Legendrian isotopy classes.

Findings

Distance is continuous w.r.t. the Hofer norm.

Distance is finite iff the contactomorphism group is orderable.

Non-degeneracy of the Chekanov-type metric for orderable Legendrian classes.

Abstract

We define a Lorentzian distance function on the group of contactomorphisms of a closed contact manifold. This distance function is continuous with respect to the Hofer norm on the group of contactomorphisms defined by Shelukhin and finite if and only if the group of contactomorphisms is orderable. To prove this we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined by Rosen and Zhang is non-degenerate. In this case similar results hold for a Lorentzian distance function on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.