Lorentz-Finsler metrics on symplectic and contact transformation groups

TL;DR

This paper explores Lorentz-Finsler metrics on symplectic and contact transformation groups, linking geometric properties to various mathematical topics like systolic problems, quasi-morphisms, and rigidity-flexibility phenomena.

Contribution

It introduces the study of Lorentz-Finsler metrics in this context and investigates their geometric properties and interrelations with multiple mathematical areas.

Findings

Connections between Lorentz-Finsler geometry and contact systolic problem

Relations to group quasi-morphisms and Monge-Ampère equations

Insights into symplectic rigidity and flexibility

Abstract

In these notes we discuss Lorentz-Finsler metrics, a notion originated in relativity theory, on certain groups of symplectic and contact transformations. Some basic geometric questions arising in this context concerning distance, geodesics and their conjugate points, and existence of a time function, turn out to be related to a variety of subjects including the contact systolic problem, group quasi-morphisms, the Monge-Amp\`ere equation, and a subtle interplay between symplectic rigidity and flexibility. We discuss these interrelations, providing necessary preliminaries, and formulate a number of open questions.