On the space of cone geodesics and positive paths of contactomorphisms

TL;DR

This paper explores the geometric structure of cone geodesics within convex cone structures, establishing a link between positive contactomorphism paths and globally hyperbolic cone geometries, generalizing Lorentzian spacetime concepts.

Contribution

It introduces a framework for viewing cone geodesics as contact manifolds and connects positive contactomorphism paths to globally hyperbolic cone structures, extending Lorentzian geometry ideas.

Findings

Space of cone geodesics can be equipped with a contact structure.

Established correspondence between positive contactomorphism paths and hyperbolic cone structures.

Generalized notions from Lorentzian spacetime to convex cone geometries.

Abstract

Often it is possible to equip the space of all cone geodesics of a strongly convex cone structure with the structure of a smooth contact manifold. This generalizes the analogous notions for the space of light rays of a Lorentzian spacetime. After reviewing these constructions on the space of cone geodesics, with a focus on the natural contact structure, we establish a correspondence between positive paths of contactomorphisms in spherical cotangent bundles and certain globally hyperbolic cone structures.