Non-negative Legendrian isotopy in $ST^*M$

TL;DR

The paper proves that in certain manifolds, different fibers of the spherical cotangent bundle cannot be connected by non-negative Legendrian isotopies, with applications to causality in spacetimes and partial order structures.

Contribution

It establishes a new non-existence result for non-negative Legendrian isotopies between fibers in manifolds with open universal covers, extending understanding in contact topology and spacetime causality.

Findings

Different fibers cannot be connected by non-negative Legendrian isotopy in manifolds with open universal cover

Application to causality in globally hyperbolic spacetimes

Strengthens partial order results on contactomorphism groups

Abstract

It is shown that if the universal cover of a manifold $M$ is an open manifold, then two different fibres of the spherical cotangent bundle $ST^*M$ cannot be connected by a non-negative Legendrian isotopy. This result is applied to the study of causality in globally hyperbolic spacetimes. It is also used to strengthen a result of Eliashberg, Kim, and Polterovich on the existence of a partial order on $\widetilde{\mathrm{Cont}}_0 (ST^*M)$.