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

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 $S{T}^{\ast }M$ 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 ${\widetilde{Cont}}_0\left(S{T}^{\ast }M\right)$.