Lorentzian distance functions in contact geometry

An important tool to analyse the causal structure of a Lorentzian manifold is given by distance function. We define class functions on group contactomorphisms closed contact depending choice form. These are continuous with respect Hofer norm for defined Shelukhin [The contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if only orderable. To prove this, we show that intervals positivity relation open topology induced norm. For orderable Legendrian isotopy classes Chekanov-type metric in [D. Rosen Zhang, Chekanov’s dichotomy topology, Math. Res. Lett. 27 (2020) 1165–1194] nondegenerate. In this case, similar results hold classes. This leads natural metrics associated globally hyperbolic such its Cauchy hypersurface has unit co-tangent bundle fibres.