Legendrian links, causality, and the Low conjecture
Vladimir Chernov, Stefan Yu Nemirovski · 2008
Let (X m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of R m. The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ∈ X are causally related if and only if the Legendrian link of spheres Sx, Sy whose points are light geodesics passing through x and y is nontrivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if Sx, Sy is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in R m.