# Legendrian links, causality, and the Low conjecture - Mathematics > Symplectic Geometry

Abstract: Let $X^{m+1}, g$ be a globally hyperbolic spacetime with Cauchy surfacediffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjectureformulated by Nat\-ario and Tod says that two events $x,y\in\ss$ are causallyrelated if and only if the Legendrian link of spheres $\mathfrak S x, \mathfrakS y$ whose points are light geodesics passing through $x$ and $y$ isnon-trivial in the contact manifold of all light geodesics in $X$. The Lowconjecture says that for $m=2$ the events $x,y$ are causally related if andonly if $\mathfrak S x, \mathfrak S y$ is non-trivial as a topological link. Weprove the Low and the Legendrian Low conjectures. We also show that similarstatements hold for any globally hyperbolic $X^{m+1}, g$ such that a cover ofits Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$