Legendrian links, causality, and the Low conjecture

Details Legendrian links, causality, and the Low conjecture Legendrian links, causality, and the Low conjecture

2008-10-28

arxiv.org/abs/0810.5091v3

Geom. Funct. Anal. 19 (2010), 1320-1333

Mathematics

Symplectic Geometry

Version 3 - minor improvements, references added 11 pages, 1 figure

Scientific paper

10.1007/s00039-009-0039-x

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events $x,y\in\ss$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial 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 $\mathfrak S_x, \mathfrak S_y$ 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 $\mathbb R^m.$

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