Causality and Legendrian linking for higher dimensional spacetimes

Let $(X^{m+1}, g)$ be an $(m+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $M^m$, and let $\widetilde M^m$ be the universal cover of the Cauchy surface. Let $\mathcal N_{X}$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent bundle $ST^*M.$ Jointly with Stefan Nemirovski we showed when $\widetilde M^m$ is {\bf not\/} a compact manifold, then two points $x, y\in X$ are causally related if and only if the Legendrian spheres $\mathfrak S_x, \mathfrak S_y$ of all light rays through $x$ and $y$ are linked in $\mathcal N_{X}.$ In this short note we use the contact Bott-Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all $X$ for which the integral cohomology ring of a closed $\widetilde M$ is {\bf not} the one of the CROSS (compact rank one symmetric space). If $M$ admits a Riemann metric $\overline g$, a point $x$ and a number $\ell>0$ such that all unit speed geodesics starting from $x$ return back to $x$ in time $\ell$, then $(M, \overline g)$ is called a $Y^x_{\ell}$ manifold. Jointly with Stefan Nemirovski we observed that causality in $(M\times \mathbb R, \overline g\oplus -t^2)$ is {\bf not} equivalent to Legendrian linking. Every $Y^x_{\ell}$-Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can {\bf not} put a $Y^x_{\ell}$ Riemann metric on a Cauchy surface $M.$