Causality and Legendrian linking for higher dimensional spacetimes

If M admits a Riemann metric g¯, a point x and a number ℓ>0 such that all unit speed geodesics starting from x return back to x in time ℓ, then (M,g¯) is called a Yℓx manifold. Jointly with Stefan Nemirovski we observed that causality in (M×R,g¯⊕−t2) is not equivalent to Legendrian linking. Every Yℓx-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 not put a Yℓx Riemann metric on a Cauchy surface M.