Generally covariant formulation of Relative Locality in curved spacetime
  We construct a theory of particles moving in curved both momentum space and spacetime, being a generalization of Relative Locality. We find that in order to construct such theory, with desired symmetries, including the general coordinate invariance, we have to use non local position variables. It turns out that free particles move on geodesics and momentum dependent translations of Relative Locality are replaced with momentum dependent geodesic deviations. 
Francesco Cianfrani, Jerzy Kowalski-Glikman, Giacomo Rosati
gr-qc hep-th