Relative-locality geometry for the Snyder model

We investigate the geometry of the energy-momentum space of the Snyder model and of its generalizations according to the definitions proposed in [G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84 (2011) 084010], in connection with the theory of relative locality. In this setting, the geometric structures of the energy-momentum space are defined in terms of the deformed composition law of momenta, and we show that in the Snyder case they describe a maximally symmetric space, with vanishing torsion and nonmetricity. However, one cannot apply straightforwardly the phenomenological relations between the geometry and the dynamics postulated in [G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D 84 (2011) 084010], because they were obtained assuming that the leading corrections to the composition law of momenta are quadratic, which is not the case with the Snyder model and its generalizations