Relativistic deformed kinematics: from flat to curved spacetimes

Doubly special relativity has been studied for the last 20 years as a way to go beyond relativistic kinematics, trying capture residual effects of quantum gravity theory. In particular, in doubly relativity, Einstenian principle is generalized, adding speed light another invariant, Planck energy. There are several papers literature showing connection between this deformed kinematics and curved momentum space. Here we review how such can be derived from geometrical ingredients rigorous way, they generalized when regarding spacetime. For aim, it mandatory consider particular geometry all phase-space variables, so-called Hamilton spaces. This construction allows us define spacetime these theories, which fact depends on momenta. Then, starting dependent metric, also revise concepts general with final aim establishing self-consistent structure phenomenology explored.