  Deformed relativistic kinematics have been considered as a way to capture
residual effects of quantum gravity. It has been shown that they can be
understood geometrically in terms of a curved momentum space on a flat
spacetime. In this article we present a systematic analysis under which
conditions and how deformed relativistic kinematics, encoded in a momentum
space metric on flat spacetime, can be lifted to curved spacetimes in terms of
a self-consistent cotangent bundle geometry, which leads to purely geometric,
geodesic motion of freely falling point particles. We comment how this
construction is connected to, and offers a new perspective on, non-commutative
spacetimes. From geometric consistency conditions we find that momentum space
metrics can be consistently lifted to curved spacetimes if they either lead to
a dispersion relation which is homogeneous in the momenta, or, if they satisfy
a specific symmetry constraint. The latter is relevant for the momentum space
metrics encoding the most studied deformed relativistic kinematics. For these,
the constraint can only be satisfied in a momentum space basis in which the
momentum space metric is invariant under linear local Lorentz transformations.
We discuss how this result can be interpreted and the consequences of relaxing
some conditions and principles of the construction from which we started.
