  In the literature, there are several papers establishing a correspondence
between a deformed kinematics and a nontrivial (momentum dependent) metric. In
this work, we study in detail the relationship between the trajectories given
by a deformed Hamiltonian and the geodesic motion obtained from a geometry in
the cotangent bundle, finding that both trajectories coincide when the
Hamiltonian is identified with the squared distance in momentum space.
Moreover, following the natural structure of the cotangent bundle geometry, we
construct the space-time curvature tensor, from which we obtain generalized
Einstein equations. Since the metric is not invariant under momentum
diffeomorphisms (changes of momentum coordinates) we note that, in order to
have a conserved Einstein tensor (in the same sense of general relativity), a
privileged momentum basis appears, a completely new result that cannot be found
in absence of space-time curvature, which settles a long standing ambiguity of
this geometric approach. After that, we consider in an expanding universe the
geodesic motion and the Raychaudhuri's equations, and we show how to construct
vacuum solutions to the Einstein equations. Finally, we make a comment about
the possible phenomenological implications of our framework.
