  Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of
many-body integrable systems. It consists of an infinite set of conservation
laws for quasi-particles traveling with effective ("dressed") velocities that
depend on the local state. We show that these equations can be recast into a
geometric dynamical problem. They are conservation equations with
state-independent quasi-particle velocities, in a space equipped with a family
of metrics, parametrized by the quasi-particles' type and speed, that depend on
the local state. In the classical hard rod or soliton gas picture, these
metrics measure the free length of space as perceived by quasi-particles, in
the quantum picture, they weigh space with the density of states available to
them. Using this geometric construction, we find a general solution to the
initial value problem of GHD, in terms of a set of integral equations where
time appears explicitly. These integral equations are solvable by iteration and
provide an extremely efficient solution algorithm for GHD.
