  We show that quantum dynamics of Bose-Einstein condensates in the weakly
interacting regime can be geometrized by a Poincar\'e disk. Each point on such
a disk represents a thermofield double state, the overlap between which equals
the metric of this hyperbolic space. This approach leads to a unique geometric
interpretation of stable and unstable modes as closed and open trajectories on
the Poincar\'e disk, respectively. The resonant modes that follow geodesics
naturally equate fundamental quantities including the time, the length, and the
temperature. Our work suggests a new geometric framework to coherently control
quantum systems and reverse their dynamics using SU(1,1) echoes. In the
presence of perturbations breaking the SU(1,1) symmetry, SU(1,1) echoes deliver
a new means to measure these perturbations such as the interactions between
excited particles.
