As I understand it, one way to describe a wave function is as a probability density distribution in phase space. The equations of motion for the wave function would describe how that density distribution evolves over time. Choice of the coordinate system in phase space should not affect the underlying behavior of the the system. If time were included as a coordinate in phase space, evolution of the system would correspond to intersection of the phase space distribution with a hyperplane perpendicular to the time dimension and moving along the time coordinate. This seems analogous to the evolution of a system as described in General Relativity.
So: If time is included in the definition of phase space, would the equations of motion in effect specify a curved geometry in phase space, where the quantity invariant under changes of coordinate system is local curvature as in General Relativity?