  We consider area-preserving deformations of the plane, acting on electronic
wavefunctions through "quantomorphisms" that change both the underlying metric
and the confining potential. We show that adiabatic sequences of such
transformations produce Berry phases that can be written in closed form in
terms of the many-body current and density, even in the presence of
interactions. For a large class of deformations that generalize squeezing and
shearing, the leading piece of the phase is a super-extensive Aharonov-Bohm
term (proportional to N$^2$ for N electrons) in the thermodynamic limit. Its
gauge-invariant subleading partner only measures the current, whose dominant
contribution to the phase stems from a jump at the edge in the limit of strong
magnetic fields. This results in a finite Berry curvature per unit area,
reminiscent of the Hall viscosity. We show that the latter is in fact included
in our formalism, bypassing its standard derivation on a torus and suggesting
realistic experimental setups for its observation in quantum simulators.
