  The Madelung transform is known to relate Schr\"odinger-type equations in
quantum mechanics and the Euler equations for barotropic-type fluids. We prove
that, more generally, the Madelung transform is a K\"ahler map (i.e. a
symplectomorphism and an isometry) between the space of wave functions and the
cotangent bundle to the density space equipped with the Fubini-Study metric and
the Fisher-Rao information metric, respectively. We also show that Fusca's
momentum map property of the Madelung transform is a manifestation of the
general approach via reduction for semi-direct product groups. Furthermore, the
Hasimoto transform for the binormal equation turns out to be the 1D case of the
Madelung transform, while its higher-dimensional version is related to the
problem of conservation of the Willmore energy in binormal flows.
