  In this paper, we generalize the famous Hasimoto's transformation by showing
that the dynamics of a closed unidimensional vortex filament embedded in a
three-dimensional manifold of constant curvature gives rise under Hasimoto's
transformation to the non-linear Schrodinger equation.
  We also give a natural interpretation of the function \psi introduced by
Hasimoto in terms of moving frames associated to a natural complex bundle over
the filament.
