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 M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrodinger equation. We also give a natural interpretation of the function. introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.