  We apply our recent formalism establishing new connections between the
geometry of moving space curves and soliton equations, to the nonlinear
  Schr\"{o}dinger equation (NLS).
  We show that any given solution of the NLS gets associated with three
distinct space curve evolutions. The tangent vector of the first of these
curves, the binormal vector of the second and the normal vector of the third,
are shown to satisfy the integrable Landau-Lifshitz (LL) equation
  ${\bf S}_u = {\bf S} \times {\bf S}_{ss}$, (${\bf S}^2=1$). These connections
enable us to find the three surfaces swept out by the moving curves associated
with the NLS. As an example, surfaces corresponding to a stationary envelope
soliton solution of the NLS are obtained.
