  Applying the loop variable proposal to a sigma model (with boundary) in a
curved target space, we give a systematic method for writing the gauge and
generally covariant interacting equations of motion for the modes of the open
string in a curved background. As in the free case described in an earlier
paper, the equations are obtained by covariantizing the flat space (gauge
invariant) interacting equations and then demanding gauge invariance in the
curved background. The resulting equation has the form of a sum of terms that
would individually be gauge invariant in flat space or at zero interaction
strength, but mix amongst themselves in curved space when interactions are
turned on. The new feature is that the loop variables are deformed so that
there is a mixing of modes. Unlike the free case, the equations are coupled,
and all the modes of the open string are required for gauge invariance.
