  The general purpose bitensorially gauge-covariant differentiation procedure
set up in the preceding article is specialised to the particular case of
bundles with nonlinear fibres that are endowed with a torsion free Riemannian
or pseudo-Riemannian structure. This formalism is used to generalize the class
of harmonic mappings between Riemannian or pseudo-Riemannian spaces to a
natural gauge coupled extension in the form of a class of field sections of a
bundle having the original image space as fibre, with a nonintegrable gauge
connection $\Amr$ belonging to the algebra of the isometry group of the fibre
space. The Bunting identity that can be used for establishing uniqueness in the
strictly positive Riemannian case with negative image space curvature is shown
to be generalizable to this gauge coupled extension.
