Harmonic sections of homogeneous fibre bundles

We show how the equations for harmonic maps into homogeneous spaces generalize to harmonic sections of homogeneous fibre bundles. Surprisingly, the generalization does not explicitly involve the curvature of the bundle. However, a number of special cases of the harmonic section equations (including the new condition of super-flatness) are studied in which the bundle curvature does appear. Some examples are given to illustrate these special cases in the non-flat environment. The bundle in question is the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M.