Seven Classes of Harmonic Diffeomorphisms

We deduce two necessary and sufficient conditions for a diffeomorphism $f : M to overline{M}$ of a Riemannian manifold (M,g) onto a Riemannian manifold $(overline{M},bar g)$ to be harmonic. Using the representation theory of groups, we define in an intrinsic way seven classes of such harmonic diffeomorphisms and partly describe the geometry of each class.