Seven classes of harmonic diffeomorphisms (Q869743)

Two characterizations for a diffeomorphism between Riemannian manifolds to be harmonic are given. The first one describes harmonicity in terms of the deformation tensor of the Levi-Civita connection under the diffeomorphism, while the second one gives equations for the metric tensors. Using basic representation theory of \(O(m)\), the authors group harmonic diffeomorphisms into seven classes, some geometry of which is subsequently discussed.