This chapter essentially describes the objects and properties that will interest us in this work. For a more detailed exposition of the general background in Riemannian geometry and in analysis on manifolds, one may refer for instance to [183] and [98]. After recalling how to associate, to each Riemannian metric on a manifold, a Laplacian operator on the same manifold, we will give a definition of smooth harmonic map between two manifolds. Very soon, we will use the variational framework, which consists in viewing harmonic maps as the critical points of the Dirichlet functional.

Next, we introduce a frequently used ingredient in this book: Noether's theorem. We present two versions of it: one related to the symmetries of the image manifold, and the other which is a consequence of an invariance of the problem under diffeomorphisms of the domain manifold (in this case it is not exactly Noether's theorem, but a “covariant” version).

These concepts may be extended to contexts where the map between the two manifolds is less regular. In fact, a relatively convenient space is that of maps with finite energy (Dirichlet integral), H1 (M, N). This space appears naturally when we try to use variational methods to construct harmonic maps, for instance the minimization of the Dirichlet integral. The price to pay is that when the domain manifold has dimension larger than or equal to 2, maps in H1 (M, N) are not smooth, in general. Moreover, H1 (M, N) does not have a differentiable manifold structure.