Let $(M^n,g), (N,h)$ be closed Riemannian manifolds. A map $u \in C^2(M,N)$ is said to be harmonic if $u$ is a critical point for the Dirichlet energy functional $E(u)$; equivalently, $\mathrm{d}^* (\mathrm{d}u) = 0$ where $\mathrm{d}u \in \Gamma(T^*M \otimes u^* TN)$ is the pushforward and $\mathrm{d}^* : \Gamma(T^*M \otimes u^*TN) \to \Gamma(u^*TN)$ is the adjoint to the exterior derivative with respect to the $L^2$ metric on sections induced by $g$ and $u^*h$.

Let $M = S^1$. Then a harmonic map $u : M \to N$ is a closed geodesic; that is, a curve that is both a geodesic and periodic. Indeed, working in a local coordinate $x$ on $S^1$ and coordinates on $N$ so that $u(x) = (u^1(x),\ldots,u^m(x))$, the harmonic map equation becomes that for each $k$, \[ \frac{\partial^2 u^k}{\partial x^2} - (\Gamma^N)_{ij}^k \frac{\partial u^i}{\partial x} \frac{\partial u^j}{\partial x} = 0, \] which is precisely the geodesic equation.

The identity $\mathrm{Id} : (M,g) \to (M,g)$ is always harmonic. Indeed in this case, the pushforward $\mathrm{d}\,\mathrm{Id}$ is the identity map $TM \to TM$ and so is given in a frame $\{e_1, \ldots, e_n\}$ with coframe $\{e^1,\ldots, e^n\}$ by $\mathrm{d}\,\mathrm{Id} = e^i \otimes e_i$. In particular, \[ e(\mathrm{d} Id) = n \] is a constant, TODO