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 $S^1$ so $u : \mathbb{R} \to N$ in this chart, the harmonic map equation becomes \[ \frac{\partial^2 u}{\partial x^2} - \Gamma^N_{ij}^k \frac{\partial u}{\partial x^i} \frac{\partial u}{\partial x^k} = 0, \] which is precisely the geodesic equation.