Harmonic map

Remarks

The tension field is a section of $u^* TN$ and does not depend on the coordinates; it is the trace with respect to $g$ of the covariant derivative of $du$. Let $M$ be compact. Then $u$ is harmonic if and only if it is a critical point of the energy $$E(u) = \frac{1}{2} \int_M \abs{du}^2 \, dV_g, \qquad \abs{du}^2 = \sum g^{ij} h_{\alpha\beta}(u) \, \partial_i u^\alpha \, \partial_j u^\beta,$$ which generalizes the Dirichlet energy; indeed $-\tau(u)$ is the gradient of $E$.

Example: let $N = \R$; then $\Gamma = 0$ and $u$ is harmonic if and only if $\Delta_g u = 0$. Let $M$ be an interval with its standard metric; then $\tau(u) = 0$ is the geodesic equation, so harmonic maps from an interval are constant-speed geodesics.

Non-example: let $M = N = \R / 2\pi\Z$ with the standard metric and $u(\theta) = \theta + \frac{1}{2}\sin\theta$. Here $\Gamma = 0$ and $\tau(u) = u''(\theta) = -\frac{1}{2}\sin\theta \ne 0$, so $u$ is not harmonic, although it is a diffeomorphism. The identity map $\theta \mapsto \theta$ is harmonic.