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  "body": "\n## Definition\n\nLet $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]] of dimensions $m$ and $n$, and let $u \\colon M \\to N$ be a smooth map.\nUse local coordinates $x^i$ on $M$ and $y^\\alpha$ on $N$.\nThe *tension field* of $u$ is\n$$\\tau(u)^\\gamma = \\Delta_g u^\\gamma + \\sum_{i,j,\\alpha,\\beta} g^{ij} \\, \\Gamma^\\gamma_{\\alpha\\beta}(u) \\, \\partial_i u^\\alpha \\, \\partial_j u^\\beta, \\qquad \\gamma = 1, \\dots, n,$$\nwhere $\\Delta_g$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $g$ and $\\Gamma^\\gamma_{\\alpha\\beta}$ are the [[Christoffel symbols|FZBA1HA9ZM]] of $h$.\nThe map $u$ is *harmonic* if $\\tau(u) = 0$ on $M$.\n\n## Remarks\n\nThe 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 $du$.\nLet $M$ be compact.\nThen $u$ is harmonic if and only if it is a critical point of the energy\n$$E(u) = \\frac{1}{2} \\int_M \\abs{du}^2 \\, dV_g, \\qquad \\abs{du}^2 = \\sum g^{ij} h_{\\alpha\\beta}(u) \\, \\partial_i u^\\alpha \\, \\partial_j u^\\beta,$$\nwhich generalizes the Dirichlet energy; indeed $-\\tau(u)$ is the gradient of $E$.\n\nExample: let $N = \\R$; then $\\Gamma = 0$ and $u$ is harmonic if and only if $\\Delta_g u = 0$.\nLet $M$ 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.\n\nNon-example: let $M = N = \\R / 2\\pi\\Z$ with the standard metric and $u(\\theta) = \\theta + \\frac{1}{2}\\sin\\theta$.\nHere $\\Gamma = 0$ and $\\tau(u) = u''(\\theta) = -\\frac{1}{2}\\sin\\theta \\ne 0$, so $u$ is not harmonic, although it is a diffeomorphism.\nThe identity map $\\theta \\mapsto \\theta$ is harmonic.\n",
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