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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $U \\subseteq M$ be an open set.\nIn local coordinates $x^1, \\dots, x^n$, write $g_{ij}$ for the metric, $g^{ij}$ for its inverse matrix and $\\abs{g} = \\det(g_{ij})$.\nThe [[Laplace–Beltrami operator|F4NWZNTRMD]] of a function $u \\in C^2(U)$ is\n$$\\Delta_g u = \\frac{1}{\\sqrt{\\abs{g}}}\\, \\partial_i \\Big( \\sqrt{\\abs{g}}\\, g^{ij}\\, \\partial_j u \\Big),$$\nwith summation over repeated indices; it equals $\\operatorname{div}_g \\operatorname{grad}_g u$ and does not depend on the coordinates.\nA function $u \\in C^2(U)$ is *harmonic* on $U$ if $\\Delta_g u = 0$ on $U$.\n\n## Remarks\n\nFor $M = \\R^n$ with the Euclidean metric, $\\Delta_g$ is the usual Laplacian, and the definition gives the classical [[harmonic functions|0DEB0JMN0W]].\nExample: on the upper half-plane $\\set{(x, y) : y \u003e 0}$ with the hyperbolic metric $g = y^{-2}(dx^2 + dy^2)$, one has $\\Delta_g = y^2(\\partial_x^2 + \\partial_y^2)$.\nSo the function $u(x, y) = x$ is harmonic, and so is the bounded function $\\arctan(x/y)$.\nNon-example: on the unit sphere $S^2 \\subset \\R^3$ with the round metric, the height function $z$ satisfies $\\Delta_g z = -2z$, so it is not harmonic.\nOn a connected closed manifold every harmonic function is constant, since $\\int_M \\abs{\\nabla u}^2 \\, dV_g = -\\int_M u\\, \\Delta_g u \\, dV_g = 0$.\nThe definition depends on the metric: a function harmonic for one metric need not be harmonic for another.\nIn dimension $2$, however, harmonicity is unchanged under conformal changes of metric.\n",
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