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  "body": "\n## Statement\n\nLet $(M, g)$ be a compact [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n$.\nFor $0 \\le k \\le n$, let $\\Omega^k(M)$ be the space of smooth $k$-[[forms|0E3G0GRPZ6]] and $d$ the [[exterior derivative|0ZHSTAG8ZY]].\nLet $\\delta = (-1)^{n(k+1)+1} \\star d \\star$ be the codifferential on $k$-forms, where $\\star$ is the [[Hodge star|1Y4WX4XQWJ]].\nLet $\\Delta = d\\delta + \\delta d$ be the Hodge Laplacian.\nLet $\\mathcal{H}^k = \\set{\\omega \\in \\Omega^k(M) : \\Delta\\omega = 0}$ be the space of harmonic $k$-forms.\nThen:\n\n1. $\\mathcal{H}^k$ is finite-dimensional.\n2. There is a decomposition, orthogonal for the $L^2$ inner product $(\\alpha, \\beta) = \\int_M \\alpha \\wedge \\star\\beta$:\n$$\\Omega^k(M) = \\mathcal{H}^k \\oplus d\\,\\Omega^{k-1}(M) \\oplus \\delta\\,\\Omega^{k+1}(M).$$\n3. Each class of the [[de Rham cohomology|GX74C07T0G]] group $H^k_{dR}(M)$ contains exactly one harmonic form, so $\\mathcal{H}^k \\cong H^k_{dR}(M)$.\n\n## Remarks\n\nThe idea of the proof: $\\delta$ is the $L^2$ adjoint of $d$, so $(\\Delta\\omega, \\omega) = \\norm{d\\omega}^2 + \\norm{\\delta\\omega}^2$, and $\\omega$ is harmonic if and only if $d\\omega = 0$ and $\\delta\\omega = 0$.\nThe operator $\\Delta$ is elliptic, and elliptic regularity with the Rellich compactness theorem gives part 1 and the decomposition $\\Omega^k = \\mathcal{H}^k \\oplus \\Delta(\\Omega^k)$, from which part 2 follows.\nFor part 3, a closed form is orthogonal to $\\delta\\Omega^{k+1}$, so it is its harmonic part plus an exact form.\n\nExample: on the flat torus $\\R^n / \\Z^n$ the harmonic $k$-forms are the forms with constant coefficients in $dx^{i_1} \\wedge \\dots \\wedge dx^{i_k}$.\nSo $\\dim H^k_{dR} = \\binom{n}{k}$.\n\nCompactness is needed: on $\\R$, every function $a x + b$ is harmonic, and the space of harmonic $0$-forms is $2$-dimensional while $H^0_{dR}(\\R)$ is $1$-dimensional.\n\n## Explanations\n\n### L1 human\n\nEach cohomology class, a kind of \"hole\" of the manifold, has many representing forms.\nThe Hodge theorem picks a best one: the unique harmonic form, which minimizes the energy in its class.\nSo topology is captured by solutions of a differential equation.\n\n### L2 human\n\nSince $\\star$ maps harmonic $k$-forms to harmonic $(n-k)$-forms, the theorem gives Poincaré duality $b_k = b_{n-k}$ for the Betti numbers.\nCombined with Weitzenböck identities, it turns curvature conditions into vanishing of cohomology, as in Bochner's theorem.\nOn compact Kähler manifolds a refinement gives the Hodge decomposition of complex cohomology into $(p, q)$-types.\n",
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