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  "body": "\n## Definition\n\nLet $(M, g)$ be an oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, and let $\\Omega^k(M)$ be the space of smooth [[differential $k$-forms|0E3G0GRPZ6]].\nLet $d \\colon \\Omega^k(M) \\to \\Omega^{k+1}(M)$ be the [[exterior derivative|0ZHSTAG8ZY]], and let $\\star$ be the [[Hodge star|1Y4WX4XQWJ]].\nThe *codifferential* $\\delta \\colon \\Omega^k(M) \\to \\Omega^{k-1}(M)$ is\n$$\\delta = (-1)^{n(k+1)+1} \\star d \\star.$$\nThe *Hodge Laplacian* on $k$-forms is\n$$\\Delta = d \\delta + \\delta d \\colon \\Omega^k(M) \\to \\Omega^k(M).$$\n\n## Remarks\n\nThe sign in $\\delta$ makes it the formal adjoint of $d$: $\\int_M \\ip{d\\alpha}{\\beta} \\, \\mathrm{vol} = \\int_M \\ip{\\alpha}{\\delta\\beta} \\, \\mathrm{vol}$ when one of the forms has compact support.\nChanging the orientation changes $\\star$ by a sign twice, so $\\delta$ and $\\Delta$ are defined on any Riemannian manifold.\nSome authors write $\\Delta_d$ or $\\Delta_H$; it is also called the Laplace–de Rham operator.\nIt is a nonnegative, formally self-adjoint, elliptic operator of order $2$, and $\\Delta = (d + \\delta)^2$.\n\nExample: on $\\R^n$ with the flat metric, let $f$ be a function.\nThen $df = \\sum_j \\partial_j f \\, dx_j$ and $\\delta(\\sum_j a_j \\, dx_j) = -\\sum_j \\partial_j a_j$, so $\\Delta f = \\delta df = -\\sum_j \\partial_j^2 f$.\nSo on functions $\\Delta = -\\operatorname{div} \\operatorname{grad}$, the [[Laplace–Beltrami operator|F4NWZNTRMD]] with the sign that makes it nonnegative.\nOn $k$-forms of $\\R^n$, $\\Delta(\\sum_I f_I \\, dx_I) = \\sum_I (-\\sum_j \\partial_j^2 f_I) \\, dx_I$.\n\nNon-example: the operator $d \\delta$ alone is not the Hodge Laplacian; on functions it is zero, since $\\delta f = 0$ for a $0$-form.\n\nOn a closed manifold, the [[Hodge theorem|A127WFVYKW]] identifies $\\ker \\Delta$ on $k$-forms with the de Rham cohomology $H^k(M; \\R)$.\nThe [[Weitzenböck identity|BE9GTWMN0B]] writes $\\Delta$ as a rough Laplacian plus a curvature term.\n\n## Explanations\n\n### L1 human\n\nThe Hodge Laplacian is the combination $d\\delta + \\delta d$ of the exterior derivative and its adjoint, acting on differential forms.\n",
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