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  "body": "\n## Statement\n\nLet $(M, g)$ be an oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, with [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$.\nOn [[differential forms|0E3G0GRPZ6]] of degree $k$, let $d$ be the [[exterior derivative|0ZHSTAG8ZY]].\nLet $\\delta = (-1)^{n(k+1)+1} \\star d \\star$ be the codifferential, where $\\star$ is the [[Hodge star|1Y4WX4XQWJ]].\nLet $\\Delta_H = d\\delta + \\delta d$ be the Hodge Laplacian.\nLet $\\nabla^* \\nabla = -\\sum_{i=1}^n \\bigl(\\nabla_{e_i} \\nabla_{e_i} - \\nabla_{\\nabla_{e_i} e_i}\\bigr)$ be the rough Laplacian, for a local orthonormal frame $e_1, \\dots, e_n$.\nThen there is a bundle endomorphism $\\mathcal{R}_k$ of $k$-forms such that\n$$\\Delta_H \\omega = \\nabla^* \\nabla \\omega + \\mathcal{R}_k(\\omega)$$\nfor every $k$-form $\\omega$.\nFor $k = 1$, the term is the [[Ricci curvature|4GTNM0WHDH]]: $\\mathcal{R}_1(\\omega)(X) = \\operatorname{Ric}(\\omega^\\sharp, X)$, where $\\omega^\\sharp$ is the vector field dual to $\\omega$.\nAt each point, $\\mathcal{R}_k$ is given by a linear expression in the Riemann curvature tensor.\n\n## Remarks\n\nThe idea of the proof: express $d$ and $\\delta$ through $\\nabla$, namely $d\\omega = \\sum_i e^i \\wedge \\nabla_{e_i} \\omega$ and $\\delta\\omega = -\\sum_i \\iota_{e_i} \\nabla_{e_i} \\omega$.\nExpanding $d\\delta + \\delta d$ gives the second-order part $\\nabla^* \\nabla$ plus commutators $[\\nabla_{e_i}, \\nabla_{e_j}]$, which are curvature terms.\n\nExample: for $\\omega = du$ the identity, paired with $du$, gives [[Bochner's formula|DF7KYMAGTC]] for the function $u$.\n\nBochner's vanishing theorem follows: let $M$ be compact.\nIf $\\omega$ is a harmonic 1-form, integrating $\\ip{\\Delta_H \\omega}{\\omega} = 0$ gives $\\int_M \\abs{\\nabla \\omega}^2 + \\operatorname{Ric}(\\omega^\\sharp, \\omega^\\sharp) = 0$.\nSo if $\\operatorname{Ric} \\ge 0$, every harmonic 1-form is parallel and the first Betti number is at most $n$; if $\\operatorname{Ric} \u003e 0$, the first Betti number is $0$.\n\n## Explanations\n\n### L1 human\n\nThere are two natural Laplacians on differential forms: one built from the exterior derivative, and one built from the covariant derivative.\nThe identity says they differ only by a curvature term with no derivatives.\nFor 1-forms that term is the Ricci curvature.\n\n### L2 human\n\nWeitzenböck identities are the engine of the Bochner technique.\nCombined with the Hodge theorem, which identifies cohomology with harmonic forms, a sign of the curvature term forces harmonic forms to vanish or to be parallel.\nThe same pattern holds for other Dirac-type operators; for the spin Dirac operator the curvature term is a quarter of the scalar curvature (the Lichnerowicz formula).\n",
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