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  "body": "\n## Statement\n\nLet $(M, g)$ be a compact oriented [[Riemannian manifold|7CBT4SM83Y]] without boundary, of even dimension $n = 2m$.\nLet $\\Omega = (\\Omega_{ij})$ be the [[curvature form|ZAJFW1S078]] of the [[Levi-Civita connection|CAZYGE1T4E]] with respect to a local oriented orthonormal frame, a skew-symmetric $n \\times n$ matrix of 2-forms.\nThen the $n$-form $\\operatorname{Pf}(\\Omega)$, where $\\operatorname{Pf}$ is the [[Pfaffian|NH5HWBPWS9]], does not depend on the frame, and\n$$\\chi(M) = \\frac{1}{(2\\pi)^m} \\int_M \\operatorname{Pf}(\\Omega),$$\nwhere $\\chi(M)$ is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.\n\n## Remarks\n\nChern gave an intrinsic proof in 1944: on the unit sphere bundle of $M$, the form $\\operatorname{Pf}(\\Omega)$ pulled back is exact, $d\\Pi$, by a transgression formula.\nTake a vector field $X$ with isolated zeros, and integrate over $M$ minus small balls around the zeros.\nStokes' theorem gives the sum of the indices of $X$, which is $\\chi(M)$ by the Poincaré–Hopf theorem.\n\nExample: for $m = 1$, $\\operatorname{Pf}(\\Omega) = \\Omega_{12} = K \\, dA$ with $K$ the Gaussian curvature, and the formula is the [[Gauss–Bonnet theorem|ESGEHKPD60]] $\\int_M K \\, dA = 2\\pi \\chi(M)$.\nFor $m = 2$ it reads\n$$\\chi(M) = \\frac{1}{32\\pi^2} \\int_M \\bigl(\\abs{\\operatorname{Rm}}^2 - 4 \\abs{\\operatorname{Ric}}^2 + R^2\\bigr) \\, dV,$$\nwith $\\operatorname{Rm}$ the Riemann tensor, $\\operatorname{Ric}$ the Ricci tensor and $R$ the scalar curvature.\nFor the unit sphere $S^4$ the integrand is $24 - 144 + 144 = 24$ and the volume is $8\\pi^2/3$, which gives $\\chi(S^4) = 2$.\n\nFor odd $n$ the Euler characteristic of a closed manifold is $0$, so there is no analogous curvature formula.\n\n## Explanations\n\n### L1 human\n\nAdd up the curvature of a closed even-dimensional space over all its points.\nThe theorem says that the result depends only on the shape up to deformation: it is the Euler characteristic.\nHowever the space is bent, the total curvature stays the same.\n\n### L2 human\n\nThe theorem generalizes the Gauss–Bonnet theorem for surfaces to all even dimensions.\nIn the language of characteristic classes, $\\operatorname{Pf}(\\Omega / 2\\pi)$ represents the Euler class of the tangent bundle in de Rham cohomology.\nThe theorem then says that the Euler number of the tangent bundle is $\\chi(M)$.\nIt is a model case of Chern–Weil theory and a special case of the Atiyah–Singer index theorem, for the operator $d + \\delta$ from even to odd forms.\n",
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