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  "body": "\n## Statement\n\n1. Let $G$ be a Lie group with Lie algebra $\\mathfrak{g}$, and let $P \\to M$ be a principal $G$-bundle with connection form $\\omega$. Let $\\Omega = d\\omega + \\frac12 [\\omega \\wedge \\omega]$ be its [[curvature form|ZAJFW1S078]]. Then\n$$d\\Omega + [\\omega \\wedge \\Omega] = 0,$$\nwhere $[\\omega \\wedge \\Omega](X, Y, Z)$ is the sum over cyclic permutations $(X, Y, Z)$ of $[\\omega(X), \\Omega(Y, Z)]$.\n2. Let $M$ be a smooth manifold with a torsion-free [[affine connection|JX65KSVHSJ]] $\\nabla$ (its [[torsion|JZNQAN00ZJ]] vanishes) and [[curvature tensor|JA29ETCM02]] $R$. Then for all vector fields $X, Y, Z, W$:\n   - (first Bianchi identity) $R(X, Y) Z + R(Y, Z) X + R(Z, X) Y = 0$;\n   - (second Bianchi identity) $(\\nabla_X R)(Y, Z) W + (\\nabla_Y R)(Z, X) W + (\\nabla_Z R)(X, Y) W = 0$, where $\\nabla_X R$ is the [[covariant derivative|5FZDQQHQKM]] of the tensor field $R$.\n\n## Remarks\n\nFor part 1, apply $d$ to the structure equation: $d\\Omega = \\frac12 d[\\omega \\wedge \\omega] = \\frac12\\big([d\\omega \\wedge \\omega] - [\\omega \\wedge d\\omega]\\big)$, substitute $d\\omega = \\Omega - \\frac12[\\omega \\wedge \\omega]$, and use the Jacobi identity to see that $[[\\omega \\wedge \\omega] \\wedge \\omega] = 0$.\nPart 1 is often written $D\\Omega = 0$, with $D$ the exterior covariant derivative.\nFor the first identity in part 2, expand $R$ and use $\\nabla_X Y - \\nabla_Y X = [X, Y]$; the cyclic sum reduces to the Jacobi identity for the Lie bracket.\nWithout the torsion-free hypothesis the identities get extra terms involving the torsion and its covariant derivative.\nExample: tracing the second identity for the Levi-Civita connection gives $\\mathrm{div}\\, \\mathrm{Ric} = \\frac12 \\, d\\,\\mathrm{scal}$, so the Einstein tensor $\\mathrm{Ric} - \\frac12 \\mathrm{scal}\\, g$ is divergence-free.\n\n## Explanations\n\n### L1 human\n\nCurvature is not an arbitrary collection of numbers: it satisfies some built-in rules.\nThe Bianchi identities are these rules.\nOne is algebraic and says that a certain cyclic sum of curvatures is zero.\nThe other says that a cyclic sum of derivatives of curvature is zero, much like the rule that the divergence of a curl is zero.\n\n### L2 human\n\nThe first identity, together with the symmetries of the Riemann tensor, cuts down its number of independent components to $n^2(n^2 - 1)/12$ in dimension $n$.\nThe contracted second identity is the reason the left side of Einstein's field equations is divergence-free, which matches conservation of energy and momentum.\nIn gauge theory, part 1 says that the field strength $F$ of a gauge field satisfies $D F = 0$; for electromagnetism this gives the two homogeneous Maxwell equations.\nPart 1 is also the key step in showing that the forms $f(\\Omega)$ of Chern–Weil theory are closed.\n",
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