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  "body": "\n## Definition\n\nLet $G$ be a Lie group with Lie algebra $\\mathfrak{g}$, let $P \\to M$ be a principal $G$-bundle, and let $\\omega$ be a [[connection form|NJ21PN5ZGJ]] on $P$.\nFor $\\mathfrak{g}$-valued $1$-forms $\\alpha, \\beta$ on $P$, let $[\\alpha \\wedge \\beta]$ be the $\\mathfrak{g}$-valued $2$-form $[\\alpha \\wedge \\beta](X, Y) = [\\alpha(X), \\beta(Y)] - [\\alpha(Y), \\beta(X)]$.\nThe *curvature form* of $\\omega$ is the $\\mathfrak{g}$-valued $2$-form\n$$\\Omega = d\\omega + \\tfrac12 [\\omega \\wedge \\omega],$$\nwhere $d$ is the [[exterior derivative|0ZHSTAG8ZY]].\nThe connection is *flat* if $\\Omega = 0$.\n\n## Remarks\n\nUnwinding the bracket, $\\Omega(X, Y) = d\\omega(X, Y) + [\\omega(X), \\omega(Y)]$.\nThis is the structure equation; an equivalent form is $\\Omega(X, Y) = d\\omega(hX, hY)$, where $h$ is the projection of $TP$ onto the horizontal bundle $\\ker \\omega$.\nThe form $\\Omega$ vanishes on vertical vectors and satisfies $R_g^* \\Omega = \\mathrm{Ad}(g^{-1}) \\Omega$.\nFor a matrix group, $\\frac12 [\\omega \\wedge \\omega] = \\omega \\wedge \\omega$ with the matrix [[wedge product|0GRAATMGP1]], so $\\Omega = d\\omega + \\omega \\wedge \\omega$.\nExample: on $M \\times G$ with $\\omega$ the pullback of the Maurer–Cartan form $\\theta$, the Maurer–Cartan equation $d\\theta + \\frac12 [\\theta \\wedge \\theta] = 0$ gives $\\Omega = 0$, so this connection is flat.\nNon-example of flatness: for $G = \\mathrm{U}(1)$ the bracket vanishes and $\\Omega = d\\omega$; for a connection on the Hopf bundle $S^3 \\to S^2$, $\\Omega$ descends to a $2$-form on $S^2$ with nonzero integral, so $\\Omega \\ne 0$.\n",
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