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  "body": "\n## Definition\n\nLet $G$ be a Lie group with Lie algebra $\\mathfrak{g}$, and let $\\pi : P \\to M$ be a smooth [[principal $G$-bundle|8E37E74XVP]] with right action $R_g(u) = u \\cdot g$.\nFor $X \\in \\mathfrak{g}$, the fundamental vector field $X^\\#$ on $P$ is $X^\\#_u = \\frac{d}{dt}\\big|_{t=0} \\, u \\cdot \\exp(tX)$, where $\\exp$ is the exponential map.\nA *connection form* on $P$ is a smooth $\\mathfrak{g}$-valued [[differential $1$-form|0E3G0GRPZ6]] $\\omega$ on $P$ such that:\n\n1. $\\omega(X^\\#) = X$ for every $X \\in \\mathfrak{g}$.\n2. $R_g^* \\omega = \\mathrm{Ad}(g^{-1}) \\, \\omega$ for every $g \\in G$, where $\\mathrm{Ad}$ is the adjoint representation.\n\nA *connection* on $P$ is a connection form on $P$.\n\n## Remarks\n\nEquivalently, a connection is an [[Ehresmann connection|TD6VEP3SCE]] $H \\subset TP$ with $dR_g(H_u) = H_{u \\cdot g}$ for all $u$ and $g$; the two notions correspond by $H = \\ker \\omega$.\nExample: on the trivial bundle $P = M \\times G$, let $\\theta$ be the left-invariant Maurer–Cartan form of $G$, so $\\theta_g(v) = dL_{g^{-1}}(v)$, and let $\\omega = \\mathrm{pr}_2^* \\theta$.\nClause 1 holds since $X^\\#_{(p, g)} = (0, dL_g X)$, and clause 2 holds since $R_h^* \\theta = \\mathrm{Ad}(h^{-1}) \\theta$.\nNon-example: $\\omega = 0$ satisfies clause 2 but fails clause 1 when $\\dim G \u003e 0$.\nFor a matrix group the trivial example reads $\\omega = g^{-1} dg$.\nThe difference of two connection forms is a $\\mathfrak{g}$-valued $1$-form that vanishes on vertical vectors and satisfies clause 2; so connections on $P$ form an affine space.\n",
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