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  "body": "\n## Definition\n\nLet $G$ be a Lie group and $H$ a closed subgroup of $G$, with Lie algebras $\\mathfrak{h} \\subset \\mathfrak{g}$.\nLet $M$ be a smooth manifold with $\\dim M = \\dim G - \\dim H$.\nA *Cartan connection on $M$ modelled on $(G, H)$* is a [[principal $H$-bundle|8E37E74XVP]] $\\pi : P \\to M$ together with a smooth $\\mathfrak{g}$-valued $1$-form $\\omega$ on $P$ such that:\n\n1. $\\omega_u : T_uP \\to \\mathfrak{g}$ is a linear isomorphism for every $u \\in P$.\n2. $R_h^* \\omega = \\mathrm{Ad}(h^{-1}) \\, \\omega$ for every $h \\in H$, where $\\mathrm{Ad}$ is the adjoint representation of $G$ restricted to $H$.\n3. $\\omega(X^\\#) = X$ for every $X \\in \\mathfrak{h}$, where $X^\\#$ is the fundamental vector field of $X$ on $P$.\n\n## Remarks\n\nThe *curvature* of $\\omega$ is $\\Omega = d\\omega + \\frac12 [\\omega \\wedge \\omega]$.\nExample: the model itself, $P = G \\to G/H$ with the left Maurer–Cartan form $\\omega = \\theta$ of $G$.\nClause 1 holds because $\\theta_g = dL_{g^{-1}}$ is an isomorphism $T_gG \\to \\mathfrak{g}$, and clauses 2 and 3 are standard properties of $\\theta$.\nIts curvature vanishes by the Maurer–Cartan equation, and a Cartan connection is locally isomorphic to this model exactly when $\\Omega = 0$.\nNon-example: a principal connection form on $P$ has values in $\\mathfrak{h}$, so it fails clause 1 unless $\\mathfrak{h} = \\mathfrak{g}$ and $M$ is a point.\nFor $G$ the Euclidean group $\\R^n \\rtimes \\mathrm{O}(n)$ and $H = \\mathrm{O}(n)$, a torsion-free Cartan connection is the same as a Riemannian metric.\n",
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