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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold with [[tangent bundle|WY6K3P4YCG]] $TM$, and let $\\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.\nAn *affine connection* on $M$ is a [[connection|5ZYRRVE8BW]] on the vector bundle $TM$.\nThat is, it is a map $\\nabla : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\mathfrak{X}(M)$, $(X, Y) \\mapsto \\nabla_X Y$, that is $C^\\infty(M)$-linear in $X$, $\\R$-linear in $Y$, and satisfies $\\nabla_X (fY) = (Xf) Y + f \\nabla_X Y$ for every smooth function $f$.\n\n## Remarks\n\nExample: on $\\R^n$ with $Y = \\sum_j Y^j \\partial_j$, the formula $\\nabla_X Y = \\sum_j (X Y^j) \\, \\partial_j$ defines an affine connection, the flat connection; the three conditions follow from the linearity and the product rule of directional derivatives.\nNon-example: $(X, Y) \\mapsto [X, Y]$ is not an affine connection, since $[fX, Y] = f [X, Y] - (Yf) X$ is not $C^\\infty(M)$-linear in $X$.\nIn a chart, $\\nabla$ is given by its Christoffel symbols $\\Gamma^k_{ij}$, defined by $\\nabla_{\\partial_i} \\partial_j = \\Gamma^k_{ij} \\partial_k$.\nEvery smooth manifold has affine connections: glue the flat connections of charts with a partition of unity.\nTwo important tensors measure an affine connection: its torsion $\\nabla_X Y - \\nabla_Y X - [X, Y]$, built with the [[Lie bracket|68FGQEBD6J]], and its curvature.\n",
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