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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold with an [[affine connection|JX65KSVHSJ]] $\\nabla$, and let $\\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.\nThe *torsion* of $\\nabla$ is the map $T : \\mathfrak{X}(M) \\times \\mathfrak{X}(M) \\to \\mathfrak{X}(M)$ given by\n$$T(X, Y) = \\nabla_X Y - \\nabla_Y X - [X, Y],$$\nwhere $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]].\nThe connection $\\nabla$ is *torsion-free* if $T = 0$.\n\n## Remarks\n\n$T$ is $C^\\infty(M)$-linear in each argument: for a smooth function $f$, $T(fX, Y) = f \\nabla_X Y - (Yf) X - f \\nabla_Y X - f [X, Y] + (Yf) X = f \\, T(X, Y)$.\nSo $T$ is a [[tensor field|WX302T93TW]] of type $(1, 2)$, and $T(Y, X) = -T(X, Y)$.\nIn a chart, $T(\\partial_i, \\partial_j) = \\sum_k (\\Gamma^k_{ij} - \\Gamma^k_{ji}) \\partial_k$, since coordinate vector fields commute.\nExample: the flat connection of $\\R^n$ has $\\Gamma^k_{ij} = 0$, so it is torsion-free.\nNon-example: on $\\R^3$ the connection $\\nabla_X Y = D_X Y + X \\times Y$, with $D$ the flat connection and $\\times$ the cross product, is a connection but not torsion-free, since $T(X, Y) = 2 \\, X \\times Y \\ne 0$ for independent $X, Y$.\n",
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