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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 *curvature tensor* of $\\nabla$ is the map $R : \\mathfrak{X}(M)^3 \\to \\mathfrak{X}(M)$ given by\n$$R(X, Y) Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z,$$\nwhere $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]].\nWhen $(M, g)$ is a [[Riemannian manifold|7CBT4SM83Y]] and $\\nabla$ is its [[Levi-Civita connection|CAZYGE1T4E]], $R$ is the *Riemann curvature tensor* of $(M, g)$.\n\n## Remarks\n\n$R$ is $C^\\infty(M)$-linear in $X$, $Y$ and $Z$, so it is a [[tensor field|WX302T93TW]] of type $(1, 3)$, and $R(Y, X) = -R(X, Y)$.\nSome books use the opposite sign; the convention here gives positive sectional curvature $g(R(X, Y) Y, X) \u003e 0$ on the sphere.\nExample: for the flat connection of $\\R^n$, $\\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z = \\nabla_{[X, Y]} Z$ by symmetry of second derivatives, so $R = 0$.\nFor the round sphere of radius $r$, $R(X, Y) Z = r^{-2} \\big( g(Y, Z) X - g(X, Z) Y \\big)$.\nNon-example of a tensor: the expression $\\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z$ alone is not $C^\\infty(M)$-linear in $X$, as replacing $X$ by $fX$ adds $-(Yf) \\nabla_X Z$; the term $\\nabla_{[X, Y]} Z$ removes this.\nIn a chart, $R(\\partial_i, \\partial_j) \\partial_k = \\sum_l R^l{}_{kij} \\partial_l$ with $R^l{}_{kij} = \\partial_i \\Gamma^l_{jk} - \\partial_j \\Gamma^l_{ik} + \\sum_m (\\Gamma^l_{im} \\Gamma^m_{jk} - \\Gamma^l_{jm} \\Gamma^m_{ik})$.\nFor the Levi-Civita connection, $R_{abcd} = g(R(X_c, X_d) X_b, X_a)$ is antisymmetric in $(a, b)$ and in $(c, d)$ and symmetric under $(a, b) \\leftrightarrow (c, d)$.\n",
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