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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \\ge 3$.\nLet $\\operatorname{Rm}(X, Y, Z, V) = g(R(X, Y)Z, V)$ be its [[Riemann curvature tensor|JA29ETCM02]], with $R(X, Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z$.\nLet $\\operatorname{Ric}$ be its [[Ricci curvature|4GTNM0WHDH]] and $S$ its [[scalar curvature|ZHX2VYQDZ3]].\nThe *Schouten tensor* is\n$$P = \\frac{1}{n - 2} \\left( \\operatorname{Ric} - \\frac{S}{2(n - 1)} \\, g \\right).$$\nFor symmetric $(0,2)$-tensors $h, k$, their Kulkarni–Nomizu product, written here $h \\odot k$, is the $(0,4)$-tensor\n$$(h \\odot k)(X, Y, Z, V) = h(X, V) k(Y, Z) + h(Y, Z) k(X, V) - h(X, Z) k(Y, V) - h(Y, V) k(X, Z).$$\nThe *Weyl tensor* of $(M, g)$ is the $(0,4)$-tensor\n$$W = \\operatorname{Rm} - P \\odot g.$$\n\n## Remarks\n\nExample: the unit sphere $S^n$, $n \\ge 3$, has $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so $\\operatorname{Rm} = \\tfrac{1}{2} \\, g \\odot g$, $\\operatorname{Ric} = (n - 1)g$ and $S = n(n - 1)$.\nThen $P = \\tfrac{1}{2} g$ and $P \\odot g = \\tfrac{1}{2} \\, g \\odot g = \\operatorname{Rm}$, so $W = 0$.\n\nNon-example: for $n = 2$ the Schouten tensor has the factor $1/(n - 2)$ and is not defined, so surfaces have no Weyl tensor in this sense.\n\nThe Weyl tensor has the symmetries of the curvature tensor and all its traces vanish.\nFor $n = 3$ it is always zero.\nFor $n \\ge 4$, $W = 0$ if and only if each point has a neighbourhood on which $g$ is a positive function times a flat metric.\nUnder a change $\\tilde g = e^{2f} g$, the Weyl tensor changes by $\\tilde W = e^{2f} W$.\nSign and index conventions for $\\operatorname{Rm}$ differ between texts, and the formula for $W$ changes accordingly.\n",
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