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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with curvature $R(X, Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]).\nThe *Ricci curvature* of $(M, g)$ is the $(0,2)$-tensor field $\\operatorname{Ric}$ defined, for $p \\in M$ and $u, v \\in T_pM$, by\n$$\\operatorname{Ric}(u, v) = \\tr\\bigl( w \\mapsto R(w, u)v \\bigr),$$\nthe [[trace|07PTPSZPME]] of the linear map $T_pM \\to T_pM$, $w \\mapsto R(w, u)v$.\n\n## Remarks\n\nIf $e_1, \\ldots, e_n$ is a $g$-orthonormal basis of $T_pM$ in the Riemannian case, then $\\operatorname{Ric}(u, v) = \\sum_i g(R(e_i, u)v, e_i)$, and for a unit vector $v = e_1$ this gives $\\operatorname{Ric}(v, v) = \\sum_{i \\ge 2} K(e_i, v)$, a sum of [[sectional curvatures|1X42JAA349]].\nIn coordinates, $R_{ij} = \\sum_k R^k{}_{kij}$ where $R(\\partial_k, \\partial_i)\\partial_j = \\sum_l R^l{}_{kij} \\partial_l$.\nThe Ricci tensor is symmetric, by the first Bianchi identity.\n\nExample: on the unit sphere $S^n$, $R(w, u)v = g(u, v)w - g(w, v)u$, whose trace in $w$ is $n \\, g(u, v) - g(u, v)$, so $\\operatorname{Ric} = (n - 1)g$.\n\nNon-example: $\\operatorname{Ric}(u, v) = \\tr(w \\mapsto R(u, v)w)$ is a different trace, and it is identically zero because $R(u, v)$ is skew-adjoint; it is not the Ricci curvature.\n\nA metric with $\\operatorname{Ric} = \\lambda g$ for a constant $\\lambda$ is an [[Einstein metric|41S2EZ0KR3]].\n",
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