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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with [[Ricci curvature|4GTNM0WHDH]] $\\operatorname{Ric}$.\nIn local coordinates at $p$, let $g_{ij}$ and $R_{ij}$ be the components of $g$ and $\\operatorname{Ric}$, and let $(g^{ij})$ be the inverse matrix of $(g_{ij})$.\nThe *scalar curvature* of $(M, g)$ is the function $S \\colon M \\to \\R$ given by\n$$S(p) = \\sum_{i,j} g^{ij}(p) \\, R_{ij}(p),$$\nthe [[trace|07PTPSZPME]] of $\\operatorname{Ric}$ with respect to $g$.\n\n## Remarks\n\nThe value does not depend on the coordinates; it is the trace of the linear map $A_p \\colon T_pM \\to T_pM$ with $g(A_p u, v) = \\operatorname{Ric}(u, v)$.\nIn the Riemannian case, for an orthonormal basis $e_1, \\ldots, e_n$ of $T_pM$, $S(p) = \\sum_i \\operatorname{Ric}(e_i, e_i) = \\sum_{i \\ne j} K(e_i, e_j)$.\n\nExample: on the unit sphere $S^n$, $\\operatorname{Ric} = (n - 1)g$, so $S = \\sum_{i,j} g^{ij} (n - 1) g_{ij} = (n - 1) n$.\n\nNon-example: the plain sum $\\sum_i R_{ii}$, without the factor $g^{ij}$, depends on the coordinates and is not the scalar curvature.\nOn the sphere of radius $r$ in $\\R^3$, $\\operatorname{Ric} = r^{-2} g$; in coordinates at $p$ with $g_{ij}(p) = r^2 \\delta_{ij}$, one gets $\\sum_i R_{ii}(p) = 2$ for every $r$, while $S = 2 / r^2$.\n\nFor a surface, $S = 2K$, where $K$ is the [[Gaussian curvature|SQVQ5DQT6W]].\n",
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