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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $\\operatorname{Ric}$ be its [[Ricci curvature|4GTNM0WHDH]] tensor.\nThe manifold $(M, g)$ is an *Einstein manifold*, and $g$ is an *Einstein metric*, if there is a real constant $\\lambda$ such that\n$$\\operatorname{Ric} = \\lambda g.$$\nThe constant $\\lambda$ is the *Einstein constant*.\n\n## Remarks\n\nExample: the unit sphere $S^n \\subset \\R^{n+1}$ with $n \\ge 2$ has sectional curvature $1$, so $\\operatorname{Ric} = (n-1) g$, and it is Einstein with $\\lambda = n - 1$.\n\nNon-example: the product $S^2 \\times S^3$ of unit spheres has $\\operatorname{Ric} = g$ on vectors tangent to $S^2$ and $\\operatorname{Ric} = 2g$ on vectors tangent to $S^3$.\nNo single constant $\\lambda$ works, so it is not Einstein.\n\nTaking the trace gives $\\lambda = R/n$, where $R$ is the [[scalar curvature|ZHX2VYQDZ3]] and $n = \\dim M$, so an Einstein metric has constant scalar curvature.\nIf $M$ is connected, $n \\ge 3$ and $\\operatorname{Ric} = f g$ for a smooth function $f$, the contracted Bianchi identity forces $f$ to be constant.\nIn dimension $2$ every metric satisfies $\\operatorname{Ric} = K g$ with $K$ the Gaussian curvature, so a surface is Einstein exactly when its curvature is constant.\nThe same definition is used for pseudo-Riemannian metrics.\n",
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