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  "body": "\n## Summary\n\nThe scalar curvature of a [[Riemannian manifold|7CBT4SM83Y]] is the function obtained by taking the trace of the [[Ricci curvature|4GTNM0WHDH]] with respect to the metric.\nIt is the simplest curvature invariant: a single number at each point.\nIt measures how the volume of a small geodesic ball differs from the volume of a Euclidean ball of the same radius.\nQuestions such as which manifolds admit metrics of positive scalar curvature are a major theme of geometry and topology.\n\n## Explanations\n\n### L1 human\n\nScalar curvature is the total amount of curving at a point, averaged over all directions.\nIf it is positive, small balls around the point have slightly less volume than flat balls of the same radius; if negative, slightly more.\nFor a surface it is twice the Gaussian curvature.\nThe unit sphere of dimension $n$ has scalar curvature $n(n - 1)$, so the ordinary unit sphere in $\\R^3$ has scalar curvature $2$.\n",
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