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  "body": "\n## Summary\n\nThe Ricci curvature of a [[Riemannian manifold|7CBT4SM83Y]] is a trace of the [[Riemann curvature tensor|JA29ETCM02]].\nIt is a symmetric bilinear form on each tangent space.\nLet $v$ be a unit vector and $e_2, \\ldots, e_n$ an orthonormal basis of $v^\\perp$.\nThen $\\operatorname{Ric}(v, v)$ is the sum of the [[sectional curvatures|1X42JAA349]] of the planes spanned by $v$ and $e_i$.\nIt controls how the volume of small balls and cones of geodesics differs from the Euclidean value.\nIt appears in Einstein's equations of general relativity, in the Ricci flow, and in theorems such as [[Myers's theorem|FPMJEX3DSJ]] and the [[Bishop–Gromov inequality|82F5Q4F1S6]].\n\n## Explanations\n\n### L1 human\n\nRicci curvature is an average of curvatures in all the planes containing a given direction.\nPositive Ricci curvature means that geodesics leaving a point in nearby directions tend to come together, so volumes grow more slowly than in flat space.\nNegative Ricci curvature means they spread apart and volumes grow faster.\nOn the unit sphere of dimension $n$, the Ricci curvature is $(n - 1)$ times the metric.\n",
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