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  "body": "\n## Summary\n\nThe Cotton tensor is defined on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension at least $3$.\nIt is a $3$-index tensor built from the covariant derivatives of the Schouten tensor, a modified [[Ricci curvature|4GTNM0WHDH]].\nIn dimension $3$ the [[Weyl tensor|SBHY31HESP]] is always zero, and the Cotton tensor takes over its role.\nA $3$-dimensional metric is locally conformally flat if and only if its Cotton tensor vanishes.\nIn higher dimensions it is, up to a constant factor, the divergence of the Weyl tensor.\n\n## Explanations\n\n### L1 human\n\nIn three dimensions all of the curvature is determined by the Ricci curvature, so the usual test for being conformally flat, the Weyl tensor, says nothing.\nThe Cotton tensor is the replacement test.\nIt looks at how the Ricci curvature changes from point to point, through its derivatives.\nA space where the Ricci curvature is a constant multiple of the metric, such as the round $3$-sphere, has zero Cotton tensor.\n",
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