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  "body": "\n## Summary\n\nAn Einstein manifold is a [[Riemannian manifold|7CBT4SM83Y]] whose [[Ricci curvature|4GTNM0WHDH]] is a constant multiple of the metric.\nThe condition is the vacuum Einstein equation of general relativity, with a cosmological constant, written for a Riemannian metric.\nEinstein metrics are natural candidates for a \"best\" metric on a manifold, and their existence is a central question of Riemannian geometry.\n\n## Explanations\n\n### L1 human\n\nThe Ricci curvature measures how the volume of small balls differs from the flat case, direction by direction.\nOn an Einstein manifold this effect is the same in every direction and at every point.\nThe round sphere is the basic example: it looks the same everywhere and in every direction.\nFlat space is also Einstein, with constant zero.\nA product of two round spheres of different dimensions and the same radius is not Einstein, because the two factors bend volumes by different amounts.\n",
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