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  "body": "\n## Summary\n\nA Sasakian manifold is an odd-dimensional Riemannian manifold whose metric cone is a Kähler manifold.\nIt is the odd-dimensional partner of Kähler geometry, and it carries a natural [[contact structure|HBQ8BZ7XB5]] and a [[Reeb vector field|GGM63KXCF2]].\nThe round odd-dimensional spheres are the basic examples.\nSasaki–Einstein manifolds are of interest in the study of Einstein metrics and in physics.\n\n## Explanations\n\n### L1 human\n\nTake a manifold $M$ and build a cone over it, like a cone over a circle in the plane.\nThe cone has one dimension more than $M$, and its metric grows with the distance $r$ from the tip.\n$M$ is Sasakian when this cone is a Kähler manifold, a complex manifold with a compatible metric whose fundamental $2$-form is closed.\nThe cone over the unit sphere $S^{2n+1}$ is $\\C^{n+1}$ with the origin removed, so the sphere is Sasakian.\n",
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