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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]].\nThe *metric cone* over $M$ is the manifold $C(M) = (0, \\infty) \\times M$ with the metric $\\bar g = dr^2 + r^2 g$, where $r$ is the coordinate on $(0, \\infty)$.\nThe manifold $(M, g)$ is *Sasakian* if $C(M)$ has an integrable almost complex structure $J$ with two properties.\nThe metric $\\bar g$ is a [[Kähler metric|CK1DCPC3D3]] for $J$, and $J$ is invariant under the dilations $(r, x) \\mapsto (\\lambda r, x)$ for $\\lambda \u003e 0$.\n\n## Remarks\n\nExample: the unit sphere $S^{2n+1} \\subset \\C^{n+1}$ with the round metric.\nThe map $(r, x) \\mapsto r x$ identifies $C(S^{2n+1})$ with $\\C^{n+1} \\setminus \\set{0}$, and it carries $dr^2 + r^2 g$ to the flat metric, which is Kähler.\n\nNon-example: the round sphere $S^2$ is not Sasakian.\nIts cone is $\\R^3 \\setminus \\set{0}$, of odd dimension $3$, so it has no almost complex structure at all.\n\nA Sasakian manifold has odd dimension $2n + 1$.\nThe vector field $\\xi = J(r \\partial_r)$, restricted to $\\set{1} \\times M$, is a unit Killing field, the [[Reeb vector field|GGM63KXCF2]] of the contact form $\\eta = g(\\xi, \\cdot)$, which makes $M$ a [[contact manifold|HBQ8BZ7XB5]].\n",
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