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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold and let $I \\subseteq \\R$ be an interval.\nA *Ricci flow* on $M$ is a family $(g(t))_{t \\in I}$ of [[Riemannian metrics|Q61VFHCH2J]] on $M$ with these properties.\nThe metric $g(t)$ depends smoothly on $t$, and\n$$\\frac{\\partial g}{\\partial t}(t) = -2 \\operatorname{Ric}\\big(g(t)\\big) \\quad \\text{for all } t \\in I,$$\nwhere $\\operatorname{Ric}(g)$ is the [[Ricci curvature|4GTNM0WHDH]] tensor of $g$.\n\n## Remarks\n\nExample: let $g_0$ be the round metric of the unit sphere $S^n$ with $n \\ge 2$, so $\\operatorname{Ric}(g_0) = (n - 1) g_0$.\nLet $g(t) = (1 - 2(n - 1)t)\\, g_0$ for $t \u003c \\frac{1}{2(n-1)}$.\nThe Ricci tensor does not change when the metric is multiplied by a positive constant, so $\\operatorname{Ric}(g(t)) = (n - 1) g_0$.\nThus $\\partial_t g = -2(n-1) g_0 = -2\\operatorname{Ric}(g(t))$, and $g$ is a Ricci flow that shrinks the sphere to a point at $t = \\frac{1}{2(n-1)}$.\nNon-example: $g(t) = (1 + 2(n - 1)t)\\, g_0$ has $\\partial_t g = +2\\operatorname{Ric}(g(t))$, which has the wrong sign.\nA flat metric, such as that of a flat torus, gives a constant Ricci flow $g(t) = g_0$.\nIn local coordinates the equation reads $\\partial_t g_{ij} = -2 R_{ij}$.\n",
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