Ricci flow

Remarks

Example: let $g_0$ be the round metric of the unit sphere $S^n$ with $n \ge 2$, so $\operatorname{Ric}(g_0) = (n - 1) g_0$. Let $g(t) = (1 - 2(n - 1)t)\, g_0$ for $t < \frac{1}{2(n-1)}$. The Ricci tensor does not change when the metric is multiplied by a positive constant, so $\operatorname{Ric}(g(t)) = (n - 1) g_0$. Thus $\partial_t g = -2(n-1) g_0 = -2\operatorname{Ric}(g(t))$, and $g$ is a Ricci flow that shrinks the sphere to a point at $t = \frac{1}{2(n-1)}$. Non-example: $g(t) = (1 + 2(n - 1)t)\, g_0$ has $\partial_t g = +2\operatorname{Ric}(g(t))$, which has the wrong sign. A flat metric, such as that of a flat torus, gives a constant Ricci flow $g(t) = g_0$. In local coordinates the equation reads $\partial_t g_{ij} = -2 R_{ij}$.