I am having some issues with the vocabulary employed by authors when they refer to some solutions of the Ricci flow equation. For instance, the shrinking sphere example.
It seems odd to me when I read "the sphere shrinks to a point since $\partial_t g = -2 \text{Ric}(g)$ yields a solution of the form $g(t) = (1-\lambda t)g_0$, where $\lambda >0$, which goes to zero as $t\to 1/\lambda$".
Indeed, the Ricci flow equation acts on the metric alone and considering the vector field $\partial_t$ to solve the Ricci flow equation implicitly requires (at least from my understanding) a fixed manifold.
Instead, my understanding of the fact that one can see this as a shrinking sphere comes from the following:
- One may allow the manifold to vary with time, then the one-parameter family $M(t) = (1-\lambda t)\mathbb{S}^n$ for $t \in [0,1/\lambda)$ admits precisely $g(t) = (1-\lambda t)g_0$ as its canonical metric. In this setting the Ricci flow equation is slightly different ($\partial_t$ is replaced by the Lie derivative of some appropriate time vector field $v$, i.e $\mathcal{L}_v$) however one can check that $g(t)$ still solves the equation.
Since I did not find this argument clearly stated, could anyone tells me if this is the right way of understanding the situation? Or else, what am I missing? Thank you for the help.