I'm new to this subject, and I'm a bit confused about how the metric changing corresponds to the manifold itself changing. For example, in Topping's note it says:
A simple example of a Ricci flow is that starting from a round sphere. This will evolve by shrinking homothetically to a point in finite time.
When I first looked at the equation $\frac{\partial g}{\partial t}=-2Ric(g)$, I thought that on a fixed manifold, we are defining the metrics $g_t$. But all the figures in chapter 1 suggest that the manifold itself is changing.
Also, if the manifold is changing, how can we define fixed vector fields (independent of time) on it, like Topping's note does in chapter 2?