What is the geometric meaning, for a metric in function of the time that is a solution of the Ricci flow ($g'(t)=-2Ric(t)$), compared to one that is not?


EXPLANATION
I'm interested to understand, being that not all metrics satisfy the equation, $g'(t)=-2Ric(t)$, what differences there are, from the geometrical point of view, between a metric that is a solution of the Ricci flow, and one that is not.
Because, for example, there may be a family of metrics within which only one is the flow solution and the others are not solution...which "quality" (pass me the term) has from the geometrical point of view this metric that the others do not have?



Geometric meaning of Ricci flow
If $g'(t) \neq -2Ric(t)$, it can be anything else. Ricci flow is in some sense a heat kernel applied to the Riemannian metric tensor, uniformizing it over time; if you start modifying the metric tensor according to whatever you see fit, you will end up with any space you like. You may want to restrict the types of things you're interested in comparing Ricci flow to; for example, a question I don't know off the top of my head: what kind of differential equations admit the same solutions in the limit as Ricci flow, for the metric tensor?


