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?