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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?

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  • $\begingroup$ As far as I know, it is supposed to evolve towards the metric of a sphere. Perhaps one could say the Ricci flow acts like some kind of 'curvature diffusion'. Warning: it's the former physicist in me who speaks... $\endgroup$
    – Sylvain JULIEN
    Commented Oct 24, 2016 at 17:30
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    $\begingroup$ Thank you Sylvain, so if a metric does not solve the equation, $g'(t)=Ric(t)$, does not evolve towards the metric of a sphere? As far as I know, evolve towards the metric of sphere if the curvature is positive everywhere... $\endgroup$
    – MathDG
    Commented Oct 24, 2016 at 18:00
  • $\begingroup$ Maybe it can do so, but not at the same pace or for very specific initial conditions. I prefer to let experts in the field give further precisions. $\endgroup$
    – Sylvain JULIEN
    Commented Oct 24, 2016 at 18:03
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    $\begingroup$ See also the earlier question, "Intuition behind the Ricci flow." $\endgroup$
    – Joseph O'Rourke
    Commented Oct 24, 2016 at 23:00

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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?

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  • $\begingroup$ ...but I do not understand your question $\endgroup$
    – MathDG
    Commented Oct 25, 2016 at 5:26
  • $\begingroup$ The link Joseph O'Rourke's provided above is useful; the question I'm asking is what other differential equations have asymptotic equivalents to the Ricci tensor. I believe (not strongly) that the answer is a subset (?) of parabolic PDEs, but I think the point is that the Ricci tensor is one of the simplest. $\endgroup$
    – Steve
    Commented Oct 25, 2016 at 20:49

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