Editing Ricci flow (section)

===Normalized Ricci flow===
Suppose that {{mvar|M}} is a compact smooth manifold, and let {{math|''g''<sub>''t''</sub>}} be a Ricci flow for {{mvar|t}} in the interval {{math|{{open-open|''a'', ''b''}}}}. Define {{math|Ψ:{{open-open|''a'', ''b''}}&nbsp;→&nbsp;{{open-open|0, ∞}}}} so that each of the Riemannian metrics {{math|Ψ(''t'')''g''<sub>''t''</sub>}} has volume 1; this is possible since {{mvar|M}} is compact. (More generally, it would be possible if each Riemannian metric {{math|''g''<sub>''t''</sub>}} had finite volume.) Then define {{math|''F'':{{open-open|''a'', ''b''}}&nbsp;→&nbsp;{{open-open|0, ∞}}}} to be the antiderivative of {{math|Ψ}} which vanishes at {{mvar|a}}. Since {{math|Ψ}} is positive-valued, {{mvar|F}} is a bijection onto its image {{math|{{open-open|0, ''S''}}}}. Now the Riemannian metrics {{math|''G''<sub>''s''</sub>&nbsp; {{=}} &nbsp;Ψ(''F''<sup> −1</sup>(''s''))''g''<sub>''F''<sup> −1</sup>(''s'')</sub>}}, defined for parameters {{math|''s''&nbsp;∈&nbsp;(0, ''S'')}}, satisfy
<math display="block">\frac{\partial}{\partial s} G_s = -2\operatorname{Ric}^{G_s} +\frac{2}{n} \frac{\int_M R^{G_s}\,d\mu_{G_s}}{\int_M d\mu_{G_s}} G_s.</math>
Here {{mvar|R}} denotes [[scalar curvature]]. This is called the '''normalized Ricci flow''' equation. Thus, with an explicitly defined change of scale {{math|Ψ}} and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.

The primary reason for considering the normalized Ricci flow is that it allows a convenient statement of the major convergence theorems for Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form. Moreover, the normalized Ricci flow is not generally meaningful on noncompact manifolds.