Editing Ricci flow (section)

== Mathematical definition ==
On a [[smooth manifold]] {{mvar|M}}, a smooth [[Riemannian manifold|Riemannian metric]] {{mvar|g}} automatically determines the [[Ricci tensor]] {{math|Ric<sup>''g''</sup>}}. For each element {{mvar|p}} of {{mvar|M}}, by definition {{math|''g''<sub>''p''</sub>}} is a [[Inner product space|positive-definite inner product]] on the [[tangent space]] {{math|''T''<sub>''p''</sub>''M''}} at {{mvar|p}}. If given a one-parameter family of Riemannian metrics {{math|''g''<sub>''t''</sub>}}, one may then consider the derivative {{math|{{sfrac|∂|∂''t''}} ''g''<sub>''t''</sub>}}, which then assigns to each particular value of {{mvar|t}} and {{mvar|p}} a [[symmetric bilinear form]] on {{math|''T''<sub>''p''</sub>''M''}}. Since the Ricci tensor of a Riemannian metric also assigns to each {{mvar|p}} a symmetric bilinear form on {{math|''T''<sub>''p''</sub>''M''}}, the following definition is meaningful.
* Given a smooth manifold {{mvar|M}} and an open real interval {{math|{{open-open|''a'', ''b''}}}}, a '''Ricci flow''' assigns, to each {{mvar|t}} in the interval {{math|{{open-open|''a'',''b''}}}}, a Riemannian metric {{math|''g''<sub>''t''</sub>}} on {{mvar|M}} such that {{math|{{sfrac|∂|∂''t''}} ''g''<sub>''t''</sub> {{=}} −2 Ric<sup>''g''<sub>''t''</sub></sup>}}.
The Ricci tensor is often thought of as an average value of the [[sectional curvature]]s, or as an algebraic [[trace (linear algebra)|trace]] of the [[Riemann curvature tensor]]. However, for the analysis of existence and uniqueness of Ricci flows, it is extremely significant that the Ricci tensor can be defined, in local coordinates, by a formula involving the first and second derivatives of the metric tensor. This makes the Ricci flow into a geometrically-defined [[partial differential equation]]. The analysis of the [[Elliptic operator|ellipticity]] of the local coordinate formula provides the foundation for the existence of Ricci flows; see the following section for the corresponding result.

Let {{mvar|k}} be a nonzero number. Given a Ricci flow {{math|''g''<sub>''t''</sub>}} on an interval {{math|{{open-open|''a'',''b''}}}}, consider {{math|''G''<sub>''t''</sub> {{=}} ''g''<sub>''kt''</sub>}} for {{mvar|t}} between {{math|{{sfrac|''a''|''k''}}}} and {{math|{{sfrac|''b''|''k''}}}}. Then {{math|{{sfrac|∂|∂''t''}} ''G''<sub>''t''</sub> {{=}} −2''k'' Ric<sup>''G''<sub>''t''</sub></sup>}}. So, with this very trivial change of parameters, the number −2 appearing in the definition of the Ricci flow could be replaced by any other nonzero number. For this reason, the use of −2 can be regarded as an arbitrary convention, albeit one which essentially every paper and exposition on Ricci flow follows. The only significant difference is that if −2 were replaced by a positive number, then the existence theorem discussed in the following section would become a theorem which produces a Ricci flow that moves backwards (rather than forwards) in parameter values from initial data.

The parameter {{mvar|t}} is usually called {{em|time}}, although this is only as part of standard informal terminology in the mathematical field of partial differential equations. It is not physically meaningful terminology. In fact, in the standard [[Quantum field theory|quantum field theoretic]] interpretation of the Ricci flow in terms of the [[renormalization group]], the parameter {{mvar|t}} corresponds to length or energy, rather than time.<ref>{{cite journal |last=Friedan|first=D.|author-link=Daniel Friedan| title=Nonlinear models in 2+ε dimensions | journal =  Physical Review Letters| volume = 45 | issue = 13| pages = 1057–1060 | year = 1980 |doi= 10.1103/PhysRevLett.45.1057 | bibcode=1980PhRvL..45.1057F | url=https://digital.library.unt.edu/ark:/67531/metadc841801/| type=Submitted manuscript}}</ref>

===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.