Editing Ricci flow (section)

===Constant-curvature and Einstein metrics===
Let <math>(M,g)</math> be a Riemannian manifold which is [[Einstein manifold|Einstein]], meaning that there is a number <math>\lambda</math> such that <math>\text{Ric}^g=\lambda g</math>. Then <math>g_t=(1-2\lambda t)g</math> is a Ricci flow with <math>g_0=g</math>, since then
:<math>\frac{\partial}{\partial t}g_t=-2\lambda g=-2\operatorname{Ric}^g=-2\operatorname{Ric}^{g_t}.</math>
If <math>M</math> is closed, then according to Hamilton's uniqueness theorem above, this is the only Ricci flow with initial data <math>g</math>. One sees, in particular, that:
* if <math>\lambda</math> is positive, then the Ricci flow "contracts" <math>g</math> since the scale factor <math>1-2\lambda t</math> is less than 1 for positive <math>t</math>; furthermore, one sees that <math>t</math> can only be less than <math>1/2\lambda</math>, in order that <math>g_t</math> is a Riemannian metric. This is the simplest examples of a "finite-time singularity". 
* if <math>\lambda</math> is zero, which is synonymous with <math>g</math> being Ricci-flat, then <math>g_t</math> is independent of time, and so the maximal interval of existence is the entire real line.
* if <math>\lambda</math> is negative, then the Ricci flow "expands" <math>g</math> since the scale factor <math>1-2\lambda t</math> is greater than 1 for all positive <math>t</math>; furthermore one sees that <math>t</math> can be taken arbitrarily large. One says that the Ricci flow, for this initial metric, is "immortal".
In each case, since the Riemannian metrics assigned to different values of <math>t</math> differ only by a constant scale factor, one can see that the normalized Ricci flow <math>G_s</math> exists for all time and is constant in <math>s</math>; in particular, it converges smoothly (to its constant value) as <math>s\to\infty</math>.

The Einstein condition has as a special case that of constant curvature; hence the particular examples of the sphere (with its standard metric) and hyperbolic space appear as special cases of the above.