Vipul: 2 revisions

2008-05-18T19:47:29Z

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Vipul: /* Ricci flow equation */

2007-03-08T08:10:31Z

Ricci flow equation

← Older revision		Revision as of 08:10, 8 March 2007
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	===Ricci flow equation===		===Ricci flow equation===

	{{~~futrher~~\|[[Ricci flow]]}}		{{further\|[[Ricci flow]]}}

	The Ricci flow equation is given by:		The Ricci flow equation is given by:

Vipul at 07:37, 8 March 2007

2007-03-08T07:37:43Z

New page

==Definition==

===Definition with symbols===

An '''intrinsic flow equation''' is a differential equation:

<math>F(g,t)=0</math>

where <math>g</math> is the metric tensor, <math>t</math> isa time-parameter, and <math>F</math> could involve taking any of the tensors arising naturally out of the metric tensor, and of course the various derivatives.

The interpretation of the intrinsic flow equation is as follows. Take any [[differential manifold]] <math>M</math> and consider a [[Riemannian metric]] <math>g(t)</math> on <math>M</math> that varies with <math>t</math>. This is a [[flow of a metric|flow]] of the Riemannian metric on the differential manifold.

The flow satisfies the intrinsic flow equation if, for any point <math>p \in M</math>, <math>F_p(g,t) = 0</math> where differentials with respect to <math>t</math> are interpreted in the natural sense.

==Examples==

===Ricci flow equation===

{{futrher|[[Ricci flow]]}}

The Ricci flow equation is given by:

<math> \frac{\partial}{\partial t} g_{ij}(x,t) = -2R_{ij} (x,t) </math>

The stationary points for the corresponding flow on any manifold are precisely the [[constant-curvature metric]]s.

===Volume-normalized Ricci flow equation===

{{further|[[volume-normalized Ricci flow]]}}

<math>\frac{\partial}{\partial t} g_{ij} = 2 (\frac{r}{n} g_{ij} - R_{ij})</math>

Intrinsic flow equation - Revision history

Vipul: 2 revisions

Vipul: /* Ricci flow equation */

Vipul at 07:37, 8 March 2007