Ricci flow - Revision history

Vipul: 10 revisions

2008-05-18T19:52:06Z

10 revisions

← Older revision	Revision as of 19:52, 18 May 2008
(No difference)

Vipul at 14:47, 2 September 2007

2007-09-02T14:47:32Z

← Older revision		Revision as of 14:47, 2 September 2007
Line 1:		Line 1:
			{{Riemannian metric flow}}

	==Definition==		==Definition==

Vipul: /* Effect on Einstein metrics */

2007-09-02T14:44:55Z

Effect on Einstein metrics

← Older revision		Revision as of 14:44, 2 September 2007
Line 26:		Line 26:

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

			Here <math>\lambda</math> is the cosmological constant of the Einstein metric.

	This simply describes an exponential decay of <math>g_{ij}(x,t)</math> with time. Hence, the [[conformal class]] of the Riemannian structure does not change. Hence, Einstein metrics, while not invariant under the Ricci flow, are invariant upto the conformal structure.		This simply describes an exponential decay of <math>g_{ij}(x,t)</math> with time. Hence, the [[conformal class]] of the Riemannian structure does not change. Hence, Einstein metrics, while not invariant under the Ricci flow, are invariant upto the conformal structure.

	On the other hand, the [[volume-normalized Ricci flow]] actually does have Einstein metrics as the stationary points.		On the other hand, the [[volume-normalized Ricci flow]] actually does have Einstein metrics as the stationary points.

Vipul: /* Definition */

2007-09-02T09:12:18Z

Definition

← Older revision		Revision as of 09:12, 2 September 2007
Line 1:		Line 1:
	==Definition==		==Definition==

	The Ricci flow is a [[flow]] on the [[space of all Riemannian metric]]s on a given [[differential manifold]]. Note that the space of all Riemannian metrics is a [[Banach manifold]].		The Ricci flow is a [[flow]] on the [[space of all Riemannian metric]]s on a given [[differential manifold]]. Note that the space of all Riemannian metrics is a [[Banach manifold]]. More precisely, the space of Riemannian metrics is contained inside the space of all sections of the second tensor power of the [[cotangent bundle]], because a Riemannian metric is after all a <math>(0,2)</math>-tensor.

	The Ricci flow is governed by the following differential equation:		The Ricci flow is governed by the following differential equation:

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

			In other words, the rate at which the Riemannian metric tensor is changing with time, is <math>-2</math> times the [[Ricci curvature tensor]] at that time (both are <math>(0,2)</math>-tensors).

	==Facts==		==Facts==

Vipul: /* Facts */

2007-09-02T09:09:07Z

Facts

← Older revision		Revision as of 09:09, 2 September 2007
Line 9:		Line 9:
	==Facts==		==Facts==

	The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a ~~metric of~~ constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature ~~metric]]s~~ are the same as the [[Einstein ~~metric]]s~~.		The motivation behind the Ricci flow is to evolve any given [[Riemannian metric]] on the [[differential manifold]] to a [[constant-curvature metric]], or at least, to an [[Einstein metric]]. In dimension 2, the constant-curvature metrics are the same as the Einstein metrics.

	{{stationarypoint\|[[Ricci-flat metric]]}}		{{stationarypoint\|[[Ricci-flat metric]]}}
Line 25:		Line 25:
	<math> \frac{\partial}{\partial t} g_{ij}(x,t) = -2\lambda g_{ij} (x,t) </math>		<math> \frac{\partial}{\partial t} g_{ij}(x,t) = -2\lambda g_{ij} (x,t) </math>

	This simply describes an exponential decay of <math>g_{ij}(x,t)</math> with time. Hence, the ''conformal class'' of the Riemannian structure does not change. Hence, Einstein metrics, while not invariant under the Ricci flow, are		This simply describes an exponential decay of <math>g_{ij}(x,t)</math> with time. Hence, the [[conformal class]] of the Riemannian structure does not change. Hence, Einstein metrics, while not invariant under the Ricci flow, are invariant upto the conformal structure.
	invariant upto the conformal structure.

	On the other hand, the [[volume-normalized Ricci flow]] actually does have Einstein metrics as the stationary points.		On the other hand, the [[volume-normalized Ricci flow]] actually does have Einstein metrics as the stationary points.

Vipul at 09:05, 2 September 2007

2007-09-02T09:05:39Z

← Older revision		Revision as of 09:05, 2 September 2007
Line 1:		Line 1:
	==~~Description~~==		==Definition==

	~~===Equation for the~~ Ricci flow~~===~~		The Ricci flow is a [[flow]] on the [[space of all Riemannian metric]]s on a given [[differential manifold]]. Note that the space of all Riemannian metrics is a [[Banach manifold]].

	The ~~equation for the~~ Ricci flow ~~in any dimension~~ is:		The Ricci flow is governed by the following differential equation:

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

Vipul at 15:36, 8 April 2007

2007-04-08T15:36:05Z

← Older revision		Revision as of 15:36, 8 April 2007
Line 5:		Line 5:
	The equation for the Ricci flow in any dimension is:		The equation for the Ricci flow in any dimension is:

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

	==Facts==		==Facts==

Vipul: /* Facts */

2007-03-07T19:43:30Z

Facts

← Older revision		Revision as of 19:43, 7 March 2007
Line 8:		Line 8:

	==Facts==		==Facts==

	~~{{stationarypoint\|[[Ricci-flat metric]]}}~~

	The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.		The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.

	~~===Effect on~~ Ricci-flat ~~metrics===~~		{{stationarypoint\|[[Ricci-flat metric]]}}

	For a Ricci-flat metric, the equation becomes:		For a Ricci-flat metric, the equation becomes:
Line 22:		Line 20:

	===Effect on Einstein metrics===		===Effect on Einstein metrics===
	Notice that in an Einstein metric, <math>R_{ij}</math> is proportional to <math>g_{ij}</math>. So if we consider the above Ricci flow for an Einstein metric, we get:
			Notice that in an [[Einstein metric]], <math>R_{ij}</math> is proportional to <math>g_{ij}</math>. So if we consider the above Ricci flow for an Einstein metric, we get:

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

Vipul at 19:42, 7 March 2007

2007-03-07T19:42:41Z

← Older revision		Revision as of 19:42, 7 March 2007
Line 8:		Line 8:

	==Facts==		==Facts==

			{{stationarypoint\|[[Ricci-flat metric]]}}

	The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.		The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.

Vipul at 19:32, 7 March 2007

2007-03-07T19:32:58Z

← Older revision		Revision as of 19:32, 7 March 2007
Line 7:		Line 7:
	<math> \frac{\partial}{\partial t} g_{ij}(x,t) = -2R_{ij} (x,t) </math>		<math> \frac{\partial}{\partial t} g_{ij}(x,t) = -2R_{ij} (x,t) </math>

	==~~Relation with Einstein manifolds~~==		==Facts==

	The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.		The motivation behind the Ricci flow is to evolve any given metric on the differential manifold to a metric of constant curvature, or at least, to an Einstein metric. In dimension 2, the [[constant-curvature metric]]s are the same as the [[Einstein metric]]s.