Ricci curvature - Revision history

Vipul: /* In terms of Ricci curvature tensor */

2012-01-06T21:05:55Z

In terms of Ricci curvature tensor

← Older revision		Revision as of 21:05, 6 January 2012
Line 3:		Line 3:
	===In terms of Ricci curvature tensor===		===In terms of Ricci curvature tensor===

	Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math>. The Ricci curvature on <math>g</math> is a function from <math>\mathbb{P}(TM)</math> (the set of tangent directions) to <math>\R</math> (~~tangent directions at points, to~~ real numbers) that associates to a particular tangent direction the value <math>Ric(X,X)</math> where <math>X</math> is a unit tangent vector in that direction.		Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math>. The Ricci curvature on <math>g</math> is a function from <math>\mathbb{P}(TM)</math> (the set of tangent directions) to <math>\R</math> (real numbers) that associates to a particular tangent direction the value <math>Ric(X,X)</math> where <math>X</math> is a unit tangent vector in that direction.

	===In terms of sectional curvature===		===In terms of sectional curvature===

Vipul: /* Facts */

2008-05-22T12:52:21Z

Facts

← Older revision		Revision as of 12:52, 22 May 2008
Line 31:		Line 31:
	''This is analogous to how the [[sectional curvature]] determined the [[Riemman curvature tensor of Levi-Civita connection\|Riemann curvature tensor]]''		''This is analogous to how the [[sectional curvature]] determined the [[Riemman curvature tensor of Levi-Civita connection\|Riemann curvature tensor]]''

	~~By the polarization trick, we can compute the~~ Ricci curvature ~~tensor from the~~ Ricci curvature. This ~~is based~~ on ~~the following facts~~:		{{further\|[[Ricci curvature determines Ricci curvature tensor]]}}

			This rests on two observations:

	* The [[Ricci curvature tensor]] is symmetric		* The [[Ricci curvature tensor]] is symmetric
Line 38:		Line 40:
	<math>b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y))</math>		<math>b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y))</math>

	* In particular, it is determined by the values taken at all pairs <math>(X,X)</math> for a unit vector <math>X</math> because every vector is a scalar multiple of a unit vector		It is also easy to see that:

	The determination of the Ricci curvature tensor from the Ricci curvature given above can be done point-wise, i.e. given the Ricci curvature of all unit tangent vectors at a point, we can compute the Ricci curvature tensor as a bilinear form on that tangent space. In particular, it is easy to see that:

	* If the Ricci curvature is constant on all unit tangent vectors at a point, then the Ricci curvature tensor at that point is that constant times the [[Riemannian metric]] restricted to that tangent space		* If the Ricci curvature is constant on all unit tangent vectors at a point, then the Ricci curvature tensor at that point is that constant times the [[Riemannian metric]] restricted to that tangent space

Vipul: 4 revisions

2008-05-18T19:51:52Z

4 revisions

← Older revision	Revision as of 19:51, 18 May 2008
(No difference)

Vipul: /* In terms of Ricci curvature tensor */

2007-08-31T03:52:06Z

In terms of Ricci curvature tensor

← Older revision		Revision as of 03:52, 31 August 2007
Line 3:		Line 3:
	===In terms of Ricci curvature tensor===		===In terms of Ricci curvature tensor===

	Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math>. The Ricci curvature on <math>g</math> is a function from <math>\mathbb{P}(TM)</math> to <math\R</math> (tangent directions at points, to real numbers) that associates to a particular tangent direction the value <math>Ric(X,X)</math> where <math>X</math> is a unit tangent vector in that direction.		Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math>. The Ricci curvature on <math>g</math> is a function from <math>\mathbb{P}(TM)</math> (the set of tangent directions) to <math>\R</math> (tangent directions at points, to real numbers) that associates to a particular tangent direction the value <math>Ric(X,X)</math> where <math>X</math> is a unit tangent vector in that direction.

	===In terms of sectional curvature===		===In terms of sectional curvature===

Vipul: /* Facts */

2007-08-31T03:51:24Z

Facts

← Older revision		Revision as of 03:51, 31 August 2007
Line 45:		Line 45:
	The converse is also true.		The converse is also true.
	* Thus, the Ricci curvature is constant on all unit tangent vectors at all points if and only if the [[Ricci curvature tensor]] is that constant times the [[Riemannian metric]]. Such Riemannian metrics are termed [[Einstein metric]]s and the constant of proportionality is termed the '''cosmological constant'''.		* Thus, the Ricci curvature is constant on all unit tangent vectors at all points if and only if the [[Ricci curvature tensor]] is that constant times the [[Riemannian metric]]. Such Riemannian metrics are termed [[Einstein metric]]s and the constant of proportionality is termed the '''cosmological constant'''.

			===Ricci curvature for constant-curvature metrics===

			The Ricci curvature at a unit tangent vector has been defined as a sum of sectional curvatures for an orthonormal basis involving that unit tangent vector. In particular, if the sectional curvature is constant for all tangent planes at the given point, then the Ricci curvature is <math>(n-1)</math> times that constant. Thus, any [[constant-curvature metric]] is an [[Einstein metric]] and the cosmological constant is <math>(n-1)</math> times the constant curvature.

Vipul: /* Facts */

2007-08-31T03:47:57Z

Facts

← Older revision		Revision as of 03:47, 31 August 2007
Line 27:		Line 27:
	==Facts==		==Facts==

	===~~The~~ Ricci curvature tensor is determined by the ~~Ricci~~ curvature~~===~~		===Ricci curvature determines Ricci curvature tensor===

			''This is analogous to how the [[sectional curvature]] determined the [[Riemman curvature tensor of Levi-Civita connection\|Riemann curvature tensor]]''

	By the polarization trick, we can compute the Ricci curvature tensor from the Ricci curvature. This is based on the following facts:		By the polarization trick, we can compute the Ricci curvature tensor from the Ricci curvature. This is based on the following facts:
Line 35:		Line 37:

	<math>b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y))</math>		<math>b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y))</math>

	* In particular, it is determined by the values taken at all pairs <math>(X,X)</math> for a unit vector <math>X</math> because every vector is a scalar multiple of a unit vector		* In particular, it is determined by the values taken at all pairs <math>(X,X)</math> for a unit vector <math>X</math> because every vector is a scalar multiple of a unit vector

			The determination of the Ricci curvature tensor from the Ricci curvature given above can be done point-wise, i.e. given the Ricci curvature of all unit tangent vectors at a point, we can compute the Ricci curvature tensor as a bilinear form on that tangent space. In particular, it is easy to see that:

			* If the Ricci curvature is constant on all unit tangent vectors at a point, then the Ricci curvature tensor at that point is that constant times the [[Riemannian metric]] restricted to that tangent space
			The converse is also true.
			* Thus, the Ricci curvature is constant on all unit tangent vectors at all points if and only if the [[Ricci curvature tensor]] is that constant times the [[Riemannian metric]]. Such Riemannian metrics are termed [[Einstein metric]]s and the constant of proportionality is termed the '''cosmological constant'''.

Vipul at 03:42, 31 August 2007

2007-08-31T03:42:40Z

New page

==Definition==

===In terms of Ricci curvature tensor===

Let <math>M</math> be a [[differential manifold]] and <math>g</math> a [[Riemannian metric]] on <math>M</math>. The Ricci curvature on <math>g</math> is a function from <math>\mathbb{P}(TM)</math> to <math\R</math> (tangent directions at points, to real numbers) that associates to a particular tangent direction the value <math>Ric(X,X)</math> where <math>X</math> is a unit tangent vector in that direction.

===In terms of sectional curvature===

Another way of defining the Ricci curvature is in terms of the [[sectional curvature]]. Let <math>(M,g)</math> be a [[Riemannian manifold]], <math>p \in M</math> and <math>X</math> a unit tangent vector at <math>p</math>. Let <math>e_1, e_2, \ldots e_n</math> be an orthonormal basis at <math>p</math> such that <math>e_1 = X</math>. Then the Ricci curvature of <math>X</math> is defined as:

<math>\sum_{i=2}^n K(e_1,e_i)</math>

By <math>K(e_1,e_i)</math> is meant the [[sectional curvature]] of the plane spanned by <math>e_1</math> and <math>e_i</math>.

===In terms of Riemann curvature tensor===

We now define the Ricci curvature directly in terms of the [[Riemann curvature tensor of Levi-Civita connection|Riemann curvature tensor]], and this definition explains both the above definitions. The Ricci curvature at a point, for a tangent direction with unit tangent vector <math>X</math>, is defined as:

<math>Tr(Z \mapsto R(X,Z)X)</math>

or equivalently, if we choose an orthonormal basis with <math>X=e_1</math> as:

<math>\sum_{i=2}^n R(X,e_i,X,e_i)</math>

This gives the above two definitions.

==Facts==

===The Ricci curvature tensor is determined by the Ricci curvature===

By the polarization trick, we can compute the Ricci curvature tensor from the Ricci curvature. This is based on the following facts:

* The [[Ricci curvature tensor]] is symmetric
* A symmetric bilinear form is completely determined by the values it takes on pairs <math>(X,X)</math> because of the identity:

<math>b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y))</math>
* In particular, it is determined by the values taken at all pairs <math>(X,X)</math> for a unit vector <math>X</math> because every vector is a scalar multiple of a unit vector