Ricci curvature tensor of Levi-Civita connection - Revision history

Vipul: /* Definition */

2009-07-24T02:28:27Z

Definition

← Older revision		Revision as of 02:28, 24 July 2009
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	Explicitly, if <math>R</math> is the Riemann curvature tensor of the Levi-Civita connection:		Explicitly, if <math>R</math> is the Riemann curvature tensor of the Levi-Civita connection:

	<math>Ric(X,Y) = -Tr(Z \mapsto R(X,Z)Y)</math>		<math>\operatorname{Ric}(X,Y) = -Tr(Z \mapsto R(X,Z)Y)</math>

	In the particular case of a Riemannian metric, we can choose an orthonormal basis <math>e_i</math> on each tangent space. For a particular tangent space, if the orthonormal basis is <math>e_i</math>, the Ricci curvature tensor evaluated at a pair of vectors <math>X,Y</math> is:		In the particular case of a Riemannian metric, we can choose an orthonormal basis <math>e_i</math> on each tangent space. For a particular tangent space, if the orthonormal basis is <math>e_i</math>, the Ricci curvature tensor evaluated at a pair of vectors <math>X,Y</math> is:

	<math>Ric(X,Y) = \sum_{i=1}^n g(R(X,e_i)Y,e_i)</math>		<math>\operatorname{Ric}(X,Y) = \sum_{i=1}^n g(R(X,e_i)Y,e_i)</math>

	Or is the language of <math>R</math> as a <math>(0,4)</math>-tensor:		Or is the language of <math>R</math> as a <math>(0,4)</math>-tensor:

	<math>Ric(X,Y) = \sum_{i=1}^n R(X,e_i,Y,e_i)</math>		<math>\operatorname{Ric}(X,Y) = \sum_{i=1}^n R(X,e_i,Y,e_i)</math>

	==Related notions==		==Related notions==

Vipul at 02:27, 24 July 2009

2009-07-24T02:27:57Z

← Older revision		Revision as of 02:27, 24 July 2009
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	==Definition==		==Definition==

	Let <math>M</math> be a [[differential manifold]] and <math>g</math> be a [[Riemannian metric]] or [[pseudo-Riemannian metric]] on <math>M</math>. Let <math>\nabla</math> be the [[Levi-Civita connection]] associated with <math>g</math>. The '''Ricci curvature tensor'' of <math>M</math> is defined as the [[Ricci curvature tensor]] of the Levi-Civita connection.		Let <math>M</math> be a [[differential manifold]] and <math>g</math> be a [[defining ingredient::Riemannian metric]] or [[defining ingredient::pseudo-Riemannian metric]] on <math>M</math>. Let <math>\nabla</math> be the [[defining ingredient::Levi-Civita connection]] associated with <math>g</math>. The '''Ricci curvature tensor''' of <math>M</math> is defined as the [[defining ingredient::Ricci curvature tensor]] of the Levi-Civita connection.

	Explicitly, if <math>R</math> is the Riemann curvature tensor of the Levi-Civita connection:		Explicitly, if <math>R</math> is the Riemann curvature tensor of the Levi-Civita connection:

	<math>Ric(X,Y) = Tr(Z \mapsto R(X,Z)Y)</math>		<math>Ric(X,Y) = -Tr(Z \mapsto R(X,Z)Y)</math>

	In the particular case of a Riemannian metric, we can choose an orthonormal basis <math>e_i</math> on each tangent space. For a particular tangent space, if the orthonormal basis is <math>e_i</math>, the Ricci curvature tensor evaluated at a pair of vectors <math>X,Y</math> is:		In the particular case of a Riemannian metric, we can choose an orthonormal basis <math>e_i</math> on each tangent space. For a particular tangent space, if the orthonormal basis is <math>e_i</math>, the Ricci curvature tensor evaluated at a pair of vectors <math>X,Y</math> is:

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==Definition==

Let <math>M</math> be a [[differential manifold]] and <math>g</math> be a [[Riemannian metric]] or [[pseudo-Riemannian metric]] on <math>M</math>. Let <math>\nabla</math> be the [[Levi-Civita connection]] associated with <math>g</math>. The '''Ricci curvature tensor'' of <math>M</math> is defined as the [[Ricci curvature tensor]] of the Levi-Civita connection.

Explicitly, if <math>R</math> is the Riemann curvature tensor of the Levi-Civita connection:

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

In the particular case of a Riemannian metric, we can choose an orthonormal basis <math>e_i</math> on each tangent space. For a particular tangent space, if the orthonormal basis is <math>e_i</math>, the Ricci curvature tensor evaluated at a pair of vectors <math>X,Y</math> is:

<math>Ric(X,Y) = \sum_{i=1}^n g(R(X,e_i)Y,e_i)</math>

Or is the language of <math>R</math> as a <math>(0,4)</math>-tensor:

<math>Ric(X,Y) = \sum_{i=1}^n R(X,e_i,Y,e_i)</math>

==Related notions==

* [[Ricci curvature]] in a direction is the Ricci curvature tensor <math>Ric(X,X)</math> where <math>X</math> is a unit vector in that direction

==Facts==

===Symmetry===

We have:

<math>Ric(X,Y) = Ric(Y,X)</math>

This follows from the fact that the [[Riemann curvature tensor of Levi-Civita connection|Riemann curvature tensor]] is symmetric in the pair of the first two and last two variables. {{proofat|[[Symmetry of Riemann curvature tensor in variable pairs]]}}