Difference between revisions of "Ricci curvature tensor of Levi-Civita connection"

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(Definition)
 
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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==

Latest revision as of 02:28, 24 July 2009

Definition

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

Explicitly, if R is the Riemann curvature tensor of the Levi-Civita connection:

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

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

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

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

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

Related notions

  • Ricci curvature in a direction is the Ricci curvature tensor Ric(X,X) where X is a unit vector in that direction

Facts

Symmetry

We have:

Ric(X,Y) = Ric(Y,X)

This follows from the fact that the Riemann curvature tensor is symmetric in the pair of the first two and last two variables. For full proof, refer: Symmetry of Riemann curvature tensor in variable pairs