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

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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

## Contents

## Definition

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

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

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

Or is the language of as a -tensor:

## Related notions

- Ricci curvature in a direction is the Ricci curvature tensor where is a unit vector in that direction

## Facts

### Symmetry

We have:

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*