Ricci curvature tensor: Difference between revisions
No edit summary |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 5: | Line 5: | ||
===Given data=== | ===Given data=== | ||
A [[differential manifold]] <math>M</math> with a [[linear connection]] <math>\nabla</math> on it. | A [[differential manifold]] <math>M</math> with a [[defining ingredient::linear connection]] <math>\nabla</math> on it. | ||
===Definition part=== | ===Definition part=== | ||
| Line 11: | Line 11: | ||
The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows. | The Ricci curvature tensor is a <math>(0,2)</math>-tensor that takes as input two vector fields and outputs a scalar function, as follows. | ||
Let <math>X</math> and <math>Y</math> be two [[vector field]]s. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[Riemann curvature tensor]]). | Let <math>X</math> and <math>Y</math> be two [[vector field]]s. Then consider the map that sends a vector field <math>Z</math> to <math>R(X,Z)Y</math> (here <math>R</math> denotes the [[defining ingredient::Riemann curvature tensor]]). | ||
This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map. | This is a linear map. The Ricci curvature function <math>Ric(X,Y)</math> is defined as the trace of this map. | ||
| Line 17: | Line 17: | ||
Explicitly, it is given by: | Explicitly, it is given by: | ||
<math>Tr(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)</math> | <math>\operatorname{Tr}(Z \mapsto \nabla_X\nabla_ZY - \nabla_Z\nabla_XY - \nabla_{[X,Z]}Y)</math> | ||
The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor. | The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor. | ||
Latest revision as of 02:29, 24 July 2009
This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (0,2)
Description
Given data
A differential manifold with a linear connection on it.
Definition part
The Ricci curvature tensor is a -tensor that takes as input two vector fields and outputs a scalar function, as follows.
Let and be two vector fields. Then consider the map that sends a vector field to (here denotes the Riemann curvature tensor).
This is a linear map. The Ricci curvature function is defined as the trace of this map.
Explicitly, it is given by:
![{\displaystyle \operatorname {Tr} (Z\mapsto \nabla _{X}\nabla _{Z}Y-\nabla _{Z}\nabla _{X}Y-\nabla _{[X,Z]}Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32120820d8e57e95eb60e7ec58d20a02691a1182)
The Ricci curvature function is a tensor because the Riemann curvature function is also a tensor.
For a Riemannian or pseudo-Riemannian manifold
Further information: Ricci curvature tensor of Levi-Civita connection
For a Riemannian manifold or pseudo-Riemannian manifold, we can give the Levi-Civita connection, a natural choice of linear connection. The Ricci curvature tensor of this connection, is termed the Ricci curvature tensor of the Riemannian or pseudo-Riemannian manifold.