Ricci curvature tensor
E8635
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Ricci tensor | 2 |
| Ricci curvature | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
(0,2)-tensor
ⓘ
geometric object ⓘ symmetric tensor ⓘ tensor ⓘ |
| appearsIn | Einstein field equations ⓘ |
| characterizes | Einstein manifold condition Ric = λg ⓘ |
| componentsDefinition |
R_{ij} = R^{k}{}_{ikj}
ⓘ
R_{ij} = R^{k}{}_{jik} ⓘ |
| componentsNotation | R_{ij} ⓘ |
| coordinateExpression | R_{ij} = \partial_k \Gamma^{k}_{ij} - \partial_j \Gamma^{k}_{ik} + \Gamma^{k}_{ij} \Gamma^{l}_{kl} - \Gamma^{k}_{il} \Gamma^{l}_{kj} ⓘ |
| definedOn |
Riemannian manifold
ⓘ
pseudo-Riemannian manifold ⓘ |
| dependsOn |
Levi-Civita connection
ⓘ
surface form:
Christoffel symbols
Levi-Civita connection ⓘ |
| derivedFrom | Riemann curvature tensor ⓘ |
| determines | scalar curvature by contraction with the metric ⓘ |
| developedWith | Tullio Levi-Civita ⓘ |
| dimensionOfComponents | n×n on an n-dimensional manifold ⓘ |
| equals | 0 in vacuum Einstein equations with zero cosmological constant ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ general relativity ⓘ pseudo-Riemannian geometry ⓘ |
| introducedBy | Gregorio Ricci-Curbastro ⓘ |
| isLocalInvariantOf | metric tensor ⓘ |
| isSymmetric | true ⓘ |
| isZeroCondition | characterizes Ricci-flat manifolds ⓘ |
| measures |
average sectional curvature
ⓘ
deviation of volume growth from Euclidean ⓘ volume distortion ⓘ |
| obtainedBy | contraction of the Riemann curvature tensor ⓘ |
| order | 2 ⓘ |
| rank | 2 ⓘ |
| relatedTo |
Einstein tensor
ⓘ
scalar curvature ⓘ sectional curvature ⓘ |
| roleInGeneralRelativity | describes how matter and energy curve spacetime on average ⓘ |
| symbol |
Ric
ⓘ
Ric(g) ⓘ |
| symmetryProperty | R_{ij} = R_{ji} ⓘ |
| traceOf | Riemann curvature tensor ⓘ |
| traceWithRespectTo | metric tensor ⓘ |
| transformationProperty | tensorial under coordinate changes ⓘ |
| usedIn |
Einstein manifolds
ⓘ
Ricci flow ⓘ Ricci flow ⓘ
surface form:
Ricci solitons
|
| usedToForm | Einstein tensor G_{ij} = R_{ij} - 1/2 R g_{ij} ⓘ |
| vanishesFor | flat manifold ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ricci curvature
this entity surface form:
Ricci tensor
this entity surface form:
Ricci tensor