Statements (27)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
gptkb:Tensor |
| gptkbp:alsoKnownAs |
gptkb:Riemann_curvature_tensor
|
| gptkbp:component |
R^i_{jkl}
|
| gptkbp:definedIn |
gptkb:Riemannian_manifold
pseudo-Riemannian manifold |
| gptkbp:describes |
curvature of a manifold
|
| gptkbp:dimensions |
n^2(n^2-1)/12 independent components in n dimensions
|
| gptkbp:field |
gptkb:general_relativity
differential geometry |
| gptkbp:form |
R^i_{jkl} = ∂_k Γ^i_{jl} - ∂_l Γ^i_{jk} + Γ^i_{km}Γ^m_{jl} - Γ^i_{lm}Γ^m_{jk}
|
| https://www.w3.org/2000/01/rdf-schema#label |
Riemann tensor
|
| gptkbp:namedAfter |
gptkb:Bernhard_Riemann
|
| gptkbp:rank |
(1,3) or (0,4)
|
| gptkbp:relatedTo |
gptkb:Christoffel_symbols
metric tensor geodesic deviation equation |
| gptkbp:symmetry |
antisymmetric in first two indices
antisymmetric in last two indices satisfies Bianchi identities |
| gptkbp:usedIn |
gptkb:Einstein_field_equations
gptkb:Weyl_tensor gptkb:Ricci_tensor sectional curvature |
| gptkbp:vanishes_for |
flat manifolds
|
| gptkbp:bfsParent |
gptkb:Weyl_tensor
|
| gptkbp:bfsLayer |
5
|