Riemann curvature tensor
E22818
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
(0,4)-tensor
→
(1,3)-tensor → geometric object → tensor → |
| appearsIn |
Cartan structure equations
→
Jacobi equation for geodesic deviation → |
| codomain |
tangent bundle of a manifold
→
|
| componentNotation |
R^i_{ jkl}
→
R_{ijkl} → |
| constructedFrom |
Christoffel symbols
→
covariant derivative → |
| definedOn |
Riemannian manifold
→
pseudo-Riemannian manifold → |
| dependsOn |
Levi-Civita connection
→
affine connection → |
| dimensionDependentProperties |
in 2D determined by a single scalar function
→
in constant curvature spaces has special algebraic form → simplifies in 2-dimensional manifolds → |
| domain |
tangent bundle of a manifold
→
|
| encodes |
failure of second covariant derivatives to commute
→
parallel transport holonomy → sectional curvature → |
| equalsZeroIf |
connection is flat
→
manifold is locally isometric to Euclidean space → |
| field |
Riemannian geometry
→
differential geometry → pseudo-Riemannian geometry → |
| generalizes |
Gaussian curvature
→
|
| hasSymmetry |
antisymmetric in first two indices
→
antisymmetric in last two indices → symmetric under pair exchange (ij)↔(kl) → |
| independentOf |
embedding in ambient space
→
|
| introducedBy |
Bernhard Riemann
→
|
| isIntrinsic |
true
→
|
| measures |
deviation from flatness of a manifold
→
intrinsic curvature → |
| namedAfter |
Bernhard Riemann
→
|
| order |
4
→
|
| rank |
4
→
|
| satisfies |
first Bianchi identity
→
second Bianchi identity → |
| symbol |
R
→
|
| usedIn |
Einstein field equations
→
classification of manifolds by curvature → definition of Ricci curvature → definition of scalar curvature → general relativity → geodesic deviation equation → |
Referenced by (9)
| Subject (surface form when different) | Predicate |
|---|---|
|
Kretschmann scalar
→
|
dependsOn |
|
Ricci curvature tensor
→
|
derivedFrom |
|
Riemannian manifold
→
|
hasComponent |
|
Ricci scalar
→
|
isContractionOf |
|
Bernhard Riemann
→
|
knownFor |
|
Friedrich Bernhard Riemann
→
|
notableConcept |
|
Christoffel symbols
→
|
relatedConcept |
|
Bianchi identities
→
|
relatesTo |
|
Ricci curvature tensor
→
|
traceOf |