Riemann curvature tensor

E22818

(0,4)-tensor (1,3)-tensor geometric object tensor

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.

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All labels observed (3)

Label	Occurrences
Riemann curvature tensor canonical	13
Riemann curvature tensor decomposition	1
Riemann tensor	1

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 ⓘ Bianchi identities ⓘ surface form: 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 (15)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemannian manifolds → hasComponent → Riemann curvature tensor ⓘ

subject surface form: Riemannian manifold

Kretschmann scalar → dependsOn → Riemann curvature tensor ⓘ

Bernhard Riemann → knownFor → Riemann curvature tensor ⓘ

Ricci curvature tensor → derivedFrom → Riemann curvature tensor ⓘ

Ricci curvature tensor → traceOf → Riemann curvature tensor ⓘ

Christoffel symbols → relatedConcept → Riemann curvature tensor ⓘ

Friedrich → notableConcept → Riemann curvature tensor ⓘ

subject surface form: Friedrich Bernhard Riemann

Ricci scalar → isContractionOf → Riemann curvature tensor ⓘ

Bianchi identities → relatesTo → Riemann curvature tensor ⓘ

Cartan structure equations → relatedTo → Riemann curvature tensor ⓘ

Ricci calculus → usesConcept → Riemann curvature tensor ⓘ

differential geometry → keyConcept → Riemann curvature tensor ⓘ

Weyl tensor → isPartOf → Riemann curvature tensor ⓘ

this entity surface form: Riemann curvature tensor decomposition

Weyl tensor → isTracelessPartOf → Riemann curvature tensor ⓘ

Weyl tensor → isConstructedFrom → Riemann curvature tensor ⓘ

this entity surface form: Riemann tensor