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gptkbp:instanceOf
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gptkb:mathematical_concept
|
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gptkbp:appearsIn
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gptkb:Einstein_field_equations
|
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gptkbp:basisFor
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gptkb:Einstein_tensor
gptkb:Weyl_tensor
gptkb:Jacobi_equation
gptkb:curvature_endomorphism
gptkb:holonomy_group
gptkb:Ricci_tensor
geodesic deviation equation
scalar curvature
sectional curvature
curvature 2-form
curvature algebra
curvature class
curvature form
curvature function
curvature invariant
curvature map
curvature operator
curvature scalar
curvature structure
curvature tensor field
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gptkbp:componentNotation
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R^l_{ijk}
R_{ijkl}
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gptkbp:describes
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curvature of a Riemannian manifold
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gptkbp:dimensions
|
depends on manifold dimension
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gptkbp:expressedIn
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gptkb:Christoffel_symbols
|
|
gptkbp:field
|
differential geometry
|
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gptkbp:firstDefined
|
19th century
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gptkbp:generalizes
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Gaussian curvature
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https://www.w3.org/2000/01/rdf-schema#label
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Riemann curvature tensor
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gptkbp:isZeroFor
|
gptkb:Euclidean_space
locally isometric spaces to Euclidean space
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gptkbp:namedAfter
|
gptkb:Bernhard_Riemann
|
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gptkbp:notation
|
Riem
|
|
gptkbp:rank
|
(0,4) tensor
(1,3) tensor
|
|
gptkbp:relatedTo
|
gptkb:Weyl_tensor
Ricci curvature
scalar curvature
|
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gptkbp:satisfies
|
gptkb:Bianchi_identities
|
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gptkbp:symmetry
|
antisymmetric in first two indices
antisymmetric in last two indices
symmetric under exchange of first and last pair of indices
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gptkbp:usedIn
|
gptkb:general_relativity
gptkb:Riemannian_geometry
pseudo-Riemannian geometry
|
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gptkbp:vanishes_for
|
flat manifolds
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gptkbp:bfsParent
|
gptkb:Ricci_tensor
gptkb:Riemannian_geometry
gptkb:Riemannian_manifold
gptkb:Bernhard_Riemann
gptkb:General_relativity
gptkb:Ricci_scalar
gptkb:Differential_geometry
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gptkbp:bfsLayer
|
5
|