Bianchi identities

E57422

concept in differential geometry concept in general relativity geometric identity tensor identity

The Bianchi identities are geometric relations in differential geometry and general relativity that express the vanishing covariant divergence of the Riemann curvature tensor, leading to conservation laws such as energy-momentum conservation via the Einstein tensor.

Aliases (1)

second Bianchi identity ×1

Statements (46)

Predicate	Object
instanceOf	concept in differential geometry → concept in general relativity → geometric identity → tensor identity →
appliesTo	Riemannian manifolds → affine connections → pseudo-Riemannian manifolds → torsion-free connections →
category	identity in Riemannian geometry → identity in gauge theory →
ensures	compatibility of curvature with connection → consistency of Einstein field equations with local conservation of energy-momentum →
expresses	covariant conservation of energy-momentum tensor in general relativity → cyclic symmetry of the Riemann tensor → vanishing covariant divergence of the Einstein tensor →
field	Riemannian geometry → differential geometry → general relativity →
hasFormulation	DΩ = 0 in Cartan’s formalism → R_{a[bcd]} = 0 → R_{abcd;e} + R_{abed;c} + R_{abce;d} = 0 →
hasType	algebraic Bianchi identity → differential Bianchi identity →
historicalPeriod	late 19th century →
holdsFor	Levi-Civita connection of any metric → curvature of any linear connection →
implies	conservation laws in general relativity → ∇_μ G^{μν} = 0 → ∇_μ T^{μν} = 0 under Einstein field equations →
involvesOperation	antisymmetrization of indices → covariant differentiation →
mathematicalNature	tensorial identity independent of coordinates →
namedAfter	Luigi Bianchi →
relatesTo	Bianchi classification → Einstein tensor → Levi-Civita connection → Riemann curvature tensor → covariant derivative → curvature 2-form →
usedIn	Yang–Mills theory → cosmological models → derivation of Einstein field equations consistency → gauge theories → numerical relativity → string theory → supergravity →

Referenced by (2)

Subject (surface form when different)	Predicate
Einstein tensor →	relatedConcept
Riemann curvature tensor ("second Bianchi identity") →	satisfies