proposition 13.157 Bianchi identities

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6866 · p. 534

Rests on

depends_on equation 13.246 eq:mfd-leibniz-forms ¶
depends_on equation 13.245 eq:mfd-nilpotency ¶
depends_on theorem 13.156 Cartan structure equations ¶
- depends_on definition 13.103 Exterior derivative ¶
  - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on equation 13.239 eq:mfd-dual-basis ¶
- depends_on definition 13.147 Torsion ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on definition 13.122 Vielbein ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.84 Tensor ¶ ↺
  - depends_on equation 13.1 eq:mfd-eta ¶
- depends_on definition 13.102 Wedge product ¶ ↺
- depends_on equation 13.303 eq:mfd-vielbein-postulate ¶
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-45 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-46 ¶

Supports

depends_on proposition 13.154 Symmetries of the curvature ¶
- depends_on definition 13.111 Linearised curvature of a symmetric field ¶
  - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
- depends_on theorem 44.23 Contracted Bianchi identity ¶
  - depends_on corollary 44.24 Conservation is an integrability condition ¶
  - depends_on proposition 44.41 Four of the ten equations are constraints ¶
  - depends_on proposition 44.30 The cosmological term ¶
    - depends_on proposition 44.31 de Sitter geometry ¶
  - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on corollary 44.35 The field equations are essentially unique ¶
- depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
  - depends_on proposition 44.31 de Sitter geometry ¶ ↺
  - depends_on proposition 15.31 Radius and cosmological constant ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
      - depends_on remark 15.33 Which branch nature is on ¶
    - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
    - depends_on remark 15.33 Which branch nature is on ¶ ↺
  - depends_on theorem 48.3 The Robertson–Walker metric ¶
    - depends_on phenomenon 48.12 The relic microwave background ¶
      - depends_on phenomenon 48.13 The anisotropy spectrum of the relic radiation ¶
      - tests experiment The Cosmic Microwave Background ¶
    - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
    - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
    - depends_on proposition 48.7 Luminosity distance ¶
      - depends_on corollary 48.8 Angular-diameter distance and the duality relation ¶
      - depends_on phenomenon 48.17 The expansion is accelerating ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:mfd-leibniz-forms	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6878
`depends_on`	→	eq:mfd-nilpotency	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6878
`depends_on`	→	Cartan structure equations	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6878
`depends_on`	←	Symmetries of the curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`proves`	←	ch:11-manifolds-tensors-curvature@proof-46	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6881