proposition 13.154 Symmetries of the curvature

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6730 · p. 533

Rests on

depends_on definition 13.149 Metric compatibility ¶
- 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 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
  - depends_on definition 13.48 Differentiable 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 13.81 Vector on a manifold ¶ ↺
- depends_on definition 13.146 Parallel transport and autoparallels ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
depends_on proposition 13.157 Bianchi identities ¶
- 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 13.84 Tensor ¶ ↺
    - 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 13.122 Vielbein ¶
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
    - 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 theorem 7.35 Mean value theorem ¶
      - 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 ¶
depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
- depends_on definition 13.145 Affine connection ¶ ↺
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
- depends_on definition 13.149 Metric compatibility ¶ ↺
- depends_on definition 13.147 Torsion ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶

Supports

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 ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶
    - tests experiment The Cosmic Microwave Background ¶
      - cites (method) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶
      - cites (evidence) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶ ↺
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶ ↺
      - cites (method) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (context) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib Planck 2018 results. VI. Cosmological parameters ¶
      - cites (method) derived bib Planck 2018 results. VI. Cosmological parameters ¶ ↺
      - … 109 more
  - 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 48.15 A standard ruler in the galaxy distribution ¶
    - depends_on phenomenon 48.17 The expansion is accelerating ¶
    - depends_on phenomenon 48.18 The Hubble tension ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	→	Bianchi identities	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	→	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6743
`depends_on`	←	Linearised curvature of a symmetric field	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5290
`depends_on`	←	Contracted Bianchi identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	←	Maximal symmetry forces constant curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984
`proves`	←	ch:11-manifolds-tensors-curvature@proof-43	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6746