theorem 13.160 Maximal symmetry forces constant curvature

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6974 · p. 535

Rests on

depends_on definition 13.153 Contractions ¶
- 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 13.44 Atlas ¶
  - 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 theorem 13.152 Riemann tensor; Ricci identity with 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 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - 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 ¶
depends_on proposition 13.140 Killing's equation ¶
- depends_on definition 13.139 Killing vector ¶
  - depends_on definition 13.120 Isometry ¶
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on equation 13.267 eq:mfd-lie-def ¶
- depends_on definition 13.149 Metric compatibility ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
- depends_on proposition 13.127 Component formulas ¶
  - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
  - depends_on definition 13.90 Mixed tensor ¶ ↺
  - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
- 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-37 ¶
depends_on proposition 13.154 Symmetries of the curvature ¶
- depends_on definition 13.149 Metric compatibility ¶ ↺
- 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 13.102 Wedge product ¶
    - depends_on definition 13.147 Torsion ¶ ↺
    - 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 ¶ ↺
    - 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 theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-48 ¶

Supports

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 103.68 The sum of the neutrino masses is bounded ¶
  - 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`	→	Contractions	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984
`depends_on`	→	Killing's equation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984
`depends_on`	→	Symmetries of the curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6984
`depends_on`	←	de Sitter geometry	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1357
`depends_on`	←	Radius and cosmological constant	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1023
`depends_on`	←	The Robertson–Walker metric	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:240
`proves`	←	ch:11-manifolds-tensors-curvature@proof-48	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6987