proposition 15.31 Radius and cosmological constant

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:998 · p. 612

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 theorem 13.160 Maximal symmetry forces constant curvature ¶
- depends_on definition 13.153 Contractions ¶ ↺
- 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 13.147 Torsion ¶ ↺
      - depends_on definition 13.122 Vielbein ¶
      - 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 ¶
proves proof ch:13-lie-algebra-expansions@proof-18 ¶

Supports

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 ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Contractions	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1023
`depends_on`	→	Maximal symmetry forces constant curvature	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1023
`depends_on`	←	The expansion parameter is the cosmological constant	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1071
`depends_on`	←	The whole chapter at $D=4$	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2407
`depends_on`	←	Which branch nature is on	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1117
`proves`	←	ch:13-lie-algebra-expansions@proof-18	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:1026