theorem 13.156 Cartan structure equations

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6824 · p. 533

Rests on

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 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.84 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 13.88 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 13.84 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 definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - 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 ¶

Supports

depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
depends_on proposition 13.157 Bianchi identities ¶
- 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 15.33 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 48.1 The Hubble–Lemaître law ¶
      - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on proposition 48.7 Luminosity distance ¶
depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
- depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
- depends_on theorem 43.3 Equivalence of the two variable sets ¶
  - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺

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`	→	Exterior derivative	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	→	Torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	→	Vielbein	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	→	Wedge product	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	→	eq:mfd-vielbein-postulate	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841
`depends_on`	←	First-order variation in vielbein-form variables	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:610
`depends_on`	←	Bianchi identities	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6878
`depends_on`	←	The connection determined by vielbein and torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6916
`proves`	←	ch:11-manifolds-tensors-curvature@proof-45	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6844