equation 13.245 eq:mfd-nilpotency

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Supports

• depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
• depends_on lemma 13.107 Poincaré lemma ¶
  • depends_on proposition 13.113 Product of a closed and an exact form ¶
  • depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
    • depends_on remark 24.11 The SI dimension of every object in this chapter ¶
      • depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
      • depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
      • depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
        • depends_on definition 24.28 Ball and cylinder ¶
        • depends_on definition 24.33 Symplectic capacity ¶
• 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 13.128 Cartan's magic formula ¶
  • depends_on lemma A.77 Poincaré lemma, converse form ¶
  • depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
  • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    • depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      • depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
    • depends_on remark 24.35 What non-squeezing does and does not say about nature ¶ ↺
    • depends_on theorem 24.21 Liouville ¶
      • depends_on example 24.29 A volume-preserving squeeze ¶
      • depends_on proposition 32.40 A Hamiltonian section preserves area ¶
      • depends_on remark 22.25 This is Liouville's theorem ¶
      • depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶ ↺
      • depends_on theorem 23.31 Liouville–Arnold ¶
        • depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
        • depends_on remark 23.33 Integrability is exceptional ¶
        • depends_on remark 23.32 What that proof imports ¶
        • depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
        • depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
      • depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
        • depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
      • depends_on theorem 24.24 Poincaré recurrence ¶
        • depends_on remark 24.26 Recurrence and irreversibility ¶
        • depends_on remark 24.25 What the recurrence proof assumes about volume ¶