equation 13.267 eq:mfd-lie-def

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Supports

• depends_on definition 13.139 Killing vector ¶
  • depends_on definition 13.142 Killing tensor ¶
    • depends_on example 13.144 The two Killing tensors every metric carries ¶
    • depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
  • depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
    • depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
      • depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
      • depends_on example A.359 The Lorentz algebra ¶
    • depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶
  • depends_on proposition 13.163 Integration of the conformal Killing equation ¶
  • depends_on proposition 13.140 Killing's equation ¶
    • depends_on example 13.144 The two Killing tensors every metric carries ¶ ↺
    • depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
    • depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶ ↺
    • depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
      • depends_on phenomenon 45.52 The shadow ¶
        • depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
      • depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
      • depends_on proposition 45.6 Radial equation and effective potential ¶
        • depends_on proposition 45.10 Circular orbits and their energy ¶
        • depends_on proposition 45.8 Orbit equation ¶
        • depends_on proposition 45.21 Radial plunge: the cycloid ¶
      • depends_on theorem 13.143 Killing tensors and geodesic invariants ¶ ↺
    • depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
    • 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 13.131 Commuting fields have commuting flows ¶
  • depends_on lemma A.548 The joint flow is a translation action on the level set ¶
    • depends_on corollary A.553 Quasi-periodic motion ¶
    • depends_on lemma A.549 The action is transitive ¶
      • depends_on proposition A.552 The component is a torus ¶
        • depends_on corollary A.553 Quasi-periodic motion ¶ ↺
        • depends_on definition A.554 Actions on the torus ¶
        • depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    • depends_on proposition A.552 The component is a torus ¶ ↺
  • depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
    • depends_on theorem 13.133 Frobenius ¶
      • depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      • depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
        • depends_on lemma A.549 The action is transitive ¶ ↺
        • depends_on lemma A.555 The actions are well defined ¶
      • depends_on theorem A.545 Liouville–Arnold ¶
    • depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
  • depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
• depends_on proposition 13.127 Component formulas ¶
  • depends_on lemma A.75 Differentiating a pullback along a flow ¶
    • depends_on lemma A.77 Poincaré lemma, converse form ¶
  • depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
  • 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 theorem 25.57 The Moyal equation, and its classical limit ¶
        • depends_on theorem 24.24 Poincaré recurrence ¶
  • depends_on proposition 13.140 Killing's equation ¶ ↺
  • depends_on theorem 44.19 Covariant conservation of stress–energy ¶
    • depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      • depends_on corollary 44.21 Dust flows on geodesics ¶