theorem A.545 Liouville–Arnold

Rests on

• depends_on equation 23.54 eq:hj-involution ¶
• depends_on equation 24.8 eq:sym-bracket-homomorphism ¶
• depends_on theorem 13.133 Frobenius ¶
  • depends_on definition 13.130 Distribution; involutive; integrable ¶
    • depends_on definition 13.54 Embedded submanifold ¶
      • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on definition 13.52 Differential; pushforward ¶
        • depends_on definition 6.7 Homeomorphism ¶
      • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.47 Differentiable structure ¶
    • depends_on definition 13.82 Vector field ¶
      • 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 equation 13.273 eq:mfd-lie-vector ¶
  • depends_on equation 13.273 eq:mfd-lie-vector ¶ ↺
  • depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
    • depends_on corollary A.292 Inverse function theorem ¶
      • depends_on proposition 7.104 Chain rule in several variables ¶
        • depends_on definition 7.13 Composite function ¶
        • depends_on definition 7.99 Differentiability at a point ¶
        • depends_on definition 7.102 Differential of a function ¶
        • proves proof ch:05-real-analysis@proof-63 ¶
      • depends_on theorem A.286 Implicit function theorem ¶
        • depends_on definition 7.99 Differentiability at a point ¶ ↺
        • depends_on proposition 7.104 Chain rule in several variables ¶ ↺
        • depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
        • proves proof app:A-long-proofs@proof-185 ¶
      • proves proof app:A-long-proofs@proof-186 ¶
    • depends_on proposition 13.131 Commuting fields have commuting flows ¶
      • depends_on equation 13.267 eq:mfd-lie-def ¶
      • depends_on equation 13.273 eq:mfd-lie-vector ¶ ↺
      • depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
        • depends_on definition 13.124 Integral curve; complete vector field ¶
        • depends_on theorem A.74 Flow of a time-dependent vector field ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-27 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-31 ¶
    • depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
    • proves proof ch:11-manifolds-tensors-curvature@proof-32 ¶
  • proves proof ch:11-manifolds-tensors-curvature@proof-33 ¶
• proves proof app:A-long-proofs@proof-338 ¶

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