theorem 13.137 Commuting complete fields on a compact manifold

Rests on

• 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 definition 13.71 Differentiable curve ¶
        • depends_on definition 13.48 Differentiable manifold ¶
      • depends_on definition 13.82 Vector field ¶
        • depends_on definition 13.81 Vector on a manifold ¶
    • depends_on theorem A.74 Flow of a time-dependent vector field ¶
      • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on definition 9.2 Linear equation; homogeneity ¶
        • depends_on remark 9.3 Normal form and first-order systems ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:07-odes-sturm-liouville@proof-3 ¶
      • depends_on lemma A.72 Iterated integral inequality ¶
        • depends_on definition 7.53 $\ee$ ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • proves proof app:A-long-proofs@proof-52 ¶
      • depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
      • proves proof app:A-long-proofs@proof-54 ¶
    • 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 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 7.9 Real function ¶
      • depends_on definition 7.99 Differentiability at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on equation 6.12 eq:top-euclidean-metric ¶
      • depends_on definition 7.102 Differential of a function ¶
        • depends_on definition 7.99 Differentiability at a point ¶ ↺
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • 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 ¶
        • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.99 Differentiability at a point ¶ ↺
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-62 ¶
      • 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 theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
  • proves proof ch:11-manifolds-tensors-curvature@proof-32 ¶
• depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
• proves proof ch:11-manifolds-tensors-curvature@proof-36 ¶

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