proposition 13.131 Commuting fields have commuting flows

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5932 · p. 524

Rests on

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 13.47 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 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 definition 9.1 Ordinary differential equation ¶
    - depends_on remark 9.3 Normal form and first-order systems ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on definition 9.4 Lipschitz condition ¶
      - depends_on equation 9.3 eq:slt-ode-system ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
    - 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 definition 7.45 Series ¶
      - depends_on proposition 7.48 Ratio test ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 ¶

Supports

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 lemma A.555 The actions are well defined ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    - 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 ¶ ↺

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`	→	eq:mfd-lie-def	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5941
`depends_on`	→	eq:mfd-lie-vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5941
`depends_on`	→	Existence, uniqueness and smoothness of the flow	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5941
`depends_on`	←	The joint flow is a translation action on the level set	declared	appendices/A-long-proofs.tex:26624
`depends_on`	←	Simultaneous straightening of commuting fields	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5983
`depends_on`	←	Commuting complete fields on a compact manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6234
`proves`	←	ch:11-manifolds-tensors-curvature@proof-31	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5944