proposition A.547 The components of the level set are the leaves of an integrable distribution

open in the book · appendices/A-long-proofs.tex:26566 · p. 3058

Rests on

depends_on lemma A.546 The Hamiltonian fields of the integrals ¶
- depends_on equation A.844 eq:app-hj-liouville-arnold-involution ¶
- depends_on equation 24.8 eq:sym-bracket-homomorphism ¶
- depends_on equation 24.5 eq:sym-hamiltonian-field ¶
- proves proof app:A-long-proofs@proof-330 ¶
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 ¶
depends_on theorem 13.59 Regular value theorem ¶
- depends_on definition 13.52 Differential; pushforward ¶ ↺
- depends_on definition 13.58 Hypersurface ¶
  - depends_on definition 13.54 Embedded submanifold ¶ ↺
- depends_on definition 13.49 Smooth map between manifolds ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
  - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
proves proof app:A-long-proofs@proof-331 ¶

Supports

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 example A.557 The check that fixes the convention ¶
  - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶ ↺
depends_on lemma A.555 The actions are well defined ¶ ↺

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`	→	The Hamiltonian fields of the integrals	declared	appendices/A-long-proofs.tex:26576
`depends_on`	→	Frobenius	declared	appendices/A-long-proofs.tex:26576
`depends_on`	→	Regular value theorem	declared	appendices/A-long-proofs.tex:26576
`depends_on`	←	The action is transitive	declared	appendices/A-long-proofs.tex:26652
`depends_on`	←	The actions are well defined	declared	appendices/A-long-proofs.tex:26802
`proves`	←	app:A-long-proofs@proof-331	declared	appendices/A-long-proofs.tex:26580