theorem 13.133 Frobenius

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6020 · p. 525

Rests on

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 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 6.7 Homeomorphism ¶
      - depends_on definition 3.47 Bijective map ¶
      - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 6.6 Continuous map ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
- depends_on definition 13.82 Vector field ¶
  - depends_on definition 13.81 Vector on a 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 7.9 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 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 13.82 Vector field ¶ ↺
    - 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 lemma A.72 Iterated integral inequality ¶
      - 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 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 ¶

Supports

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 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 lemma A.555 The actions are well defined ¶ ↺
depends_on theorem A.545 Liouville–Arnold ¶

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`	→	Distribution; involutive; integrable	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6030
`depends_on`	→	eq:mfd-lie-vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6030
`depends_on`	→	Simultaneous straightening of commuting fields	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6030
`depends_on`	←	Frobenius for a Pfaffian system	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6114
`depends_on`	←	The components of the level set are the leaves of an integrable distribution	declared	appendices/A-long-proofs.tex:26576
`depends_on`	←	Liouville–Arnold	declared	appendices/A-long-proofs.tex:26521
`proves`	←	ch:11-manifolds-tensors-curvature@proof-33	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6034