proposition 13.132 Simultaneous straightening of commuting fields

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5975 · p. 525

Rests on

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 3.43 Map ¶
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - 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 ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - 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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - 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 13.48 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 theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
proves proof ch:11-manifolds-tensors-curvature@proof-32 ¶

Supports

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 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 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 ¶
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`	→	Inverse function theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5983
`depends_on`	→	Commuting fields have commuting flows	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5983
`depends_on`	→	Existence, uniqueness and smoothness of the flow	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5983
`depends_on`	←	Frobenius	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6030
`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-32	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5986