definition A.554 Actions on the torus

open in the book · appendices/A-long-proofs.tex:26784 · p. 3060

ground object -- no derivation owed

Rests on

depends_on equation 23.49 eq:hj-action-variable ¶
depends_on equation 24.3 eq:sym-canonical-forms ¶
depends_on proposition A.552 The component is a torus ¶
- depends_on lemma A.548 The joint flow is a translation action on the level set ¶
  - 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 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 app:A-long-proofs@proof-332 ¶
- depends_on lemma A.549 The action is transitive ¶
  - 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 lemma A.548 The joint flow is a translation action on the level set ¶ ↺
  - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
    - depends_on lemma A.546 The Hamiltonian fields of the integrals ¶ ↺
    - depends_on theorem 13.133 Frobenius ¶
      - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - depends_on equation 13.273 eq:mfd-lie-vector ¶ ↺
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - 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 13.49 Smooth map between manifolds ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
    - proves proof app:A-long-proofs@proof-331 ¶
  - proves proof app:A-long-proofs@proof-333 ¶
- depends_on theorem A.550 Quoted: discrete subgroups of a real vector space ¶
  - depends_on definition 4.21 Group ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- proves proof app:A-long-proofs@proof-334 ¶

Supports

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 ¶

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:hj-action-variable	declared	appendices/A-long-proofs.tex:26793
`depends_on`	→	eq:sym-canonical-forms	declared	appendices/A-long-proofs.tex:26793
`depends_on`	→	The component is a torus	declared	appendices/A-long-proofs.tex:26793
`depends_on`	←	The actions are well defined	declared	appendices/A-long-proofs.tex:26802
`depends_on`	←	The Jacobian of the actions is the lattice matrix	declared	appendices/A-long-proofs.tex:26838