definition 13.124 Integral curve; complete vector field

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5675 · p. 521

ground object -- no derivation owed

Rests on

depends_on definition 13.71 Differentiable curve ¶
- depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
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 13.44 Atlas ¶ ↺
    - depends_on definition 13.43 Coordinate system ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
- 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 proposition A.556 The Jacobian of the actions is the lattice matrix ¶
  - depends_on proposition A.552 The component is a torus ¶ ↺
- depends_on lemma A.567 The differential of the momentum map along the orbit ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- depends_on proposition 13.131 Commuting fields have commuting flows ¶
  - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
  - 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 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 ¶ ↺
- depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
- 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`	→	Differentiable curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5681
`depends_on`	→	Vector field	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5681
`depends_on`	←	Existence, uniqueness and smoothness of the flow	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5704