proposition A.570 Existence and uniqueness of the reduced form

open in the book · appendices/A-long-proofs.tex:27348 · p. 3066

Rests on

depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
- depends_on definition A.561 Coadjoint action and equivariance ¶
  - 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 14.2 Lie group ¶
  - depends_on definition 24.43 Momentum map ¶
    - depends_on definition 14.2 Lie group ¶ ↺
    - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on equation 22.4 eq:ham-pdot ¶
      - depends_on equation 22.3 eq:ham-qdot ¶
    - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on definition A.70 Pullback ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition A.565 Freeness makes every value regular ¶
  - depends_on lemma A.564 Dimension and double orthogonal ¶
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - proves proof app:A-long-proofs@proof-339 ¶
  - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
    - depends_on definition 24.43 Momentum map ¶ ↺
    - 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 ch:07-symplectic-geometry@proof-15 ¶
  - depends_on theorem 13.59 Regular value theorem ¶ ↺
  - proves proof app:A-long-proofs@proof-340 ¶
- depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on definition 13.66 Smooth action; free; proper; orbit ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.49 Smooth map between manifolds ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - 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 theorem 13.63 Constant rank theorem ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem A.286 Implicit function theorem ¶
      - proves proof app:A-long-proofs@proof-186 ¶
    - 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 theorem A.286 Implicit function theorem ¶ ↺
    - proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
  - proves proof app:A-long-proofs@proof-329 ¶
  - proves proof ch:11-manifolds-tensors-curvature@prooflink-2 ¶
- proves proof app:A-long-proofs@proof-341 ¶
depends_on theorem A.568 The radical is the isotropy orbit ¶
- depends_on lemma A.567 The differential of the momentum map along the orbit ¶
  - depends_on definition A.561 Coadjoint action and equivariance ¶ ↺
  - depends_on proposition A.565 Freeness makes every value regular ¶ ↺
  - 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 app:A-long-proofs@proof-342 ¶
- depends_on lemma A.564 Dimension and double orthogonal ¶ ↺
- depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
- proves proof app:A-long-proofs@proof-343 ¶
depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
- depends_on definition 24.14 Hamiltonian vector field ¶ ↺
- depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition 13.128 Cartan's magic formula ¶
  - depends_on definition 13.103 Exterior derivative ¶
    - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
  - depends_on equation 13.246 eq:mfd-leibniz-forms ¶
  - depends_on equation 13.245 eq:mfd-nilpotency ¶
  - depends_on proposition 13.127 Component formulas ¶
    - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
      - depends_on definition 13.88 General coordinate transformation ¶ ↺
    - depends_on equation 13.267 eq:mfd-lie-def ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
- proves proof ch:07-symplectic-geometry@proof-5 ¶
proves proof app:A-long-proofs@proof-344 ¶

Supports

depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The isotropy group acts, and the quotient is smooth	declared	appendices/A-long-proofs.tex:27353
`depends_on`	→	The radical is the isotropy orbit	declared	appendices/A-long-proofs.tex:27353
`depends_on`	→	The Hamiltonian flow preserves the symplectic form	declared	appendices/A-long-proofs.tex:27353
`depends_on`	←	Invariant Hamiltonians descend with their flows	declared	appendices/A-long-proofs.tex:27413
`proves`	←	app:A-long-proofs@proof-344	declared	appendices/A-long-proofs.tex:27357