proposition A.566 The isotropy group acts, and the quotient is smooth

open in the book · appendices/A-long-proofs.tex:27208 · p. 3065

Rests on

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 4.3 Algebraic structure ¶
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
- 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 definition 13.105 Closed form ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 24.2 Symplectic vector space ¶
    - 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 7.97 Partial derivative; gradient ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
    - 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 theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - 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 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.58 Hypersurface ¶
      - depends_on definition 13.54 Embedded submanifold ¶
    - 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 13.44 Atlas ¶
  - 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 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on definition 13.54 Embedded submanifold ¶ ↺
- 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 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 definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - 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 ¶

Supports

depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
- depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶

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`	→	Coadjoint action and equivariance	declared	appendices/A-long-proofs.tex:27215
`depends_on`	→	Freeness makes every value regular	declared	appendices/A-long-proofs.tex:27215
`depends_on`	→	Quotient manifold theorem	declared	appendices/A-long-proofs.tex:27215
`depends_on`	←	Existence and uniqueness of the reduced form	declared	appendices/A-long-proofs.tex:27353
`proves`	←	app:A-long-proofs@proof-341	declared	appendices/A-long-proofs.tex:27219