proposition A.565 Freeness makes every value regular

open in the book · appendices/A-long-proofs.tex:27174 · p. 3064

Rests on

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 definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-54 ¶
- 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - 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 13.105 Closed form ¶
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on definition 24.2 Symplectic vector space ¶ ↺
- 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 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 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 13.44 Atlas ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - 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 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.58 Hypersurface ¶
    - depends_on definition 13.54 Embedded submanifold ¶
      - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - 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 ¶

Supports

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 A.566 The isotropy group acts, and the quotient is smooth ¶
- depends_on proposition A.570 Existence and uniqueness of the reduced form ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Dimension and double orthogonal	declared	appendices/A-long-proofs.tex:27181
`depends_on`	→	The level set of the momentum map is the symplectic orthogonal of the orbit	declared	appendices/A-long-proofs.tex:27181
`depends_on`	→	Regular value theorem	declared	appendices/A-long-proofs.tex:27181
`depends_on`	←	The differential of the momentum map along the orbit	declared	appendices/A-long-proofs.tex:27253
`depends_on`	←	The isotropy group acts, and the quotient is smooth	declared	appendices/A-long-proofs.tex:27215
`proves`	←	app:A-long-proofs@proof-340	declared	appendices/A-long-proofs.tex:27185