definition 13.54 Embedded submanifold

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2637 · p. 484

ground object -- no derivation owed

Rests on

depends_on definition 13.53 Immersion, submersion, embedding ¶
- 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 13.43 Coordinate system ¶
    - 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 ¶
      - depends_on definition 13.47 Differentiable structure ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
    - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 6.7 Homeomorphism ¶
  - depends_on definition 3.47 Bijective map ¶
  - depends_on definition 3.51 Inverse map ¶
    - depends_on definition 3.47 Bijective map ¶ ↺
    - depends_on definition 3.49 Composition map ¶
    - depends_on definition 3.50 Identity map ¶
  - depends_on definition 6.6 Continuous map ¶
    - depends_on definition 3.53 Preimage ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.1 Topological space ¶ ↺
depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

depends_on corollary 13.64 The image of a constant-rank map, locally ¶
depends_on definition 13.130 Distribution; involutive; integrable ¶
- depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
- 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 lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.555 The actions are well defined ¶
  - depends_on theorem A.545 Liouville–Arnold ¶
depends_on definition 13.58 Hypersurface ¶
- depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
  - depends_on example A.294 The sphere, made explicit ¶
  - depends_on example 13.87 The $2$-sphere ¶
- depends_on theorem 13.59 Regular value theorem ¶
  - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶ ↺
  - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
    - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
    - depends_on theorem A.563 Marsden–Weinstein reduction ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
      - depends_on example A.572 Rotational reduction of the central-force problem ¶
    - 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 theorem 24.47 Marsden–Weinstein reduction ¶
      - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
  - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶ ↺
  - depends_on proposition A.565 Freeness makes every value regular ¶ ↺
  - depends_on theorem 24.47 Marsden–Weinstein reduction ¶ ↺
depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
depends_on theorem 13.67 Quotient manifold theorem ¶
- depends_on example 13.68 Why each hypothesis is there ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶
- depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶ ↺
- depends_on theorem A.563 Marsden–Weinstein reduction ¶ ↺
depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
- depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺

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`	→	Immersion, submersion, embedding	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2643
`depends_on`	→	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2643
`depends_on`	←	The image of a constant-rank map, locally	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3002
`depends_on`	←	Distribution; involutive; integrable	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5929
`depends_on`	←	Hypersurface	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2757
`depends_on`	←	Every orbit is an embedded copy of $G$	declared	appendices/A-long-proofs.tex:26057
`depends_on`	←	Quotient manifold theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3082
`depends_on`	←	Regular value theorem in codimension $k$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879