definition 13.53 Immersion, submersion, embedding

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2622 · p. 484

ground object -- no derivation owed

Rests on

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 13.44 Atlas ¶ ↺
- 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 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on definition 6.1 Topological space ¶ ↺

Supports

depends_on definition 13.54 Embedded submanifold ¶
- 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 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 theorem A.563 Marsden–Weinstein reduction ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - 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 proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  - 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 ¶ ↺
depends_on example 13.55 An injective immersion that is not an embedding ¶
depends_on lemma A.533 The orbit map has constant rank $d$ ¶
- depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
- depends_on proposition A.535 Existence of a slice ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶
    - depends_on theorem A.537 The smooth structure on the orbit space ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
depends_on lemma A.538 Submersions have smooth local sections ¶
- depends_on proposition A.539 Universal property ¶
  - depends_on corollary A.540 Uniqueness of the smooth structure ¶
depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺

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`	→	Differential; pushforward	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2634
`depends_on`	→	Homeomorphism	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2634
`depends_on`	←	Embedded submanifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2643
`depends_on`	←	An injective immersion that is not an embedding	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2690
`depends_on`	←	The orbit map has constant rank $d$	declared	appendices/A-long-proofs.tex:26013
`depends_on`	←	Submersions have smooth local sections	declared	appendices/A-long-proofs.tex:26279
`depends_on`	←	Every orbit is an embedded copy of $G$	declared	appendices/A-long-proofs.tex:26057
`depends_on`	←	The smooth structure on the orbit space	declared	appendices/A-long-proofs.tex:26224