definition 13.49 Smooth map between manifolds

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2546 · p. 483

ground object -- no derivation owed

Rests on

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 ¶ ↺

Supports

depends_on definition 13.50 Diffeomorphism ¶
depends_on definition 13.52 Differential; pushforward ¶
- depends_on definition 13.53 Immersion, submersion, embedding ¶
  - 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 definition 13.58 Hypersurface ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on theorem 13.59 Regular value theorem ¶
    - 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 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 ¶ ↺
  - 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 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 ¶
- depends_on example 13.55 An injective immersion that is not an embedding ¶ ↺
- depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
  - depends_on remark 24.11 The SI dimension of every object in this chapter ¶
    - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
    - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
    - depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
- depends_on theorem 13.59 Regular value theorem ¶ ↺
- depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶ ↺
depends_on definition 13.66 Smooth action; free; proper; orbit ¶
- depends_on example 13.68 Why each hypothesis is there ¶ ↺
- depends_on lemma A.531 The projection is open ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
  - depends_on proposition A.532 Separation and countability of the quotient ¶
    - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
    - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
- depends_on lemma A.533 The orbit map has constant rank $d$ ¶ ↺
- depends_on proposition A.535 Existence of a slice ¶ ↺
- depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
- depends_on theorem 13.67 Quotient manifold theorem ¶ ↺
depends_on lemma A.538 Submersions have smooth local sections ¶ ↺
depends_on proposition A.539 Universal property ¶ ↺
depends_on theorem 13.59 Regular value theorem ¶ ↺

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`	→	Atlas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2557
`depends_on`	→	Differentiable map on a topological space	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2557
`depends_on`	→	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2557
`depends_on`	←	Diffeomorphism	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2571
`depends_on`	←	Differential; pushforward	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2617
`depends_on`	←	Smooth action; free; proper; orbit	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	←	Submersions have smooth local sections	declared	appendices/A-long-proofs.tex:26279
`depends_on`	←	Universal property	declared	appendices/A-long-proofs.tex:26300
`depends_on`	←	Regular value theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2772