definition 13.52 Differential; pushforward

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2595 · p. 483

ground object -- no derivation owed

Rests on

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 ¶ ↺

Supports

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 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 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.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 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 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 ¶ ↺
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 example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - 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 $k$ ¶ ↺

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`	→	Smooth map between manifolds	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2617
`depends_on`	→	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2617
`depends_on`	←	Immersion, submersion, embedding	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2634
`depends_on`	←	An injective immersion that is not an embedding	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2690
`depends_on`	←	The canonical form is symplectic and intrinsic	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:280
`depends_on`	←	Regular value theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2772
`depends_on`	←	Regular value theorem in codimension $k$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879