definition 13.66 Smooth action; free; proper; orbit

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3046 · p. 489

ground object -- no derivation owed

Rests on

depends_on definition 13.48 Differentiable manifold ¶
- depends_on definition 13.47 Differentiable structure ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
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 13.44 Atlas ¶ ↺
  - depends_on definition 13.43 Coordinate system ¶ ↺
- depends_on definition 13.48 Differentiable manifold ¶ ↺
depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - 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 equation 3.51 eq:set-indexed ¶ ↺

Supports

depends_on example 13.68 Why each hypothesis is there ¶
- depends_on remark A.542 Where each hypothesis is spent ¶
depends_on lemma A.531 The projection is open ¶
- 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 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.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 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 example 13.68 Why each hypothesis is there ¶ ↺
- 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 example A.573 The abelian case, and eliminating a cyclic coordinate ¶
  - depends_on example A.572 Rotational reduction of the central-force problem ¶

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`	→	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	→	Smooth map between manifolds	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	←	Why each hypothesis is there	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3114
`depends_on`	←	The projection is open	declared	appendices/A-long-proofs.tex:25947
`depends_on`	←	The orbit map has constant rank $d$	declared	appendices/A-long-proofs.tex:26013
`depends_on`	←	Existence of a slice	declared	appendices/A-long-proofs.tex:26106
`depends_on`	←	Separation and countability of the quotient	declared	appendices/A-long-proofs.tex:25968
`depends_on`	←	Quotient manifold theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3082