lemma A.533 The orbit map has constant rank $d$

open in the book · appendices/A-long-proofs.tex:26008 · p. 3052

Rests on

depends_on definition 13.66 Smooth action; free; proper; orbit ¶
- 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 equation 3.51 eq:set-indexed ¶
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 13.81 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 theorem 13.63 Constant rank theorem ¶
- depends_on corollary A.292 Inverse function theorem ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - proves proof ch:05-real-analysis@proof-63 ¶
  - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-62 ¶
    - proves proof app:A-long-proofs@proof-185 ¶
  - proves proof app:A-long-proofs@proof-186 ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
- depends_on theorem A.286 Implicit function theorem ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
proves proof app:A-long-proofs@proof-321 ¶

Supports

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 ¶

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 action; free; proper; orbit	declared	appendices/A-long-proofs.tex:26013
`depends_on`	→	Immersion, submersion, embedding	declared	appendices/A-long-proofs.tex:26013
`depends_on`	→	Constant rank theorem	declared	appendices/A-long-proofs.tex:26013
`depends_on`	←	Every orbit is an embedded copy of $G$	declared	appendices/A-long-proofs.tex:26057
`depends_on`	←	Existence of a slice	declared	appendices/A-long-proofs.tex:26106
`proves`	←	app:A-long-proofs@proof-321	declared	appendices/A-long-proofs.tex:26017