example 13.68 Why each hypothesis is there

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3088 · p. 489

example -- no derivation owed by its kind

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 theorem 13.67 Quotient manifold theorem ¶
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
- depends_on definition 13.54 Embedded submanifold ¶
  - 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 6.7 Homeomorphism ¶
      - depends_on definition 3.47 Bijective map ¶
      - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 6.6 Continuous map ¶
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
- 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 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - 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 ¶
      - 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 definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on theorem A.286 Implicit function theorem ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
- proves proof app:A-long-proofs@proof-329 ¶
- proves proof ch:11-manifolds-tensors-curvature@prooflink-2 ¶

Supports

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	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3114
`depends_on`	→	Quotient manifold theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3114
`depends_on`	←	Where each hypothesis is spent	declared	appendices/A-long-proofs.tex:26414