theorem A.537 The smooth structure on the orbit space

Rests on

• 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 13.44 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 13.81 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 definition 6.1 Topological space ¶ ↺
• depends_on lemma A.536 A slice is a chart domain downstairs ¶
  • depends_on definition 6.7 Homeomorphism ¶ ↺
  • depends_on lemma A.531 The projection is open ¶
    • depends_on definition 13.66 Smooth action; free; proper; orbit ¶
      • depends_on definition 13.48 Differentiable manifold ¶ ↺
      • depends_on definition 13.49 Smooth map between manifolds ¶ ↺
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
    • depends_on definition 6.6 Continuous map ¶ ↺
    • depends_on definition 6.2 Open set ¶ ↺
    • proves proof app:A-long-proofs@proof-319 ¶
  • depends_on proposition A.535 Existence of a slice ¶
    • 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 13.66 Smooth action; free; proper; orbit ¶ ↺
    • depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      • depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
      • depends_on definition 13.53 Immersion, submersion, embedding ¶ ↺
      • depends_on theorem 13.63 Constant rank theorem ¶
        • depends_on corollary A.292 Inverse function theorem ¶ ↺
        • 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 ¶
    • proves proof app:A-long-proofs@proof-323 ¶
  • proves proof app:A-long-proofs@proof-324 ¶
• depends_on proposition A.532 Separation and countability of the quotient ¶
  • depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
  • depends_on lemma A.530 Two elementary facts about compactness ¶
    • depends_on definition A.529 Hausdorff; second countable; locally compact ¶
      • depends_on definition 6.9 Compact set ¶ ↺
      • depends_on definition 6.2 Open set ¶ ↺
      • depends_on definition 6.1 Topological space ¶ ↺
    • depends_on definition 6.9 Compact set ¶ ↺
    • depends_on proposition 6.10 Continuous images of compact sets ¶
      • depends_on definition 6.9 Compact set ¶ ↺
      • depends_on definition 6.6 Continuous map ¶ ↺
      • proves proof ch:04-topology@proof-1 ¶
    • proves proof app:A-long-proofs@proof-318 ¶
  • depends_on lemma A.531 The projection is open ¶ ↺
  • proves proof app:A-long-proofs@proof-320 ¶
• proves proof app:A-long-proofs@proof-325 ¶

Supports

Nothing declares a dependency on this node yet.