Smooth manifold, atlas, charts. Tangent space T_pM, derivative. Submersion, immersion.
graph TD
D1["Def: Smooth manifold\natlas, charts φ:U→ℝⁿ"]
D2["Def: Tangent space\nT_pM derivations"]
D3["Def: df pushforward\nmap on T_p"]
D4["Def: Submersion\nrank df = dim N"]
T1["Thm: T_pM ≅ ℝⁿ\ndim M"]
T2["Thm: Immersion theorem\nrank k ⇒ local coords"]
T3["Thm: Submersion thm\nlocal onto"]
T4["Thm: Regular value\nf⁻¹(c) submanifold"]
T5["Thm: Whitney embed\nM↪ℝ²ⁿ"]
D1 --> D2
D2 --> D3
D3 --> D4
D2 --> T1
D4 --> T3
D3 --> T4
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 12
Frontier: math.DG