Smooth Manifolds

graph TD D1["D1 Smooth manifold\nHausdorff, 2nd countable, charts"] D2["D2 Atlas\nMaximal collection of compatible charts"] D3["D3 Tangent space T_p M\nDerivations at a point"] D4["D4 Smooth map\nInfinitely differentiable between manifolds"] T1["T1 Tangent space is vector space\ndim T_p M = n"] T2["T2 Inverse function theorem\nLocal diffeomorphism criterion"] T3["T3 Submersion theorem\nRegular value gives submanifold"] T4["T4 Whitney embedding\nCompact M embeds in R^{2n}"] T5["T5 Partition of unity\nExistence on paracompact manifolds"] D1 --> D2 D1 --> D3 D4 --> D3 D2 --> D1 D3 --> T1 D4 --> T2 D4 --> T3 D1 --> T4 D1 --> T5 D2 --> T5 T2 --> T3 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem