Riemannian Geometry

graph TD D1["D1 Riemannian metric g\nPositive definite smooth 2-tensor"] D2["D2 Levi-Civita connection ∇\nMetric compatibility, torsion-free"] D3["D3 Riemann curvature tensor R\nR(X,Y)Z = ∇_X∇_Y Z − ∇_Y∇_X Z − ∇_[X,Y] Z"] D4["D4 Geodesic\n∇_γ' γ' = 0"] T1["T1 Fundamental theorem\nUnique Levi-Civita connection"] T2["T2 Gauss-Bonnet\n∫ K dA = 2πχ for surfaces"] T3["T3 Comparison: K ≥ 0 ⇒ geodesics diverge\nBonnet-Myers for K ≥ κ > 0"] T4["T4 Cartan-Hadamard\nSimply connected, K ≤ 0 ⇒ diffeo to R^n"] T5["T5 Parallel transport\nPreserves inner product along curves"] D1 --> D2 D1 --> T1 D2 --> D3 D2 --> D4 D2 --> T5 D3 --> T2 D3 --> T3 D3 --> T4 D4 --> T3 T1 --> D2 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