Hyperbolic Geometry

graph TD D1["D1 Hyperbolic space H^n\nComplete simply connected K ≡ −1"] D2["D2 Poincaré disk model\nUnit ball with ds² = 4(dx²)/(1−|x|²)²"] D3["D3 Ideal boundary ∂∞ H^n\nAsymptotic geodesic classes"] D4["D4 Hyperbolic isometry\nMöbius or fractional linear"] T1["T1 Geodesics are lines/circles\nPerpendicular to boundary in disk"] T2["T2 Mostow rigidity\nn ≥ 3: isom π₁M ≅ π₁N ⇒ M ≅ N"] T3["T3 Triangle area\nArea = π − (angles) for geodesic triangle"] T4["T4 Thin triangles\nSides within δ of each other"] T5["T5 Classification of isometries\nElliptic, parabolic, hyperbolic"] D1 --> D2 D1 --> D3 D1 --> D4 D2 --> T1 D1 --> T2 D2 --> T3 D1 --> T4 D4 --> T5 T1 --> T3 T1 --> T4 T2 --> D4 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