Geodesics and Length Spaces

graph TD D1["D1 Length of curve γ\nL(γ) = sup Σ d(γ(t_i), γ(t_{i+1}))"] D2["D2 Length space\nd(x,y) = inf L(γ) over curves from x to y"] D3["D3 Geodesic\nCurve realizing d(x,y)"] D4["D4 Intrinsic metric\nMetric induced by length structure"] T1["T1 Geodesic space\nAny two points joined by geodesic"] T2["T2 Hopf-Rinow\nComplete + Geodesic ⇒ bounded closed balls compact"] T3["T3 Length space completion\nCompletions preserve length structure"] T4["T4 Alexandrov triangle\nComparison with model space"] T5["T5 Local geodesic extends\nIn complete space, local geodesic ⇒ geodesic"] D1 --> D2 D2 --> D3 D2 --> D4 D3 --> T1 D2 --> T2 D2 --> T3 D3 --> T4 D3 --> T5 T1 --> T2 T2 --> T5 T4 --> T2 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