Connections on Manifolds

graph TD D1["D1 Affine connection ∇\nKoszul axioms: C^∞-linear in X, Leibniz in Y"] D2["D2 Parallel transport\nP_γ: T_p M → T_q M along γ"] D3["D3 Holonomy group Hol_p\nCurvature at p from parallel transport"] D4["D4 Torsion tensor T\nT(X,Y) = ∇_X Y − ∇_Y X − [X,Y]"] T1["T1 Existence of parallel sections\nFlat connection ⇒ local parallel frames"] T2["T2 Ambrose-Singer\nHolonomy algebra = curvature derivatives"] T3["T3 Frobenius for distributions\nInvolutive ⇒ integrable"] T4["T4 Ricci identity\n[∇_X, ∇_Y]σ − ∇_[X,Y] σ = R(X,Y)σ"] T5["T5 Geodesic equation\nConnection defines autoparallel curves"] D1 --> D2 D1 --> D4 D2 --> D3 D1 --> T1 D3 --> T2 D1 --> T3 D1 --> T4 D1 --> T5 D2 --> T1 D4 --> T4 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