Ivancevic_Applied-Diff-Geom

500 Applied Differential Geometry: A Modern IntroductionEvery splittingof the exact sequence (4.13) andY × T X ↩→ T Y, ∂ α ↦→ ∂ α + Γ i α(y)∂ i ,V ∗ Y → T ∗ Y, d i ↦→ dy i − Γ i α(y)dx α ,of the exact sequence (4.14), by definition, corresponds to a certain connectionon the bundle Y → X, and vice versa.Let Φ be a fibred map of a bundle Y → X to a bundle Y ′ → X ′ overf : X → X ′ . The tangent map T Φ : T Y → T Y ′ to Φ reads(ẋ ′ α , ẏ′i ) ◦ T Φ = (∂µ f α ẋ µ , ∂ µ Φ i ẋ µ + ∂ j Φ i ẏ j ). (4.15)It is both the linear bundle map over Φ, given by the commutativity diagramT ΦT Y ✲ T Y ′π Yπ Y ′❄Y ✲ ❄YΦ′as well as the fibred map over the tangent map T f to f, given by thecommutativity diagramT ΦT Y ✲ T Y ′4.3.4 Affine Bundles❄T X ✲ ❄T XT f′Given a vector bundle Y → X, an affine bundle modelled over Y is afibre bundle Y → X whose fibres Y x , (for all x ∈ X), are affine spacesmodelled over the corresponding fibres Y x of the vector bundle Y , and Yadmits a bundle atlas Ψ Y (4.8) whose trivialization morphisms ψ ξ (x) andtransition functions functions ρ ξζ (x) are affine maps. The correspondingbundle coordinates (y i ) possess an affine coordinate transformation lawy ′ i = ρij (x α )y j + ρ i (x α ).