Ivancevic_Applied-Diff-Geom

808 Applied Differential Geometry: A Modern IntroductionLet Y → X be an affine bundle modelled on a vector bundle Y → X. Anaffine connection on Y readsΓ = dx α ⊗ [∂ α + (−Γ i jα(x)y j + Γ i α(x))∂ i ],whereΓ = dx α ⊗ [∂ α − Γ i jα(x)y j ∂ i ] is a linear connection on Y .Since the affine jet bundle J 1 (X, Y ) −→ Y is modelled on the vectorbundle Y −→ X, Ehresmann connections on Y −→ X constitute an affinespace modelled on the linear space of soldering forms on Y . If Γ is aconnection and σ is a soldering form (4.138) on Y , its sumΓ + σ = dx α ⊗ [∂ α + (Γ i α + σ i α)∂ i ]is a connection on Y . Conversely, if Γ and Γ ′ are connections on Y , thenΓ − Γ ′ = (Γ i α − Γ ′ iα)dx α ⊗ ∂ iis a soldering form.Given a connection Γ, a vector–field u on a fibre bundle Y −→ X iscalled horizontal if it lives in the horizontal distribution HY , i.e., takes theformu = u α (y)(∂ α + Γ i α(y)∂ i ). (5.22)Any vector–field τ on the base X of a fibre bundle Y −→ X admits thehorizontal liftΓτ = τ⌋Γ = τ α (∂ α + Γ i α∂ i ) (5.23)onto Y by means of a connection Γ on Y −→ X.Given the splitting (5.17), the dual splitting of the exact sequence (4.14)isΓ : V ∗ Y → T ∗ Y, dy i ↦→ Γ⌋dy i = dy i − Γ i αdx α , (5.24)where Γ is the vertical-valued form (5.20).There is 1–1 correspondence between the connections on a fibre bundleY → X and the jet fields, i.e., global sections of the affine jet bundleJ 1 (X, Y ) → Y . Indeed, given a global section Γ of J 1 (X, Y ) → Y , thetangent–valued formλ ◦ Γ = dx α ⊗ (∂ α + Γ i α∂ i )