Ivancevic_Applied-Diff-Geom

814 Applied Differential Geometry: A Modern IntroductionY → R, defines a trivialization Y ≃ R×M.Let us consider a bundle π : Y → X which admits a composite fibrationY → Σ → X, (5.38)where Y → Σ and Σ → X are bundles. It is equipped with the bundlecoordinates (x α , σ m , y i ) together with the transition functionsx α → x ′ α (x µ ), σ m → σ ′ m (x µ , σ n ), y i → y ′ i (x µ , σ n , y j ),where (x µ , σ m ) are bundle coordinates of Σ → X. For example, we havethe composite bundlesLetT Y → Y → X, V Y → Y → X, J 1 (X, Y ) → Y → X.A = dx α ⊗ (∂ α + A i α∂ i ) + dσ m ⊗ (∂ m + A i m∂ i ) (5.39)be a connection on the bundle Y −→ Σ andΓ = dx α ⊗ (∂ α + Γ m α ∂ m )a connection on the bundle Σ −→ X. Given a vector–field τ on X, let usconsider its horizontal lift τ Γ onto Σ by Γ and then the horizontal lift (τ Γ ) Aof τ Γ onto Y by the connection (5.39).There exists the connectionγ = dx α ⊗ [∂ α + Γ m α ∂ m + (A i mΓ m α + A i α)∂ i ]. (5.40)on Y → X such that the horizontal lift τ γ onto Y of any vector–field τ onX consists with the above lift (τ Γ ) A [Sardanashvily (1993); Sardanashvily(1995)]. It is called the composite connection.Given a composite bundle Y (5.38), the exact sequence0 → V Y Σ ↩→ V Y → Y × V Σ → 0over Y take place, where V Y Σ is the vertical tangent bundle of Y → Σ.Every connection (5.39) on the bundle Y → Σ induces the splittingV Y = V Y Σ ⊕ (Y × V Σ),given byẏ i ∂ i + ˙σ m ∂ m = (ẏ i − A i m ˙σ m )∂ i + ˙σ m (∂ m + A i m∂ i ).