Ivancevic_Applied-Diff-Geom

810 Applied Differential Geometry: A Modern IntroductionGiven a fibre bundle Y → X, let f : X ′ → X be a manifold map andf ∗ Y the pull–back of Y over X ′ . Any connection Γ (5.20) on Y → Xinduces the pull–back connectionf ∗ Γ = (dy i − Γ i α(f µ (x ′ ν), y j ) ∂f α∂x ′ µ dx′µ ) ⊗ ∂ i (5.28)on the pull–back fibre bundle f ∗ Y → X ′ .Since the affine jet bundle J 1 (X, Y ) −→ Y is modelled on the vectorbundle Y −→ X, connections on a fibre bundle Y constitute the affine spacemodelled on the linear space of soldering forms on Y . It follows that, if Γis a connection andσ = σ i αdx α ⊗ ∂ iis a soldering form on a fibre bundle 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 a fibrebundle Y , thenΓ − Γ ′ = (Γ i α − Γ ′ iα)dx α ⊗ ∂ iis a soldering form on Y .The key point for physical applications lies in the fact that every connectionΓ on a fibre bundle Y −→ X induces the first–order differentialoperatorD Γ : J 1 (X, Y ) → T ∗ X ⊗ V Y, D Γ = λ − Γ ◦ π 1 0 = (y i α − Γ i α)dx α ⊗ ∂ i ,(5.29)called the covariant differential relative to the connection Γ. If s : X → Yis a section, one defines its covariant differentialand its covariant derivative∇ Γ s = D Γ ◦ j 1 s = (∂ α s i − Γ i α ◦ s)dx α ⊗ ∂ i (5.30)∇ Γ τ s = τ⌋∇ Γ s (5.31)along a vector–field τ on X. A (local) section s of Y → X is said to be anintegral section of a connection Γ (or parallel with respect to Γ) if s obeysthe equivalent conditions∇ Γ s = 0 or j 1 s = Γ ◦ s. (5.32)