Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 815Due to this splitting, one can construct the first–order differential operator˜D = π 1 ◦ D γ : J 1 (X, Y ) → T ∗ X ⊗ V Y → T ∗ X ⊗ V Y Σ ,˜D = dx α ⊗ (y i α − A i α − A i mσ m α )∂ i , (5.41)on the composite manifold Y , where D γ is the covariant differential (5.29)relative to the composite connection (5.40). We call ˜D the vertical covariantdifferential.5.3.1 Principal ConnectionsThe above general approach to connections as jet fields is suitable to formulatethe classical concept of principal connections. In this section, astructure group G of a principal bundle is assumed to be a real finite–dimensional Lie group (of positive dimension dim G > 0).A principal connection A on a principal bundle P → Q is defined to bea G−equivariant global jet field on P such thatj 1 R g ◦ A = A ◦ R gfor each canonical map (4.31). We haveA ◦ R g = j 1 R g ◦ A, (g ∈ G), (5.42)A = dq α ⊗ (∂ α + A m α (p)e m ), (p ∈ P ),A m α (qg) = A m α (p)adg −1 (e m ).A principal connection A determines splitting T Q ↩→ T G P of the exactsequence (4.38). We will refer toA = A − θ Q = A m α dq α ⊗ e m (5.43)as a local connection form.Let J 1 (Q, P ) be the 1–jet space of a principal bundle P → Q with astructure Lie group G. The jet prolongationJ 1 (Q, P ) × J 1 (Q × G) → J 1 (Q, P )of the canonical action (4.31) brings the jet bundle J 1 (Q, P ) → Q into ageneral affine bundle modelled on the group bundleJ 1 (Q × G) = G × (T ∗ Q ⊗ g l ) (5.44)over Q. However, the jet bundle J 1 (Q, P ) → Q fails to be a principal bundlesince the group bundle (5.44) is not a trivial bundle over Q in general. At