Ivancevic_Applied-Diff-Geom

812 Applied Differential Geometry: A Modern IntroductionLet Y and Y ′ be vector bundles over X. Given linear connections Γ andΓ ′ on Y and Y ′ respectively, there is the unique linear connection Γ ⊗ Γ ′ onthe tensor product Y ⊗Y ′ → X, such that the following diagram commutes:J 1 (X, Y ) × J 1 (X, Y ) ′ J 1 ⊗ ✲ J 1 (Y ⊗ Y ′ )Γ × Γ ′❄Y × Y ′ ✲ ❄⊗Y ⊗ Y ′Γ ⊗ Γ ′It is called the tensor–product connection and has the coordinate expression(Γ ⊗ Γ ′ ) ikα = Γ i jαy jk + Γ ′ kjαy ij .Every connection Γ on Y → X, by definition, induces the horizontaldistribution on Y ,Γ : T X ↩→ T Y, locally given by ∂ α ↦→ ∂ α + Γ i α(y)∂ i .It is generated by horizontal liftsτ Γ = τ α (∂ α + Γ i α∂ i )onto Y of vector–fields τ = τ α ∂ α on X. The associated Pfaffian system islocally generated by the forms (dy i − Γ i αdx α ).The horizontal distribution Γ(T X) is involutive iff Γ is a curvature–freeconnection. As a proof, straightforward calculations show that [τ Γ , τ ′ Γ] =([τ, τ ′ ]) Γ iff the curvature R (5.34) of Γ vanishes everywhere.Not every bundle admits a curvature–free connection. If a principal bundleover a simply–connected base (i.e., its first homotopy group is trivial)admits a curvature–free connection, this bundle is trivializable [Kobayashiand Nomizu (1963/9)].The horizontal distribution defined by a curvature–free connection iscompletely integrable. The corresponding foliation on Y is transversal tothe foliation defined by the fibration π : Y −→ X. It is called the horizontalfoliation. Its leaf through a point y ∈ Y is defined locally by the integralsection s y of the connection Γ through y. Conversely, let Y admits a horizontalfoliation such that, for each point y ∈ Y , the leaf of this foliationthrough y is locally defined by some section s y of Y −→ X through y. Then,the following map is well definedΓ : Y −→ J 1 (X, Y ), Γ(y) = j 1 s s y , π(y) = x.