Connection (affine bundle)

Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J1Y of the jet bundle J1Y → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the connection on an affine bundle is an example of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use of the adjective "affine".) With respect to affine bundle coordinates (xλ, yi) on Y, an affine connection Γ on Y → X is given by the tangent-valued connection form Γ = d x λ ⊗ ( ∂ λ + Γ λ i ∂ i ) , Γ λ i = Γ λ i j ( x ν ) y j + σ λ i ( x ν ) .

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