Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 803It is convenient to call π 1 (5.1) the jet bundle, while π 1 0 (5.2) is called theaffine jet bundle. Note that, if Y → X is a vector or an affine bundle, italso holds for the jet bundle π 1 (5.1) [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)].There exist several equivalent ways in order to give the 1–jet spaceJ 1 (X, Y ) with the smooth manifold structure. Let Y → X be a fibrebundle with fibred coordinate atlases (4.4). The 1–jet space J 1 (X, Y ) ofthe bundle Y → X admits the adapted coordinate atlases(x α , y i , y i α), (x α , y i , y i α)(j 1 xs) = (x α , s i (x), ∂ α s i ), (5.3)y ′iα = ( ∂y′ i∂y j yj µ + ∂y′ i∂x µ ) ∂xµ∂x ′ α , (5.4)and thus satisfies the conditions which are required of a manifold. Thesurjection (5.1) is a bundle. The surjection (5.2) is a bundle. If Y → X isa bundle, so is the surjection (5.1).The transformation law (5.4) shows that the jet bundle J 1 (X, Y ) → Yis an affine bundle. It is modelled on the vector bundle T ∗ X ⊗ V Y → Y.In particular, if Y is the trivial bundleπ 2 : V × R m −→ R m ,the corresponding jet bundle J 1 (X, Y ) −→ R m (5.1) is a trivial bundle.There exist the following two canonical bundle monomorphisms of thejet bundle J 1 (X, Y ) −→ Y [Sardanashvily (1993); Sardanashvily (1995);Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a); Sardanashvily(2002a)]:• the contact mapλ : J 1 (X, Y ) ↩→ T ∗ X ⊗ T Y, λ = dx α ⊗ ̂∂ α = dx α ⊗ (∂ α + y i α∂ i ),(5.5)• the complementary mapθ : J 1 (X, Y ) ↩→ T ∗ Y ⊗V Y, θ = ̂dy i ⊗∂ i = (dy i −y i αdx α )⊗∂ i . (5.6)These canonical maps enable us to express the jet–space machinery in termsof tangent–valued differential forms (see section 4.10 above).The operatorŝ∂ α = ∂ α + y i α∂ i