Ivancevic_Applied-Diff-Geom

804 Applied Differential Geometry: A Modern Introductionare usually called the total derivatives, or the formal derivatives, while theformŝdy i = dy i − y i αdx αare conventionally called the contact forms.Identifying the 1–jet space J 1 (X, Y ) to its images under the canonicalmaps (5.5) and (5.6), one can represent 1–jets j 1 xs ≡ (x α , y i , y i α) by tangent–valued formsdx α ⊗ (∂ α + y i α∂ i ), and (dy i − y i αdx α ) ⊗ ∂ i . (5.7)There exists a jet functor J : Bun → Jet, from the category Bunof fibre bundles to the category Jet of jet spaces. It implies the naturalprolongation of maps of bundles to maps of jet spaces.Every bundle map Φ : Y −→ Y ′ over a diffeomorphism f of X has the1–jet prolongation to the bundle map j 1 Φ : J 1 (X, Y ) −→ J 1 (X, Y ) ′ , givenbyj 1 Φ : j 1 xs ↦→ j 1 f(x) (Φ ◦ s ◦ f −1 ), (5.8)y ′ iα ◦ j 1 Φ = ∂ α (Φ i ◦ f −1 ) + ∂ j (Φ i y j α ◦ f −1 ).It is both an affine bundle map over Φ and a fibred map over the diffeomorphismf. The 1–jet prolongations (5.8) of fibred maps satisfy the chainrulesj 1 (Φ ◦ Φ ′ ) = j 1 Φ ◦ j 1 Φ ′ , j 1 (Id Y ) = Id J 1 (X,Y ) .If Φ is a surjection (resp. an injection), so is j 1 Φ.In particular, every section s of a bundle Y → X admits the 1–jetprolongation to the section j 1 xs of the jet bundle J 1 (X, Y ) → X, given byWe have(y i , y i α) ◦ j 1 xs = (s i (x), ∂ α s i ).λ ◦ j 1 xs = T s,where λ is the contact map (5.5).Every projectable vector–field u on a fibre bundle Y → X,u = u α (x)∂ α + u i (y)∂ i