Ivancevic_Applied-Diff-Geom

820 Applied Differential Geometry: A Modern IntroductionX. It takes its values into J 2 (X, Y ) and consists with the 2–jet prolongationj 2 s of s:j 1 j 1 s(x) = j 2 s(x) = j 2 xs.Given a 2–jet space J 2 (X, Y ) of the fibre bundle Y → X, we have(i) the fibred map r 2 : J 2 (Y, T Y ) → T J 2 (X, Y ), given locally by(ẏ i α, ẏ i αµ) ◦ r 2 = ((ẏ i ) α − y i µẋ µ α, (ẏ i ) αµ − y i µẋ µ αµ − y i αµẋ µ α),where J 2 (Y, T Y ) is the 2–jet space of the tangent bundle T Y, and(ii) the canonical isomorphism V J 2 (X, Y ) = J 2 (X, V Y ), where V J 2 (X, Y )is the vertical tangent bundle of the fibre bundle J 2 (X, Y ) → X, andJ 2 (X, V Y ) is the 2–jet space of the fibre bundle V Y → X.As a consequence, every vector–field u on a fibre bundle Y → X admitsthe 2−jet lift to the projectable vector–fieldj 2 u = r 2 ◦ j 2 u : J 2 (X, Y ) → T J 2 (X, Y ).In particular, if2–jet lift readsu = u α ∂ α + u i ∂ i is a projectable vector–field on Y , itsj 2 u = u α ∂ α + u i ∂ i + (∂ α u i + y j α∂ j u i − y i µ∂ α u µ )∂ α i (5.52)+ [(∂ α + y j α∂ j + y j βα ∂β j )(∂ α + y k α∂ k )u i − y i µẋ µ αβ − yi µβẋ µ α]∂ αβi .Generalizations of the contact and complementary maps (5.5–5.6) tothe 2–jet space J 2 (X, Y ) readλ : J 2 (X, Y ) → T ∗ X ⊗ T J 1 (X, Y )is locally given byλ = dx α ⊗ ̂∂ α = dx α ⊗ (∂ α + yα∂ i i + yµα∂ i µ i ), while (5.53)θ : J 2 (X, Y ) → T ∗ J 1 (X, Y ) ⊗ V J 1 (X, Y ) is locally given byθ = (dy i − y i αdx α ) ⊗ ∂ i + (dy i µ − y i µαdx α ) ⊗ ∂ µ i . (5.54)The contact map (5.53) defines the canonical horizontal splitting of theexact sequence0 → V J 1 (X, Y ) ↩→ T J 1 (X, Y ) → J 1 (X, Y ) × T X → 0.Hence, we get the canonical horizontal splitting of a projectable vector–fieldj 1 u ≡ u on J 1 (X, Y ) over J 2 (X, Y ):j 1 u = u H + u V = u α [∂ α + y i α + y i µα] + [(u i − y i αu α )∂ i + (u i µ − y i µαu α )∂ µ i ].