Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 805has the 1−jet lift to the projectable vector–field j 1 u on the 1–jet spaceJ 1 (X, Y ), given byj 1 u ≡ u = r 1 ◦ j 1 u : J 1 (X, Y ) → T J 1 (X, Y ),j 1 u ≡ u = u α ∂ α + u i ∂ i + (d α u i − y i µ∂ α u µ )∂ α i . (5.9)Geometrical applications of jet spaces are based on the canonical mapover J 1 (X, Y ),J 1 (X, Y ) × T X → J 1 (X, Y ) × T Y,which means the canonical horizontal splitting of the tangent bundle T Ydetermined over J 1 (X, Y ) as follows [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a);Sardanashvily (2002a)].The canonical maps (5.5) and (5.6) induce the bundle monomorphismŝλ : J 1 (X, Y ) × T X → J 1 (X, Y ) × T Y, ∂ α ↦→ ̂∂ α = ∂ α ⌋λ (5.10)̂θ : J 1 (X, Y ) × V ∗ Y → J 1 (X, Y ) × T ∗ Y, dy i ↦→ ̂dy i = θ⌋dy i (5.11)The map (5.10) determines the canonical horizontal splitting of the pull–backJ 1 (X, Y ) × T Y = ̂λ(T X) ⊕ V Y, (5.12)ẋ α ∂ α + ẏ i ∂ i = ẋ α (∂ α + y i α∂ i ) + (ẏ i − ẋ α y i α)∂ i .Similarly, the map (5.11) induces the dual canonical horizontal splitting ofthe pull–backJ 1 (X, Y ) × T ∗ Y = T ∗ X ⊕ ̂θ(V ∗ Y ), (5.13)ẋ α dx α + ẏ i dy i = (ẋ α + ẏ i y i α)dx α + ẏ i (dy i − y i αdx α ).Building on the canonical splittings (5.12) and (5.13), one gets the followingcanonical horizontal splittings of• a projectable vector–field on a fibre bundle Y → X,u = u α ∂ α + u i ∂ i = u H + u V = u α (∂ α + y i α∂ i ) + (u i − u α y i α)∂ i , (5.14)• an exterior 1–formσ = σ α dx α + σ i dy i = (σ α + y i ασ i )dx α + σ i (dy i − y i αdx α ),