Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 821Building on the maps (5.53) and (5.54), one can get the horizontalsplittings of the canonical tangent–valued 1–form on J 1 (X, Y ),θ J 1 (X,Y ) = dx α ⊗ ∂ α + dy i ⊗ ∂ i + dy i µ ⊗ ∂ µ i= α + θand the associated exterior differentiald = d θJ 1 (X,Y )= d α + d θ = d H + d V . (5.55)They are similar to the horizontal splittings (5.15) and (5.16).A 2–jet field (resp. a 2–connection) Γ on a fibre bundle Y → X is definedto be a 1–jet field (resp. a 1–connection) on the jet bundle J 1 (X, Y ) → X,i.e., Γ is a section (resp. a global section) of the bundle J 1 (X, J 1 (X, Y )) →J 1 (X, Y ).In the coordinates (y i α, y i (µ) , yi αµ) of the repeated jet spaceJ 1 (X, J 1 (X, Y )), a 2–jet field Γ is given by the expression(y i α, y i (µ) , yi αµ) ◦ Γ = (y i α, Γ i (µ), Γ i αµ).Using the contact map (5.53), one can represent it by the tangent–valuedhorizontal 1–form on the jet bundle J 1 (X, Y ) → X,Γ = dx µ ⊗ (∂ µ + Γ i (µ)∂ i + Γ i αµ∂ α i ). (5.56)A 2–jet field Γ on a fibre bundle Y → X is called a sesquiholonomic(resp. holonomic) 2–jet field if it takes its values into the subbundleĴ 2 (X, Y ) (resp. J 2 (X, Y )) of J 1 (X, J 1 (X, Y )). We have the coordinateequality Γ i (µ) = yµ i for a sesquiholonomic 2–jet field and the additionalequality Γ i αµ = Γ i µα for a holonomic 2–jet field.Given a symmetric connection K on the cotangent bundle T ∗ X, everyconnection Γ on a fibre bundle Y → X induces the connectionjΓ = dx µ ⊗ [∂ µ + Γ i µ∂ i + (∂ α Γ i µ + ∂ j Γ i µy j α − K α αµ(y i α − Γ i α))∂ α i ]on the jet bundle J 1 (X, Y ) → X. Note that the curvature R of a connectionΓ on a fibre bundle Y → X induces the soldering form σ R on J 1 (X, Y ) →X,σ R = R i αµdx µ ⊗ ∂ α i .