Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 823on J ∞ (X, Y ), with suitable notation for vector–fields, derivatives and differentialforms just as like as in the finite order case (see [Sardanashvily (1993);Sardanashvily (1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily(2000a); Sardanashvily (2002a)]).A vector–field u k on the k−-jet space J k (X, Y ) is called projectablevector–field if for any l < k there exists a vector–field u k on J l (X, Y ) → Xsuch thatu l ◦ π k l = T π k l ◦ u k .The tangent map T π k lsends projectable vector–fields on J k (X, Y ) onto theprojectable vector–fields on J l (X, Y ).Now consider projectable vector–fields u k which are extension to thehigher–order jet spaces of infinitesimal transformations of the fibre bundleY → X. The linear space of projectable vector–fields on J ∞ (X, Y )is defined as the limit of the inverse system of projectable vector–fieldson k−jet spaces. As a consequence, every projectable vector–field on thebundle Y → X,u = u α ∂ α + u i ∂ i ,induces a projectable vector–field u ∞ on J ∞ (X, Y ). We have its canonicaldecompositionu ∞ = u ∞ H + u ∞ V ,u ∞ H = u α ̂∂∞ α = u α (∂ α + yα∂ i i + ...), (5.59)∞∑u ∞ V = ̂∂ α k k...̂∂ α 1 1u i V ∂ α1...α ki ,k=0where u V is the vertical part of the splitting (5.14) of π 1∗0 u. In particular,u ∞ H is the canonical lift of the vector–field τ = uα ∂ α on X onto J ∞ (X, Y ).By the same limiting process we can introduce the notions of innerproduct of exterior forms and projectable vector–fields, the Lie bracket ofprojectable vector–fields and the Lie derivative of exterior forms by projectablevector–fields on J ∞ (X, Y ). All the usual identities are satisfied.In particular, the notion of contact forms is extended to the formŝdy i α 1...α r= dy i α 1...α r− y i α 1...α rνdx ν .Let Ω r,k denote the space of exterior forms on J ∞ (X, Y ) which are of theorder r in the horizontal forms dx ν and of the order k in the contact forms.