Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 893for every critical section s of Y −→ X. In particular, if ɛ = dε is an exactform, we get the weak conservation lawd(s ∗ (u⌋Ξ L − ε)) ≈ 0.In particular, gauge transformations in gauge theory on a 3D base X arethe invariant transformations if L is the Yang–Mills Lagrangian density andthey are the generalized invariant transformations if L is the Chern–SimonsLagrangian density.5.9.2 General Covariance ConditionNow we consider the class of bundles T −→ X which admit the canonicallift of vector–fields τ on X. They are called the bundles of geometricalobjects. In fact, such canonical lift is the particular case of the horizontallift of a field τ with respect to the suitable connection on the bundle T−→ X [Giachetta et. al. (2005)].Let τ = τ α ∂ α be a vector–field on the manifold X. There exists thecanonical lift˜τ = T τ = τ α ∂ α + ∂ ν τ α ẋ ν ∂∂ẋ α (5.253)of τ onto the tangent bundle T X of X. This lift consists with the horizontallift of τ by means the symmetric connection K on the tangent bundle whichhas τ as the integral section or as the geodesic field:∂ ν τ α + K α αντ α = 0.Generalizing the canonical lift (5.253), one can construct the canonicallifts of a vector–field τ on X onto the following bundles over X. For thesake of simplicity, we denote all these lifts by the same symbol ˜τ. We have:• the canonical lift of τ onto the cotangent bundle T ∗ X, given by˜τ = τ α ∂ α − ∂ β τ ν ∂ ẋ ν ;∂ẋ β• the canonical lift of τ onto the tensor bundle T k mX = (⊗ m T X) ⊗(⊗ k T ∗ X), given by˜τ = τ α ∂ α + [∂ ν τ α1 ẋ να2···αmβ 1···β k+ . . . − ∂ β1 τ ν ẋ α1···αmνβ 2···β k− . . .]∂;∂ẋ α1···αmβ 1···β k