Ivancevic_Applied-Diff-Geom

894 Applied Differential Geometry: A Modern Introduction• the canonical lift of τ onto the bundle C of the linear connections onT X, given by˜τ = τ α ∂ α + [∂ ν τ α k ν βα − ∂ β τ ν k α να − ∂ α τ ν k α βν − ∂ βα τ α ]∂∂k α βαOne can think of the vector–fields ˜τ on a bundle of geometrical objectsT as being the vector–fields associated with local 1–parameter groups ofthe holonomic isomorphisms of T induced by diffeomorphisms of its baseX. In particular, if T = T X they are the tangent isomorphisms. We callthese isomorphisms the general covariant transformations.Let T be the bundle of geometrical objects and L a Lagrangian densityon the configuration space J 1 (X, T ). Given a vector–field τ on the base Xand its canonical lift ˜τ onto T , one may use the first variational formula(5.247) in order to get the corresponding SEM transformation law. Theleft side of this formula can be simplified if the Lagrangian density satisfiesthe general covariance condition.Note that, if the Lagrangian density L depends on background fields, weshould consider the corresponding total bundle (5.238) and the Lagrangiandensity L tot on the total configuration space J 1 (X, T ) tot . We say that theLagrangian density L satisfies the general covariance condition if L tot isinvariant under 1–parameter groups of general covariant transformations ofT tot induced by diffeomorphisms of the base X. It takes place iff, for anyvector–field τ on X, the Lagrangian density L tot obeys the equalityL j 10 eτ L tot = 0 (5.254)where ˜τ is the canonical lift of τ onto T tot and j 1 0˜τ is the jet lift of ˜τ ontoJ 1 (X, T ) tot .If the Lagrangian density L does not depend on background fields, theequality (5.254) becomesL j 10 eτ L = 0. (5.255)Substituting it in the first variational formula (5.247), we get the weekconservation lawd H h 0 (˜τ⌋Ξ L ) ≈ 0. (5.256)One can show that the conserved quantity is reduced to a superpotentialterm.Here, we verify this fact in case of a tensor bundle T −→ X. Let it becoordinated by (x α , y A ) where the collective index A is employed. Given a.