Ivancevic_Applied-Diff-Geom

896 Applied Differential Geometry: A Modern IntroductionOn the other hand, in gauge theory (see subsection 5.11 below), severaltypes of gauge transformations are considered. To get the Noether conservationlaws, we restrict our consideration to vertical isomorphisms of theprincipal bundle P . These are the G−equivariant isomorphism Φ of P overId X , that is,r g ◦ Φ = Φ ◦ r g , (g ∈ G). (5.263)We call them the gauge isomorphisms. As is well–known, they yield thevertical isomorphisms of the bundle of principal connections C and theP −associated bundle E.For example, let P −→ X be a principal bundle with the structure Liegroup G. Let us consider general gauge isomorphisms Φ of this principalbundle over diffeomorphisms of the base X. They satisfy the relation(5.263). We denote by u G the projectable vector–fields on P correspondingto local 1–parameter groups of such isomorphisms. There is the 1–1 correspondence between these vector–fields and sections of the bundleT G P = T P/G. They are called the general principal vector–fields (see [Giachettaand Mangiarotti (1990)]). In particular, one can show that, given avector–field τ on the base X, its horizontal lift onto the principal bundle Pby means of a principal connection on P is a general principal vector–field.General gauge isomorphisms of the principal bundle P , as like as itsvertical isomorphisms, yield the corresponding isomorphisms of the associatedbundles E and the bundle of principal connections C. We denote bythe same symbol u G the corresponding general principal vector–fields onthese bundles.Consider the product S = C × E × T, where T → X is a bundle ofgeometrical objects. Let a Lagrangian density L on the corresponding configurationspace J 1 (T, S) be invariant under the isomorphisms of the bundleS which are general gauge isomorphisms of C × E over diffeomorphisms ofthe base X and the general covariant transformations of T induced by thesediffeomorphisms of X. In particular, vertical isomorphisms of S consist ofvertical isomorphisms of C × E only. It should be emphasized that thegeneral gauge isomorphisms of the bundle C × E and those of the bundleT taken separately are not the bundle isomorphisms of the product S becausethey must covering the same diffeomorphisms of the base X of Y .At the same time, one can say that the Lagrangian density L satisfies thegeneral covariance condition in the sense that it is invariant under generalisomorphisms of the bundle S [Giachetta and Mangiarotti (1990)].