Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 957the conventional Einstein’s gravitation theory. We are not concerned herewith the general problem of equivalence of metric, affine and affine–metrictheories of gravity [Ferraris and Kijowski (1982)]. At the same time, whenK is the Levi–Civita connection of h, the Lagrangian density of fermionfields which uses the covariant derivative (5.470) becomes that in the Einstein’sgravitation theory. It follows that the configuration space of metric(or tetrad) gravitational fields and general linear connections may play therole of the universal configuration space of realistic gravitational models.In particular, one then can think of the generalized Komar superpotentialas being the universal superpotential of energy–momentum of gravity[Giachetta and Sardanashvily (1995)].We follow [Giachetta and Sardanashvily (1997)] in the geometrical approachto field theory when classical fields are described by global sectionsof a fibre bundle Y −→ X over a smooth world space–time manifold X.Their dynamics is phrased in terms of jet spaces [Sardanashvily (1993);Saunders (1989)]. Recall that a kth–order differential operator on sectionsof a fibre bundle Y −→ X is defined to be a bundle morphism of the jetbundle J k (X, Y ) −→ X to a vector bundle over X.In particular, given bundle coordinates (x µ , y i ) of a fibre bundle Y −→X, the 1–jet space J 1 (X, Y ) of Y has the adapted coordinates (x µ , y i , yµ),iwhere yµ(j i xs) 1 = ∂ µ s i (x).There is the 1–1 correspondence between the connections on the fibrebundle Y → X and the global sections Γ = dx α ⊗ (∂ α + Γ i α∂ i ) of the affinejet bundle J 1 (X, Y ) → Y . Every connection Γ on Y → X induces thefirst–order differential operator on Y ,D Γ : J 1 (X, Y ) −→ T ∗ X ⊗ V Y, D Γ = (y i α − Γ i α)dx α ⊗ ∂ i ,which is called the covariant differential relative to the connection Γ.Recall that in the first–order Lagrangian formalism, the 1–jet spaceJ 1 (X, Y ) of Y plays the role of the finite–dimensional configuration spaceof fields represented by sections s of a bundle Y → X. A first–order Lagrangiandensity L : J 1 (X, Y ) −→ ∧ n T ∗ X is defined to be a horizontaldensity L = L(x µ , y i , yµ)ω i on the jet bundle J 1 (X, Y ) → X, whereω = dx 1 ∧ ... ∧ dx n , (n = dim X). Since the jet bundle J 1 (X, Y ) → Y isaffine, every polynomial Lagrangian density of field theory factors throughL : J 1 D(X, Y ) −→ T ∗ X ⊗ V Y → ∧ n T ∗ X, where D is the covariant differentialon Y , and V Y is the vertical tangent bundle of Y .Let us consider the gauge theory of gravity and fermion fields. By X