Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 943In the Lagrangian formalism, the metric SEM–tensor is defined to be√ −gtαβ = 2 ∂L∂g αβ .In case of a background world metric g, this object is well–behaved. In theframework of the multimomentum Hamiltonian formalism, one can introducethe similar tensor√ −gtH αβ = 2 ∂H∂g αβ. (5.420)If a Hamiltonian form H is associated with a semiregular Lagrangiandensity L, there are the equalitiest H αβ (q) = −g αµ g βν t µν (x α , y i , ∂ i αH(q)),(q ∈ Q),t H αβ (x α , y i , π α i (z)) = −g αµ g βν t µν (z), Ĥ ◦ ̂L(z) = z.In view of these equalities, we can think of the tensor (5.420) restrictedto the Lagrangian constraint space Q as being the Hamiltonian metricSEM–tensor. On Q, the tensor (5.420) does not depend upon choice of aHamiltonian form H associated with L. Therefore, we shall denote it bythe common symbol t. Sett λ α = g αν t λν .In the presence of a background world metric g, the identity (5.398) takesthe formt λ α{ α λµ} √ −g + (Γ i µ∂ i − ∂ i Γ j µr α j ∂ i α) ˜H Γ ≈ddx α T Γ α µ + r α i R i αµ , (5.421)where by { α λµ} are meant the Christoffel symbols of the world metric g.SEM Tensors in Gauge TheoryIn this subsection, following [Sardanashvily (1998)] we consider thegauge theory of principal connections treated as gauge potentials. Here,the manifold X is assumed to be oriented and provided with a nondegeneratefibre metric g µν in the tangent bundle of X. We denote g = det(g µν ).Let P → X be a principal bundle with a structure Lie group G whichacts freely and transitively on P on the right: r g : p ↦→ pg, (p ∈ P, g ∈G).A principal connection A on P → X is defined to be a G-equivariant connectionon P such that j 1 r g ◦A = A◦r g for each canonical morphism r g .