Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 945Let τ be a vector–field on the base X andτ BK = τ α {∂ α + [∂ µ B m α − c m nlk n µB l α − K β µλ(B m β − k m β )]∂ µ m} (5.428)its horizontal lift onto C by means of the connection (5.427). For everyvector–field τ, one can choose the connection K on the tangent bundle T Xwhich has τ as the geodesic field. In this case, the horizontal lift (5.428) ofthe vector–field τ becomes its canonical liftτ B = τ α ∂ α + [τ α (∂ µ B m α − c m nlk n µB l α) + ∂ µ τ α (B m α − k m α )]∂ µ m , (5.429)by means of the principal connection B on the principal bundle P [Giachettaand Mangiarotti (1990)]. The vector–field (5.429) is just the generalprincipal vector–field on C that has been mentioned in the previous Section.Hence, the Lie derivative of the Lagrangian density (5.424) by the jetlift τ B of the field τ B becomesL τ BL Y M = (∂ α τ α L Y M + τ α ∂ α L Y M − F m µν∂ α τ µ π νλm )ω.The corresponding SEM transformation law takes the form√∂ α τ α L Y M − τ µ t α β |g|Γβµα − Fµν∂ m α τ µ π νλm ≈ (5.430)̂∂ α [π νλm (τ µ (∂ ν Bµ m − c m nlkν n Bµ) l + ∂ ν τ µ (Bµ m − kµ m ) − τ µ kνµ) m + δ α µτ µ L Y M ],wheret α β = 1 √|g|(π ναm F m βν − δ α βL Y M )is the metric SEM–tensor of gauge potentials.Note that, in general case of the principal connection B, the correspondingSEM transformation law (5.430) differs from the covariant conservationlaw in the Noether conservation laŵ∂ α (π νλm u gmν ) ≈ 0,whereu g = (∂ ν α m + c m nlk l να n )∂ ν m, α m = τ µ (B m µ − A m µ )is the principal vector–field (5.353) on C.Following the general procedure [Sardanashvily (1993); Sardanashvily(1994)], let us consider connections on the fibre bundle C −→ X which taketheir values into Ker ̂L Y M :Γ : C → C + , Γ m µλ − Γ m λµ − c m nlk n αk l µ = 0. (5.431)