Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 927a principal connection on P and A the corresponding connection form, wehave Γ ∗ A θ = A. If Ω and R A are the curvature 2–forms of the connectionsθ and A respectively, then Γ ∗ A Ω = R A.Local connection 1–forms on C associated with the canonical connectionθ are given by the coordinate expressions k m µ dx µ ⊗ I m . The correspondingcurvature two–form on C readsΩ C = (dk m µ ∧ dx µ − 1 2 cm nlk n µk l νdx ν ∧ dx µ ) ⊗ I m .Let I(g) be the algebra of real G−invariant polynomial on the Lie algebrag of the group G. Then, there is the well–known Weyl homomorphismof I(g) into the de Rham cohomology algebra H ∗ (C, R). Using this isomorphism,every k−linear element r ∈ I(g) is represented by the cohomologyclass of the closed characteristic 2k−form r(Ω C ) on C. If A is a section ofC, we have A ∗ r(Ω C ) = r(F ), where F is the strength of A and r(F ) isthe corresponding characteristic form on X.Let dim X be even and a characteristic n−form r(Ω C ) on C exist. Thisis a Lepagian form which defines a gauge–invariant Lagrangian densityL r = h 0 (r(Ω C ))on the jet space J 1 (Q, C). The Euler–Lagrangian operator associated withL r is equal to zero. Then, for any projectable vector–field u on C, we havethe strong relation (5.251):L u h 0 (r(Ω C )) = h 0 (du⌋r(Ω C )).If u is a general principal vector–field on C, this relation takes the formd H (u⌋r(Ω C )) = 0.For example, let dim X = 4 and the group G be semisimple. Then, thecharacteristic Chern–Pontryagin 4–formr(Ω C ) = a G mnΩ n C ∧ Ω m C .It is the Lepagian equivalent of the Chern–Pontryagin Lagrangian densityL = 1 k aG mnε αβµν F n αβF m µνd 4 xof the topological Yang–Mills theory.