Ivancevic_Applied-Diff-Geom

916 Applied Differential Geometry: A Modern IntroductionSince C (5.45) is an affine bundle modelled over the vector bundleC = T ∗ Q ⊗ V G P → Q,the vertical tangent bundle of C possesses the canonical trivializationV C = C × T ∗ Q ⊗ V G P, where (5.339)V G P C = V G (C × P ) = C × V G P.Then the strength F A of the connection (5.338) is the V G P −valued horizontal2–form on C,F A = 1 2 d AA = 1 2 [A, A] F N ∈ ∧ 2 (C) ⊗ V G P (Q),F A = (da r µ ∧ dq µ + 1 2 cr pqa p αa q µdq α ∧ dq µ ) ⊗ e r . (5.340)Note that, given a global section connection A of the connection bundleC → Q, the pull–back A ∗ F A = F A is the strength (5.325) of the principalconnection A.Let us take the pull–back of the form F A onto J 1 (Q, C) with respect tothe fibration (5.336), and consider the V G P −valued horizontal 2–formF = h 0 (F A ) = 1 2 F r λµdq α ∧ dq µ ⊗ e r ,F r λµ = a r λµ − a r µλ + c r pqa p αa q µ, (5.341)where h 0 is the horizontal projection (5.58). Note thatF/2 : J 1 (Q, C) → C × ∧ 2 T ∗ Q ⊗ V G P (5.342)is an affine map over C of constant rank. Hence, its kernel C + = Ker Fis the affine subbundle of J 1 (Q, C) → C, and we have a desired canonicalsplittingJ 1 (Q, C) = C + ⊕ C − = C + ⊕ (C × ∧ 2 T ∗ Q⊗V G P ), (5.343)a r λµ = 1 2 (ar λµ + a r µλ − c r pqa p αa q µ) + 1 2 (ar λµ − a r µλ + c r pqa p αa q µ), (5.344)over C of the jet space J 1 (Q, C). The corresponding canonical projectionsareπ 1 = S : J 1 (Q, C) → C + , S r λµ = 1 2 (ar λµ + a r µλ − c r pqa p αa q µ), (5.345)and π 2 = F/2 given by (5.342).