Operators on Differential Forms for Lie Transformation Groups

For any Lie group action S:G×P → P , we introduce C∞(P )linear operators S • , that transform n-forms ωn ∈ An(P, V ) into (n−i)-forms S •ωn ∈ An−i(P,Alti(g, V )) . We compute the exterior derivative of these generated forms and their behavior under interior products with vector fields and Lie differentiation. By combination with Lie algebra valued forms θ ∈ A1(P,g) and φp ∈ Ap(P,g) , we recover V -valued forms ω◦S θ ∈ An(P, V ) , resp., (χsn◦S θ)•φp for χn ∈ An(P,Hom(s g, V )) and compute their exterior derivative. The derived formulae play an important role for local evaluations of connections on fiber bundles. Math. Subj. Class. 1991: Primary: 58A10, Secondary: 57S20, 22E15, 53C05 1. Motivation Let P (M,G) denote a principal bundle with base manifold M = ⋃ α∈A Uα , projection π:P →M , Lie group G, right action R:P ×G→ P and local trivializations ψα: π −1(Uα)→ Uα×G with local projections πα = prG ◦ψα . Recall that any connection Γ on P defines horizontal and vertical projections of vector fields, not only on P , but also on every associated fiber bundle B(M,F,G) = P ×G F with fiber F and left action L:G× F → F , such that the vertical fields are tangential to the fiber. We thus obtain projections h, v of differential forms via ωh(. . . ,X , . . .) := ω(. . . , hX , . . .), ωv(. . . ,X , . . .) := ω(. . . , vX , . . .) (1) for all V -valued forms ω ∈ A(B, V ). If we compute the vertical projections locally on the bundle charts Uα × F , we obtain with L :G → F defined by L (g) := L(g, f) and its differential (dL )e: g→ Tf (F ) at the neutral element e ∈ G: (φv)(x,f)(. . . , (X , F ), . . .) = φ(x,f)(. . . , (0, (dL )eA α x(X ) + F ), . . .) (2) ∗Supported by a grant of the Studienstiftung des deutschen Volkes ISSN 0949–5932 / $2.50 C © Heldermann Verlag