Prolongation of linear semibasic tangent valued forms to product preserving gauge bundles of vector bundles

Abstract

Let 𝐴 be a Weil algebra and V be an 𝐴 -module with dimR V < ∞. Let E → M be a vector bundle and let 𝑇 ᴬ՚ᵛ c→ 𝑇 ᴬ M be the vector bundle corresponding to (𝐴, V). We construct canonically a linear semibasic tangent valued p-form 𝑇 ᴬ՚ᵛφ: 𝑇 ᴬ՚ᵛ E → ∧p 𝑇 ∗ 𝑇 ᴬM ⊗T AM 𝑇 𝑇 ᴬ՚ᵛE on 𝑇 A, V E → 𝑇 A M from a linear semibasic tangent valued p-form φ: E → ∧p 𝑇 ∗ M ⊗ 𝑇 E on E → M . For the Frolicher-Nijenhuis bracket we prove that [[𝑇 A,V φ, 𝑇 A,V ψ]] = 𝑇 A,V ([[φ, ψ]]) for any linear semibasic tangent valued p- and q- forms φ and ψ on → M . We apply these results to linear general connections on E → M.