Ivancevic_Applied-Diff-Geom

700 Applied Differential Geometry: A Modern IntroductionSince the operator Dβα and its concatenations will have an importantrole in the following discussions we develop notation for their target spaces.First let F ρ be defined as follows,Then we writeF ρ := Ker(Y Aρ :: E ρ → E A ).F ρ···σα···β := F ρ ⊗ · · · ⊗ F σ ⊗ E α ⊗ · · · ⊗ E β ,and F ρ···σρ···σα···β[w] = Fα···β ⊗ E[w]. Finally letS ρ···σ[w] := (⊙α · · · βk Fα) ρ ⊗ E[w].} {{ }kNote that sections of Fρα (= Sρ α ) are not generally trace–free, but that Fραis in a complement to the trace–part of Eρ α .Now if f ∈ E[w] then Dαf ρ ∈ Fα[w]. ρ Similarly observe that if v σ ∈ F σthenis in Fαρσ . ThusD ρ αv σ − δ σ αv ρgives an invariant operatorD ρσαβ := 1 2 (Dρ αD σ β + D σ βD ρ α − δ σ αD ρ β − δρ β Dσ α)D ρσSimilarly we define D ρσµαβγ byαβ : E µ···νγ···δ[w] → Sρσαβ ⊗ E µ···νγ···δ [w].D ρσµαβγ := 1 3 ((Dρ αD σµβγ + Dσ βD ρµαγ + D µ γ D ρσαβ − δρ αD σµβγ − δσ βD ρµαγ − δ µ γD ρσαβ )and so on for Dρ···ν α···δ . Notice that the construction of these is designed in sucha way that the resulting operators are annihilated if composed (contracted)with YνB on any index.The Splitting MachineryIn terms of the algebraic projectors and embeddings introduced in the last