Ivancevic_Applied-Diff-Geom

696 Applied Differential Geometry: A Modern Introductionthese bundles as a twistor bundle and sections of such bundles as localtwistors. In particular observe that there is a canonical completely skewlocal–twistor (p + q)−form h αβ···γ on E α which is equivalent to the isomorphism(4.165). We write h αβ···γ for the dual completely skew twistorsatisfying h αβ···γ h αβ···γ = (p + q)!.All finite dimensional P −modules enjoy filtrations which split completelyas G 0 −modules. V α and V α , give the simplest cases and, asP −modules, admit filtrationsV α = V A + V A′ , V α = V A ′ + V A .(Our notational convention is that the ‘right ends’ in the formal sums aresubmodules while the ‘left ends’ are quotients.) These determine filtrationsof the twistor bundlesE α = E A + E A′ , E α = E A ′ + E A .We write XA α for the canonical section of E α ′ A ′ which gives the injectingmorphism E A′ → E α viav A′ ↦→ X α A ′vA′ . (4.173)Similarly Y A αdescribes the injection of E A into dual twistors,E A ∋ u A ↦→ Y A α u A ∈ E α . (4.174)It follows from standard representation theory that a choice of splittingof the exact sequence,0 → V A′ → V α → V A → 0is equivalent to the choice of subgroup of P which is isomorphic to G 0 . Itfollows immediately that a choice of splitting of the twistor bundle E α isequivalent to a reduction from G to G 0 . Such a splitting is a G 0 −equivarianthomomorphism ξ : E α → E A′ . We can regard ξ here as a section of E α ⊗E A′ = EαA′ and then in our index notation the homomorphism is determinedby v α ↦→ ξ A′α v α , for any section v α of E α . The composition of ξ with themonomorphism E A′ → E α must be the identity so we have,ξ A′β X β B = ′ δA′ B ′.A splitting ξ A′α of E α determines a dual splitting λ α A of E α , λ α A : E α → E A .Given such splittings we have E α = E A ⊕ E A′ and E α = E A ′ ⊕ E A , so we