Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 693Since T ab c is skew on its lower indices, T ab c = T [ab] c , it can be writtenas a sum of two termsT ab c = F ab c + ˜F ab c ,whereF ab c = : F A′ B ′ CABC ′ = F [A′ B ′ ]C(AB)C ′ , ˜Fab c =: ˜F A′ B ′ CABC ′ = F (A′ B ′ )C[AB]C ′ .The Cartan bundle G over the manifold M has the quotient G 0 , a principalfibre bundle with structure group G 0 . By the general theory, eachG 0 −equivariant section σ : G 0 → G of the quotient projection defines thedistinguished principal connection on G 0 , the pullback of the g 0 −part of ω.The whole class of these connections consists precisely of connections on G 0with the unique torsion taking values in the kernel of ∂ ∗ . A straightforwardcomputation shows that the latter condition is equivalent to the conditionthat both ˜F and F be completely trace–free. Each principal connectionon G 0 induces the induced connection on the bundle E[1] \ {0} which isassociated to G 0 and, moreover, the resulting correspondence between thesections σ and the latter connections is bijective. In particular, each sectionξ of the bundle E[1] \ {0} defines uniquely a reduction σ, such that thecorresponding distinguished connection leaves ξ horizontal.Therefore, given a scale ξ on an AG–structure there are unique connectionson E A and E A ′ such that F A′ B ′ CABC and ˜F ′A′ B ′ CABC are totally trace–free,′the induced covariant derivative preserves the isomorphism h of (4.165),and ∇ a ξ = 0. The torsion components F c ab and ˜F c ab of the induced connectionon T M are invariants of the so–called AG–structures [Bailey andEastwood (1991)].Note that in the special case of the four-dimensional conformal geometries,there is always a connection with vanishing torsion on G 0 and so bothF and ˜F are zero. The scales correspond to a choice of metric from theconformal class while the general distinguished connections (correspondingto the reduction parameter σ being not necessarily exact) are just the Weylgeometries.We may write ∇ ξ a to indicate a connection as determined by the Theorem,although mostly we omit the ξ. Thus we might write ∇ˆξa or simplyˆ∇ a to indicate the connection corresponding to a scale ˆξ and similar conventionswill be used for other operators and tensors that depend on ξ.In what follows, for the purpose of explicit calculations, we shall oftenchoose a scale and work with the corresponding connections. Objects arethen well defined, or invariant, (on the AG-structure) if they are independentof the choice of scale. Note that if we change the scale according to