Ivancevic_Applied-Diff-Geom

694 Applied Differential Geometry: A Modern Introductionξ ↦→ ˆξ = Ω −1 ξ, where Ω is a smooth non–vanishing function, then theconnection transforms as follows [Gover and Slovak (1999)]:E A :E A ′ :E B :E B′ :ˆ∇A′A u C = ∇ A′A u C + δ C AΥ A′B uBˆ∇A′A u C ′ = ∇ A′A u C ′ + δ A′C ′ΥB′ A u B ′ˆ∇A′A v B = ∇ A′A v B − Υ A′B v Aˆ∇A′A v B′ = ∇ A′A v B′ − Υ B′where Υ a := Ω −1 ∇ a Ω. ConsequentlyA vA′ (4.167)̂∇ a f = ∇ a f + wΥ a f if f ∈ E[w]. (4.168)CGiven a choice of scale ξ, we write R abD (or R (ξ)ab C D to emphasise thechoice of scale) for the curvature of ∇ a on E A Cand R ′ab D for the curvature′of ∇ a on E A ′, that is(2∇ [a ∇ b] − T ab e ∇ e )v C = R abCD v D , (2∇ [a ∇ b] − T ab e ∇ e )w D ′ = −R abC ′D ′w C ′.Then the curvature of the induced linear connection on T M isR abcd = R abC ′D ′δC D + R abCD δ C′D ′.Note that since ∇ a preserves the volume forms ɛ ξ A ′···C and ′ ɛξ D···Eit followsC Cthat R abD and R ′ab D are trace–free on the spinor indices displayed. Thus′the equationsC CR abD = U abD −δ C BP A′ B ′AD +δ C AP B′ A ′CBD , R ′ab D = U ′ ab C′D ′+δB′ D ′P A′ C ′AB −δ A′D ′P B′ C ′BAC Cdetermine the objects U abD , U ′ab D and the Rho–tensor, P ab, if we require′that U A′ B ′ CACD = 0 = U A′ D ′ C ′ABD . In this notation we have,′c cR abd = U abd + δ D′C ′ δC AP B′ A ′BD − δ D′C ′ δC BP A′ B ′AD − δ C Dδ A′C ′P B′ D ′BA + δ C Dδ B′C ′P A′ D ′AB (4.169) ,c Cwhere U abd = U abD δ D′C + U ′ ab D′C ′ δC D. (4.170)In the case of p = 2 = q this agrees with the usual decomposition ofthe curvature of the Levi–Civita connection into the conformally invariant(and trace–free) Weyl tensor part and the remaining part given by the Rho–tensor (see e.g., [Bailey et. al. (1994)]). Note that U’s are 2–forms valuedin g 0 coming from the curvature of the canonical Cartan connection andso they are in the kernel of ∂ ∗ . This is the source of the condition on thetrace, but they are not trace–free in general:U abCC = −U abC ′C ′ = 2P [ab]. (4.171)