eldorado.tu-dortmund.de/server/api/core/bitstreams/f7043a77-8f49-4038-95d6-3fec1dc5f1c7/content
calculation
ζ(k−1v1, . . . , k −1vq) = ζ(k−1Φ−1(Φ(v1)), . . . , k−1Φ−1(Φ(vq)))
= Ψ−1(Ψζ)(k−1Φ−1(v1), . . . , k−1Φ−1(vq))
= (Ψ−1ξ)(k−1Φ−1(v1), . . . , k−1Φ−1(vq)) (1.6)= ξ((ΦkΦ−1)−1(v1) . . . , (ΦkΦ−1)−1(vq)),
where [...] have
ps([δ1,0, δ1,0∗])(x, θ)ζx = θ1,0 ⊙ (θ0,1)# ⌟ ζx − (θ0,1)# ⌟ θ1,0 ⊙ ζx
= −g(θ1,0, θ0,1) ζx
= −1
2 g(θ, θ)ζx
ps([δ0,1, δ0,1∗])(x, θ)ζx = θ0,1 ⊙ (θ1,0)# ⌟ ζx − (θ1,0)# ⌟ θ0,1 ⊙ ζx
= −g(θ0,1, θ̄1,0) ζx
= [...]
∈u(n)⊥
. (1.20)
Since R̃ ∈ Λ2 ⊗ u(Vj), its corresponding 4−tensor lies in
Λ2 ⊗ λ1,1 = (λ2,0 ⊕ λ1,1)⊗ λ1,1 = S2(λ1,1)⊕Λ2(λ1,1)⊕ (λ2,0 ⊗ λ1,1)
and it therefore splits accordingly as R̃ = R̃1 + R̃2 + R̃3 …
