eldorado.tu-dortmund.de/server/api/core/bitstreams/f7043a77-8f49-4038-95d6-3fec1dc5f1c7/content
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
= −1
2 g(θ, θ)ζx.
Lemma 2 [...] J(J∗(dω)3,0)(X,Y,Z) = J∗(dω)3,0(JX,Y,Z) + J∗(dω)3,0(X,JY,Z) + J∗(dω)3,0(X,Y, JZ)
= −(dω)3,0(X,JY, JZ) − (dω)3,0(JX,Y, JZ) − (dω)3,0(JX,JY,Z)
= 3(dω)3,0(X,Y,Z)
and alt(N ) ∈ λ3,0, it follows
4J∗(dω)3,0 = 3alt(N [...] The operators ∆1,0 ∶= 2[δ1,0, δ1,0∗] and ∆0,1 ∶= 2[δ0,1, δ0,1∗] define Laplacians on each Sp,q(M).
Corollary 2.2.15. The Lichnerowicz Laplacians ∆1,0 and ∆0,1 can be locally written as
∆1,0 = 4 n
∑ j=1
∇̂Z̄j …
