Page 180 - Curvature and Homology

1 62 V. COMPLEX MANIFOLDS

Thus, if we put

- @, = R',, dzZ A dzm + RjiZm* dzz A drm
where

R;lm + p<mz= 0
we have

and

Its only non-vanishing components are

For, substituting (5.3.1 1) into (5.3.16) and (5.3.17) and applying the
relation d(gij* giek) = 0, we derive

Since Pik = rfZk* the curvature tensor is self adjoint.
Transvecting (5.3.18) with gjr. we obtain

Hence, the only non-vanishing 'covariant' components of the curvature
tensor are
Rij*kl*, &j*k* 1, Rf* jkl*, &jk* l*
Again, by virtue of the given splitting the Bianchi identities have the
form

together with the conjugate relations.