Page 184 - Curvature and Homology

1 66 V. COMPLEX MANIFOLDS

and from (5.3.33)
gti = doi+

It follows that c l dOii is a (real) closed 2-form in the bundle of
frames over M. Moreover,

Since the operator d is real (that is, it sends real forms into real forms),
2/=i Oi, defines a real 1-form (which we denote by 2q) on the bundle
B of unitary frames. Let ?r : B + M denote the projection map and put

Then, ?r*+ = - dx. The 2-form + defines the lat Chwn class of M
(cf. § 6.12).
In contrast with Kaehler geometry there are three distinct contractions
of the curvature tensor in an hermitian manifold with non-vanishing
torsion. They are called the Ricci tensors and are defined as follows:

kl*
RU* = gkl* Ripkj*, Sue = gkl* R,*~,*, Ti,* = g Rkpij*.
If the contracted torsion tensor vanishes, that is if Tjgi = 0, Tij, = Rij,.
This is one of two rather natural conditions that can be imposed on
the torsion, the other being that the torsion forms be holomorphic.
From (5.3.21) we see that the latter condition implies the symmetry
relation
Rijkl* = Rikjl*. (5.3.39)
Since the curvature tensor is skew-symmetric in its last two indices the
symmetry relation (5.3.39) shows that Sij* = Rij*.
Now, from (5.3.21) we obtain

where Ti& = g,,, Ti,'. Hence, the conditions i3Tiki/aZ1 = 0 imply
the symmetry relations
Rij*kl* = Rkpij* (5.3.41)
as in a Riemannian manifold. We conclude that Sij. = T,., that is
the Ricci curvature tensors coincide as in a Kaehler manifold. That
they need not be the same may be seen by the following example [15].