Page 185 - Curvature and Homology

5.3. LOCAL HERMITIAN GEOMETRY 167

Consider the cartesian product of a 1-sphere and a 3-sphere:
M = S1 x S3. In example 6 of 5 5.1 it was shown that M is a complex
manifold. A natural metric is given by

so that

A computation yields

from which we obtain

Summarizing, we see that the curvature tensor defined by a con-
nection with holomorphic torsion has the same symmetry properties
as the curvature tensor defined by a Kaehler metric.
The condition that the torsion be holomorphic is a rigidity restriction
on the manifold. Indeed, if the manifold is compact, it is actually
Kaehlerian [32].
One may also consider a connection which carries holomorphic
tensor fields into holomorphic tensor fields (cf. 5 6.5). Such a connection
must satisfy

and, for this reason, the connection is said to be holomorphic. From
(5.3.17) it follows that the curvature tensor of a holomorphic connection
must vanish.
In an hermitian manifold M with non-vanishing torsion, if the Ricci
tensor Ri5* defines a positive definite quadratic form, then it defines an
hermetian metric g on M. Erom the second of equations (5.3.20) it
follows that the form 8ii is closed, and hence g is a Kaehler metric.
A complex manifold M of complex dimension n is said to be complex
parallelisable if there are n linearly independent holomorphic vector fields
defined everywhere over M (cf. p. 247). In an hermitian manifold, it is not
difficult to prove that the vanishing of the curvature tensor is a necessary
condition for the manifold to be complex parallelisable. (Hence, the