Page 181 - Curvature and Homology

5.3. LOCAL HERMITIAN GEOMETRY 163
In a complex coordinate system the first of (5.3.20) are given by

and their conjugates together with the Jacobi identities

and their conjugates where as usual Di denotes covariant differentiation
with respect to the connection (5.3.11). From the second Bianchi
identity we derive the relations

together with their conjugates.
Since the connection is a metrical connection

Dk gij* = Dk* gij* = 0. (5.3.24)
Hence, from (5.3.23)

Dm Ri* jkl* - Dk Rt* jml+ = Ri* jrg* Tm: (5.3.25)
together with the conjugate relations.
In terms of the hermitian metric, the torsion tensor has the com-
ponents

Thus, a necessary and sufficient condition that the torsion forms vanish
may be given in terms of the hermitian metric tensor g by the system
of differential equations

In this case, g is said to defi-ne a Kaehler metric. A complex manifold
endowed with this particular metric is called a Kaehler manifold.
If the metric of an hermitian manifold is given by

(locally) for some real-valued function f, then, clearly, from (5.3.27)