Page 183 - Curvature and Homology

5.3. LOCAL HERMITIAN GEOMETRY 165

If g is a Kaehler metric, the real 2-form

canonically defined by this metric, is closed. conversely, if Q is closed,
g is a Kaehler metric.
In an hermitian manifold, the 2-form Q is called the fundamental form.
We remark that the tensor field g as well as the fundamental form can be
given a particularly simple representation in terms of the 2n forms
(ai, Ci) on M. For, from (5.3.2) and (5.3.8)

and

From the equations (5.3.4) and (5.3.13) we deduce the equations of
structure of a Kaehler manifold M:

dei = ek A e:
and

where the 2-forms Bf define the curvature of the manifold. They are
locally expressible in terms of local complex coordinates by

The Ricci tensor of M is given locally by

and so from (5.3.17) it may be expressed explicitly in terms of the
metric g by
aa log det G
Rkl* = - &+ a,$ 9 G = (&*I* (5.3.36)

Now, from (5.3.34)
nd, = Rkl* dzk A d2'