Page 182 - Curvature and Homology

1 64 V. COMPLEX MANIFOLDS
it is a Kaehler metric. Conversely, the metric of a Kaehler manifold is
locally expressible in this form. For, since the equations (5.3.27) must
be satisfied, the equations

are completely integrable. If p, is a solution, the general solution is
given by

where the t,b6 are arbitrary functions of the variables (2). Consider the
system of first order equations

The integrability conditions of this system are given by

Differentiating these equations with respect to Zj we find, after applying
the conditions (5.3.27), that functions t,bi can be chosen satisfying the
integrability conditions. That f may be taken to be real is a consequence
of the fact that f is also a solution of the system.
We remark that an even-dimensional analytic Riemannian manifold M
with a locally Kaehlerian metric, that is, whose metric in local complex
coordinates satisfies the equations (5.3.27) is not necessarily a Kaehler
manifold. For, consider the cartesian product of a circle with a compact
3-dimensional Euclidean space form whose first betti number is zero [24.
It can be shown that such a space form exists; in fact, there is only one.
This manifold is compact, orientable, and has a locally flat metric. The
last property implies that its metric is locally Kaehlerian. (We have
invoked the theorem that an even-dimensional locally flat analytic
Riemannian manifold is locally Kaehlerian). Since its first betti number
is one it cannot be a Kaehler manifold (cf. theorem 5.6.2).
An hermitian metric g is expressible in the local coordinates (z,, 9)
by means of the positive definite quadratic form