Page 192 - Curvature and Homology

1 74 V. COMPLEX MANIFOLDS

This latter condition is the requirement that in the sought after coor-
dinates, the coefficients of connection vanish at P, that is, in terms of
the metric tensor g, dgij.(P) = 0 (cf. 5.3.3, 5.3.32 and 5.3.10).
Let (zi) be a system of local complex coordinates at P such that
zi(P) = 0, i = 1, -.a, n and ei(P) = dzi(P). We put

and look for the relations satisfied by the coefficients aitk and bijk in
order that (i), (ii), and (iii) hold. For condition (ii) to hold it is necessary
and sufficient that
aijk + 6kjf = 0. (5.5.3)
Now, put

Then, (iii) is satisfied, if and only if

Substituting in (5.5.3), we derive

These are the necessary conditions that a complex geodesic local
coordinate system exists at P.
Conversely, assume that there exist cijk, ctijk satisfying cijk = Etjn -
E'yi. If we put aUk = - Elkj, and bijk = ctijk the relations (5.5.3) and
(5.5.4) are satisfied. If we define the forms Oi by (5.5.2), the conditions (i),
(ii), and (iii) for a complex geodesic local coordinate system are satisfied.
We recall that an hermitian metric is a Kaehler metric if the associated
2-form SZ = 1/---1 Xi@{ A 8, is closed and, in this case, M is a Kaehler
manifold. Hence, at iaih point of a Kaehler manifold th&e exists a system
of local complex coordinates which is geodesic. This property of the Kaehler
metric leads to many significant topological properties of compact
Kaehler manifolds which we now pursue.