Page 179 - Curvature and Homology

5.3. LOCAL HERMITIAN GEOMETRY 161

Evaluating the differential of the metric tensor g as in 5 1.9 we obtain

This is precisely the condition that the wij must satisfy in order to
define a metrical connection. Hence, for a metrical connection

Substituting from (5.3.9) into (5.3.6), applying (5.3.11) and observing
that

we obtain the desired relation.
The second of the equations of structure (1.8.8)

dOB, - 6 19% = @%
A
splits into

by virtue of the decomposition TC = TIJJ @ P1.
Denote the curvature forms in the local coordinates (xi, 2) by LP,,
that is, the Qj, are the forms @ji pulled down to M by means of the
cross-section M -t {( a/ ( a/ aii),) . Consequently, in much the
same way as above, it may be shown that if they are locally given by

then, in the bundle of unitary frames, the curvature forms are the 83,.
Since cd, = r:, dzk, the equations (5.3.14) become