Page 178 - Curvature and Homology

160 V. COMPLEX MANIFOLDS

covering in such a way that in the overlap of two coordinate neighbor-
hoods related by holomorphic functions the equations (1.7.5) are satisfied
by the n2 forms of given below. We then insist that the 2n2 forms
oi, 4: be given by

from which it follows that ck = I&; the remaining 2n2 forms are
set equal to zero.
In terms of this connection we take the covariant differential of each
of the vectors 5,;) thereby obtaining as in 5 1.8 the forms a? . Their
images in B will be denoted by O8,. By (1.8.6) and (1.8.5)
4 = (d&d + &(:I) I!! (5.3.3)

from which, by (5.3.2)
dei = ek A eik + ei (5.3.4)

-the torsion forms being given by

@is- * [(;) [O &:I ~~j oZ A em, ~~j = r; - T:~. (5.3.5)

This is the first of the equations of structure. The forms pB are not
independent, but rather, are related by
-
e: +B: =O, 8:= e;.

For, from (5.3.3)

Since,

(5.3.7) becomes

5'; f dt(:) 5';' + f(i t(h &k1* = 0' (5.3.9)