Page 177 - Curvature and Homology

5.3. LOCAL HERMITIAN GEOMETRY 1 59
local complex coordinates (ai) at P by the natural basis {a/az?,
i = 1, ..., n of T>O and the group U(n). In the notation of 5 1.8, we put

Since the vector Xi. E q v l is the conjugate of Xi E TioO, k:) = f$)
where we have written fk:,, for c,. By putting f:; = t&., = 0 these
equations may be written in the abbreviated form
a
XA = f(21 A, B = 1, -, n, 1 *, -, n*.

The coefficients f,;, are the elements of a matrix in Gqn, C). However,
they are not independent. For, they must satisfy the relation

where gkl. = g( a/ a/ a9).
Let (('A,)) denote the inverse matrix of (f&). AS in 5 1.8 it defines 2n
linearly independent differential forms 8A in B: In the overlap of the
coordinate neighborhoods with the local coordinates (ad, f&) and
(atA, we have by (1.8.3)

Hence, by (5.2.17)

The 2n covariant vector fields f';) therefore define 2n independent
I-forms BA = (P, P*) in B with Bi* = & (i = 1, -em, n). In terms of the
local coordinates (ai), they may be expressed by

where T: B + M is the projection map.
By analogy they form a 'frame' in T$ and for this reason this frame is
called a coframe.
There are several ways of defining a metrical connection in M. We
propose to do this in a manner compatible with the complex and
hermitian structures since this approach seems to be natural for
hermitian manifolds. Indeed, as in 5 1.7 we prescribe (2n)Z linear
differential forms U; = I'&dzc in each coordinate neighborhood of a