Page 233 - Curvature and Homology

6.7. Zero curvature

In this section we examine the effect of zero curvature on the properties
of hermitian manifolds-the curvature being defined as in 5 5.3.

Theorem 6.7.1. The curvature of an hermitian manifold.vanishes, if and
only if, it is possible to choose a parallel field of orthonormal holomorphic
frames in a naghborhood of each point of the manifold.
By a field of frames on the manifold M or an open subset S of M
is meant a cross section in the (principal) bundle of frames over M or S,
respectively. The field is said to be parallel if each of the vector fields
is parallel.
We first prove the sufficiency. If the curvature is zero, the system of
equations w5, = 0 is completely integrable. Therefore, in a suitably
chosen coordinate neighborhood U of each point P it is possible to
introduce a field of orthonormal frames P, (el, .--, e,, ll, --., which
are parallel and are uniquely determined by the initially chosen frame
at P. For, by 5 1.9 the vector fields e, satisfy the differential system

de, = di e,.
(The metric being locally flat, the e, may be thought of as covectors.)
Of course, we also have the conjugate relations. Since the e, are of
bidegree (l,O), the condition &, = 0 implies d"e, = 0, that is the e,
are holomorphic vector fields. Hence, the condition that the curvature
is zero implies the existence of a field of parallel orthonormal holo-
morphic frames in U.
Conversely, with respect to any parallel field of orthonormal frames
the equations

imply Rjikl, = 0. The curvature tensor must therefore vanish for all
frames.
Let us call the neighborhoods U of the theorem admissible neighbor-
hoods. Parallel displacement of a frame at P along any path in such a
neighborhood IJ of P is independent of the path since the system
wij = 0 has a unique solution through U coinciding with the given
frame at P E U. (In the remainder of this section we shall write U(P)
in place of U).
Now, given any two points Po and PI of the manifold and a path C
joining them, there is a neighborhood U(C) = U U(Q) of C such
QEC