Page 234 - Curvature and Homology

that the displacement of a frame from Po to P, is the same along any
path from Po to P, in U(C). We call U(C) an admissible neighborhood.
Let Co and C, be any two homotopic paths joining Po to PI and denote
by {C,) (0 5 t 5 1) the class of curves defining the homotopy. Let S
be the subset of the unit interval I corresponding to those paths C,
for which parallel displacement of a frame from Po to PI is identical
with that along Co. Hence, 0 E S. That S is an open subset of 1 is clear.
We show that S is closed. If S # I, it has a least upper bound s'.
Consequently, since U(C,e) is of finite width we have a contradiction.
For, S is both open and closed, and so S = I. We have proved

Theorem 6.7.2. In an hermitian manifold of zero curvature, parallel
dikplacemettt along a path depends only on the homotopy class of the
path [Iq.

Corollary. A simply connected hermitian manifold of zero curvature is
(complex) parallelisable by means of parallel orthonormal frames.
It is shown next that a complex parallelisable manifold has a canonically
defined hermitian metric g with respect to which the curvature vanishes.
Indeed, in the notation of theorem 6.6.1 let

with respect to the system (zi) of local complex coordinates. In terms
of the inverse matrix (P:)) of (&:,) the n 1-forms

define a basis of the space of covectors of bidegree (1,O). We define the
metric g by means of the matrix of coefficients

Hence, since (X,., Be) = 8,d,