Page 237 - Curvature and Homology

equations (5.3.3) - (5.3.5) by restricting to parallel orthonormal frames
(cf. proof of theorem 6.7.1). Consequently,

Equations (6.8.1) and (6.8.2) imply that the Bi (i = 1, a*., n) define a
local Lie group. This group cannot, in general, be extended to the
whole of M. For this reason we consider the universal covering space
A? of M. For, A? is simply connected and has a naturally induced
hermitian metric of zero curvature (cf. theorem 6.7.2, cor., and prop.
5.8.3). We prove

Theorem 6.8.1. The universd covering space M of a compact complex
parallelisable manifold M is a complex Lie group [69].
In the first place, since the projection rr: i@ -t M is a holomorphic
map, i@ has a naturally induced complex structure (cf. prop. 5.8.3).
On the other hand, rr is a local homeomorphism; hence, it is (1-1).
Consequently, the n forms
0' = ,*(@)
are linearly independent and holomorphic, the latter property being
due to the fact thatrr is holomorphic (cf. lemma 5.8.2). Moreover,
dB' = d(,*@) = w*(dOi)

Hence, the & define a local Lie group. The BL being independent we
define the (hepitian) metric

on I@. That this metric is not, in general Kaehlerian follows from the
fact that the Bi are not necessarily df-closed.
With respect to this metric, A? may be shown to be complete (cf. 5 7.7).
To see this, since M is compact it is complete with respect to the metric

The completeness of fl now follows from that of M.