Page 238 - Curvature and Homology

220 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

For,
R" = IT*&
Hence, (!@, ,f) is a 'hermitian covering space' of (M, g), that is, the
holomorphic projection map n induces the metric of M.
The universal covering space AT of a compact complex parallelisable
manifold therefore has the properties:
(i) there are n independent abelian differentials of the first kind on a;
(ii) they satisfy the equations of Maurer-Cartan ;
(iii) l@ is simply connected, and
(iv) AT is complete (with respect to 2).

Under the circun~stances, i@ can be given a group structure in such
a way that multiplication in the group is holomorphic. Moreover, the
abelian differentials are left invariant pfaffian forms. We conclude
therefore that A?! is a complex Lie group.
That compactness is essential to the argument may be seen from the
following example:
Let M = C2 - 0. Define the holomorphic pfaffian forms e1 and e2
on M as follows:

Denote by XI and X2 their duals in TC. The components of the torsion
tensor with respect to this basis are given by (6.8.2), namely,

Although Xl and X2 form parallel frames, these components are not
constant.

6.9. A topological characterization of compact complex parallelisable
manifolds

In this section, a compact complex parallelisahle manifold M is
characterized as the quotient space of a complex Lie group. In fact, it is
shown that M is holomorphically isomorphic with l@/D where D is the
fundamental group of lq. As a consequence of this, it follows that M
is Kaehlerian, if and only if, it is a multi-torus.
Let D be the fundamental group of the universal covering space
(i@, w) of the compact complex parallelisable manifold M, that is, the