Page 216 - Curvature and Homology

from a geometrical standpoint in describing its homology. More
generally, a compact hermitian manifold M of zero curvature has as its
universal covering space l@ a complex Lie group. If D (the fundamental
group) is a discrete group of covering transformations of M whose
elements are isometries acting without fixed points, then M is homeo-
morphic with WD. If M is simply connected, a necessary condition
for zero curvature is complex parallelisability by means of a parallel
field of orthonormal frames, that is, the existence of n globally defined
linearly independent holomorphic vector fields which are parallel with
respect to the connection defined in fj 5.3. On the other hand, a complex
parallelisable manifold has a natural hermitian metric of zero curvature.
The existence of a metric with zero curvature is consequently a weaker
property than parallelisability. The problem of determining those
manifolds with a locally flat hermitian metric is considered. It is shown
that a compact hermitian manifold of zero curvature is homeomorphic
with a quotient space of a complex Lie group modulo a discrete
subgroup. It is Kaehlerian, if and only if, it is a multi-torus [69].
The hyperbolic spaces will be considered from the point of view of
the problem of imbedding into a locally flat space. Our interest lies in
the local properties of a manifold for which a holomorphic imbedding
which induces the metric is possible. If the Ricci curvature is positive,
it is not possible to define such an imbedding. On the other hand, negative
Ricci curvature is not sufficient to guarantee this. For, one need only
consider the classical hyperbolic space defined by the metric g(x, 2) =
in
(1 - ~2)-~ the unit circle I z I < 1. Such imbeddings consequently
appear rather remote and can only occur if the Ricci curvature is not
positive [5].
Whereas positive Ricci curvature yields information on homology,
negative curvature is of interest in the study of groups of transformations
(cf. Chap. 111). Chapter VII is concerned essentially with the study of
groups of holomorphic and conformal homeomorphisms of Kaehler
manifolds, and so some of the results for negative curvature are post-
poned until then. In any case, the elliptic and parabolic spaces are
particularly interesting from our point of view in that their homology
properties may be described by the methods of Chapters I11 and V.
For negative curvature no holomorphic contravariant tensor fields of
bidegree (p, 0) can exist. Hence, in particular (as already observed),
the manifold is not complex parallelisable. A generalization may be
obtained by assuming that the 1" Chern class is negative definite
(cf. VIi.A.4).
The Gauss-Bonnet formbla is also particularly interesting from our
point of view. In fact, if M is a compact Kaehler manifold on which there