  This thesis is concerned with extending the idea of geodesic completeness
from pseudo-Riemannian to complex geometry: we take, however a completely
holomorphicpoint of view; that is to say, a 'metric' will be a (meromorphic)
symmetric section of the twice covariant holomorphic tensor bundle. We shall
hint at the need of reformulating some aspects of the theory of differential
equations in the complex domain, originating in the interpenetration betwixt
differential and algebraic aspects when analytical continuation is pursued. The
notion itself of path should be reformulated: we remark that geodesics will be
defined on Riemann surfaces which are domains over regions in the complex
plane. Of course geodesics will be eventually defined to be auto-parallel
paths, but we shall focus our attention on the fact that the Levi-Civita
connection will be meromorphic if the metric from which it is induced is
allowed to have poles or even simply to lower somewhere in its rank. We shall
study rather more deeply a class of manifolds, namely warped products of
Riemann surfaces; some hypotheses concerning their metrics will be done,
(metrics will be suppose to be 'coercive', in a sense that will be defined) but
we shall show that the range of applicability of the yielded completeness
theorems will not be exceedingly restricted.
