  This is the author's Ph.D. thesis, submitted to the University of Leipzig. It
deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian
metrics on a fixed closed, finite-dimensional manifold.
  The main body of the thesis is a description of the completion manifold of
metrics with respect to the $L^2$ metric. The primary motivation for studying
this problem comes from Teichmueller theory, where similar considerations lead
to a completion of the well-known Weil-Petersson metric. We give an application
of the main theorem to the completions of Teichmueller space with respect to a
class of metrics that generalize the Weil-Petersson metric.
  We also prove that the $L^2$ metric induces a metric space structure on the
manifold of metrics. As the $L^2$ metric is a weak Riemannian metric, this fact
does not follow from general results. In addition, we prove several results on
the exponential mapping and distance function of a weak Riemannian metric on a
Hilbert/Frechet manifold. The statements are analogous to, but weaker than,
what is known in the case of a Riemannian metric on a finite-dimensional
manifold or a strong Riemannian metric on a Hilbert manifold.
