  The space of full-ranked one-forms on a smooth, orientable, compact manifold
(possibly with boundary) is metrically incomplete with respect to the induced
geodesic distance of the generalized Ebin metric. We show a distance equality
between the induced geodesic distances of the generalized Ebin metric on the
space of full-ranked one-forms and the corresponding Riemannian metric defined
on each fiber. Using this result we immediately have a concrete description of
the metric completion of the space of full-ranked one-forms. Additionally, we
study the relationship between the space of full-ranked one-forms and the space
of all Riemannian metrics, leading to quotient structures for the space of
Riemannian metrics and its completion.

Differential Geometry