The metric completion of the space of vector-valued one-forms

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