The Metric Completion of the Space of Vector-Valued One-Forms

TL;DR

This paper characterizes the metric completion of the space of full-ranked one-forms on a manifold, establishing a distance equality and exploring the relationship with Riemannian metrics, thus providing a concrete description of the completion.

Contribution

It proves a distance equality between geodesic distances of the generalized Ebin metric and fiberwise Riemannian metrics, and describes the metric completion of the space of full-ranked one-forms.

Findings

Distance equality between Ebin metric and fiberwise Riemannian metric

Concrete description of the metric completion of full-ranked one-forms

Quotient structures relating one-forms and Riemannian metrics

Abstract

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.