The metric completion of Diff([0,1]) with the H^1 right-invariant metric is the space of increasing maps of [0,1] fixing the endpoints.
Michor and David Mumford
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Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric
The metric completion of Diff([0,1]) with the H^1 right-invariant metric is the space of increasing maps of [0,1] fixing the endpoints.