*preprint*

**Inserted:** 11 feb 2022

**Last Updated:** 8 jul 2024

**Year:** 2006

**Abstract:**

We define a manifold $M$ where objects $c\in M$ are curves, which we
parameterize as $c:S^1\to R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve
$c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all
deformations $h:S^1\to R^n$ of $c$. In this paper we study geometries on the
manifold of curves, provided by Sobolev--type metrics $H^j$. We study $H^j$
type metrics for the cases $j=1,2$; we prove estimates, and characterize the
completion of the space of smooth curves. As a bonus, we prove that the
Fr\'echet distance of curves (see arXiv:math.DG*0312384) coincides with the
distance induced by the ``Finsler $L^\infinity$ metric'' defined in \S2.2 in
arXiv:math.DG*0412454.