Properties of Sobolev-type metrics in the space of curves

We define a manifold M where objects c∈M are curves, which we parameterize as c:S1→\realn (n≥2, S1 is the circle). We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics Hj. These metrics have been shown to regularize gradient flows used in Computer Vision applications, see \cite{ganesh:SAC07, ganesh:new_sobol_activ_contour08} and references therein. We provide some basic results of Hj metrics; and, for the cases j=1,2, we characterize the completion of the space of smooth curves. We call these completions \emph{``H1 and H2 Sobolev--type Riemannian Manifolds of Curves''}. \gsinsert{This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics.} As a byproduct, we prove that the Fr\'echet distance of curves (see \cite{Michor-Mumford}) coincides with the distance induced by the ``Finsler L\infinity metric'' defined in \S2.2 in \cite{YM:metrics04}