Properties of Sobolev-type metrics in the space of curves

We define a manifold M where objects c 2 M are curves, which we parameterize as c : S-1 -> R-n (n >= 2, S-1 is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics H-j. These metrics have been shown to regularize gradient flows used in computer vision applications (see [13, 14, 16] and references therein). We provide some basic results on Hj metrics; and, for the cases j = 1; 2, we characterize the completion of the space of smooth curves. We call these completions "H-1 and H-2 Sobolev-type Riemannian manifolds of curves". 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 Frechet distance of curves (see [7]) coincides with the distance induced by the "Finsler L-infinity metric" defined in 2.2 of [18].